@Model.Header
3 || @Model.TableHeader | Description |
12 | @{ var id = Html.UniqueID().ToString(); }
13 |
14 | @Html.Encode(it.Details.FormatUsage(40))
15 |
16 |
17 | Signature: @Html.Encode(it.Details.Signature)
25 | 18 | @if (!it.Details.Modifiers.IsEmpty) { 19 | Modifiers: @it.Details.FormatModifiers 20 | } 21 | @if (!it.Details.TypeArguments.IsEmpty) { 22 | Type parameters: @it.Details.FormatTypeArguments 23 | } 24 | |
26 |
27 | @if (!String.IsNullOrEmpty(it.Details.FormatSourceLocation))
28 | {
29 |
30 | 
31 | 
32 |
33 | }
34 | @it.Comment.FullText
35 | |
36 |
@Model.Name
8 | 9 | @{ var nsIndex = 0; } 10 | @foreach (var ns in Model.Namespaces) 11 | { 12 | nsIndex++; 13 | var typedNs = (Namespace)ns; 14 | var allCategories = 15 | typedNs.Types.Select(t => t.Category) 16 | .Concat(typedNs.Modules.Select(m => m.Category)) 17 | .Distinct() 18 | .OrderBy(s => String.IsNullOrEmpty(s) ? "ZZZ" : s); 19 | var allByCategory = 20 | allCategories 21 | .Select((c, i) => new { 22 | Name = String.IsNullOrEmpty(c) ? "Other namespace members" : c, 23 | Index = String.Format("{0}_{1}", nsIndex, i), 24 | Types = typedNs.Types.Where(t => t.Category == c).ToArray(), 25 | Modules = typedNs.Modules.Where(m => m.Category == c).ToArray() }) 26 | .Where(c => c.Types.Length + c.Modules.Length > 0).ToArray(); 27 | 28 |@ns.Name Namespace
29 | if (allByCategory.Length > 1) 30 | { 31 | 32 |-
33 | @foreach (var g in allByCategory)
34 | {
35 |
- @g.Name 36 | } 37 |
@g.Name
44 | } 45 |
46 | @RenderPart("part-nested", new
47 | {
48 | Types = g.Types,
49 | Modules = g.Modules
50 | })
51 |
52 | }
53 | }
54 |
--------------------------------------------------------------------------------
/src/FsAlg/AssemblyInfo.fs:
--------------------------------------------------------------------------------
1 | #light
2 |
3 | namespace FsAlg
4 |
5 | open System.Reflection;
6 | open System.Runtime.CompilerServices;
7 | open System.Runtime.InteropServices;
8 |
9 | // General Information about an assembly is controlled through the following
10 | // set of attributes. Change these attribute values to modify the information
11 | // associated with an assembly.
12 | [
20 |
21 |
29 |
30 | Quick Usage Example
31 | -------------------
32 | *)
33 |
34 | open FsAlg.Generic
35 |
36 | let A = matrix [[3.; 1.]; [2.; -1.]]
37 | let b = vector [5.; 0.]
38 |
39 | // Solve a matrix equation
40 | let x = Matrix.solve A b
41 |
42 | // Find inverse of A
43 | let Ainv = Matrix.inverse A
44 |
45 | // Initialize a vector
46 | let w = Vector.init 2 (fun i -> float (i * i))
47 |
48 | // Map a function to a vector
49 | let y = Vector.map (fun x -> sin (exp x)) w
50 |
51 | // Matrix-vector operations
52 | let z = A * (x - y)
--------------------------------------------------------------------------------
/src/FsAlg/Util.fs:
--------------------------------------------------------------------------------
1 | //
2 | // This file is part of
3 | // FsAlg: Generic Linear Algebra Library
4 | //
5 | // Copyright (c) 2015, National University of Ireland Maynooth (Atilim Gunes Baydin, Barak A. Pearlmutter)
6 | //
7 | // FsAlg is released under the BSD license.
8 | // (See accompanying LICENSE file.)
9 | //
10 | // Written by:
11 | //
12 | // Atilim Gunes Baydin
13 | // atilimgunes.baydin@nuim.ie
14 | //
15 | // Barak A. Pearlmutter
16 | // barak@cs.nuim.ie
17 | //
18 | // Brain and Computation Lab
19 | // Hamilton Institute & Department of Computer Science
20 | // National University of Ireland Maynooth
21 | // Maynooth, Co. Kildare
22 | // Ireland
23 | //
24 | // www.bcl.hamilton.ie
25 | //
26 |
27 | /// Various utility functions for internal use
28 | module FsAlg.Generic.Util
29 |
30 | /// Checks whether the 2d array `m` has the same number of elements in both dimensions
31 | let (|Square|) (m:_[,]) =
32 | match m with
33 | | m when m.GetLength 0 = m.GetLength 1 -> m
34 | | _ -> invalidArg "m" "Expecting a square 2d array"
35 |
36 | /// Gets the transpose of the 2d array `m`
37 | let inline transpose (m:_[,]) = Array2D.init (m.GetLength 1) (m.GetLength 0) (fun i j -> m.[j, i])
38 |
39 | /// Gets an array containing the diagonal elements of the square 2d array `m`
40 | let inline diagonal (Square m:_[,]) = Array.init (m.GetLength 0) (fun i -> m.[i, i])
41 |
42 | /// Gets the trace of the square matrix given in the 2d array `m`
43 | let inline trace (m:_[,]) = Array.sum (diagonal m)
44 |
45 | /// Copies the upper triangular elements of the square matrix given in the 2d array `m` to the lower triangular part
46 | let inline copyUpperToLower (Square m:_[,]) =
47 | let r = Array2D.copy m
48 | let rows = r.GetLength 0
49 | if rows > 1 then
50 | for i = 1 to rows - 1 do
51 | for j = 0 to i - 1 do
52 | r.[i, j] <- r.[j, i]
53 | r
54 |
55 | /// Finds an array that, when multiplied by an LU matrix `lu`, gives array `b`
56 | let inline matrixSolveHelper (lu:'a[,]) (b:'a[]) =
57 | let n = lu.GetLength 0
58 | let x = Array.copy b
59 | for i = 1 to n - 1 do
60 | let mutable sum = x.[i]
61 | for j = 0 to i - 1 do
62 | sum <- sum - lu.[i, j] * x.[j]
63 | x.[i] <- sum
64 | x.[n - 1] <- x.[n - 1] / lu.[n - 1, n - 1]
65 | for i in (n - 2) .. -1 .. 0 do
66 | let mutable sum = x.[i]
67 | for j = i + 1 to n - 1 do
68 | sum <- sum - lu.[i, j] * x.[j]
69 | x.[i] <- sum / lu.[i, i]
70 | x
71 |
72 | /// Checks whether a float contains an integer value
73 | let isInteger a = a = float (int a)
--------------------------------------------------------------------------------
/.gitattributes:
--------------------------------------------------------------------------------
1 | ###############################################################################
2 | # Set default behavior to automatically normalize line endings.
3 | ###############################################################################
4 | * text=auto
5 |
6 | ###############################################################################
7 | # Set default behavior for command prompt diff.
8 | #
9 | # This is need for earlier builds of msysgit that does not have it on by
10 | # default for csharp files.
11 | # Note: This is only used by command line
12 | ###############################################################################
13 | #*.cs diff=csharp
14 |
15 | ###############################################################################
16 | # Set the merge driver for project and solution files
17 | #
18 | # Merging from the command prompt will add diff markers to the files if there
19 | # are conflicts (Merging from VS is not affected by the settings below, in VS
20 | # the diff markers are never inserted). Diff markers may cause the following
21 | # file extensions to fail to load in VS. An alternative would be to treat
22 | # these files as binary and thus will always conflict and require user
23 | # intervention with every merge. To do so, just uncomment the entries below
24 | ###############################################################################
25 | #*.sln merge=binary
26 | #*.csproj merge=binary
27 | #*.vbproj merge=binary
28 | #*.vcxproj merge=binary
29 | #*.vcproj merge=binary
30 | #*.dbproj merge=binary
31 | #*.fsproj merge=binary
32 | #*.lsproj merge=binary
33 | #*.wixproj merge=binary
34 | #*.modelproj merge=binary
35 | #*.sqlproj merge=binary
36 | #*.wwaproj merge=binary
37 |
38 | ###############################################################################
39 | # behavior for image files
40 | #
41 | # image files are treated as binary by default.
42 | ###############################################################################
43 | #*.jpg binary
44 | #*.png binary
45 | #*.gif binary
46 |
47 | ###############################################################################
48 | # diff behavior for common document formats
49 | #
50 | # Convert binary document formats to text before diffing them. This feature
51 | # is only available from the command line. Turn it on by uncommenting the
52 | # entries below.
53 | ###############################################################################
54 | #*.doc diff=astextplain
55 | #*.DOC diff=astextplain
56 | #*.docx diff=astextplain
57 | #*.DOCX diff=astextplain
58 | #*.dot diff=astextplain
59 | #*.DOT diff=astextplain
60 | #*.pdf diff=astextplain
61 | #*.PDF diff=astextplain
62 | #*.rtf diff=astextplain
63 | #*.RTF diff=astextplain
64 |
--------------------------------------------------------------------------------
/docs/BuildDocs.fsx:
--------------------------------------------------------------------------------
1 |
2 |
3 | #I "../packages/FSharp.Compiler.Service.0.0.86/lib/net40"
4 | #r "FSharp.Compiler.Service.dll"
5 | #I "../packages/FSharpVSPowerTools.Core.1.7.0/lib/net45"
6 | #r "FSharpVSPowerTools.Core.dll"
7 | #I "../packages/FSharp.Formatting.2.8.0/lib/net40"
8 | #r "CSharpFormat.dll"
9 | #r "FSharp.CodeFormat.dll"
10 | #r "FSharp.Literate.dll"
11 | #r "FSharp.MetadataFormat.dll"
12 | #r "FSharp.Markdown.dll"
13 |
14 | open System.IO
15 | open FSharp.Literate
16 | open FSharp.MetadataFormat
17 |
18 | //
19 | // Setup output directory structure and copy static files
20 | //
21 |
22 | let source = __SOURCE_DIRECTORY__
23 | let docs = Path.Combine(source, "")
24 | let relative subdir = Path.Combine(docs, subdir)
25 |
26 | if not (Directory.Exists(relative "output")) then
27 | Directory.CreateDirectory(relative "output") |> ignore
28 | if not (Directory.Exists(relative "output/img")) then
29 | Directory.CreateDirectory (relative "output/img") |> ignore
30 | if not (Directory.Exists(relative "output/misc")) then
31 | Directory.CreateDirectory (relative "output/misc") |> ignore
32 | if not (Directory.Exists(relative "output/reference")) then
33 | Directory.CreateDirectory (relative "output/reference") |> ignore
34 |
35 | for fileInfo in DirectoryInfo(relative "input/files/misc").EnumerateFiles() do
36 | fileInfo.CopyTo(Path.Combine(relative "output/misc", fileInfo.Name), true) |> ignore
37 |
38 | for fileInfo in DirectoryInfo(relative "input/files/img").EnumerateFiles() do
39 | fileInfo.CopyTo(Path.Combine(relative "output/img", fileInfo.Name), true) |> ignore
40 |
41 | //
42 | // Generate documentation
43 | //
44 |
45 | let tags = ["project-name", "FsAlg"; "project-author", "Atılım Güneş Baydin"; "project-github", "http://github.com/gbaydin/FsAlg"; "project-nuget", "https://www.nuget.org/packages/fsalg"; "root", ""]
46 |
47 | Literate.ProcessScriptFile(relative "input/index.fsx", relative "input/templates/template.html", relative "output/index.html", replacements = tags, fsiEvaluator = FsiEvaluator())
48 | Literate.ProcessScriptFile(relative "input/vector-operations.fsx", relative "input/templates/template.html", relative "output/vector-operations.html", replacements = tags, fsiEvaluator = FsiEvaluator())
49 | Literate.ProcessScriptFile(relative "input/matrix-operations.fsx", relative "input/templates/template.html", relative "output/matrix-operations.html", replacements = tags, fsiEvaluator = FsiEvaluator())
50 |
51 |
52 | //
53 | // Generate API reference
54 | //
55 |
56 | let library = relative "../src/FsAlg/bin/Debug/FsAlg.dll"
57 | let layoutRoots = [relative "input/templates"; relative "input/templates/reference" ]
58 |
59 | MetadataFormat.Generate(library, relative "output/reference", layoutRoots, tags, markDownComments = true)
60 |
--------------------------------------------------------------------------------
/.gitignore:
--------------------------------------------------------------------------------
1 | ## Ignore Visual Studio temporary files, build results, and
2 | ## files generated by popular Visual Studio add-ons.
3 |
4 | # User-specific files
5 | *.suo
6 | *.user
7 | *.sln.docstates
8 |
9 | # Build results
10 |
11 | [Dd]ebug/
12 | [Rr]elease/
13 | x64/
14 | build/
15 | [Bb]in/
16 | [Oo]bj/
17 |
18 | # Enable "build/" folder in the NuGet Packages folder since NuGet packages use it for MSBuild targets
19 | !packages/*/build/
20 |
21 | # MSTest test Results
22 | [Tt]est[Rr]esult*/
23 | [Bb]uild[Ll]og.*
24 |
25 | *_i.c
26 | *_p.c
27 | *.ilk
28 | *.meta
29 | *.obj
30 | *.pch
31 | *.pdb
32 | *.pgc
33 | *.pgd
34 | *.rsp
35 | *.sbr
36 | *.tlb
37 | *.tli
38 | *.tlh
39 | *.tmp
40 | *.tmp_proj
41 | *.log
42 | *.vspscc
43 | *.vssscc
44 | .builds
45 | *.pidb
46 | *.log
47 | *.scc
48 |
49 | # Visual C++ cache files
50 | ipch/
51 | *.aps
52 | *.ncb
53 | *.opensdf
54 | *.sdf
55 | *.cachefile
56 |
57 | # Visual Studio profiler
58 | *.psess
59 | *.vsp
60 | *.vspx
61 |
62 | # Guidance Automation Toolkit
63 | *.gpState
64 |
65 | # ReSharper is a .NET coding add-in
66 | _ReSharper*/
67 | *.[Rr]e[Ss]harper
68 |
69 | # TeamCity is a build add-in
70 | _TeamCity*
71 |
72 | # DotCover is a Code Coverage Tool
73 | *.dotCover
74 |
75 | # NCrunch
76 | *.ncrunch*
77 | .*crunch*.local.xml
78 |
79 | # Installshield output folder
80 | [Ee]xpress/
81 |
82 | # DocProject is a documentation generator add-in
83 | DocProject/buildhelp/
84 | DocProject/Help/*.HxT
85 | DocProject/Help/*.HxC
86 | DocProject/Help/*.hhc
87 | DocProject/Help/*.hhk
88 | DocProject/Help/*.hhp
89 | DocProject/Help/Html2
90 | DocProject/Help/html
91 |
92 | # Click-Once directory
93 | publish/
94 |
95 | # Publish Web Output
96 | *.Publish.xml
97 |
98 | # NuGet Packages Directory
99 | ## TODO: If you have NuGet Package Restore enabled, uncomment the next line
100 | packages/
101 |
102 | # Windows Azure Build Output
103 | csx
104 | *.build.csdef
105 |
106 | # Windows Store app package directory
107 | AppPackages/
108 |
109 | # Others
110 | sql/
111 | *.Cache
112 | ClientBin/
113 | [Ss]tyle[Cc]op.*
114 | ~$*
115 | *~
116 | *.dbmdl
117 | *.[Pp]ublish.xml
118 | *.pfx
119 | *.publishsettings
120 |
121 | # RIA/Silverlight projects
122 | Generated_Code/
123 |
124 | # Backup & report files from converting an old project file to a newer
125 | # Visual Studio version. Backup files are not needed, because we have git ;-)
126 | _UpgradeReport_Files/
127 | Backup*/
128 | UpgradeLog*.XML
129 | UpgradeLog*.htm
130 |
131 | # SQL Server files
132 | App_Data/*.mdf
133 | App_Data/*.ldf
134 |
135 |
136 | #LightSwitch generated files
137 | GeneratedArtifacts/
138 | _Pvt_Extensions/
139 | ModelManifest.xml
140 |
141 | # =========================
142 | # Windows detritus
143 | # =========================
144 |
145 | # Windows image file caches
146 | Thumbs.db
147 | ehthumbs.db
148 |
149 | # Folder config file
150 | Desktop.ini
151 |
152 | # Recycle Bin used on file shares
153 | $RECYCLE.BIN/
154 |
155 | # Mac desktop service store files
156 | .DS_Store
157 |
--------------------------------------------------------------------------------
/docs/input/templates/reference/type.cshtml:
--------------------------------------------------------------------------------
1 | @using FSharp.MetadataFormat
2 | @{
3 | Layout = "template";
4 | Title = Model.Type.Name + " - " + Properties["project-name"];
5 | }
6 |
7 | @{
8 | // Get all the members & comment for the type
9 | var members = (IEnumerable
22 |
27 |
28 |
23 | The FsAlg library is available on NuGet. To install, run the following command in the Package Manager Console:
24 |
26 | PM> Install-Package FsAlg25 |
@Model.Type.Name
31 |
32 | @foreach (var sec in comment.Sections) {
33 | // XML comment for the type has multiple sections that can be labelled
34 | // with categories (to give comment for an individual category). Here,
35 | // we print only those that belong to the section.
36 | if (!byCategory.Any(g => g.GroupKey == sec.Key)) {
37 | if (sec.Key != "") {
38 |
44 | @if (byCategory.Count() > 1)
45 | {
46 |
47 | @sec.Key
39 | } 40 | @sec.Value 41 | } 42 | } 43 |Table of contents
48 |-
49 | @foreach (var g in byCategory)
50 | {
51 |
- @g.Name 52 | } 53 |
) and add XML comment from the type
58 | // that is related to this specific category.
59 | if (byCategory.Count() > 1) {
60 | @g.Name
61 | var info = comment.Sections.FirstOrDefault(kvp => kvp.Key == g.GroupKey);
62 | if (info.Key != null) {
63 |
64 | @info.Value
65 |
66 | }
67 | }
68 |
69 | @RenderPart("part-members", new {
70 | Header = "Union Cases",
71 | TableHeader = "Union Case",
72 | Members = g.Members.Where(m => m.Kind == MemberKind.UnionCase)
73 | })
74 |
75 | @RenderPart("part-members", new {
76 | Header = "Record Fields",
77 | TableHeader = "Record Field",
78 | Members = g.Members.Where(m => m.Kind == MemberKind.RecordField)
79 | })
80 |
81 | @RenderPart("part-members", new {
82 | Header = "Static parameters",
83 | TableHeader = "Static parameters",
84 | Members = g.Members.Where(m => m.Kind == MemberKind.StaticParameter)
85 | })
86 |
87 | @RenderPart("part-members", new {
88 | Header = "Constructors",
89 | TableHeader = "Constructor",
90 | Members = g.Members.Where(m => m.Kind == MemberKind.Constructor)
91 | })
92 |
93 | @RenderPart("part-members", new {
94 | Header = "Instance members",
95 | TableHeader = "Instance member",
96 | Members = g.Members.Where(m => m.Kind == MemberKind.InstanceMember)
97 | })
98 |
99 | @RenderPart("part-members", new {
100 | Header = "Static members",
101 | TableHeader = "Static member",
102 | Members = g.Members.Where(m => m.Kind == MemberKind.StaticMember)
103 | })
104 | }
105 |
--------------------------------------------------------------------------------
/docs/input/templates/reference/module.cshtml:
--------------------------------------------------------------------------------
1 | @using FSharp.MetadataFormat
2 | @{
3 | Layout = "template";
4 | Title = Model.Module.Name + " - " + Properties["project-name"];
5 | }
6 |
7 | @{
8 | // Get all the members & comment for the type
9 | var members = (IEnumerable)Model.Module.AllMembers;
10 | var comment = (Comment)Model.Module.Comment;
11 |
12 | // Group all members by their category which is an inline annotation
13 | // that can be added to members using special XML comment:
14 | //
15 | // /// [category:Something]
16 | //
17 | // ...and can be used to categorize members in large modules or types
18 | // (but if this is not used, then all members end up in just one category)
19 | var byCategory = members
20 | .GroupBy(m => m.Category)
21 | .OrderBy(g => String.IsNullOrEmpty(g.Key) ? "ZZZ" : g.Key)
22 | .Select((g, n) => new {
23 | Index = n,
24 | GroupKey = g.Key,
25 | Members = g.OrderBy(m => m.Name),
26 | Name = String.IsNullOrEmpty(g.Key) ? "Other module members" : g.Key
27 | });
28 |
29 | // Get nested modules and nested types as statically typed collections
30 | var nestModules = (IEnumerable)Model.Module.NestedModules;
31 | var nestTypes = (IEnumerable)Model.Module.NestedTypes;
32 | }
33 |
34 | @Model.Module.Name
35 |
36 | @foreach (var sec in comment.Sections) {
37 | // XML comment for the type has multiple sections that can be labelled
38 | // with categories (to give comment for an individual category). Here,
39 | // we print only those that belong to the section.
40 | if (!byCategory.Any(g => g.GroupKey == sec.Key))
41 | {
42 | if (sec.Key != "") {
43 | @sec.Key
44 | }
45 | @sec.Value
46 | }
47 | }
48 |
49 | @if (byCategory.Count() > 1)
50 | {
51 |
52 | Table of contents
53 |
54 | @foreach (var g in byCategory)
55 | {
56 | - @g.Name
57 | }
58 |
59 | }
60 |
61 |
62 | @if (nestTypes.Count() + nestModules.Count() > 0)
63 | {
64 | Nested types and modules
65 |
66 | @RenderPart("part-nested", new {
67 | Types = nestTypes,
68 | Modules = nestModules
69 | })
70 |
71 | }
72 |
73 | @foreach (var g in byCategory)
74 | {
75 | // Iterate over all the categories and print members. If there are more than one
76 | // categories, print the category heading (as ) and add XML comment from the type
77 | // that is related to this specific category.
78 | if (byCategory.Count() > 1)
79 | {
80 | @g.Name
81 | var info = comment.Sections.FirstOrDefault(kvp => kvp.Key == g.GroupKey);
82 | if (info.Key != null)
83 | {
84 |
85 | @info.Value
86 |
87 | }
88 | }
89 |
90 | @RenderPart("part-members", new {
91 | Header = "Functions and values",
92 | TableHeader = "Function or value",
93 | Members = g.Members.Where(m => m.Kind == MemberKind.ValueOrFunction)
94 | })
95 |
96 | @RenderPart("part-members", new {
97 | Header = "Type extensions",
98 | TableHeader = "Type extension",
99 | Members = g.Members.Where(m => m.Kind == MemberKind.TypeExtension)
100 | })
101 |
102 | @RenderPart("part-members", new {
103 | Header = "Active patterns",
104 | TableHeader = "Active pattern",
105 | Members = g.Members.Where(m => m.Kind == MemberKind.ActivePattern)
106 | })
107 | }
--------------------------------------------------------------------------------
/src/FsAlg/FsAlg.fsproj:
--------------------------------------------------------------------------------
1 |
2 |
3 |
5 | Debug
6 | AnyCPU
7 | 2.0
8 | 8715ea25-3b32-4cf5-a595-d4034442fe58
9 | Library
10 | FsAlg
11 | FsAlg
12 | v4.5
13 | 4.3.1.0
14 | FsAlg
15 |
17 |
18 | true
19 | full
20 | false
21 | false
22 | bin\Debug\
23 | DEBUG;TRACE
24 | 3
25 | bin\Debug\FsAlg.XML
26 |
27 |
28 | pdbonly
29 | true
30 | true
31 | bin\Release\
32 | TRACE
33 | 3
34 | bin\Release\FsAlg.XML
35 |
36 |
37 | 11
38 |
39 |
40 |
41 |
42 | $(MSBuildExtensionsPath32)\..\Microsoft SDKs\F#\3.0\Framework\v4.0\Microsoft.FSharp.Targets
43 |
44 |
45 |
46 |
47 | $(MSBuildExtensionsPath32)\Microsoft\VisualStudio\v$(VisualStudioVersion)\FSharp\Microsoft.FSharp.Targets
48 |
49 |
50 |
51 |
53 |
62 |
63 |
65 | True
66 |
67 |
71 |
78 |
--------------------------------------------------------------------------------
/docs/input/templates/template.html:
--------------------------------------------------------------------------------
1 |
2 |
3 |
4 |
5 | {page-title}
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
22 |
30 |
31 |
32 |
33 |
34 |
38 | {project-name}
39 |
40 |
41 |
42 |
43 | {document}
44 | {tooltips}
45 |
46 |
47 |
48 | 
49 |
50 |
68 |
69 |
70 |
71 |
72 |
82 | 
87 |
88 |
89 |
--------------------------------------------------------------------------------
/docs/input/templates/template.cshtml:
--------------------------------------------------------------------------------
1 |
2 |
3 |
4 |
5 | @Title
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
22 |
30 |
31 |
32 |
33 |
34 |
38 | @Properties["project-name"]
39 |
40 |
41 |
42 |
43 | @RenderBody()
44 |
45 |
46 |
47 |
48 | 
49 |
50 |
68 |
69 |
70 |
71 |
72 |
82 |
87 |
88 |
89 |
90 |
--------------------------------------------------------------------------------
/docs/input/files/misc/style_light.css:
--------------------------------------------------------------------------------
1 | @import url(https://fonts.googleapis.com/css?family=Droid+Sans|Droid+Sans+Mono|Open+Sans:400,600,700);
2 |
3 | /*--------------------------------------------------------------------------
4 | Formatting for F# code snippets
5 | /*--------------------------------------------------------------------------*/
6 |
7 | /* identifier */
8 | span.i { color:#000000; }
9 | /* string */
10 | span.s { color:#a31515; }
11 | /* keywords */
12 | span.k { color:#0000ff; }
13 | /* comment */
14 | span.c { color:#008000; }
15 | /* operators */
16 | span.o { color:#000000; }
17 | /* numbers */
18 | span.n { color:#000000; }
19 | /* line number */
20 | span.l { color:#96c2cd; }
21 | /* type or module */
22 | span.t { color:#2b91af; }
23 | /* function */
24 | span.f { color:#0000a0; }
25 | /* DU case or active pattern */
26 | span.p { color:#800080; }
27 | /* mutable var or ref cell */
28 | span.v { color:#000000; font-weight: bold; }
29 | /* printf formatters */
30 | span.pf { color:#2b91af; }
31 | /* escaped chars */
32 | span.e { color:#ff0080; }
33 | /* mutable var or ref cell */
34 |
35 |
36 | /* inactive code */
37 | span.inactive { color:#808080; }
38 | /* preprocessor */
39 | span.prep { color:#0000ff; }
40 | /* fsi output */
41 | span.fsi { color:#808080; }
42 |
43 | /* omitted */
44 | span.omitted {
45 | background:#3c4e52;
46 | border-radius:5px;
47 | color:#808080;
48 | padding:0px 0px 1px 0px;
49 | }
50 | /* tool tip */
51 | div.tip {
52 | background:#e5e5e5;
53 | border-radius:4px;
54 | font:9pt 'Droid Sans', arial, sans-serif;
55 | padding:6px 8px 6px 8px;
56 | display:none;
57 | color:#000000;
58 | }
59 | table.pre pre {
60 | padding:0px;
61 | margin:0px;
62 | border:none;
63 | }
64 | table.pre, pre.fssnip, pre {
65 | line-height:13pt;
66 | border:1px solid #d8d8d8;
67 | border-collapse:separate;
68 | white-space:pre;
69 | font: 10pt consolas,monospace;
70 | width:90%;
71 | margin:10px 20px 20px 20px;
72 | background-color:#fdfdfd;
73 | padding:10px;
74 | border-radius:5px;
75 | color:#000000;
76 | }
77 | table.pre pre {
78 | padding:0px;
79 | margin:0px;
80 | border-radius:0px;
81 | width: 100%;
82 | }
83 | table.pre td {
84 | padding:0px;
85 | white-space:normal;
86 | margin:0px;
87 | }
88 | table.pre td.lines {
89 | width:30px;
90 | }
91 |
92 | /*--------------------------------------------------------------------------
93 | Formatting for page & standard document content
94 | /*--------------------------------------------------------------------------*/
95 |
96 | body {
97 | font-family: 'Open Sans', serif;
98 | padding-top: 0px;
99 | padding-bottom: 40px;
100 | }
101 |
102 | pre {
103 | word-wrap: inherit;
104 | }
105 |
106 | /* Format the heading - nicer spacing etc. */
107 | .masthead {
108 | overflow: hidden;
109 | }
110 | .masthead .muted a {
111 | text-decoration:none;
112 | color:#999999;
113 | }
114 | .masthead ul, .masthead li {
115 | margin-bottom:0px;
116 | }
117 | .masthead .nav li {
118 | margin-top: 15px;
119 | font-size:110%;
120 | }
121 | .masthead h3 {
122 | margin-bottom:5px;
123 | font-size:170%;
124 | }
125 | hr {
126 | margin:0px 0px 20px 0px;
127 | }
128 |
129 | /* Make table headings and td.title bold */
130 | td.title, thead {
131 | font-weight:bold;
132 | }
133 |
134 | /* Format the right-side menu */
135 | #menu {
136 | margin-top:50px;
137 | font-size:11pt;
138 | padding-left:20px;
139 | }
140 |
141 | #menu .nav-header {
142 | font-size:12pt;
143 | color:#606060;
144 | margin-top:20px;
145 | }
146 |
147 | #menu li {
148 | line-height:25px;
149 | }
150 |
151 | /* Change font sizes for headings etc. */
152 | #main h1 { font-size: 26pt; margin:10px 0px 15px 0px; font-weight:400; }
153 | #main h2 { font-size: 20pt; margin:20px 0px 0px 0px; font-weight:400; }
154 | #main h3 { font-size: 14pt; margin:15px 0px 0px 0px; font-weight:600; }
155 | #main p { font-size: 11pt; margin:5px 0px 15px 0px; }
156 | #main ul { font-size: 11pt; margin-top:10px; }
157 | #main li { font-size: 11pt; margin: 5px 0px 5px 0px; }
158 | #main strong { font-weight:700; }
159 |
160 | /*--------------------------------------------------------------------------
161 | Formatting for API reference
162 | /*--------------------------------------------------------------------------*/
163 |
164 | .type-list .type-name, .module-list .module-name {
165 | width:25%;
166 | font-weight:bold;
167 | }
168 | .member-list .member-name {
169 | width:35%;
170 | }
171 | #main .xmldoc h2 {
172 | font-size:14pt;
173 | margin:10px 0px 0px 0px;
174 | }
175 | #main .xmldoc h3 {
176 | font-size:12pt;
177 | margin:10px 0px 0px 0px;
178 | }
179 | .github-link {
180 | float:right;
181 | text-decoration:none;
182 | }
183 | .github-link img {
184 | border-style:none;
185 | margin-left:10px;
186 | }
187 | .github-link .hover { display:none; }
188 | .github-link:hover .hover { display:block; }
189 | .github-link .normal { display: block; }
190 | .github-link:hover .normal { display: none; }
191 |
192 | /*--------------------------------------------------------------------------
193 | Links
194 | /*--------------------------------------------------------------------------*/
195 |
196 | h1 a, h1 a:hover, h1 a:focus,
197 | h2 a, h2 a:hover, h2 a:focus,
198 | h3 a, h3 a:hover, h3 a:focus,
199 | h4 a, h4 a:hover, h4 a:focus,
200 | h5 a, h5 a:hover, h5 a:focus,
201 | h6 a, h6 a:hover, h6 a:focus { color : inherit; text-decoration : inherit; outline:none }
202 |
203 | /*--------------------------------------------------------------------------
204 | Additional formatting for the homepage
205 | /*--------------------------------------------------------------------------*/
206 |
207 | #nuget {
208 | margin-top:20px;
209 | font-size: 11pt;
210 | padding:20px;
211 | }
212 |
213 | #nuget pre {
214 | font-size:11pt;
215 | -moz-border-radius: 0px;
216 | -webkit-border-radius: 0px;
217 | border-radius: 0px;
218 | background: #404040;
219 | border-style:none;
220 | color: #e0e0e0;
221 | margin-top:15px;
222 | }
--------------------------------------------------------------------------------
/docs/input/files/misc/style.css:
--------------------------------------------------------------------------------
1 | @import url(https://fonts.googleapis.com/css?family=Droid+Sans|Droid+Sans+Mono|Open+Sans:400,600,700);
2 |
3 | /*--------------------------------------------------------------------------
4 | Formatting for F# code snippets
5 | /*--------------------------------------------------------------------------*/
6 |
7 | /* strings --- and stlyes for other string related formats */
8 | span.s { color:#E0E268; }
9 | /* printf formatters */
10 | span.pf { color:#E0C57F; }
11 | /* escaped chars */
12 | span.e { color:#EA8675; }
13 |
14 | /* identifiers --- and styles for more specific identifier types */
15 | span.i { color:#d1d1d1; }
16 | /* type or module */
17 | span.t { color:#43AEC6; }
18 | /* function */
19 | span.f { color:#e1e1e1; }
20 | /* DU case or active pattern */
21 | span.p { color:#4ec9b0; }
22 |
23 | /* keywords */
24 | span.k { color:#FAB11D; }
25 | /* comment */
26 | span.c { color:#808080; }
27 | /* operators */
28 | span.o { color:#af75c1; }
29 | /* numbers */
30 | span.n { color:#96C71D; }
31 | /* line number */
32 | span.l { color:#80b0b0; }
33 | /* mutable var or ref cell */
34 | span.v { color:#d1d1d1; font-weight: bold; }
35 | /* inactive code */
36 | span.inactive { color:#808080; }
37 | /* preprocessor */
38 | span.prep { color:#af75c1; }
39 | /* fsi output */
40 | span.fsi { color:#808080; }
41 |
42 | /* omitted */
43 | span.omitted {
44 | background:#3c4e52;
45 | border-radius:5px;
46 | color:#808080;
47 | padding:0px 0px 1px 0px;
48 | }
49 | /* tool tip */
50 | div.tip {
51 | background:#475b5f;
52 | border-radius:4px;
53 | font:11pt 'Droid Sans', arial, sans-serif;
54 | padding:6px 8px 6px 8px;
55 | display:none;
56 | color:#d1d1d1;
57 | }
58 | table.pre pre {
59 | padding:0px;
60 | margin:0px;
61 | border:none;
62 | }
63 | table.pre, pre.fssnip, pre {
64 | line-height:13pt;
65 | border:1px solid #d8d8d8;
66 | border-collapse:separate;
67 | white-space:pre;
68 | font: 9pt 'Droid Sans Mono',consolas,monospace;
69 | width:90%;
70 | margin:10px 20px 20px 20px;
71 | background-color:#212d30;
72 | padding:10px;
73 | border-radius:5px;
74 | color:#d1d1d1;
75 | }
76 | table.pre pre {
77 | padding:0px;
78 | margin:0px;
79 | border-radius:0px;
80 | width: 100%;
81 | }
82 | table.pre td {
83 | padding:0px;
84 | white-space:normal;
85 | margin:0px;
86 | }
87 | table.pre td.lines {
88 | width:30px;
89 | }
90 |
91 | /*--------------------------------------------------------------------------
92 | Formatting for page & standard document content
93 | /*--------------------------------------------------------------------------*/
94 |
95 | body {
96 | font-family: 'Open Sans', serif;
97 | padding-top: 0px;
98 | padding-bottom: 40px;
99 | }
100 |
101 | pre {
102 | word-wrap: inherit;
103 | }
104 |
105 | /* Format the heading - nicer spacing etc. */
106 | .masthead {
107 | overflow: hidden;
108 | }
109 | .masthead .muted a {
110 | text-decoration:none;
111 | color:#999999;
112 | }
113 | .masthead ul, .masthead li {
114 | margin-bottom:0px;
115 | }
116 | .masthead .nav li {
117 | margin-top: 15px;
118 | font-size:110%;
119 | }
120 | .masthead h3 {
121 | margin-bottom:5px;
122 | font-size:170%;
123 | }
124 | hr {
125 | margin:0px 0px 20px 0px;
126 | }
127 |
128 | /* Make table headings and td.title bold */
129 | td.title, thead {
130 | font-weight:bold;
131 | }
132 |
133 | /* Format the right-side menu */
134 | #menu {
135 | margin-top:50px;
136 | font-size:11pt;
137 | padding-left:20px;
138 | }
139 |
140 | #menu .nav-header {
141 | font-size:12pt;
142 | color:#606060;
143 | margin-top:20px;
144 | }
145 |
146 | #menu li {
147 | line-height:25px;
148 | }
149 |
150 | /* Change font sizes for headings etc. */
151 | #main h1 { font-size: 26pt; margin:10px 0px 15px 0px; font-weight:400; }
152 | #main h2 { font-size: 20pt; margin:20px 0px 0px 0px; font-weight:400; }
153 | #main h3 { font-size: 14pt; margin:15px 0px 0px 0px; font-weight:600; }
154 | #main p { font-size: 11pt; margin:5px 0px 15px 0px; }
155 | #main ul { font-size: 11pt; margin-top:10px; }
156 | #main li { font-size: 11pt; margin: 5px 0px 5px 0px; }
157 | #main strong { font-weight:700; }
158 |
159 | /*--------------------------------------------------------------------------
160 | Formatting for API reference
161 | /*--------------------------------------------------------------------------*/
162 |
163 | .type-list .type-name, .module-list .module-name {
164 | width:25%;
165 | font-weight:bold;
166 | }
167 | .member-list .member-name {
168 | width:35%;
169 | }
170 | #main .xmldoc h2 {
171 | font-size:14pt;
172 | margin:10px 0px 0px 0px;
173 | }
174 | #main .xmldoc h3 {
175 | font-size:12pt;
176 | margin:10px 0px 0px 0px;
177 | }
178 | .github-link {
179 | float:right;
180 | text-decoration:none;
181 | }
182 | .github-link img {
183 | border-style:none;
184 | margin-left:10px;
185 | }
186 | .github-link .hover { display:none; }
187 | .github-link:hover .hover { display:block; }
188 | .github-link .normal { display: block; }
189 | .github-link:hover .normal { display: none; }
190 |
191 | /*--------------------------------------------------------------------------
192 | Links
193 | /*--------------------------------------------------------------------------*/
194 |
195 | h1 a, h1 a:hover, h1 a:focus,
196 | h2 a, h2 a:hover, h2 a:focus,
197 | h3 a, h3 a:hover, h3 a:focus,
198 | h4 a, h4 a:hover, h4 a:focus,
199 | h5 a, h5 a:hover, h5 a:focus,
200 | h6 a, h6 a:hover, h6 a:focus { color : inherit; text-decoration : inherit; outline:none }
201 |
202 | /*--------------------------------------------------------------------------
203 | Additional formatting for the homepage
204 | /*--------------------------------------------------------------------------*/
205 |
206 | #nuget {
207 | margin-top:20px;
208 | font-size: 11pt;
209 | padding:20px;
210 | }
211 |
212 | #nuget pre {
213 | font-size:11pt;
214 | -moz-border-radius: 0px;
215 | -webkit-border-radius: 0px;
216 | border-radius: 0px;
217 | background: #404040;
218 | border-style:none;
219 | color: #e0e0e0;
220 | margin-top:15px;
221 | }
--------------------------------------------------------------------------------
/docs/input/vector-operations.fsx:
--------------------------------------------------------------------------------
1 | (*** hide ***)
2 | #r "../../src/FsAlg/bin/Debug/FsAlg.dll"
3 |
4 | (**
5 | Vector Operations
6 | =================
7 |
8 | This page lists only a selection of commonly used vector operations provided by FsAlg.
9 |
10 | Please refer to [API Reference](reference/index.html) for a full list of supported operations.
11 |
12 | The **FsAlg.Generic.Vector** module provides functionality similar to the F# [Collecitons.Array](https://msdn.microsoft.com/en-us/library/ee370273.aspx) module, for creating and manipulating vectors.
13 |
14 | Creating Vectors
15 | ----------------
16 | *)
17 |
18 | open FsAlg.Generic
19 |
20 | let v1 = vector [1.; 2.; 3.]
21 | let v2 = Vector.create 3 1.
22 | let v3 = Vector.init 3 (fun i -> exp (float i))
23 |
24 | (*** hide, define-output: o ***)
25 | printf "val v1 : Vector = Vector [|1.0; 2.0; 3.0|]
26 | val v2 : Vector = Vector [|1.0; 1.0; 1.0|]
27 | val v3 : Vector = Vector [|1.0; 2.718281828; 7.389056099|]"
28 | (*** include-output: o ***)
29 |
30 | (**
31 | Basic Operations
32 | ----------------
33 | *)
34 |
35 | let v4 = v1 + v2 // Vector addition
36 | let v5 = v1 - v2 // Vector substraction
37 | let v6 = v1 * v2 // Vector inner product (dot / scalar product)
38 | let v7 = v1 %* v2 // Vector cross product
39 | let v8 = v1 .* v2 // Vector element-wise (Hadamard) product
40 | let v9 = v1 ./ v2 // Vector element-wise (Hadamard) division
41 | let v10 = v1 + 2. // Add scalar to vector
42 | let v11 = v1 - 2. // Subtract scalar from vector
43 | let v12 = 2. - v1 // Subtract each element of vector from scalar
44 | let v13 = v1 * 2. // Vector-scalar multiplication
45 | let v14 = v1 / 2. // Vector-scalar division
46 | let v15 = 2. / v1 // Divides each element of vector by scalar
47 | let v16 = -v1 // Unary negation
48 |
49 | (*** hide, define-output: o2 ***)
50 | printf "val v4 : Vector = Vector [|2.0; 3.0; 4.0|]
51 | val v5 : Vector = Vector [|0.0; 1.0; 2.0|]
52 | val v6 : float = 6.0
53 | val v7 : Vector = Vector [|-1.0; 2.0; -1.0|]
54 | val v8 : Vector = Vector [|1.0; 2.0; 3.0|]
55 | val v9 : Vector = Vector [|1.0; 2.0; 3.0|]
56 | val v10 : Vector = Vector [|3.0; 4.0; 5.0|]
57 | val v11 : Vector = Vector [|-1.0; 0.0; 1.0|]
58 | val v12 : Vector = Vector [|1.0; 0.0; -1.0|]
59 | val v13 : Vector = Vector [|2.0; 4.0; 6.0|]
60 | val v14 : Vector = Vector [|0.5; 1.0; 1.5|]
61 | val v15 : Vector = Vector [|2.0; 1.0; 0.6666666667|]
62 | val v16 : Vector = Vector [|-1.0; -2.0; -3.0|]"
63 | (*** include-output: o2 ***)
64 |
65 | (**
66 | Vector Norms
67 | ------------
68 | *)
69 |
70 | let n1 = Vector.l1norm v1 // L1 norm
71 | let n2 = Vector.l2norm v1 // L2 norm
72 | let n3 = Vector.l2normSq v1 // Squared L2 norm
73 | let n4 = Vector.lpnorm 3. v1 // Lp norm
74 |
75 | (*** hide, define-output: o3 ***)
76 | printf "val n1 : float = 6.0
77 | val n2 : float = 3.741657387
78 | val n3 : float = 14.0
79 | val n4 : float = 3.301927249"
80 | (*** include-output: o3 ***)
81 |
82 | (**
83 | Accessing Elements & Conversions
84 | --------------------------------
85 | *)
86 |
87 | let e1 = v1.[0] // 1st element of v1
88 | let e2 = Vector.get v1 1 // 2nd element of v1
89 | let e3 = v1.[..1] // Slice, until 2nd element
90 | let e4 = v1.[1..2] // Slice, between 2nd and 3rd elements
91 | let v20 = Vector.ofSeq [1.; 2.; 3.] // Convert sequence to vector
92 | let v21 = vector [1.; 2.; 3.] // Same with above, supports lists,
93 | let v22 = vector [|1.; 2.; 3.|] // arrays,
94 | let v23 = vector (seq {yield 1.}) // and sequences.
95 | let aa1 = Vector.toArray v1 // Convert vector to array
96 | let sq1 = Vector.toSeq v1 // Convert vector to sequence
97 |
98 | (*** hide, define-output: o4 ***)
99 | printf "val e1 : float = 1.0
100 | val e2 : float = 2.0
101 | val e3 : Vector = Vector [|1.0; 2.0|]
102 | val e4 : Vector = Vector [|2.0; 3.0|]
103 | val v20 : Vector = Vector [|1.0; 2.0; 3.0|]
104 | val v21 : Vector = Vector [|1.0; 2.0; 3.0|]
105 | val v22 : Vector = Vector [|1.0; 2.0; 3.0|]
106 | val v23 : Vector = Vector [|1.0|]
107 | val aa1 : float [] = [|1.0; 2.0; 3.0|]
108 | val sq1 : seq"
109 | (*** include-output: o4 ***)
110 |
111 | (**
112 | Mutating/Replacing Elements
113 | ------------------
114 | *)
115 | v1.[0] <- 4. // Mutate an element
116 | Vector.replace (fun x -> x + 2.) v1 // Replace by mapping a function, mutating in place
117 | Vector.replacei (fun i x -> x + float i) v1 // Replace in place, with index
118 | Vector.replace2 (fun x y -> x - y) v1 v2 // Replace v1 in place, using a function of v1 and v2
119 | Vector.replacei2 (fun i x y -> x - y + float i) v1 v2 // Replace v1 in place, with index
120 | Vector.replaceWith v1 v2 // Replace elements of v1 with v2, mutating in place
121 |
122 | (**
123 | Splitting and Concatenating
124 | ---------------------------
125 | *)
126 |
127 | let ss1 = Vector.splitEqual 3 (vector [1.; 2.; 3.; 4.; 5.; 6.]) // Split into 3 vectors of equal length
128 | let ss2 = Vector.split [2; 4] (vector [1.; 2.; 3.; 4.; 5.; 6.]) // Split into vectors of given lengths
129 | let cc1 = Vector.concat ss1 // Concatenate vectors into one
130 |
131 | (*** hide, define-output: o5 ***)
132 | printf "val ss1 : seq>
133 | val ss2 : seq>
134 | val cc1 : Vector = Vector [|1.0; 2.0; 3.0; 4.0; 5.0; 6.0|]"
135 | (*** include-output: o5 ***)
136 |
137 | (**
138 | Mathematica and MATLAB Strings
139 | ------------------------------
140 |
141 | You can generate string representations of vectors that you can copy and paste into Mathematica notebooks or MATLAB.
142 | *)
143 |
144 | let s1 = v1.ToMathematicaString()
145 | let s2 = v1.ToMatlabString()
146 |
147 | (*** hide, define-output: o6 ***)
148 | printf "val s1 : string = \"{1.00, 2.00, 3.00}\"
149 | val s2 : string = \"[1.00 2.00 3.00]\""
150 | (*** include-output: o6 ***)
151 |
152 | (**
153 | Other Operations
154 | ---------------------
155 | The following are just a selection of other operations. Please refer to [API Reference](reference/index.html) for a full list of supported operations.
156 | *)
157 |
158 | let len = Vector.length v1 // Length of vector
159 | let min = Vector.min v1 // Minimum element of vector
160 | let max = Vector.max v1 // Maximum element of vector
161 | let sum = Vector.sum v1 // Sum elements of vector
162 | let v17 = Vector.unitVector v1 // Get unit vector codirectional with v1
163 | let v18 = Vector.map (fun x -> sin x) v1 // Map function to vector
164 | let v19 = Vector.fold (fun s x -> s + sin x) 0. v1 // Fold a vector
165 |
166 | (*** hide, define-output: o7 ***)
167 | printf "val len : int = 3
168 | val min : float = 1.0
169 | val max : float = 3.0
170 | val sum : float = 6.0
171 | val v17 : Vector = Vector [|0.2672612419; 0.5345224838; 0.8017837257|]
172 | val v18 : Vector = Vector [|0.8414709848; 0.9092974268; 0.1411200081|]
173 | val v19 : float = 1.89188842"
174 | (*** include-output: o7 ***)
--------------------------------------------------------------------------------
/docs/input/matrix-operations.fsx:
--------------------------------------------------------------------------------
1 | (*** hide ***)
2 | #r "../../src/FsAlg/bin/Debug/FsAlg.dll"
3 |
4 | (**
5 | Matrix Operations
6 | =================
7 |
8 | This page lists only a selection of commonly used matrix operations provided by FsAlg.
9 |
10 | Please refer to [API Reference](reference/index.html) for a full list of supported operations.
11 |
12 | The **FsAlg.Generic.Matrix** module provides functionality similar to the F# [Collecitons.Array2D](https://msdn.microsoft.com/en-us/library/ee353794.aspx) module, for creating and manipulating matrices.
13 |
14 | Creating Matrices
15 | -----------------
16 | *)
17 |
18 | open FsAlg.Generic
19 |
20 | let m1 = matrix [[1.; 2.]; [3.; 4.]]
21 | let m2 = Matrix.create 2 2 1.
22 | let m3 = Matrix.init 2 2 (fun i j -> exp (float (i + j)))
23 |
24 | (*** hide, define-output: o ***)
25 | printf "val m1 : Matrix = Matrix [[1.0; 1.0]
26 | [1.0; 1.0]]
27 | val m2 : Matrix = Matrix [[1.0; 1.0]
28 | [1.0; 1.0]]
29 | val m3 : Matrix = Matrix [[1.0; 2.718281828]
30 | [2.718281828; 7.389056099]]"
31 | (*** include-output: o ***)
32 |
33 | (**
34 | Basic Operations
35 | ----------------
36 | *)
37 |
38 | let m4 = m1 + m2 // Matrix addition
39 | let m5 = m1 - m2 // Matrix subtraction
40 | let m6 = m1 * m2 // Matrix product
41 | let m7 = m1 .* m2 // Matrix element-wise (Hadamard) product
42 | let m8 = m1 ./ m2 // Matrix element-wise (Hadamard) division
43 |
44 | let v = vector [1.; 2.]
45 | let m9 = m1 * v // Matrix-vector product
46 | let m10 = v * m1 // Vector-matrix product
47 |
48 | let m11 = m1 + 2. // Add scalar to matrix
49 | let m12 = m1 - 2. // Subtract scalar from matrix
50 | let m13 = 2. - m1 // Subtract each element of matrix from scalar
51 | let m14 = m1 * 2. // Matrix-scalar multiplication
52 | let m15 = m1 / 2. // Matrix-scalar division
53 | let m16 = 2. / m1 // Divides each element of matrix by scalar
54 | let m17 = -m1 // Unary negation
55 |
56 | (*** hide, define-output: o2 ***)
57 | printf "val m4 : Matrix = Matrix [[2.0; 3.0]
58 | [4.0; 5.0]]
59 | val m5 : Matrix = Matrix [[0.0; 1.0]
60 | [2.0; 3.0]]
61 | val m6 : Matrix = Matrix [[3.0; 3.0]
62 | [7.0; 7.0]]
63 | val m7 : Matrix = Matrix [[1.0; 2.0]
64 | [3.0; 4.0]]
65 | val m8 : Matrix = Matrix [[1.0; 2.0]
66 | [3.0; 4.0]]
67 | val v : Vector = Vector [|1.0; 2.0|]
68 | val m9 : Vector = Vector [|5.0; 11.0|]
69 | val m10 : Vector = Vector [|7.0; 10.0|]
70 | val m11 : Matrix = Matrix [[3.0; 4.0]
71 | [5.0; 6.0]]
72 | val m12 : Matrix = Matrix [[-1.0; 0.0]
73 | [1.0; 2.0]]
74 | val m13 : Matrix = Matrix [[1.0; 0.0]
75 | [-1.0; -2.0]]
76 | val m14 : Matrix = Matrix [[2.0; 4.0]
77 | [6.0; 8.0]]
78 | val m15 : Matrix = Matrix [[0.5; 1.0]
79 | [1.5; 2.0]]
80 | val m16 : Matrix = Matrix [[2.0; 1.0]
81 | [0.6666666667; 0.5]]
82 | val m17 : Matrix = Matrix [[-1.0; -2.0]
83 | [-3.0; -4.0]]"
84 | (*** include-output: o2 ***)
85 |
86 | (**
87 | Matrix Operations
88 | -----------------
89 | *)
90 |
91 | let m1det = Matrix.det m1 // Determinant
92 | let m1inv = Matrix.inverse m1 // Inverse
93 | let m1eig = Matrix.eigenvalues m1 // Eigenvalues
94 | let m1t = Matrix.transpose m1 // Transpose
95 | let m1tr = Matrix.trace m1 // Trace
96 | let m1dia = Matrix.diagonal m1 // Diagonal
97 |
98 | (*** hide, define-output: o3 ***)
99 | printf "val m1det : float = -2.0
100 | val m1inv : Matrix = Matrix [[-2.0; 1.0]
101 | [1.5; -0.5]]
102 | val m1eig : Vector = Vector [|5.372281312; -0.3722813117|]
103 | val m1t : Matrix = Matrix [[1.0; 3.0]
104 | [2.0; 4.0]]
105 | val m1tr : float = 5.0
106 | val m1dia : Vector = Vector [|1.0; 4.0|]"
107 | (*** include-output: o3 ***)
108 |
109 | (**
110 | Solving Systems of Linear Equations
111 | -----------------------------------
112 | We can solve a [system of linear equations](http://en.wikipedia.org/wiki/System_of_linear_equations) represented in matrix form $\mathbf{A} \mathbf{x} = \mathbf{b} \;,$ which has solution $\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}\;$.
113 |
114 | For example, the system
115 |
116 | $$$
117 | \begin{eqnarray*}
118 | 3x + 2y - z &=& 1\\
119 | 2x - 2y +4z &=& -2\\
120 | -x + 0.5y -z &=& 0\\
121 | \end{eqnarray*}
122 |
123 | can be represented as
124 |
125 | $$$
126 | \begin{bmatrix}
127 | 3 & 2 & -1\\
128 | 2 & -2 & 4\\
129 | -1& 0.5 &-1\\
130 | \end{bmatrix} \begin{bmatrix}
131 | x\\
132 | y\\
133 | z\\
134 | \end{bmatrix} = \begin{bmatrix}
135 | 1\\
136 | -2\\
137 | 0\\
138 | \end{bmatrix}
139 |
140 | and has solution
141 |
142 | $$$
143 | \begin{eqnarray*}
144 | x &=& 1\\
145 | y &=& -2\\
146 | z &=& -2\\
147 | \end{eqnarray*}\; .
148 | *)
149 |
150 | let A = matrix [[3.; 2.; -1.]; [2.; -2.; 4.]; [-1.; 0.5; -1.]]
151 | let b = vector [1.; -2.; 0.]
152 |
153 | let x = Matrix.solve A b
154 |
155 | (*** hide, define-output: o4 ***)
156 | printf "val A : Matrix = Matrix [[3.0; 2.0; -1.0]
157 | [2.0; -2.0; 4.0]
158 | [-1.0; 0.5; -1.0]]
159 | val b : Vector = Vector [|1.0; -2.0; 0.0|]
160 | val x : Vector = Vector [|1.0; -2.0; -2.0|]"
161 | (*** include-output: o4 ***)
162 |
163 | (**
164 | Matrix Decompositions
165 | ---------------------
166 | *)
167 |
168 | let lu, piv, togg = Matrix.decomposeLU m1 // LU decomposition
169 | let q, r = Matrix.decomposeQR m1 // QR decomposition
170 |
171 | (*** hide, define-output: o5 ***)
172 | printf "val togg : float = -1.0
173 | val piv : int [] = [|1; 0|]
174 | val lu : Matrix = Matrix [[3.0; 4.0]
175 | [0.3333333333; 0.6666666667]]
176 | val r : Matrix = Matrix [[3.16227766; 4.427188724]
177 | [4.440892099e-16; 0.632455532]]
178 | val q : Matrix = Matrix [[0.316227766; 0.9486832981]
179 | [0.9486832981; -0.316227766]]"
180 | (*** include-output: o5 ***)
181 |
182 | (**
183 | Accessing Elements & Conversions
184 | --------------------------------
185 | *)
186 |
187 | let el1 = m1.[0, 0] // Element at 0, 0
188 | let el2 = Matrix.get m1 1 1 // Element at 1, 1
189 | let sl1 = m1.[0..1, 1..] // Slice, between rows 0 and 1, columns 1 and beyond
190 | let sl2 = m1.[*, 0..1] // Slice, all rows, between columns 0 and 1
191 | let sl3 = m1.[0, *] // Return row 0
192 | let sl4 = Matrix.row 0 m1 // Return row 0
193 | let sl5 = m1.[*, 1] // Return column 1
194 | let sl6 = Matrix.col 1 m1 // Return column 1
195 | let sl7 = Matrix.toRows m1 // Return rows as a sequence
196 | let m18 = Matrix.ofRows [vector [1.;2.]; vector [3.;4.]] // Create matrix from row vectors
197 | let sl8 = Matrix.toCols m1 // Return columns as a sequence
198 | let m19 = Matrix.ofCols [vector [1.;2.]; vector [3.;4.]] // Create matrix from column vectors
199 | let m20 = Matrix.ofSeqSeq [[1.; 2.]; [3.; 4.]] // Convert sequence of sequences to matrix
200 | let m21 = matrix [[1.; 2.]; [3.; 4.]] // Same with above, supports lists,
201 | let m22 = matrix [|[|1.; 2.|]; [|3.; 4.|]|] // arrays,
202 | let m23 = matrix (seq {yield seq {yield 1.}}) // and sequences.
203 | let aa1 = Matrix.toArrayArray m1 // Convert matrix to jagged array
204 | let aa2 = Matrix.toArray2D m1 // Convert matrix to 2d array
205 | let aa3 = Matrix.toArray m1 // Convert matrix to 1d array, scanning left to right, top to bottom
206 | let ma3 = Matrix.ofArray 2 aa3 // Convert 1d array to matrix, populating two rows
207 | let vv1 = Matrix.toVector m1 // Convert matrix to vector, scanning left to right, top to bottom
208 | let mv1 = Matrix.ofVector 2 vv1 // Convert vector to matrix, populating two rows
209 |
210 | (*** hide, define-output: o6 ***)
211 | printf "val el1 : float = 1.0
212 | val el2 : float = 4.0
213 | val sl1 : Matrix = Matrix [[2.0]
214 | [4.0]]
215 | val sl2 : Matrix = Matrix [[1.0; 2.0]
216 | [3.0; 4.0]]
217 | val sl3 : Matrix = Matrix [[1.0; 2.0]]
218 | val sl4 : Matrix = Matrix [[1.0; 2.0]]
219 | val sl5 : Matrix = Matrix [[2.0]
220 | [4.0]]
221 | val sl6 : Matrix = Matrix [[2.0]
222 | [4.0]]
223 | val sl7 : seq>
224 | val m18 : Matrix = Matrix [[1.0; 2.0]
225 | [3.0; 4.0]]
226 | val sl8 : seq>
227 | val m19 : Matrix = Matrix [[1.0; 3.0]
228 | [2.0; 4.0]]
229 | val m20 : Matrix = Matrix [[1.0; 2.0]
230 | [3.0; 4.0]]
231 | val m21 : Matrix = Matrix [[1.0; 2.0]
232 | [3.0; 4.0]]
233 | val m22 : Matrix = Matrix [[1.0; 2.0]
234 | [3.0; 4.0]]
235 | val m23 : Matrix = Matrix [[1.0]]
236 | val aa1 : float [] [] = [|[|1.0; 2.0|]; [|3.0; 4.0|]|]
237 | val aa2 : float [,] = [[1.0; 1.0]
238 | [1.0; 1.0]]
239 | val aa3 : float [] = [|1.0; 2.0; 3.0; 4.0|]
240 | val ma3 : Matrix = Matrix [[1.0; 2.0]
241 | [3.0; 4.0]]
242 | val vv1 : Vector = Vector [|1.0; 2.0; 3.0; 4.0|]
243 | val mv1 : Matrix = Matrix [[1.0; 2.0]
244 | [3.0; 4.0]]"
245 | (*** include-output: o6 ***)
246 |
247 |
248 | (**
249 | Mutating/Replacing Elements
250 | ------------------
251 | *)
252 |
253 | m1.[0, 0] <- 4. // Mutate an element
254 | Matrix.replace (fun x -> x + 2.) m1 // Replace by mapping a function, mutating in place
255 | Matrix.replacei (fun i j x -> x + float (i * j)) m1 // Replace in place, with index
256 | Matrix.replace2 (fun x y -> x - y) m1 m2 // Replace m1 in place, using a function of m1 and m2
257 | Matrix.replacei2 (fun i j x y -> x - y + float (i * j)) m1 m2 // Replace m1 in place, with index
258 | Matrix.replaceWith m1 m2 // Replace elements of m1 with m2, mutating in place
259 | Matrix.appendRow v m1 // Append a vector to a matrix as a new row
260 | Matrix.appendCol v m1 // Append a vector to a matrix as a new column
261 | Matrix.prependRow v m1 // Prepend a vector to a matrix as a new row
262 | Matrix.prependCol v m1 // Prepend a vector to a matrix as a new column
263 |
264 | (**
265 | Mathematica and MATLAB Strings
266 | ------------------------------
267 | You can generate string representations of matrices that you can copy and paste into Mathematica notebooks or MATLAB.
268 | *)
269 |
270 | let s1 = m1.ToMathematicaString()
271 | let s2 = m2.ToMatlabString()
272 |
273 | (*** hide, define-output: o7 ***)
274 | printf "val s1 : string = \"{{1.00, 2.00}, {3.00, 4.00}}\"
275 | val s2 : string = \"[1.00 1.00; 1.00 1.00]\""
276 | (*** include-output: o7 ***)
277 |
278 | (**
279 | Other Operations
280 | ---------------------
281 | The following are just a selection of other operations. Please refer to [API Reference](reference/index.html) for a full list of supported operations.
282 | *)
283 |
284 | let l1 = Matrix.rows m1 // Number of rows
285 | let l2 = Matrix.cols m1 // Number of columns
286 | let m24 = Matrix.map (fun x -> sin x) m1 // Map function to matrix
287 |
288 | (*** hide, define-output: o8 ***)
289 | printf "val l1 : int = 2
290 | val l2 : int = 2
291 | val m24 : Matrix = Matrix [[0.8414709848; 0.8414709848]
292 | [0.8414709848; 0.8414709848]]"
293 | (*** include-output: o8 ***)
294 |
--------------------------------------------------------------------------------
/src/FsAlg/Vector.fs:
--------------------------------------------------------------------------------
1 | //
2 | // This file is part of
3 | // FsAlg: Generic Linear Algebra Library
4 | //
5 | // Copyright (c) 2015, National University of Ireland Maynooth (Atilim Gunes Baydin, Barak A. Pearlmutter)
6 | //
7 | // FsAlg is released under the BSD license.
8 | // (See accompanying LICENSE file.)
9 | //
10 | // Written by:
11 | //
12 | // Atilim Gunes Baydin
13 | // atilimgunes.baydin@nuim.ie
14 | //
15 | // Barak A. Pearlmutter
16 | // barak@cs.nuim.ie
17 | //
18 | // Brain and Computation Lab
19 | // Hamilton Institute & Department of Computer Science
20 | // National University of Ireland Maynooth
21 | // Maynooth, Co. Kildare
22 | // Ireland
23 | //
24 | // www.bcl.hamilton.ie
25 | //
26 |
27 | namespace FsAlg.Generic
28 |
29 | open FsAlg.Generic.Util
30 |
31 | /// Generic vector type
32 | []
33 | type Vector<'T when 'T : (static member Zero : 'T)
34 | and 'T : (static member One : 'T)
35 | and 'T : (static member (+) : 'T * 'T -> 'T)
36 | and 'T : (static member (-) : 'T * 'T -> 'T)
37 | and 'T : (static member (*) : 'T * 'T -> 'T)
38 | and 'T : (static member (/) : 'T * 'T -> 'T)
39 | and 'T : (static member (~-) : 'T -> 'T)
40 | and 'T : (static member Abs : 'T -> 'T)
41 | and 'T : (static member Pow : 'T * 'T -> 'T)
42 | and 'T : (static member Sqrt : 'T -> 'T)
43 | and 'T : (static member op_Explicit : 'T -> float)
44 | and 'T : comparison> =
45 | | ZeroVector of 'T
46 | | Vector of 'T[]
47 | /// ZeroVector
48 | static member inline Zero = ZeroVector LanguagePrimitives.GenericZero<'T>
49 | /// Converts vector `v` to float[]
50 | static member inline op_Explicit(v:Vector<'T>) =
51 | match v with
52 | | Vector v -> Array.map float v
53 | | ZeroVector _ -> [||]
54 | /// The element of this vector at the given position `i`
55 | member inline v.Item
56 | with get i =
57 | match v with
58 | | Vector v -> v.[i]
59 | | ZeroVector z -> z
60 | and set i vv =
61 | match v with
62 | | Vector v -> v.[i] <- vv
63 | | ZeroVector _ -> ()
64 | /// Gets a subvector between bounds `lower` and `upper`
65 | member inline v.GetSlice(lower, upper) =
66 | match v with
67 | | Vector v ->
68 | let l = defaultArg lower 0
69 | let u = defaultArg upper (v.Length - 1)
70 | if l > u then invalidArg "" "Given slice bounds are invalid."
71 | Vector v.[l..u]
72 | | ZeroVector _ -> invalidArg "" "Cannot get slice of a ZeroVector."
73 | /// Gets the first element of this vector
74 | member inline v.FirstItem =
75 | match v with
76 | | Vector v -> v.[0]
77 | | ZeroVector z -> z
78 | /// Gets the total number of elements of this vector
79 | member inline v.Length =
80 | match v with
81 | | Vector v -> v.Length
82 | | ZeroVector _ -> 0
83 | /// Gets the L1 (Manhattan) norm of this vector
84 | member inline v.GetL1Norm() =
85 | match v with
86 | | Vector v -> Array.sumBy abs v
87 | | ZeroVector z -> z
88 | /// Gets the L2 (Euclidean) norm of this vector
89 | member inline v.GetL2Norm() =
90 | match v with
91 | | Vector v -> sqrt (Array.sumBy (fun x -> x * x) v)
92 | | ZeroVector z -> z
93 | /// Gets the squared L2 (Euclidean) norm of this vector
94 | member inline v.GetL2NormSq() =
95 | match v with
96 | | Vector v -> Array.sumBy (fun x -> x * x) v
97 | | ZeroVector z -> z
98 | /// Gets the Lp norm (or p-norm) of this vector, with the given `p`
99 | member inline v.GetLPNorm(p:'T):'T =
100 | match v with
101 | | Vector v -> (Array.sumBy (fun x -> (abs x) ** p) v) ** (LanguagePrimitives.GenericOne<'T> / p)
102 | | ZeroVector z -> z
103 | /// Gets the minimum element of this vector
104 | member inline v.GetMin() =
105 | match v with
106 | | Vector v -> Array.min v
107 | | ZeroVector z -> z
108 | /// Gets the minimum element of this vector, compared by using Operators.min on the result of function `f`
109 | member inline v.GetMinBy(f) =
110 | match v with
111 | | Vector v -> Array.minBy f v
112 | | ZeroVector z -> z
113 | /// Gets the maximum element of this vector
114 | member inline v.GetMax() =
115 | match v with
116 | | Vector v -> Array.max v
117 | | ZeroVector z -> z
118 | /// Gets the maximum element of this vector, compared by using Operators.max on the result of function `f`
119 | member inline v.GetMaxBy(f) =
120 | match v with
121 | | Vector v -> Array.maxBy f v
122 | | ZeroVector z -> z
123 | /// Gets the unit vector codirectional with this vector
124 | member inline v.GetUnitVector() =
125 | match v with
126 | | Vector vv -> let n = v.GetL2Norm() in Vector (Array.map (fun x -> x / n) vv)
127 | | ZeroVector z -> ZeroVector z
128 | /// Returns a sequence of vectors that are obtained by splitting this vector into subvectors whose lengths are given in sequence `n`
129 | member inline v.Split(n:seq) =
130 | match v with
131 | | Vector v ->
132 | let i = ref 0
133 | seq {for j in n do
134 | yield Array.sub v !i j |> Vector
135 | i := !i + j}
136 | | ZeroVector _ -> seq {yield v}
137 | /// Returns a sequence of vectors that are obtained by splitting this vector into `n` subvectors of equal length. The length of this vector must be an integer multiple of `n`, otherwise ArgumentException is raised.
138 | member inline v.SplitEqual(n:int) =
139 | match v with
140 | | Vector v ->
141 | if n <= 0 then invalidArg "" "For splitting this vector, n should be a positive integer."
142 | let l = (float v.Length) / (float n)
143 | if not (isInteger l) then invalidArg "" "Cannot split vector into n equal pieces when length of vector is not an integer multiple of n."
144 | seq {for i in 0 .. (int l) .. (v.Length - 1) do yield Vector (Array.sub v i (int l))}
145 | | ZeroVector _ -> seq {yield v}
146 | /// Gets a string representation of this vector that can be pasted into a Mathematica notebook
147 | member inline v.ToMathematicaString() =
148 | let sb = System.Text.StringBuilder()
149 | sb.Append("{") |> ignore
150 | for i = 0 to v.Length - 1 do
151 | sb.Append(sprintf "%.2f" (float v.[i])) |> ignore
152 | if i < v.Length - 1 then sb.Append(", ") |> ignore
153 | sb.Append("}") |> ignore
154 | sb.ToString()
155 | /// Gets a string representation of this vector that can be pasted into MATLAB
156 | member inline v.ToMatlabString() =
157 | let sb = System.Text.StringBuilder()
158 | sb.Append("[") |> ignore
159 | for i = 0 to v.Length - 1 do
160 | sb.Append(sprintf "%.2f" (float v.[i])) |> ignore
161 | if i < v.Length - 1 then sb.Append(" ") |> ignore
162 | sb.Append("]") |> ignore
163 | sb.ToString()
164 | /// Converts the elements of this vector to another type, using the given conversion function `f`
165 | member inline v.Convert(f:'T->'a):Vector<'a> =
166 | match v with
167 | | Vector v -> Vector (Array.map f v)
168 | | ZeroVector _ -> ZeroVector LanguagePrimitives.GenericZero<'a>
169 | /// Creates a copy of this vector
170 | member inline v.Copy() =
171 | match v with
172 | | Vector v -> Vector (Array.copy v)
173 | | ZeroVector z -> ZeroVector z
174 | /// Converts this vector to an array
175 | member inline v.ToArray() =
176 | match v with
177 | | Vector v -> v
178 | | ZeroVector _ -> [||]
179 | /// Converts this vector to a sequence
180 | member inline v.ToSeq() =
181 | match v with
182 | | Vector v -> Array.toSeq v
183 | | ZeroVector _ -> Seq.empty
184 | /// Creates a new vector that contains the given subrange of elements, specified by start index `s` and count `c`
185 | member inline v.GetSubVector(s, c) =
186 | match v with
187 | | Vector v -> Vector (Array.sub v s c)
188 | | ZeroVector _ -> Vector.Zero
189 | /// Returns an enumerator that iterates through the elements of this vector
190 | member inline v.GetEnumerator() =
191 | v.ToSeq().GetEnumerator()
192 | /// Adds vector `a` to vector `b`
193 | static member inline (+) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
194 | match a, b with
195 | | Vector a, Vector b -> try Vector (Array.map2 (+) a b) with | _ -> invalidArg "" "Cannot add two vectors of different dimensions."
196 | | Vector _, ZeroVector _ -> a
197 | | ZeroVector _, Vector _ -> b
198 | | ZeroVector _, ZeroVector _ -> Vector.Zero
199 | /// Subtracts vector `b` from vector `a`
200 | static member inline (-) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
201 | match a, b with
202 | | Vector a, Vector b -> try Vector (Array.map2 (-) a b) with | _ -> invalidArg "" "Cannot subtract two vectors of different dimensions."
203 | | Vector _, ZeroVector _ -> a
204 | | ZeroVector _, Vector b -> Vector (Array.map (~-) b)
205 | | ZeroVector _, ZeroVector _ -> Vector.Zero
206 | /// Computes the inner product (dot / scalar product) of vector `a` and vector `b`
207 | static member inline (*) (a:Vector<'T>, b:Vector<'T>):'T =
208 | match a, b with
209 | | Vector a, Vector b -> try Array.map2 (*) a b |> Array.sum with | _ -> invalidArg "" "Cannot multiply two vectors of different dimensions."
210 | | Vector _, ZeroVector _ -> LanguagePrimitives.GenericZero<'T>
211 | | ZeroVector _, Vector _ -> LanguagePrimitives.GenericZero<'T>
212 | | ZeroVector _, ZeroVector _ -> LanguagePrimitives.GenericZero<'T>
213 | /// Computes the cross product of vector `a` and vector `b` (three-dimensional)
214 | static member inline (%*) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
215 | match a, b with
216 | | Vector va, Vector vb ->
217 | if (a.Length <> 3) || (b.Length <> 3) then invalidArg "" "The cross product is only defined for three-dimensional vectors."
218 | Vector [|va.[1] * vb.[2] - va.[2] * vb.[1]; va.[2] * vb.[0] - va.[0] * vb.[2]; va.[0] * vb.[1] - va.[1] * vb.[0]|]
219 | | Vector _, ZeroVector _ -> Vector.Zero
220 | | ZeroVector _, Vector _ -> Vector.Zero
221 | | ZeroVector _, ZeroVector _ -> Vector.Zero
222 | /// Multiplies vector `a` and vector `b` element-wise (Hadamard product)
223 | static member inline (.*) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
224 | match a, b with
225 | | Vector a, Vector b -> try Vector (Array.map2 (*) a b) with | _ -> invalidArg "" "Cannot multiply two vectors of different dimensions."
226 | | Vector _, ZeroVector _ -> Vector.Zero
227 | | ZeroVector _, Vector _ -> Vector.Zero
228 | | ZeroVector _, ZeroVector _ -> Vector.Zero
229 | /// Divides vector `a` by vector `b` element-wise (Hadamard division)
230 | static member inline (./) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
231 | match a, b with
232 | | Vector a, Vector b -> try Vector (Array.map2 (/) a b) with | _ -> invalidArg "" "Cannot divide two vectors of different dimensions."
233 | | Vector _, ZeroVector _-> raise (new System.DivideByZeroException("Attempted to divide a Vector by a ZeroVector."))
234 | | ZeroVector _, Vector _ -> Vector.Zero
235 | | ZeroVector _, ZeroVector _ -> raise (new System.DivideByZeroException("Attempted to divide a ZeroVector by a ZeroVector."))
236 | /// Adds scalar `b` to each element of vector `a`
237 | static member inline (+) (a:Vector<'T>, b:'T):Vector<'T> =
238 | match a with
239 | | Vector a -> Vector (Array.map ((+) b) a)
240 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroVector."
241 | /// Adds scalar `a` to each element of vector `b`
242 | static member inline (+) (a:'T, b:Vector<'T>):Vector<'T> =
243 | match b with
244 | | Vector b -> Vector (Array.map ((+) a) b)
245 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroVector."
246 | /// Subtracts scalar `b` from each element of vector `a`
247 | static member inline (-) (a:Vector<'T>, b:'T):Vector<'T> =
248 | match a with
249 | | Vector a -> Vector (Array.map (fun x -> x - b) a)
250 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot subtract a scalar from a ZeroVector."
251 | /// Subtracts each element of vector `b` from scalar `a`
252 | static member inline (-) (a:'T, b:Vector<'T>):Vector<'T> =
253 | match b with
254 | | Vector b -> Vector (Array.map ((-) a) b)
255 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot add subtract a ZeroVector from a scalar."
256 | /// Multiplies each element of vector `a` by scalar `b`
257 | static member inline (*) (a:Vector<'T>, b:'T):Vector<'T> =
258 | match a with
259 | | Vector a -> Vector (Array.map ((*) b) a)
260 | | ZeroVector _ -> Vector.Zero
261 | /// Multiplies each element of vector `b` by scalar `a`
262 | static member inline (*) (a:'T, b:Vector<'T>):Vector<'T> =
263 | match b with
264 | | Vector b -> Vector (Array.map ((*) a) b)
265 | | ZeroVector _ -> Vector.Zero
266 | /// Divides each element of vector `a` by scalar `b`
267 | static member inline (/) (a:Vector<'T>, b:'T):Vector<'T> =
268 | match a with
269 | | Vector a -> Vector (Array.map (fun x -> x / b) a)
270 | | ZeroVector _ -> Vector.Zero
271 | /// Divides scalar `a` by each element of vector `b`
272 | static member inline (/) (a:'T, b:Vector<'T>):Vector<'T> =
273 | match b with
274 | | Vector b -> Vector (Array.map ((/) a) b)
275 | | ZeroVector _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroVector."))
276 | /// Gets the negative of Vector `a`
277 | static member inline (~-) (a:Vector<'T>):Vector<'T> =
278 | match a with
279 | | Vector a -> Vector (Array.map (~-) a)
280 | | ZeroVector _ -> Vector.Zero
281 |
282 |
283 | /// Operations on Vector type. (Implementing functionality similar to Microsoft.FSharp.Collections.Array)
284 | []
285 | module Vector =
286 | /// Createsa vector from array `v`
287 | let inline ofArray (v:'T[]) = Vector v
288 | /// Converts vector `v` to an array
289 | let inline toArray (v:Vector<'T>):'T[] = v.ToArray()
290 | /// Creates a vector from sequence `s`
291 | let inline ofSeq (s:seq<'T>):Vector<'T> = Vector (Array.ofSeq s)
292 | /// Returns vector `v` as a sequence
293 | let inline toSeq (v:Vector<'T>):seq<'T> = v.ToSeq()
294 | /// Creates a vector that contains the elements of vector `v1` followed by the elements of vector `v2`
295 | let inline append (v1:Vector<'T>) (v2:Vector<'T>):Vector<'T> = Array.append (v1 |> toArray) (v2 |> toArray) |> Vector
296 | /// Builds a new vector that contains the elements of each of the given sequence of vectors `v`
297 | let inline concat (v:seq>):Vector<'T> = Seq.map toArray v |> Array.concat |> ofSeq
298 | /// Creates a copy of vector `v`
299 | let inline copy (v:Vector<'T>):Vector<'T> = v.Copy()
300 | /// Creates a vector with `n` elements, all having value `v`
301 | let inline create (n:int) (v:'T):Vector<'T> = Vector (Array.create n v)
302 | /// Creates a vector with `n` elements, where the element with index `i` has value `v` and the rest of the elements have value 0
303 | let inline createBasis (n:int) (i:int) (v:'T):Vector<'T> = Vector (Array.init n (fun j -> if j = i then v else LanguagePrimitives.GenericZero))
304 | /// Tests if any element of vector `v` satisfies predicate `p`
305 | let inline exists (p:'T->bool) (v:Vector<'T>):bool = v |> toArray |> Array.exists p
306 | /// Tests if any pair of corresponding elements of vectors `v1` and `v2` satisfies predicate `p`
307 | let inline exists2 (p:'T1->'T2->bool) (v1:Vector<'T1>) (v2:Vector<'T2>):bool = Array.exists2 p (v1 |> toArray) (v2 |> toArray)
308 | /// Fills a range of elements of vector `v` with value `a` starting with index `s` and counting `c` elements
309 | let inline fill (v:Vector<'T>) (s:int) (c:int) (a:'T):unit = Array.fill (v |> toArray) s c a
310 | /// Returns the first element of vector `v` for which predicate `p` is true
311 | let inline find (p:'T->bool) (v:Vector<'T>):'T = v |> toArray |> Array.find p
312 | /// Returns the index of the first element of vector `v` for which predicate `p` is true
313 | let inline findIndex (p:'T->bool) (v:Vector<'T>):int = v |> toArray |> Array.findIndex p
314 | /// Applies function `f` to each element of vector `v`, threading an accumulator (with initial state `s`) through the computation. If the input function is f and the elements are i0...iN then computes f (... (f s i0)...) iN.
315 | let inline fold (f:'S->'T->'S) (s:'S) (v:Vector<'T>):'S = v |> toArray |> Array.fold f s
316 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2`, threading an accumulator (with initial state `s`) through the computation. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
317 | let inline fold2 (f:'S->'T1->'T2->'S) (s:'S) (v1:Vector<'T1>) (v2:Vector<'T2>):'S = Array.fold2 f s (v1 |> toArray) (v2 |> toArray)
318 | /// Applies function `f` to each element of vector `v`, threading an accumulator (with initial state `s`) through the computation. If the input function is f and the elements are i0...iN then computes f i0 (...(f iN s)).
319 | let inline foldBack (f:'T->'S->'S) (v:Vector<'T>) (s:'S):'S = Array.foldBack f (v |> toArray) s
320 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2`, threading an accumulator (with initial state `s`) through the computation. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
321 | let inline foldBack2 (f:'T1->'T2->'S->'S) (v1:Vector<'T1>) (v2:Vector<'T2>) (s:'S):'S = Array.foldBack2 f (v1 |> toArray) (v2 |> toArray) s
322 | /// Tests if all elements of vector `v` satisfy predicate `p`
323 | let inline forall (p:'T->bool) (v:Vector<'T>):bool = v |> toArray |> Array.forall p
324 | /// Tests if all corresponding elements of vectors `v1` and `v2` satisfy predicate `p` pairwise
325 | let inline forall2 (p:'T1->'T2->bool) (v1:Vector<'T1>) (v2:Vector<'T2>):bool = Array.forall2 p (v1 |> toArray) (v2 |> toArray)
326 | /// Gets the element of vector `v` with index `i`
327 | let inline get (v:Vector<'T>) (i:int):'T = v.[i]
328 | /// Creates a vector with dimension `n` and a generator function `f` to compute the elements
329 | let inline init (n:int) (f:int->'T):Vector<'T> = Vector (Array.init n f)
330 | /// Applies function `f` to each element of vector `v`
331 | let inline iter (f:'T->unit) (v:Vector<'T>):unit = v |> toArray |> Array.iter f
332 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
333 | let inline iter2 (f:'T1->'T2->unit) (v1:Vector<'T1>) (v2:Vector<'T2>):unit = Array.iter2 f (v1 |> toArray) (v2 |> toArray)
334 | /// Applies function `f` to each element of vector `v`. The integer passed to function `f` indicates the index of element.
335 | let inline iteri (f:int->'T->unit) (v:Vector<'T>):unit = v |> toArray |> Array.iteri f
336 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The integer passed to function `f` indicates the index of element. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
337 | let inline iteri2 (f:int->'T1->'T2->unit) (v1:Vector<'T1>) (v2:Vector<'T2>):unit = Array.iteri2 f (v1 |> toArray) (v2 |> toArray)
338 | /// Gets the L1 (Manhattan) norm of vector `v`
339 | let inline l1norm (v:Vector<'T>):'T = v.GetL1Norm()
340 | /// Gets the L2 (Euclidean) norm of vector `v`. This is the same with `Vector.norm`.
341 | let inline l2norm (v:Vector<'T>):'T = v.GetL2Norm()
342 | /// Gets the squared L2 (Euclidean) norm of vector `v`. This is the same with `Vector.normSq`.
343 | let inline l2normSq (v:Vector<'T>):'T = v.GetL2NormSq()
344 | /// Returns the length of vector `v`
345 | let inline length (v:Vector<'T>):int = v.Length
346 | /// Gets the Lp norm (or p-norm) of vector `v`, with the given `p`
347 | let inline lpnorm (p:'T) (v:Vector<'T>):'T = v.GetLPNorm(p)
348 | /// Creates a vector whose elements are the results of applying function `f` to each element of vector `v`
349 | let inline map (f:'T->'U) (v:Vector<'T>):Vector<'U> = v |> toArray |> Array.map f |> Vector
350 | /// Creates a vector whose elements are the results of applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
351 | let inline map2 (f:'T1->'T2->'U) (v1:Vector<'T1>) (v2:Vector<'T2>):Vector<'U> = Array.map2 f (v1 |> toArray) (v2 |> toArray) |> Vector
352 | /// Creates a vector whose elements are the results of applying function `f` to each element of vector `v`. An element index is also supplied to function `f`.
353 | let inline mapi (f:int->'T->'U) (v:Vector<'T>):Vector<'U> = v |> toArray |> Array.mapi f |> Vector
354 | /// Creates a vector whose elements are the results of applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The integer passed to function `f` indicates the index of element. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
355 | let inline mapi2 (f:int->'T1->'T2->'U) (v1:Vector<'T1>) (v2:Vector<'T2>):Vector<'U> = Array.mapi2 f (v1 |> toArray) (v2 |> toArray) |> Vector
356 | /// Returns the maximum of all elements of vector `v`
357 | let inline max (v:Vector<'T>):'T = v.GetMax()
358 | /// Returns the maximum of all elements of vector `v`, compared by using Operators.max on the result of function `f`
359 | let inline maxBy (f:'T->'U) (v:Vector<'T>):'T = v.GetMaxBy(f)
360 | /// Returns the minimum of all elements of vector `v`
361 | let inline min (v:Vector<'T>):'T = v.GetMin()
362 | /// Returns the minimum of all elements of vector `v`, compared by using Operators.min on the result of function `f`
363 | let inline minBy (f:'T->'U) (v:Vector<'T>):'T = v.GetMinBy(f)
364 | /// Gets the L2 (Euclidean) norm of vector `v`. This is the same with `Vector.l2norm`.
365 | let inline norm (v:Vector<'T>):'T = l2norm v
366 | /// Gets the squared L2 (Euclidean) norm of vector `v`. This is the same with `Vector.l2normSq`.
367 | let inline normSq (v:Vector<'T>):'T = l2normSq v
368 | /// Applies function `f` to each element of vector `v`, threading an accumulator argument through the computation. If the input function is f and the elements are i0...iN, then computes f (... (f i0 i1)...) iN.
369 | let inline reduce (f:'T->'T->'T) (v:Vector<'T>):'T = v |> toArray |> Array.reduce f
370 | /// Applies function `f` to each element of vector `v`, threading an accumulator argument through the computation. If the input function is f and the elements are i0...iN then computes f i0 (...(f iN-1 iN)).
371 | let inline reduceBack (f:'T->'T->'T) (v:Vector<'T>):'T = v |> toArray |> Array.reduceBack f
372 | /// Replaces the elements of vector `v` by mutating them in place, passing them through function `f`
373 | let inline replace (f:'T->'T) (v:Vector<'T>):unit = for i in 0..(v.Length - 1) do v.[i] <- f v.[i]
374 | /// Replaces the elements of vector `v1` by mutating them in place. The new values are computed by applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The two input vectors must have the same lengths, otherwie ArgumentException is raised.
375 | let inline replace2 (f:'T1->'T2->'T1) (v1:Vector<'T1>) (v2:Vector<'T2>):unit =
376 | if v1.Length <> v2.Length then invalidArg "" "The vectors should have the same length."
377 | for i in 0..(v1.Length - 1) do v1.[i] <- f v1.[i] v2.[i]
378 | /// Replaces the elements of vector `v` by mutating them in place, passing them through function `f`. An element index is also supplied to function `f`.
379 | let inline replacei (f:int->'T->'T) (v:Vector<'T>):unit = for i in 0..(v.Length - 1) do v.[i] <- f i v.[i]
380 | /// Replaces the elements of vector `v1` by mutating them in place. The new values are computed by applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. An element index is also supplied to function `f`. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
381 | let inline replacei2 (f:int->'T1->'T2->'T1) (v1:Vector<'T1>) (v2:Vector<'T2>):unit =
382 | if v1.Length <> v2.Length then invalidArg "" "The vectors should have the same length."
383 | for i in 0..(v1.Length - 1) do v1.[i] <- f i v1.[i] v2.[i]
384 | /// Replaces the elements of vector `v1` with the elements of vector `v2`, by mutating them in place. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
385 | let inline replaceWith (v1:Vector<'T>) (v2:Vector<'T>):unit =
386 | if v1.Length <> v2.Length then invalidArg "" "The vectors should have the same length."
387 | Array.blit (v2 |> toArray) 0 (v1 |> toArray) 0 v1.Length
388 | /// Like Vector.fold, but returns the intermediate and final results
389 | let inline scan (f:'S->'T->'S) (s:'S) (v:Vector<'T>):Vector<'S> = v |> toArray |> Array.scan f s |> ofSeq
390 | /// Like Vector.foldBack, but returns both the intermediate and final results
391 | let inline scanBack (f:'T->'S->'S) (s:'S) (v:Vector<'T>):Vector<'S> = Array.scanBack f (v |> toArray) s |> ofSeq
392 | /// Sets the element of vector`v` with index `i` to value `a`
393 | let inline set (v:Vector<'T>) (i:int) (a:'T):unit = v.[i] <- a
394 | /// Returns a sequence of vectors that are obtained by splitting vector `v` into subvectors whose lengths are given in the sequence `n`
395 | let inline split (n:seq) (v:Vector<'T>):seq> = v.Split(n)
396 | /// Returns a sequence of vectors that are obtained by splitting vector `v` into `n` subvectors of equal length. The length of vector `v` must be an integer multiple of `n`, otherwise ArgumentException is raised.
397 | let inline splitEqual (n:int) (v:Vector<'T>):seq> = v.SplitEqual(n)
398 | /// Creates a vector with `n` elements, where the `i`-th element is 1 and the rest of the elements are 0
399 | let inline standardBasis (n:int) (i:int):Vector<'T> = createBasis n i LanguagePrimitives.GenericOne
400 | /// Creates a new vector that contains the given subrange of vector `v`, specified by start index `s` and count `c`
401 | let inline sub (v:Vector<'T>) (s:int) (c:int):Vector<'T> = v.GetSubVector(s, c)
402 | /// Returns the sum of all the elements in vector `v`
403 | let inline sum (v:Vector<'T>):'T = v |> toArray |> Array.sum
404 | /// Returns the sum of the results generated by applying function `f` to each element of vector `v`
405 | let inline sumBy (f:'T->'U) (v:Vector<'T>):'U = v |> toArray |> Array.sumBy f
406 | /// Gets the unit vector codirectional with vector `v`
407 | let inline unitVector (v:Vector<'T>) = v.GetUnitVector()
--------------------------------------------------------------------------------
/src/FsAlg/Matrix.fs:
--------------------------------------------------------------------------------
1 | //
2 | // This file is part of
3 | // FsAlg: Generic Linear Algebra Library
4 | //
5 | // Copyright (c) 2015, National University of Ireland Maynooth (Atilim Gunes Baydin, Barak A. Pearlmutter)
6 | //
7 | // FsAlg is released under the BSD license.
8 | // (See accompanying LICENSE file.)
9 | //
10 | // Written by:
11 | //
12 | // Atilim Gunes Baydin
13 | // atilimgunes.baydin@nuim.ie
14 | //
15 | // Barak A. Pearlmutter
16 | // barak@cs.nuim.ie
17 | //
18 | // Brain and Computation Lab
19 | // Hamilton Institute & Department of Computer Science
20 | // National University of Ireland Maynooth
21 | // Maynooth, Co. Kildare
22 | // Ireland
23 | //
24 | // www.bcl.hamilton.ie
25 | //
26 |
27 | namespace FsAlg.Generic
28 |
29 | open FsAlg.Generic.Util
30 |
31 | /// Generic matrix type
32 | []
33 | type Matrix<'T when 'T : (static member Zero : 'T)
34 | and 'T : (static member One : 'T)
35 | and 'T : (static member (+) : 'T * 'T -> 'T)
36 | and 'T : (static member (-) : 'T * 'T -> 'T)
37 | and 'T : (static member (*) : 'T * 'T -> 'T)
38 | and 'T : (static member (/) : 'T * 'T -> 'T)
39 | and 'T : (static member (~-) : 'T -> 'T)
40 | and 'T : (static member Abs : 'T -> 'T)
41 | and 'T : (static member Pow : 'T * 'T -> 'T)
42 | and 'T : (static member Sqrt : 'T -> 'T)
43 | and 'T : (static member op_Explicit : 'T -> float)
44 | and 'T : comparison> =
45 | | ZeroMatrix of 'T
46 | | Matrix of 'T[,]
47 | /// ZeroMatrix
48 | static member inline Zero = ZeroMatrix LanguagePrimitives.GenericZero<'T>
49 | /// Converts Matrix `m` to float[,]
50 | static member inline op_Explicit(m:Matrix<'T>) =
51 | match m with
52 | | Matrix m -> Array2D.map float m
53 | | ZeroMatrix _ -> Array2D.zeroCreate 0 0
54 | /// Gets the number of rows of this matrix
55 | member inline m.Rows =
56 | match m with
57 | | Matrix m -> m.GetLength 0
58 | | ZeroMatrix _ -> 0
59 | /// Gets the number of columns of this matrix
60 | member inline m.Cols =
61 | match m with
62 | | Matrix m -> m.GetLength 1
63 | | ZeroMatrix _ -> 0
64 | /// The entry of this matrix at row `i` and column `j`
65 | member inline m.Item
66 | with get (i, j) =
67 | match m with
68 | | Matrix m -> m.[i, j]
69 | | ZeroMatrix z -> z
70 | and set (i, j) v =
71 | match m with
72 | | Matrix m -> m.[i, j] <- v
73 | | ZeroMatrix _ -> ()
74 | /// Gets a submatrix of this matrix with the bounds given in `rowStart`, `rowFinish`, `colStart`, `colFinish`
75 | member inline m.GetSlice(rowStart, rowFinish, colStart, colFinish) =
76 | match m with
77 | | Matrix mm ->
78 | let rowStart = defaultArg rowStart 0
79 | let rowFinish = defaultArg rowFinish (m.Rows - 1)
80 | if rowStart > rowFinish then invalidArg "" "Given row slice bounds are invalid."
81 | let colStart = defaultArg colStart 0
82 | let colFinish = defaultArg colFinish (m.Cols - 1)
83 | if colStart > colFinish then invalidArg "" "Given column slice bounds are invalid."
84 | Matrix mm.[rowStart..rowFinish, colStart..colFinish]
85 | | ZeroMatrix _ -> invalidArg "" "Cannot get slice of a ZeroMatrix."
86 | /// Gets a row of this matrix with the given row index `row` and column bounds `colStart` and `colFinish`
87 | member inline m.GetSlice(row, colStart, colFinish) =
88 | match m with
89 | | Matrix mm ->
90 | let colStart = defaultArg colStart 0
91 | let colFinish = defaultArg colFinish (m.Cols - 1)
92 | if colStart > colFinish then invalidArg "" "Given column slice bounds are invalid."
93 | [mm.[row, colStart..colFinish]] |> array2D |> Matrix
94 | | ZeroMatrix _ -> invalidArg "" "Cannot get slice of a ZeroMatrix."
95 | /// Gets a column of this matrix with the given column index `col` and row bounds `rowStart` and `rowFinish`
96 | member inline m.GetSlice(rowStart, rowFinish, col) =
97 | match m with
98 | | Matrix mm ->
99 | let rowStart = defaultArg rowStart 0
100 | let rowFinish = defaultArg rowFinish (m.Rows - 1)
101 | if rowStart > rowFinish then invalidArg "" "Given row slice bounds are invalid."
102 | [mm.[rowStart..rowFinish, col]] |> array2D |> transpose |> Matrix
103 | | ZeroMatrix _ -> invalidArg "" "Cannot get slice of a ZeroMatrix."
104 | /// Gets a string representation of this matrix that can be pasted into a Mathematica notebook
105 | member inline m.ToMathematicaString() =
106 | let sb = System.Text.StringBuilder()
107 | sb.Append("{") |> ignore
108 | for i = 0 to m.Rows - 1 do
109 | sb.Append("{") |> ignore
110 | for j = 0 to m.Cols - 1 do
111 | sb.Append(sprintf "%.2f" (float m.[i, j])) |> ignore
112 | if j <> m.Cols - 1 then sb.Append(", ") |> ignore
113 | sb.Append("}") |> ignore
114 | if i <> m.Rows - 1 then sb.Append(", ") |> ignore
115 | sb.Append("}") |> ignore
116 | sb.ToString()
117 | /// Gets a string representation of this matrix that can be pasted into MATLAB
118 | member inline m.ToMatlabString() =
119 | let sb = System.Text.StringBuilder()
120 | sb.Append("[") |> ignore
121 | for i = 0 to m.Rows - 1 do
122 | for j = 0 to m.Cols - 1 do
123 | sb.Append(sprintf "%.2f" (float m.[i, j])) |> ignore
124 | if j < m.Cols - 1 then sb.Append(" ") |> ignore
125 | if i < m.Rows - 1 then sb.Append("; ") |> ignore
126 | sb.Append("]") |> ignore
127 | sb.ToString()
128 | /// Converts this matrix into a 2d array
129 | member inline m.ToArray2D() =
130 | match m with
131 | | Matrix m -> m
132 | | ZeroMatrix _ -> Array2D.zeroCreate 0 0
133 | /// Converts this matrix into a jagged array, e.g. from Matrix to float[][]
134 | member inline m.ToArray() =
135 | let a = m.ToArray2D()
136 | [|for i = 0 to m.Rows - 1 do yield [|for j = 0 to m.Cols - 1 do yield a.[i, j]|]|]
137 | /// Converts this matrix into a one dimensional sequence, scanning columns from left to right and rows from top to bottom
138 | member inline m.ToSeq() =
139 | let a = m.ToArray()
140 | seq {for aa in a do yield! Seq.ofArray aa}
141 | /// Returns an enumerator that iterates through this matrix
142 | member inline m.GetEnumerator() =
143 | m.ToSeq().GetEnumerator()
144 | /// Creates a copy of this matrix
145 | member inline m.Copy() =
146 | match m with
147 | | Matrix m -> Matrix (Array2D.copy m)
148 | | ZeroMatrix z -> ZeroMatrix z
149 | /// Gets the trace of this matrix
150 | member inline m.GetTrace() =
151 | match m with
152 | | Matrix m -> trace m
153 | | ZeroMatrix z -> z
154 | /// Gets the transpose of this matrix
155 | member inline m.GetTranspose() =
156 | match m with
157 | | Matrix m -> Matrix (transpose m)
158 | | ZeroMatrix z -> ZeroMatrix z
159 | /// Gets a vector of the diagonal elements of this matrix
160 | member inline m.GetDiagonal() =
161 | match m with
162 | | Matrix mm ->
163 | if m.Rows <> m.Cols then invalidArg "" "Cannot get the diagonal entries of a nonsquare matrix."
164 | Array.init m.Rows (fun i -> mm.[i, i])
165 | | ZeroMatrix z -> [||]
166 | /// Returns the LU decomposition of this matrix. The return values are the LU matrix, pivot indices, and a toggle value indicating the number of row exchanges during the decomposition, which is +1 if the number of exchanges were even, -1 if odd.
167 | member inline m.GetLUDecomposition() =
168 | match m with
169 | | Matrix mm ->
170 | if (m.Rows <> m.Cols) then invalidArg "" "Cannot compute the LU decomposition of a nonsquare matrix."
171 | let res = Array2D.copy mm
172 | let perm = Array.init m.Rows (fun i -> i)
173 | let mutable toggle = LanguagePrimitives.GenericOne<'T>
174 | for j = 0 to m.Rows - 2 do
175 | let mutable colmax:'T = abs res.[j, j]
176 | let mutable prow = j
177 | for i = j + 1 to m.Rows - 1 do
178 | let absresij = abs res.[i, j]
179 | if absresij > colmax then
180 | colmax <- absresij
181 | prow <- i
182 | if prow <> j then
183 | let tmprow = res.[prow, 0..]
184 | res.[prow, 0..] <- res.[j, 0..]
185 | res.[j, 0..] <- tmprow
186 | let tmp = perm.[prow]
187 | perm.[prow] <- perm.[j]
188 | perm.[j] <- tmp
189 | toggle <- -toggle
190 | for i = j + 1 to m.Rows - 1 do
191 | res.[i, j] <- res.[i, j] / res.[j, j]
192 | for k = j + 1 to m.Rows - 1 do
193 | res.[i, k] <- res.[i, k] - res.[i, j] * res.[j, k]
194 | Matrix res, perm, toggle
195 | | ZeroMatrix z -> ZeroMatrix z, [||], LanguagePrimitives.GenericZero<'T>
196 | /// Gets the determinant of this matrix
197 | member inline m.GetDeterminant() =
198 | match m with
199 | | Matrix _ ->
200 | if (m.Rows <> m.Cols) then invalidArg "" "Cannot compute the determinant of a nonsquare matrix."
201 | let lu, _, toggle = m.GetLUDecomposition()
202 | toggle * Array.fold (fun s x -> s * x) LanguagePrimitives.GenericOne<'T> (lu.GetDiagonal())
203 | | ZeroMatrix z -> z
204 | /// Gets the inverse of this matrix
205 | member inline m.GetInverse() =
206 | match m with
207 | | Matrix mm ->
208 | if (m.Rows <> m.Cols) then invalidArg "" "Cannot compute the inverse of a nonsquare matrix."
209 | let res = Array2D.copy mm
210 | let lu, perm, _ = m.GetLUDecomposition()
211 | let b:'T[] = Array.zeroCreate m.Rows
212 | for i = 0 to m.Rows - 1 do
213 | for j = 0 to m.Rows - 1 do
214 | if i = perm.[j] then
215 | b.[j] <- LanguagePrimitives.GenericOne<'T>
216 | else
217 | b.[j] <- LanguagePrimitives.GenericZero<'T>
218 | let x = matrixSolveHelper (lu.ToArray2D()) b
219 | res.[0.., i] <- x
220 | Matrix res
221 | | ZeroMatrix z -> ZeroMatrix z
222 | /// Adds matrix `a` to matrix `b`
223 | static member inline (+) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
224 | match a, b with
225 | | Matrix ma, Matrix mb ->
226 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot add matrices of different sizes."
227 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] + mb.[i, j]))
228 | | Matrix _, ZeroMatrix _ -> a
229 | | ZeroMatrix _, Matrix _ -> b
230 | | ZeroMatrix z, ZeroMatrix _ -> ZeroMatrix z
231 | /// Subtracts matrix `b` from matrix `a`
232 | static member inline (-) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
233 | match a, b with
234 | | Matrix ma, Matrix mb ->
235 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot subtract matrices of different sizes."
236 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] - mb.[i, j]))
237 | | Matrix _, ZeroMatrix _ -> a
238 | | ZeroMatrix _, Matrix b -> Matrix (Array2D.map (~-) b)
239 | | ZeroMatrix _, ZeroMatrix _ -> Matrix.Zero
240 | /// Multiplies matrix `a` and matrix `b` (matrix product)
241 | static member inline (*) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
242 | match a, b with
243 | | Matrix ma, Matrix mb ->
244 | if (a.Cols <> b.Rows) then invalidArg "" "Cannot multiply two matrices of incompatible sizes."
245 | Matrix (Array2D.init a.Rows b.Cols (fun i j -> Array.sumBy (fun k -> ma.[i, k] * mb.[k, j]) [|0..(b.Rows - 1)|] ))
246 | | Matrix _, ZeroMatrix _ -> Matrix.Zero
247 | | ZeroMatrix _, Matrix _ -> Matrix.Zero
248 | | ZeroMatrix _, ZeroMatrix _ -> Matrix.Zero
249 | /// Multiplies matrix `a` and matrix `b` element-wise (Hadamard product)
250 | static member inline (.*) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
251 | match a, b with
252 | | Matrix ma, Matrix mb ->
253 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot multiply matrices of different sizes."
254 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] * mb.[i, j]))
255 | | Matrix _, ZeroMatrix _ -> Matrix.Zero
256 | | ZeroMatrix _, Matrix _ -> Matrix.Zero
257 | | ZeroMatrix _, ZeroMatrix _ -> Matrix.Zero
258 | /// Divides matrix `a` by matrix `b` element-wise (Hadamard division)
259 | static member inline (./) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
260 | match a, b with
261 | | Matrix ma, Matrix mb ->
262 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot divide matrices of different sizes."
263 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] / mb.[i, j]))
264 | | Matrix _, ZeroMatrix _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroMatrix."))
265 | | ZeroMatrix _, Matrix _ -> Matrix.Zero
266 | | ZeroMatrix _, ZeroMatrix _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroMatrix."))
267 | /// Computes the matrix-vector product of matrix `a` and vector `b`
268 | static member inline (*) (a:Matrix<'T>, b:Vector<'T>):Vector<'T> =
269 | match a, b with
270 | | Matrix ma, Vector vb ->
271 | if (a.Cols <> b.Length) then invalidArg "" "Cannot compute the matrix-vector product of a matrix and a vector of incompatible sizes."
272 | Vector (Array.init a.Rows (fun i -> Array.sumBy (fun j -> ma.[i, j] * vb.[j]) [|0..(b.Length - 1)|] ))
273 | | Matrix _, ZeroVector _ -> Vector.Zero
274 | | ZeroMatrix _, Vector _ -> Vector.Zero
275 | | ZeroMatrix _, ZeroVector _ -> Vector.Zero
276 | /// Computes the vector-matrix product of vector `a` and matrix `b`
277 | static member inline (*) (a:Vector<'T>, b:Matrix<'T>):Vector<'T> =
278 | match a, b with
279 | | Vector va, Matrix mb ->
280 | if (a.Length <> b.Rows) then invalidArg "" "Cannot compute the vector-matrix product of a vector and matrix of incompatible sizes."
281 | Vector (Array.init b.Cols (fun i -> Array.sumBy (fun j -> va.[j] * mb.[j, i]) [|0..(a.Length - 1)|]))
282 | | Vector _, ZeroMatrix _ -> Vector.Zero
283 | | ZeroVector _, Matrix _ -> Vector.Zero
284 | | ZeroVector _, ZeroMatrix _ -> Vector.Zero
285 | /// Adds scalar `b` to each element of matrix `a`
286 | static member inline (+) (a:Matrix<'T>, b:'T):Matrix<'T> =
287 | match a with
288 | | Matrix a -> Matrix (Array2D.map ((+) b) a)
289 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroMatrix."
290 | /// Adds scalar `a` to each element of matrix `b`
291 | static member inline (+) (a:'T, b:Matrix<'T>):Matrix<'T> =
292 | match b with
293 | | Matrix b -> Matrix (Array2D.map ((+) a) b)
294 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroMatrix."
295 | /// Subtracts scalar `b` from each element of matrix `a`
296 | static member inline (-) (a:Matrix<'T>, b:'T):Matrix<'T> =
297 | match a with
298 | | Matrix a -> Matrix (Array2D.map (fun x -> x - b) a)
299 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot subtract a scalar from a ZeroMatrix."
300 | /// Subtracts each element of of matrix `b` from scalar `a`
301 | static member inline (-) (a:'T, b:Matrix<'T>):Matrix<'T> =
302 | match b with
303 | | Matrix b -> Matrix (Array2D.map ((-) a) b)
304 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot subtract a ZeroMatrix from a scalar."
305 | /// Multiplies each element of matrix `a` by scalar `b`
306 | static member inline (*) (a:Matrix<'T>, b:'T):Matrix<'T> =
307 | match a with
308 | | Matrix a -> Matrix (Array2D.map ((*) b) a)
309 | | ZeroMatrix _ -> Matrix.Zero
310 | /// Multiplies each element of matrix `b` by scalar `a`
311 | static member inline (*) (a:'T, b:Matrix<'T>):Matrix<'T> =
312 | match b with
313 | | Matrix b -> Matrix (Array2D.map ((*) a) b)
314 | | ZeroMatrix _ -> Matrix.Zero
315 | /// Divides each element of matrix `a` by scalar `b`
316 | static member inline (/) (a:Matrix<'T>, b:'T):Matrix<'T> =
317 | match a with
318 | | Matrix a -> Matrix (Array2D.map (fun x -> x / b) a)
319 | | ZeroMatrix _ -> Matrix.Zero
320 | /// Creates a matrix whose elements are scalar `a` divided by each element of matrix `b`
321 | static member inline (/) (a:'T, b:Matrix<'T>):Matrix<'T> =
322 | match b with
323 | | Matrix b -> Matrix (Array2D.map ((/) a) b)
324 | | ZeroMatrix _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroMatrix."))
325 | /// Gets the negative of matrix `a`
326 | static member inline (~-) (a:Matrix<'T>) =
327 | match a with
328 | | Matrix a -> Matrix (Array2D.map (~-) a)
329 | | ZeroMatrix _ -> Matrix.Zero
330 | /// Returns the QR decomposition of this matrix
331 | member inline m.GetQRDecomposition() =
332 | match m with
333 | | ZeroMatrix z -> failwith "Cannot compute the QR decomposition of ZeroMatrix."
334 | | Matrix mm ->
335 | let minor (m:_[,]) (d) =
336 | let rows = Array2D.length1 m
337 | let cols = Array2D.length2 m
338 | let ret = Array2D.zeroCreate rows cols
339 | for i = 0 to d - 1 do
340 | ret.[i, i] <- LanguagePrimitives.GenericOne
341 | Array2D.blit m d d ret d d (rows - d) (cols - d)
342 | ret
343 | let identity d = Array2D.init d d (fun i j -> if i = j then LanguagePrimitives.GenericOne else LanguagePrimitives.GenericZero)
344 | // Householder
345 | let kmax = -1 + min (m.Rows - 1) m.Cols
346 | let mutable z = m.Copy()
347 | let q = Array.create m.Rows Matrix.Zero
348 | for k = 0 to kmax do
349 | z <- Matrix (minor (z.ToArray2D()) k)
350 | let x = z.[*, k].ToSeq() |> Array.ofSeq |> Vector
351 | let mutable a = x.GetL2Norm()
352 | if mm.[k, k] > LanguagePrimitives.GenericZero then a <- -a
353 | let e = (x + Vector.createBasis m.Rows k a).GetUnitVector()
354 | q.[k] <- Matrix (identity m.Rows) + Matrix (Array2D.init m.Rows m.Rows (fun i j -> -(e.[i] * e.[j] + e.[i] * e.[j])))
355 | z <- q.[k] * z
356 | let mutable q' = q.[0]
357 | for i = 1 to kmax do
358 | q' <- q.[i] * q'
359 | q'.GetTranspose(), q' * m
360 | /// Returns the eigenvalues of this matrix. (Experimental code, complex eigenvalues are not supported.)
361 | member inline m.GetEigenvalues() =
362 | let mutable m' = m.Copy()
363 | for i = 0 to 20 do
364 | let q, r = m'.GetQRDecomposition()
365 | m' <- r * q
366 | m'.GetDiagonal()
367 |
368 | /// Operations on Matrix type. (Implementing functionality similar to Microsoft.FSharp.Collections.Array2D)
369 | []
370 | module Matrix =
371 | /// Creates a matrix from 2d array `m`
372 | let inline ofArray2D (m:'T[,]):Matrix<'T> = Matrix m
373 | /// Converts matrix `m` to a 2d array, e.g. from Matrix to float[,]
374 | let inline toArray2D (m:Matrix<'T>):'T[,] = m.ToArray2D()
375 | /// Creates a matrix from a jagged array, e.g. from float[][] to Matrix
376 | let inline ofArrayArray (m:'T[][]):Matrix<'T> = m |> array2D |> Matrix
377 | /// Converts matrix `m` to a jagged array, e.g. from Matrix to float[][]
378 | let inline toArrayArray (m:Matrix<'T>):'T[][] = m.ToArray()
379 | /// Creates a matrix from sequence of sequences `s`
380 | let inline ofSeqSeq (s:seq>):Matrix<'T> = s |> array2D |> Matrix
381 | /// Converts matrix `m` to a sequence of sequences
382 | let inline toSeqSeq (m:Matrix<'T>):seq> = m.ToArray() |> Array.map Seq.ofArray |> Seq.ofArray
383 | /// Creates a matrix with `m` rows from the one dimensional sequence `s`, filling columns from left to right and rows from top to bottom. The number of columns will be deduced from `m` and the length of the sequence `s`. The length of `s` must be an integer multiple of `m`.
384 | let inline ofSeq (m:int) (s:seq<'T>):Matrix<'T> =
385 | let n = Seq.length s / m
386 | Array2D.init m n (fun i j -> Seq.nth (i * n + j) s) |> Matrix
387 | /// Converts matrix `m` to a one dimensional sequence, scanning columns from left to right and rows from top to bottom
388 | let inline toSeq (m:Matrix<'T>):seq<'T> = m.ToSeq()
389 | /// Creates a matrix with `m` rows from the one dimensional array `a`, filling columns from left to right and rows from top to bottom. The number of columns will be deduced from `m` and the length of the array `a`. The length of `a` must be an integer multiple of `m`.
390 | let inline ofArray (m:int) (a:'T[]):Matrix<'T> = ofSeq m a
391 | /// Converts matrix `m` to a one dimensional array, scanning columns from left to right and rows from top to bottom
392 | let inline toArray (m:Matrix<'T>):'T[] = m |> toSeq |> Array.ofSeq
393 | /// Creates a matrix with `m` rows from the vector `v`, filling columns from left to right and rows from top to bottom. The number of columns will be deduced from `m` and the length of the vector `v`. The length of `v` must be an integer multiple of `m`.
394 | let inline ofVector (m:int) (v:Vector<'T>):Matrix<'T> = v |> Vector.toSeq |> ofSeq m
395 | /// Converts matrix `m` to a vector, scanning columns from left to right and rows from top to bottom
396 | let inline toVector (m:Matrix<'T>):Vector<'T> = m |> toSeq |> Vector.ofSeq
397 | /// Returns the j-th column of matrix `m`
398 | let inline col (j:int) (m:Matrix<'T>):Matrix<'T> = m.[*,j]
399 | /// Returns the number of columns in matrix `m`. This is the same with `Matrix.length2`.
400 | let inline cols (m:Matrix<'T>):int = m.Cols
401 | /// Creates a copy of Matrix `m`
402 | let inline copy (m:Matrix<'T>):Matrix<'T> = m.Copy()
403 | /// Creates a matrix with `m` rows, `n` columns, and all entries having value `v`
404 | let inline create (m:int) (n:int) (v:'T):Matrix<'T> = Matrix (Array2D.create m n v)
405 | /// Creates a matrix with `m` rows and all rows equal to array `v`
406 | let inline createRows (m:int) (v:'T[]):Matrix<'T> = Matrix (array2D (Array.init m (fun _ -> v)))
407 | /// Gets the LU decomposition of matrix `m`. The return values are the LU matrix, pivot indices, and a toggle value indicating the number of row exchanges during the decomposition, which is +1 if the number of exchanges were even, -1 if odd.
408 | let inline decomposeLU (m:Matrix<'T>):(Matrix<'T>*int[]*'T) = m.GetLUDecomposition()
409 | /// Gets the QR decomposition of matrix `m`
410 | let inline decomposeQR (m:Matrix<'T>):(Matrix<'T>*Matrix<'T>) = m.GetQRDecomposition()
411 | /// Gets the determinant of matrix `m`
412 | let inline det (m:Matrix<'T>):'T = m.GetDeterminant()
413 | /// Gets the diagonal elements of matrix `m`
414 | let inline diagonal (m:Matrix<'T>):Vector<'T> = m.GetDiagonal() |> Vector.ofSeq
415 | /// Creates the identity matrix with `m` rows and columns
416 | let inline identity (m:int):Matrix<'T> =
417 | Matrix (Array2D.init m m (fun i j -> if i = j then LanguagePrimitives.GenericOne<'T> else LanguagePrimitives.GenericZero<'T>))
418 | /// Gets the eigenvalues of matrix `m`
419 | let inline eigenvalues (m:Matrix<'T>):Vector<'T> = m.GetEigenvalues() |> Vector.ofSeq
420 | /// Gets the entry of matrix `m` with indices `i` and `j`
421 | let inline get (m:Matrix<'T>) (i:int) (j:int):'T = m.[i, j]
422 | /// Creates a matrix with `m` rows, `n` columns and a generator function `f` to compute the entries
423 | let inline init (m:int) (n:int) (f:int->int->'T):Matrix<'T> = Matrix (Array2D.init m n f)
424 | /// Creates a square matrix with `m` rows and columns and a generator function `f` to compute the elements. Function `f` is used only for populating the diagonal and the upper triangular part of the matrix, the lower triangular part will be the reflection.
425 | let inline initSymmetric (m:int) (f:int->int->'T):Matrix<'T> =
426 | if m = 0 then
427 | Matrix.Zero
428 | else
429 | let s = Array2D.zeroCreate<'T> m m
430 | for i = 0 to m - 1 do
431 | for j = i to m - 1 do
432 | s.[i, j] <- f i j
433 | Matrix (copyUpperToLower s)
434 | /// Gets the inverse of matrix `m`
435 | let inline inverse (m:Matrix<'T>):Matrix<'T> = m.GetInverse()
436 | /// Applies function `f` to each element of matrix `m`
437 | let inline iter (f:'T->unit) (m:Matrix<'T>):unit = m |> toArray2D |> Array2D.iter f
438 | /// Applies function `f` to each element of matrix `m`. Element indices are also supplied to function `f`.
439 | let inline iteri (f:int->int->'T->unit) (m:Matrix<'T>):unit = m |> toArray2D |> Array2D.iteri f
440 | /// Returns the number of rows in matrix `m`. This is the same with `Matrix.rows`.
441 | let inline length1 (m:Matrix<'T>):int = m.Rows
442 | /// Returns the number of columns in matrix `m`. This is the same with `Matrix.cols`.
443 | let inline length2 (m:Matrix<'T>):int = m.Cols
444 | /// Creates a matrix whose entries are the results of applying function `f` to each entry of matrix `m`
445 | let inline map (f:'T->'U) (m:Matrix<'T>):Matrix<'U> = m |> toArray2D |> Array2D.map f |> Matrix
446 | /// Creates a matrix whose entries are the results of applying function `f` to each entry of matrix `m`. Element indices are also supplied to function `f`.
447 | let inline mapi (f:int->int->'T->'U) (m:Matrix<'T>):Matrix<'U> = m |> toArray2D |> Array2D.mapi f |> Matrix
448 | /// Replaces the elements of matrix `m` by mutating them in place, passing them through function `f`
449 | let inline replace (f:'T->'T) (m:Matrix<'T>):unit =
450 | for i in 0..(m.Rows - 1) do
451 | for j in 0..(m.Cols - 1) do
452 | m.[i,j] <- f m.[i, j]
453 | /// Replaces the elements of matrix `m1` by mutating them in place. The new values are computed by applying function `f` to the corresponding elements of matrices `m1` and `m2` pairwise. The two input matrices should have the same dimensions, otherwise ArgumentException is raised.
454 | let inline replace2 (f:'T1->'T2->'T1) (m1:Matrix<'T1>) (m2:Matrix<'T2>):unit =
455 | if (m1.Rows <> m2.Rows) || (m1.Cols <> m2.Cols) then invalidArg "" "The matrices should have the same dimensions."
456 | for i in 0..(m1.Rows - 1) do
457 | for j in 0..(m1.Cols - 1) do
458 | m1.[i, j] <- f m1.[i, j] m2.[i, j]
459 | /// Replaces the elements of matrix `m` by mutating them in place, passing them through function` f`. Element indices are also supplied to function `f`.
460 | let inline replacei (f:int->int->'T->'T) (m:Matrix<'T>):unit =
461 | for i in 0..(m.Rows - 1) do
462 | for j in 0..(m.Cols - 1) do
463 | m.[i,j] <- f i j m.[i, j]
464 | /// Replaces the elements of matrix `m1` by mutating them in place. The new values are computed by applying function `f` to the corresponding elements of matrices `m1` and `m2` pairwise. Element indices are also supplied to function `f`. The two input matrices should have the same dimensions, otherwise ArgumentException is raised.
465 | let inline replacei2 (f:int->int->'T1->'T2->'T1) (m1:Matrix<'T1>) (m2:Matrix<'T2>):unit =
466 | if (m1.Rows <> m2.Rows) || (m1.Cols <> m2.Cols) then invalidArg "" "The matrices should have the same dimensions."
467 | for i in 0..(m1.Rows - 1) do
468 | for j in 0..(m1.Cols - 1) do
469 | m1.[i, j] <- f i j m1.[i, j] m2.[i, j]
470 | /// Replaces the elements of matrix `m1` with the elements of matrix `m2`, by mutating them in place. The two input matrices should have the same dimensions, otherwise ArgumentException is raised.
471 | let inline replaceWith (m1:Matrix<'T>) (m2:Matrix<'T>) =
472 | if (m1.Rows <> m2.Rows) || (m1.Cols <> m2.Cols) then invalidArg "" "The matrices should have the same dimensions."
473 | Array2D.blit (m2 |> toArray2D) 0 0 (m1 |> toArray2D) 0 0 m1.Rows m1.Cols
474 | /// Returns the i-th row of matrix `m`
475 | let inline row (i:int) (m:Matrix<'T>):Matrix<'T> = m.[i,*]
476 | /// Returns the number of rows in matrix `m`. This is the same with `Matrix.length1`.
477 | let inline rows (m:Matrix<'T>):int = m.Rows
478 | /// Sets the entry of matrix `m` with indices `i` and `j` to value `a`
479 | let inline set (m:Matrix<'T>) (i:int) (j:int) (a:'T) = m.[i, j] <- a
480 | /// Solves a system of linear equations ax = b, where the coefficients are given in matrix `a` and the result vector is vector `b`. The returned vector will correspond to x.
481 | let inline solve (a:Matrix<'T>) (b:Vector<'T>):Vector<'T> =
482 | if a.Cols <> b.Length then invalidArg "" "Cannot solve the system of equations using a matrix and a vector of incompatible sizes."
483 | let lu, perm, _ = a.GetLUDecomposition()
484 | let bp = Array.init a.Rows (fun i -> b.[perm.[i]])
485 | Vector (matrixSolveHelper (lu.ToArray2D()) bp)
486 | /// Gets the trace of matrix `m`
487 | let inline trace (m:Matrix<'T>):'T = m.GetTrace()
488 | /// Gets the transpose of matrix `m`
489 | let inline transpose (m:Matrix<'T>):Matrix<'T> = m.GetTranspose()
490 | /// Constructs a matrix out of a sequence of row vectors `v`. The row vectors should be of equal length.
491 | let inline ofRows (v:seq>):Matrix<'T> = v |> Seq.map Vector.toSeq |> ofSeqSeq
492 | /// Returns the rows of matrix `m` as a sequence of matrices
493 | let inline toRows (m:Matrix<'T>):seq> = Seq.init m.Rows (fun i -> m.[i,*])
494 | /// Constructs a matrix out of a sequence of column vectors `v`. The column vectors should be of equal length.
495 | let inline ofCols (v:seq>):Matrix<'T> = v |> ofRows |> transpose
496 | /// Returns the columns of matrix `m` as a sequence of matrices
497 | let inline toCols (m:Matrix<'T>):seq> = m |> transpose |> toRows
498 | /// Returns a matrix where vector `v` is appended as a new row to matrix `m`
499 | let inline appendRow (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
500 | if m.Cols <> v.Length then invalidArg "" "The length of the appended row should be the same with the number of the columns in the matrix."
501 | let mm = m |> toArray2D
502 | let ret = Array2D.zeroCreate (m.Rows + 1) m.Cols
503 | Array2D.blit mm 0 0 ret 0 0 m.Rows m.Cols
504 | for i = 0 to m.Cols - 1 do
505 | ret.[m.Rows, i] <- v.[i]
506 | Matrix ret
507 | /// Returns a matrix where vector `v` is appended as a new column to matrix `m`
508 | let inline appendCol (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
509 | if m.Rows <> v.Length then invalidArg "" "The length of the appended column should be the same with the number of rows in the matrix."
510 | let mm = m |> toArray2D
511 | let ret = Array2D.zeroCreate m.Rows (m.Cols + 1)
512 | Array2D.blit mm 0 0 ret 0 0 m.Rows m.Cols
513 | for i = 0 to m.Rows - 1 do
514 | ret.[i, m.Cols] <- v.[i]
515 | Matrix ret
516 | /// Returns a matrix where vector `v` is appended as a new row to matrix `m`
517 | let inline prependRow (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
518 | if m.Cols <> v.Length then invalidArg "" "The length of the prepended row should be the same with the number of the columns in the matrix."
519 | let mm = m |> toArray2D
520 | let ret = Array2D.zeroCreate (m.Rows + 1) m.Cols
521 | Array2D.blit mm 0 0 ret 1 0 m.Rows m.Cols
522 | for i = 0 to m.Cols - 1 do
523 | ret.[0, i] <- v.[i]
524 | Matrix ret
525 | /// Returns a matrix where vector `v` is prepended as a new column to matrix `m`
526 | let inline prependCol (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
527 | if m.Rows <> v.Length then invalidArg "" "The length of the prepended column should be the same with the number of rows in the matrix."
528 | let mm = m |> toArray2D
529 | let ret = Array2D.zeroCreate m.Rows (m.Cols + 1)
530 | Array2D.blit mm 0 0 ret 0 1 m.Rows m.Cols
531 | for i = 0 to m.Rows - 1 do
532 | ret.[i, 0] <- v.[i]
533 | Matrix ret
534 | /// Creates a matrix with `n` columns and a generator function `f` that gives each column as a vector
535 | let inline initCols (n:int) (f:int->Vector<'T>):Matrix<'T> = Matrix (array2D (Array.init n (f >> Vector.toArray))) |> transpose
536 | /// Creates a matrix with `m` rows and a generator function `f` that gives each row as a a vector
537 | let inline initRows (m:int) (f:int->Vector<'T>):Matrix<'T> = Matrix (array2D (Array.init m (f >> Vector.toArray)))
538 |
--------------------------------------------------------------------------------
@Model.Module.Name
35 |
36 | @foreach (var sec in comment.Sections) {
37 | // XML comment for the type has multiple sections that can be labelled
38 | // with categories (to give comment for an individual category). Here,
39 | // we print only those that belong to the section.
40 | if (!byCategory.Any(g => g.GroupKey == sec.Key))
41 | {
42 | if (sec.Key != "") {
43 |
49 | @if (byCategory.Count() > 1)
50 | {
51 |
52 | @sec.Key
44 | } 45 | @sec.Value 46 | } 47 | } 48 |Table of contents
53 |-
54 | @foreach (var g in byCategory)
55 | {
56 |
- @g.Name 57 | } 58 |
Nested types and modules
65 |
66 | @RenderPart("part-nested", new {
67 | Types = nestTypes,
68 | Modules = nestModules
69 | })
70 |
71 | }
72 |
73 | @foreach (var g in byCategory)
74 | {
75 | // Iterate over all the categories and print members. If there are more than one
76 | // categories, print the category heading (as ) and add XML comment from the type
77 | // that is related to this specific category.
78 | if (byCategory.Count() > 1)
79 | {
80 | @g.Name
81 | var info = comment.Sections.FirstOrDefault(kvp => kvp.Key == g.GroupKey);
82 | if (info.Key != null)
83 | {
84 |
85 | @info.Value
86 |
87 | }
88 | }
89 |
90 | @RenderPart("part-members", new {
91 | Header = "Functions and values",
92 | TableHeader = "Function or value",
93 | Members = g.Members.Where(m => m.Kind == MemberKind.ValueOrFunction)
94 | })
95 |
96 | @RenderPart("part-members", new {
97 | Header = "Type extensions",
98 | TableHeader = "Type extension",
99 | Members = g.Members.Where(m => m.Kind == MemberKind.TypeExtension)
100 | })
101 |
102 | @RenderPart("part-members", new {
103 | Header = "Active patterns",
104 | TableHeader = "Active pattern",
105 | Members = g.Members.Where(m => m.Kind == MemberKind.ActivePattern)
106 | })
107 | }
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/src/FsAlg/FsAlg.fsproj:
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1 |
2 |
3 |
5 | Debug
6 | AnyCPU
7 | 2.0
8 | 8715ea25-3b32-4cf5-a595-d4034442fe58
9 | Library
10 | FsAlg
11 | FsAlg
12 | v4.5
13 | 4.3.1.0
14 | FsAlg
15 |
17 |
18 | true
19 | full
20 | false
21 | false
22 | bin\Debug\
23 | DEBUG;TRACE
24 | 3
25 | bin\Debug\FsAlg.XML
26 |
27 |
28 | pdbonly
29 | true
30 | true
31 | bin\Release\
32 | TRACE
33 | 3
34 | bin\Release\FsAlg.XML
35 |
36 |
37 | 11
38 |
39 |
40 |
41 |
42 | $(MSBuildExtensionsPath32)\..\Microsoft SDKs\F#\3.0\Framework\v4.0\Microsoft.FSharp.Targets
43 |
44 |
45 |
46 |
47 | $(MSBuildExtensionsPath32)\Microsoft\VisualStudio\v$(VisualStudioVersion)\FSharp\Microsoft.FSharp.Targets
48 |
49 |
50 |
51 |
53 |
62 |
63 |
65 | True
66 |
67 |
71 |
78 |
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/docs/input/templates/template.html:
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1 |
2 |
3 |
4 |
5 | {page-title}
6 |
7 |
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11 |
12 |
13 |
14 |
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16 |
17 |
18 |
19 |
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34 |
38 | {project-name}
39 |
40 |
41 |
42 |
43 | {document}
44 | {tooltips}
45 |
46 |
47 |
48 |
49 |
50 |
68 |
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/docs/input/templates/template.cshtml:
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1 |
2 |
3 |
4 |
5 | @Title
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
22 |
30 |
31 |
32 |
33 |
34 |
38 | @Properties["project-name"]
39 |
40 |
41 |
42 |
43 | @RenderBody()
44 |
45 |
46 |
47 |
48 |
49 |
50 |
68 |
69 |
70 |
71 |
72 |
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87 |
88 |
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90 |
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/docs/input/files/misc/style_light.css:
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1 | @import url(https://fonts.googleapis.com/css?family=Droid+Sans|Droid+Sans+Mono|Open+Sans:400,600,700);
2 |
3 | /*--------------------------------------------------------------------------
4 | Formatting for F# code snippets
5 | /*--------------------------------------------------------------------------*/
6 |
7 | /* identifier */
8 | span.i { color:#000000; }
9 | /* string */
10 | span.s { color:#a31515; }
11 | /* keywords */
12 | span.k { color:#0000ff; }
13 | /* comment */
14 | span.c { color:#008000; }
15 | /* operators */
16 | span.o { color:#000000; }
17 | /* numbers */
18 | span.n { color:#000000; }
19 | /* line number */
20 | span.l { color:#96c2cd; }
21 | /* type or module */
22 | span.t { color:#2b91af; }
23 | /* function */
24 | span.f { color:#0000a0; }
25 | /* DU case or active pattern */
26 | span.p { color:#800080; }
27 | /* mutable var or ref cell */
28 | span.v { color:#000000; font-weight: bold; }
29 | /* printf formatters */
30 | span.pf { color:#2b91af; }
31 | /* escaped chars */
32 | span.e { color:#ff0080; }
33 | /* mutable var or ref cell */
34 |
35 |
36 | /* inactive code */
37 | span.inactive { color:#808080; }
38 | /* preprocessor */
39 | span.prep { color:#0000ff; }
40 | /* fsi output */
41 | span.fsi { color:#808080; }
42 |
43 | /* omitted */
44 | span.omitted {
45 | background:#3c4e52;
46 | border-radius:5px;
47 | color:#808080;
48 | padding:0px 0px 1px 0px;
49 | }
50 | /* tool tip */
51 | div.tip {
52 | background:#e5e5e5;
53 | border-radius:4px;
54 | font:9pt 'Droid Sans', arial, sans-serif;
55 | padding:6px 8px 6px 8px;
56 | display:none;
57 | color:#000000;
58 | }
59 | table.pre pre {
60 | padding:0px;
61 | margin:0px;
62 | border:none;
63 | }
64 | table.pre, pre.fssnip, pre {
65 | line-height:13pt;
66 | border:1px solid #d8d8d8;
67 | border-collapse:separate;
68 | white-space:pre;
69 | font: 10pt consolas,monospace;
70 | width:90%;
71 | margin:10px 20px 20px 20px;
72 | background-color:#fdfdfd;
73 | padding:10px;
74 | border-radius:5px;
75 | color:#000000;
76 | }
77 | table.pre pre {
78 | padding:0px;
79 | margin:0px;
80 | border-radius:0px;
81 | width: 100%;
82 | }
83 | table.pre td {
84 | padding:0px;
85 | white-space:normal;
86 | margin:0px;
87 | }
88 | table.pre td.lines {
89 | width:30px;
90 | }
91 |
92 | /*--------------------------------------------------------------------------
93 | Formatting for page & standard document content
94 | /*--------------------------------------------------------------------------*/
95 |
96 | body {
97 | font-family: 'Open Sans', serif;
98 | padding-top: 0px;
99 | padding-bottom: 40px;
100 | }
101 |
102 | pre {
103 | word-wrap: inherit;
104 | }
105 |
106 | /* Format the heading - nicer spacing etc. */
107 | .masthead {
108 | overflow: hidden;
109 | }
110 | .masthead .muted a {
111 | text-decoration:none;
112 | color:#999999;
113 | }
114 | .masthead ul, .masthead li {
115 | margin-bottom:0px;
116 | }
117 | .masthead .nav li {
118 | margin-top: 15px;
119 | font-size:110%;
120 | }
121 | .masthead h3 {
122 | margin-bottom:5px;
123 | font-size:170%;
124 | }
125 | hr {
126 | margin:0px 0px 20px 0px;
127 | }
128 |
129 | /* Make table headings and td.title bold */
130 | td.title, thead {
131 | font-weight:bold;
132 | }
133 |
134 | /* Format the right-side menu */
135 | #menu {
136 | margin-top:50px;
137 | font-size:11pt;
138 | padding-left:20px;
139 | }
140 |
141 | #menu .nav-header {
142 | font-size:12pt;
143 | color:#606060;
144 | margin-top:20px;
145 | }
146 |
147 | #menu li {
148 | line-height:25px;
149 | }
150 |
151 | /* Change font sizes for headings etc. */
152 | #main h1 { font-size: 26pt; margin:10px 0px 15px 0px; font-weight:400; }
153 | #main h2 { font-size: 20pt; margin:20px 0px 0px 0px; font-weight:400; }
154 | #main h3 { font-size: 14pt; margin:15px 0px 0px 0px; font-weight:600; }
155 | #main p { font-size: 11pt; margin:5px 0px 15px 0px; }
156 | #main ul { font-size: 11pt; margin-top:10px; }
157 | #main li { font-size: 11pt; margin: 5px 0px 5px 0px; }
158 | #main strong { font-weight:700; }
159 |
160 | /*--------------------------------------------------------------------------
161 | Formatting for API reference
162 | /*--------------------------------------------------------------------------*/
163 |
164 | .type-list .type-name, .module-list .module-name {
165 | width:25%;
166 | font-weight:bold;
167 | }
168 | .member-list .member-name {
169 | width:35%;
170 | }
171 | #main .xmldoc h2 {
172 | font-size:14pt;
173 | margin:10px 0px 0px 0px;
174 | }
175 | #main .xmldoc h3 {
176 | font-size:12pt;
177 | margin:10px 0px 0px 0px;
178 | }
179 | .github-link {
180 | float:right;
181 | text-decoration:none;
182 | }
183 | .github-link img {
184 | border-style:none;
185 | margin-left:10px;
186 | }
187 | .github-link .hover { display:none; }
188 | .github-link:hover .hover { display:block; }
189 | .github-link .normal { display: block; }
190 | .github-link:hover .normal { display: none; }
191 |
192 | /*--------------------------------------------------------------------------
193 | Links
194 | /*--------------------------------------------------------------------------*/
195 |
196 | h1 a, h1 a:hover, h1 a:focus,
197 | h2 a, h2 a:hover, h2 a:focus,
198 | h3 a, h3 a:hover, h3 a:focus,
199 | h4 a, h4 a:hover, h4 a:focus,
200 | h5 a, h5 a:hover, h5 a:focus,
201 | h6 a, h6 a:hover, h6 a:focus { color : inherit; text-decoration : inherit; outline:none }
202 |
203 | /*--------------------------------------------------------------------------
204 | Additional formatting for the homepage
205 | /*--------------------------------------------------------------------------*/
206 |
207 | #nuget {
208 | margin-top:20px;
209 | font-size: 11pt;
210 | padding:20px;
211 | }
212 |
213 | #nuget pre {
214 | font-size:11pt;
215 | -moz-border-radius: 0px;
216 | -webkit-border-radius: 0px;
217 | border-radius: 0px;
218 | background: #404040;
219 | border-style:none;
220 | color: #e0e0e0;
221 | margin-top:15px;
222 | }
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/docs/input/files/misc/style.css:
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1 | @import url(https://fonts.googleapis.com/css?family=Droid+Sans|Droid+Sans+Mono|Open+Sans:400,600,700);
2 |
3 | /*--------------------------------------------------------------------------
4 | Formatting for F# code snippets
5 | /*--------------------------------------------------------------------------*/
6 |
7 | /* strings --- and stlyes for other string related formats */
8 | span.s { color:#E0E268; }
9 | /* printf formatters */
10 | span.pf { color:#E0C57F; }
11 | /* escaped chars */
12 | span.e { color:#EA8675; }
13 |
14 | /* identifiers --- and styles for more specific identifier types */
15 | span.i { color:#d1d1d1; }
16 | /* type or module */
17 | span.t { color:#43AEC6; }
18 | /* function */
19 | span.f { color:#e1e1e1; }
20 | /* DU case or active pattern */
21 | span.p { color:#4ec9b0; }
22 |
23 | /* keywords */
24 | span.k { color:#FAB11D; }
25 | /* comment */
26 | span.c { color:#808080; }
27 | /* operators */
28 | span.o { color:#af75c1; }
29 | /* numbers */
30 | span.n { color:#96C71D; }
31 | /* line number */
32 | span.l { color:#80b0b0; }
33 | /* mutable var or ref cell */
34 | span.v { color:#d1d1d1; font-weight: bold; }
35 | /* inactive code */
36 | span.inactive { color:#808080; }
37 | /* preprocessor */
38 | span.prep { color:#af75c1; }
39 | /* fsi output */
40 | span.fsi { color:#808080; }
41 |
42 | /* omitted */
43 | span.omitted {
44 | background:#3c4e52;
45 | border-radius:5px;
46 | color:#808080;
47 | padding:0px 0px 1px 0px;
48 | }
49 | /* tool tip */
50 | div.tip {
51 | background:#475b5f;
52 | border-radius:4px;
53 | font:11pt 'Droid Sans', arial, sans-serif;
54 | padding:6px 8px 6px 8px;
55 | display:none;
56 | color:#d1d1d1;
57 | }
58 | table.pre pre {
59 | padding:0px;
60 | margin:0px;
61 | border:none;
62 | }
63 | table.pre, pre.fssnip, pre {
64 | line-height:13pt;
65 | border:1px solid #d8d8d8;
66 | border-collapse:separate;
67 | white-space:pre;
68 | font: 9pt 'Droid Sans Mono',consolas,monospace;
69 | width:90%;
70 | margin:10px 20px 20px 20px;
71 | background-color:#212d30;
72 | padding:10px;
73 | border-radius:5px;
74 | color:#d1d1d1;
75 | }
76 | table.pre pre {
77 | padding:0px;
78 | margin:0px;
79 | border-radius:0px;
80 | width: 100%;
81 | }
82 | table.pre td {
83 | padding:0px;
84 | white-space:normal;
85 | margin:0px;
86 | }
87 | table.pre td.lines {
88 | width:30px;
89 | }
90 |
91 | /*--------------------------------------------------------------------------
92 | Formatting for page & standard document content
93 | /*--------------------------------------------------------------------------*/
94 |
95 | body {
96 | font-family: 'Open Sans', serif;
97 | padding-top: 0px;
98 | padding-bottom: 40px;
99 | }
100 |
101 | pre {
102 | word-wrap: inherit;
103 | }
104 |
105 | /* Format the heading - nicer spacing etc. */
106 | .masthead {
107 | overflow: hidden;
108 | }
109 | .masthead .muted a {
110 | text-decoration:none;
111 | color:#999999;
112 | }
113 | .masthead ul, .masthead li {
114 | margin-bottom:0px;
115 | }
116 | .masthead .nav li {
117 | margin-top: 15px;
118 | font-size:110%;
119 | }
120 | .masthead h3 {
121 | margin-bottom:5px;
122 | font-size:170%;
123 | }
124 | hr {
125 | margin:0px 0px 20px 0px;
126 | }
127 |
128 | /* Make table headings and td.title bold */
129 | td.title, thead {
130 | font-weight:bold;
131 | }
132 |
133 | /* Format the right-side menu */
134 | #menu {
135 | margin-top:50px;
136 | font-size:11pt;
137 | padding-left:20px;
138 | }
139 |
140 | #menu .nav-header {
141 | font-size:12pt;
142 | color:#606060;
143 | margin-top:20px;
144 | }
145 |
146 | #menu li {
147 | line-height:25px;
148 | }
149 |
150 | /* Change font sizes for headings etc. */
151 | #main h1 { font-size: 26pt; margin:10px 0px 15px 0px; font-weight:400; }
152 | #main h2 { font-size: 20pt; margin:20px 0px 0px 0px; font-weight:400; }
153 | #main h3 { font-size: 14pt; margin:15px 0px 0px 0px; font-weight:600; }
154 | #main p { font-size: 11pt; margin:5px 0px 15px 0px; }
155 | #main ul { font-size: 11pt; margin-top:10px; }
156 | #main li { font-size: 11pt; margin: 5px 0px 5px 0px; }
157 | #main strong { font-weight:700; }
158 |
159 | /*--------------------------------------------------------------------------
160 | Formatting for API reference
161 | /*--------------------------------------------------------------------------*/
162 |
163 | .type-list .type-name, .module-list .module-name {
164 | width:25%;
165 | font-weight:bold;
166 | }
167 | .member-list .member-name {
168 | width:35%;
169 | }
170 | #main .xmldoc h2 {
171 | font-size:14pt;
172 | margin:10px 0px 0px 0px;
173 | }
174 | #main .xmldoc h3 {
175 | font-size:12pt;
176 | margin:10px 0px 0px 0px;
177 | }
178 | .github-link {
179 | float:right;
180 | text-decoration:none;
181 | }
182 | .github-link img {
183 | border-style:none;
184 | margin-left:10px;
185 | }
186 | .github-link .hover { display:none; }
187 | .github-link:hover .hover { display:block; }
188 | .github-link .normal { display: block; }
189 | .github-link:hover .normal { display: none; }
190 |
191 | /*--------------------------------------------------------------------------
192 | Links
193 | /*--------------------------------------------------------------------------*/
194 |
195 | h1 a, h1 a:hover, h1 a:focus,
196 | h2 a, h2 a:hover, h2 a:focus,
197 | h3 a, h3 a:hover, h3 a:focus,
198 | h4 a, h4 a:hover, h4 a:focus,
199 | h5 a, h5 a:hover, h5 a:focus,
200 | h6 a, h6 a:hover, h6 a:focus { color : inherit; text-decoration : inherit; outline:none }
201 |
202 | /*--------------------------------------------------------------------------
203 | Additional formatting for the homepage
204 | /*--------------------------------------------------------------------------*/
205 |
206 | #nuget {
207 | margin-top:20px;
208 | font-size: 11pt;
209 | padding:20px;
210 | }
211 |
212 | #nuget pre {
213 | font-size:11pt;
214 | -moz-border-radius: 0px;
215 | -webkit-border-radius: 0px;
216 | border-radius: 0px;
217 | background: #404040;
218 | border-style:none;
219 | color: #e0e0e0;
220 | margin-top:15px;
221 | }
--------------------------------------------------------------------------------
/docs/input/vector-operations.fsx:
--------------------------------------------------------------------------------
1 | (*** hide ***)
2 | #r "../../src/FsAlg/bin/Debug/FsAlg.dll"
3 |
4 | (**
5 | Vector Operations
6 | =================
7 |
8 | This page lists only a selection of commonly used vector operations provided by FsAlg.
9 |
10 | Please refer to [API Reference](reference/index.html) for a full list of supported operations.
11 |
12 | The **FsAlg.Generic.Vector** module provides functionality similar to the F# [Collecitons.Array](https://msdn.microsoft.com/en-us/library/ee370273.aspx) module, for creating and manipulating vectors.
13 |
14 | Creating Vectors
15 | ----------------
16 | *)
17 |
18 | open FsAlg.Generic
19 |
20 | let v1 = vector [1.; 2.; 3.]
21 | let v2 = Vector.create 3 1.
22 | let v3 = Vector.init 3 (fun i -> exp (float i))
23 |
24 | (*** hide, define-output: o ***)
25 | printf "val v1 : Vector = Vector [|1.0; 2.0; 3.0|]
26 | val v2 : Vector = Vector [|1.0; 1.0; 1.0|]
27 | val v3 : Vector = Vector [|1.0; 2.718281828; 7.389056099|]"
28 | (*** include-output: o ***)
29 |
30 | (**
31 | Basic Operations
32 | ----------------
33 | *)
34 |
35 | let v4 = v1 + v2 // Vector addition
36 | let v5 = v1 - v2 // Vector substraction
37 | let v6 = v1 * v2 // Vector inner product (dot / scalar product)
38 | let v7 = v1 %* v2 // Vector cross product
39 | let v8 = v1 .* v2 // Vector element-wise (Hadamard) product
40 | let v9 = v1 ./ v2 // Vector element-wise (Hadamard) division
41 | let v10 = v1 + 2. // Add scalar to vector
42 | let v11 = v1 - 2. // Subtract scalar from vector
43 | let v12 = 2. - v1 // Subtract each element of vector from scalar
44 | let v13 = v1 * 2. // Vector-scalar multiplication
45 | let v14 = v1 / 2. // Vector-scalar division
46 | let v15 = 2. / v1 // Divides each element of vector by scalar
47 | let v16 = -v1 // Unary negation
48 |
49 | (*** hide, define-output: o2 ***)
50 | printf "val v4 : Vector = Vector [|2.0; 3.0; 4.0|]
51 | val v5 : Vector = Vector [|0.0; 1.0; 2.0|]
52 | val v6 : float = 6.0
53 | val v7 : Vector = Vector [|-1.0; 2.0; -1.0|]
54 | val v8 : Vector = Vector [|1.0; 2.0; 3.0|]
55 | val v9 : Vector = Vector [|1.0; 2.0; 3.0|]
56 | val v10 : Vector = Vector [|3.0; 4.0; 5.0|]
57 | val v11 : Vector = Vector [|-1.0; 0.0; 1.0|]
58 | val v12 : Vector = Vector [|1.0; 0.0; -1.0|]
59 | val v13 : Vector = Vector [|2.0; 4.0; 6.0|]
60 | val v14 : Vector = Vector [|0.5; 1.0; 1.5|]
61 | val v15 : Vector = Vector [|2.0; 1.0; 0.6666666667|]
62 | val v16 : Vector = Vector [|-1.0; -2.0; -3.0|]"
63 | (*** include-output: o2 ***)
64 |
65 | (**
66 | Vector Norms
67 | ------------
68 | *)
69 |
70 | let n1 = Vector.l1norm v1 // L1 norm
71 | let n2 = Vector.l2norm v1 // L2 norm
72 | let n3 = Vector.l2normSq v1 // Squared L2 norm
73 | let n4 = Vector.lpnorm 3. v1 // Lp norm
74 |
75 | (*** hide, define-output: o3 ***)
76 | printf "val n1 : float = 6.0
77 | val n2 : float = 3.741657387
78 | val n3 : float = 14.0
79 | val n4 : float = 3.301927249"
80 | (*** include-output: o3 ***)
81 |
82 | (**
83 | Accessing Elements & Conversions
84 | --------------------------------
85 | *)
86 |
87 | let e1 = v1.[0] // 1st element of v1
88 | let e2 = Vector.get v1 1 // 2nd element of v1
89 | let e3 = v1.[..1] // Slice, until 2nd element
90 | let e4 = v1.[1..2] // Slice, between 2nd and 3rd elements
91 | let v20 = Vector.ofSeq [1.; 2.; 3.] // Convert sequence to vector
92 | let v21 = vector [1.; 2.; 3.] // Same with above, supports lists,
93 | let v22 = vector [|1.; 2.; 3.|] // arrays,
94 | let v23 = vector (seq {yield 1.}) // and sequences.
95 | let aa1 = Vector.toArray v1 // Convert vector to array
96 | let sq1 = Vector.toSeq v1 // Convert vector to sequence
97 |
98 | (*** hide, define-output: o4 ***)
99 | printf "val e1 : float = 1.0
100 | val e2 : float = 2.0
101 | val e3 : Vector = Vector [|1.0; 2.0|]
102 | val e4 : Vector = Vector [|2.0; 3.0|]
103 | val v20 : Vector = Vector [|1.0; 2.0; 3.0|]
104 | val v21 : Vector = Vector [|1.0; 2.0; 3.0|]
105 | val v22 : Vector = Vector [|1.0; 2.0; 3.0|]
106 | val v23 : Vector = Vector [|1.0|]
107 | val aa1 : float [] = [|1.0; 2.0; 3.0|]
108 | val sq1 : seq"
109 | (*** include-output: o4 ***)
110 |
111 | (**
112 | Mutating/Replacing Elements
113 | ------------------
114 | *)
115 | v1.[0] <- 4. // Mutate an element
116 | Vector.replace (fun x -> x + 2.) v1 // Replace by mapping a function, mutating in place
117 | Vector.replacei (fun i x -> x + float i) v1 // Replace in place, with index
118 | Vector.replace2 (fun x y -> x - y) v1 v2 // Replace v1 in place, using a function of v1 and v2
119 | Vector.replacei2 (fun i x y -> x - y + float i) v1 v2 // Replace v1 in place, with index
120 | Vector.replaceWith v1 v2 // Replace elements of v1 with v2, mutating in place
121 |
122 | (**
123 | Splitting and Concatenating
124 | ---------------------------
125 | *)
126 |
127 | let ss1 = Vector.splitEqual 3 (vector [1.; 2.; 3.; 4.; 5.; 6.]) // Split into 3 vectors of equal length
128 | let ss2 = Vector.split [2; 4] (vector [1.; 2.; 3.; 4.; 5.; 6.]) // Split into vectors of given lengths
129 | let cc1 = Vector.concat ss1 // Concatenate vectors into one
130 |
131 | (*** hide, define-output: o5 ***)
132 | printf "val ss1 : seq>
133 | val ss2 : seq>
134 | val cc1 : Vector = Vector [|1.0; 2.0; 3.0; 4.0; 5.0; 6.0|]"
135 | (*** include-output: o5 ***)
136 |
137 | (**
138 | Mathematica and MATLAB Strings
139 | ------------------------------
140 |
141 | You can generate string representations of vectors that you can copy and paste into Mathematica notebooks or MATLAB.
142 | *)
143 |
144 | let s1 = v1.ToMathematicaString()
145 | let s2 = v1.ToMatlabString()
146 |
147 | (*** hide, define-output: o6 ***)
148 | printf "val s1 : string = \"{1.00, 2.00, 3.00}\"
149 | val s2 : string = \"[1.00 2.00 3.00]\""
150 | (*** include-output: o6 ***)
151 |
152 | (**
153 | Other Operations
154 | ---------------------
155 | The following are just a selection of other operations. Please refer to [API Reference](reference/index.html) for a full list of supported operations.
156 | *)
157 |
158 | let len = Vector.length v1 // Length of vector
159 | let min = Vector.min v1 // Minimum element of vector
160 | let max = Vector.max v1 // Maximum element of vector
161 | let sum = Vector.sum v1 // Sum elements of vector
162 | let v17 = Vector.unitVector v1 // Get unit vector codirectional with v1
163 | let v18 = Vector.map (fun x -> sin x) v1 // Map function to vector
164 | let v19 = Vector.fold (fun s x -> s + sin x) 0. v1 // Fold a vector
165 |
166 | (*** hide, define-output: o7 ***)
167 | printf "val len : int = 3
168 | val min : float = 1.0
169 | val max : float = 3.0
170 | val sum : float = 6.0
171 | val v17 : Vector = Vector [|0.2672612419; 0.5345224838; 0.8017837257|]
172 | val v18 : Vector = Vector [|0.8414709848; 0.9092974268; 0.1411200081|]
173 | val v19 : float = 1.89188842"
174 | (*** include-output: o7 ***)
--------------------------------------------------------------------------------
/docs/input/matrix-operations.fsx:
--------------------------------------------------------------------------------
1 | (*** hide ***)
2 | #r "../../src/FsAlg/bin/Debug/FsAlg.dll"
3 |
4 | (**
5 | Matrix Operations
6 | =================
7 |
8 | This page lists only a selection of commonly used matrix operations provided by FsAlg.
9 |
10 | Please refer to [API Reference](reference/index.html) for a full list of supported operations.
11 |
12 | The **FsAlg.Generic.Matrix** module provides functionality similar to the F# [Collecitons.Array2D](https://msdn.microsoft.com/en-us/library/ee353794.aspx) module, for creating and manipulating matrices.
13 |
14 | Creating Matrices
15 | -----------------
16 | *)
17 |
18 | open FsAlg.Generic
19 |
20 | let m1 = matrix [[1.; 2.]; [3.; 4.]]
21 | let m2 = Matrix.create 2 2 1.
22 | let m3 = Matrix.init 2 2 (fun i j -> exp (float (i + j)))
23 |
24 | (*** hide, define-output: o ***)
25 | printf "val m1 : Matrix = Matrix [[1.0; 1.0]
26 | [1.0; 1.0]]
27 | val m2 : Matrix = Matrix [[1.0; 1.0]
28 | [1.0; 1.0]]
29 | val m3 : Matrix = Matrix [[1.0; 2.718281828]
30 | [2.718281828; 7.389056099]]"
31 | (*** include-output: o ***)
32 |
33 | (**
34 | Basic Operations
35 | ----------------
36 | *)
37 |
38 | let m4 = m1 + m2 // Matrix addition
39 | let m5 = m1 - m2 // Matrix subtraction
40 | let m6 = m1 * m2 // Matrix product
41 | let m7 = m1 .* m2 // Matrix element-wise (Hadamard) product
42 | let m8 = m1 ./ m2 // Matrix element-wise (Hadamard) division
43 |
44 | let v = vector [1.; 2.]
45 | let m9 = m1 * v // Matrix-vector product
46 | let m10 = v * m1 // Vector-matrix product
47 |
48 | let m11 = m1 + 2. // Add scalar to matrix
49 | let m12 = m1 - 2. // Subtract scalar from matrix
50 | let m13 = 2. - m1 // Subtract each element of matrix from scalar
51 | let m14 = m1 * 2. // Matrix-scalar multiplication
52 | let m15 = m1 / 2. // Matrix-scalar division
53 | let m16 = 2. / m1 // Divides each element of matrix by scalar
54 | let m17 = -m1 // Unary negation
55 |
56 | (*** hide, define-output: o2 ***)
57 | printf "val m4 : Matrix = Matrix [[2.0; 3.0]
58 | [4.0; 5.0]]
59 | val m5 : Matrix = Matrix [[0.0; 1.0]
60 | [2.0; 3.0]]
61 | val m6 : Matrix = Matrix [[3.0; 3.0]
62 | [7.0; 7.0]]
63 | val m7 : Matrix = Matrix [[1.0; 2.0]
64 | [3.0; 4.0]]
65 | val m8 : Matrix = Matrix [[1.0; 2.0]
66 | [3.0; 4.0]]
67 | val v : Vector = Vector [|1.0; 2.0|]
68 | val m9 : Vector = Vector [|5.0; 11.0|]
69 | val m10 : Vector = Vector [|7.0; 10.0|]
70 | val m11 : Matrix = Matrix [[3.0; 4.0]
71 | [5.0; 6.0]]
72 | val m12 : Matrix = Matrix [[-1.0; 0.0]
73 | [1.0; 2.0]]
74 | val m13 : Matrix = Matrix [[1.0; 0.0]
75 | [-1.0; -2.0]]
76 | val m14 : Matrix = Matrix [[2.0; 4.0]
77 | [6.0; 8.0]]
78 | val m15 : Matrix = Matrix [[0.5; 1.0]
79 | [1.5; 2.0]]
80 | val m16 : Matrix = Matrix [[2.0; 1.0]
81 | [0.6666666667; 0.5]]
82 | val m17 : Matrix = Matrix [[-1.0; -2.0]
83 | [-3.0; -4.0]]"
84 | (*** include-output: o2 ***)
85 |
86 | (**
87 | Matrix Operations
88 | -----------------
89 | *)
90 |
91 | let m1det = Matrix.det m1 // Determinant
92 | let m1inv = Matrix.inverse m1 // Inverse
93 | let m1eig = Matrix.eigenvalues m1 // Eigenvalues
94 | let m1t = Matrix.transpose m1 // Transpose
95 | let m1tr = Matrix.trace m1 // Trace
96 | let m1dia = Matrix.diagonal m1 // Diagonal
97 |
98 | (*** hide, define-output: o3 ***)
99 | printf "val m1det : float = -2.0
100 | val m1inv : Matrix = Matrix [[-2.0; 1.0]
101 | [1.5; -0.5]]
102 | val m1eig : Vector = Vector [|5.372281312; -0.3722813117|]
103 | val m1t : Matrix = Matrix [[1.0; 3.0]
104 | [2.0; 4.0]]
105 | val m1tr : float = 5.0
106 | val m1dia : Vector = Vector [|1.0; 4.0|]"
107 | (*** include-output: o3 ***)
108 |
109 | (**
110 | Solving Systems of Linear Equations
111 | -----------------------------------
112 | We can solve a [system of linear equations](http://en.wikipedia.org/wiki/System_of_linear_equations) represented in matrix form $\mathbf{A} \mathbf{x} = \mathbf{b} \;,$ which has solution $\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}\;$.
113 |
114 | For example, the system
115 |
116 | $$$
117 | \begin{eqnarray*}
118 | 3x + 2y - z &=& 1\\
119 | 2x - 2y +4z &=& -2\\
120 | -x + 0.5y -z &=& 0\\
121 | \end{eqnarray*}
122 |
123 | can be represented as
124 |
125 | $$$
126 | \begin{bmatrix}
127 | 3 & 2 & -1\\
128 | 2 & -2 & 4\\
129 | -1& 0.5 &-1\\
130 | \end{bmatrix} \begin{bmatrix}
131 | x\\
132 | y\\
133 | z\\
134 | \end{bmatrix} = \begin{bmatrix}
135 | 1\\
136 | -2\\
137 | 0\\
138 | \end{bmatrix}
139 |
140 | and has solution
141 |
142 | $$$
143 | \begin{eqnarray*}
144 | x &=& 1\\
145 | y &=& -2\\
146 | z &=& -2\\
147 | \end{eqnarray*}\; .
148 | *)
149 |
150 | let A = matrix [[3.; 2.; -1.]; [2.; -2.; 4.]; [-1.; 0.5; -1.]]
151 | let b = vector [1.; -2.; 0.]
152 |
153 | let x = Matrix.solve A b
154 |
155 | (*** hide, define-output: o4 ***)
156 | printf "val A : Matrix = Matrix [[3.0; 2.0; -1.0]
157 | [2.0; -2.0; 4.0]
158 | [-1.0; 0.5; -1.0]]
159 | val b : Vector = Vector [|1.0; -2.0; 0.0|]
160 | val x : Vector = Vector [|1.0; -2.0; -2.0|]"
161 | (*** include-output: o4 ***)
162 |
163 | (**
164 | Matrix Decompositions
165 | ---------------------
166 | *)
167 |
168 | let lu, piv, togg = Matrix.decomposeLU m1 // LU decomposition
169 | let q, r = Matrix.decomposeQR m1 // QR decomposition
170 |
171 | (*** hide, define-output: o5 ***)
172 | printf "val togg : float = -1.0
173 | val piv : int [] = [|1; 0|]
174 | val lu : Matrix = Matrix [[3.0; 4.0]
175 | [0.3333333333; 0.6666666667]]
176 | val r : Matrix = Matrix [[3.16227766; 4.427188724]
177 | [4.440892099e-16; 0.632455532]]
178 | val q : Matrix = Matrix [[0.316227766; 0.9486832981]
179 | [0.9486832981; -0.316227766]]"
180 | (*** include-output: o5 ***)
181 |
182 | (**
183 | Accessing Elements & Conversions
184 | --------------------------------
185 | *)
186 |
187 | let el1 = m1.[0, 0] // Element at 0, 0
188 | let el2 = Matrix.get m1 1 1 // Element at 1, 1
189 | let sl1 = m1.[0..1, 1..] // Slice, between rows 0 and 1, columns 1 and beyond
190 | let sl2 = m1.[*, 0..1] // Slice, all rows, between columns 0 and 1
191 | let sl3 = m1.[0, *] // Return row 0
192 | let sl4 = Matrix.row 0 m1 // Return row 0
193 | let sl5 = m1.[*, 1] // Return column 1
194 | let sl6 = Matrix.col 1 m1 // Return column 1
195 | let sl7 = Matrix.toRows m1 // Return rows as a sequence
196 | let m18 = Matrix.ofRows [vector [1.;2.]; vector [3.;4.]] // Create matrix from row vectors
197 | let sl8 = Matrix.toCols m1 // Return columns as a sequence
198 | let m19 = Matrix.ofCols [vector [1.;2.]; vector [3.;4.]] // Create matrix from column vectors
199 | let m20 = Matrix.ofSeqSeq [[1.; 2.]; [3.; 4.]] // Convert sequence of sequences to matrix
200 | let m21 = matrix [[1.; 2.]; [3.; 4.]] // Same with above, supports lists,
201 | let m22 = matrix [|[|1.; 2.|]; [|3.; 4.|]|] // arrays,
202 | let m23 = matrix (seq {yield seq {yield 1.}}) // and sequences.
203 | let aa1 = Matrix.toArrayArray m1 // Convert matrix to jagged array
204 | let aa2 = Matrix.toArray2D m1 // Convert matrix to 2d array
205 | let aa3 = Matrix.toArray m1 // Convert matrix to 1d array, scanning left to right, top to bottom
206 | let ma3 = Matrix.ofArray 2 aa3 // Convert 1d array to matrix, populating two rows
207 | let vv1 = Matrix.toVector m1 // Convert matrix to vector, scanning left to right, top to bottom
208 | let mv1 = Matrix.ofVector 2 vv1 // Convert vector to matrix, populating two rows
209 |
210 | (*** hide, define-output: o6 ***)
211 | printf "val el1 : float = 1.0
212 | val el2 : float = 4.0
213 | val sl1 : Matrix = Matrix [[2.0]
214 | [4.0]]
215 | val sl2 : Matrix = Matrix [[1.0; 2.0]
216 | [3.0; 4.0]]
217 | val sl3 : Matrix = Matrix [[1.0; 2.0]]
218 | val sl4 : Matrix = Matrix [[1.0; 2.0]]
219 | val sl5 : Matrix = Matrix [[2.0]
220 | [4.0]]
221 | val sl6 : Matrix = Matrix [[2.0]
222 | [4.0]]
223 | val sl7 : seq>
224 | val m18 : Matrix = Matrix [[1.0; 2.0]
225 | [3.0; 4.0]]
226 | val sl8 : seq>
227 | val m19 : Matrix = Matrix [[1.0; 3.0]
228 | [2.0; 4.0]]
229 | val m20 : Matrix = Matrix [[1.0; 2.0]
230 | [3.0; 4.0]]
231 | val m21 : Matrix = Matrix [[1.0; 2.0]
232 | [3.0; 4.0]]
233 | val m22 : Matrix = Matrix [[1.0; 2.0]
234 | [3.0; 4.0]]
235 | val m23 : Matrix = Matrix [[1.0]]
236 | val aa1 : float [] [] = [|[|1.0; 2.0|]; [|3.0; 4.0|]|]
237 | val aa2 : float [,] = [[1.0; 1.0]
238 | [1.0; 1.0]]
239 | val aa3 : float [] = [|1.0; 2.0; 3.0; 4.0|]
240 | val ma3 : Matrix = Matrix [[1.0; 2.0]
241 | [3.0; 4.0]]
242 | val vv1 : Vector = Vector [|1.0; 2.0; 3.0; 4.0|]
243 | val mv1 : Matrix = Matrix [[1.0; 2.0]
244 | [3.0; 4.0]]"
245 | (*** include-output: o6 ***)
246 |
247 |
248 | (**
249 | Mutating/Replacing Elements
250 | ------------------
251 | *)
252 |
253 | m1.[0, 0] <- 4. // Mutate an element
254 | Matrix.replace (fun x -> x + 2.) m1 // Replace by mapping a function, mutating in place
255 | Matrix.replacei (fun i j x -> x + float (i * j)) m1 // Replace in place, with index
256 | Matrix.replace2 (fun x y -> x - y) m1 m2 // Replace m1 in place, using a function of m1 and m2
257 | Matrix.replacei2 (fun i j x y -> x - y + float (i * j)) m1 m2 // Replace m1 in place, with index
258 | Matrix.replaceWith m1 m2 // Replace elements of m1 with m2, mutating in place
259 | Matrix.appendRow v m1 // Append a vector to a matrix as a new row
260 | Matrix.appendCol v m1 // Append a vector to a matrix as a new column
261 | Matrix.prependRow v m1 // Prepend a vector to a matrix as a new row
262 | Matrix.prependCol v m1 // Prepend a vector to a matrix as a new column
263 |
264 | (**
265 | Mathematica and MATLAB Strings
266 | ------------------------------
267 | You can generate string representations of matrices that you can copy and paste into Mathematica notebooks or MATLAB.
268 | *)
269 |
270 | let s1 = m1.ToMathematicaString()
271 | let s2 = m2.ToMatlabString()
272 |
273 | (*** hide, define-output: o7 ***)
274 | printf "val s1 : string = \"{{1.00, 2.00}, {3.00, 4.00}}\"
275 | val s2 : string = \"[1.00 1.00; 1.00 1.00]\""
276 | (*** include-output: o7 ***)
277 |
278 | (**
279 | Other Operations
280 | ---------------------
281 | The following are just a selection of other operations. Please refer to [API Reference](reference/index.html) for a full list of supported operations.
282 | *)
283 |
284 | let l1 = Matrix.rows m1 // Number of rows
285 | let l2 = Matrix.cols m1 // Number of columns
286 | let m24 = Matrix.map (fun x -> sin x) m1 // Map function to matrix
287 |
288 | (*** hide, define-output: o8 ***)
289 | printf "val l1 : int = 2
290 | val l2 : int = 2
291 | val m24 : Matrix = Matrix [[0.8414709848; 0.8414709848]
292 | [0.8414709848; 0.8414709848]]"
293 | (*** include-output: o8 ***)
294 |
--------------------------------------------------------------------------------
/src/FsAlg/Vector.fs:
--------------------------------------------------------------------------------
1 | //
2 | // This file is part of
3 | // FsAlg: Generic Linear Algebra Library
4 | //
5 | // Copyright (c) 2015, National University of Ireland Maynooth (Atilim Gunes Baydin, Barak A. Pearlmutter)
6 | //
7 | // FsAlg is released under the BSD license.
8 | // (See accompanying LICENSE file.)
9 | //
10 | // Written by:
11 | //
12 | // Atilim Gunes Baydin
13 | // atilimgunes.baydin@nuim.ie
14 | //
15 | // Barak A. Pearlmutter
16 | // barak@cs.nuim.ie
17 | //
18 | // Brain and Computation Lab
19 | // Hamilton Institute & Department of Computer Science
20 | // National University of Ireland Maynooth
21 | // Maynooth, Co. Kildare
22 | // Ireland
23 | //
24 | // www.bcl.hamilton.ie
25 | //
26 |
27 | namespace FsAlg.Generic
28 |
29 | open FsAlg.Generic.Util
30 |
31 | /// Generic vector type
32 | []
33 | type Vector<'T when 'T : (static member Zero : 'T)
34 | and 'T : (static member One : 'T)
35 | and 'T : (static member (+) : 'T * 'T -> 'T)
36 | and 'T : (static member (-) : 'T * 'T -> 'T)
37 | and 'T : (static member (*) : 'T * 'T -> 'T)
38 | and 'T : (static member (/) : 'T * 'T -> 'T)
39 | and 'T : (static member (~-) : 'T -> 'T)
40 | and 'T : (static member Abs : 'T -> 'T)
41 | and 'T : (static member Pow : 'T * 'T -> 'T)
42 | and 'T : (static member Sqrt : 'T -> 'T)
43 | and 'T : (static member op_Explicit : 'T -> float)
44 | and 'T : comparison> =
45 | | ZeroVector of 'T
46 | | Vector of 'T[]
47 | /// ZeroVector
48 | static member inline Zero = ZeroVector LanguagePrimitives.GenericZero<'T>
49 | /// Converts vector `v` to float[]
50 | static member inline op_Explicit(v:Vector<'T>) =
51 | match v with
52 | | Vector v -> Array.map float v
53 | | ZeroVector _ -> [||]
54 | /// The element of this vector at the given position `i`
55 | member inline v.Item
56 | with get i =
57 | match v with
58 | | Vector v -> v.[i]
59 | | ZeroVector z -> z
60 | and set i vv =
61 | match v with
62 | | Vector v -> v.[i] <- vv
63 | | ZeroVector _ -> ()
64 | /// Gets a subvector between bounds `lower` and `upper`
65 | member inline v.GetSlice(lower, upper) =
66 | match v with
67 | | Vector v ->
68 | let l = defaultArg lower 0
69 | let u = defaultArg upper (v.Length - 1)
70 | if l > u then invalidArg "" "Given slice bounds are invalid."
71 | Vector v.[l..u]
72 | | ZeroVector _ -> invalidArg "" "Cannot get slice of a ZeroVector."
73 | /// Gets the first element of this vector
74 | member inline v.FirstItem =
75 | match v with
76 | | Vector v -> v.[0]
77 | | ZeroVector z -> z
78 | /// Gets the total number of elements of this vector
79 | member inline v.Length =
80 | match v with
81 | | Vector v -> v.Length
82 | | ZeroVector _ -> 0
83 | /// Gets the L1 (Manhattan) norm of this vector
84 | member inline v.GetL1Norm() =
85 | match v with
86 | | Vector v -> Array.sumBy abs v
87 | | ZeroVector z -> z
88 | /// Gets the L2 (Euclidean) norm of this vector
89 | member inline v.GetL2Norm() =
90 | match v with
91 | | Vector v -> sqrt (Array.sumBy (fun x -> x * x) v)
92 | | ZeroVector z -> z
93 | /// Gets the squared L2 (Euclidean) norm of this vector
94 | member inline v.GetL2NormSq() =
95 | match v with
96 | | Vector v -> Array.sumBy (fun x -> x * x) v
97 | | ZeroVector z -> z
98 | /// Gets the Lp norm (or p-norm) of this vector, with the given `p`
99 | member inline v.GetLPNorm(p:'T):'T =
100 | match v with
101 | | Vector v -> (Array.sumBy (fun x -> (abs x) ** p) v) ** (LanguagePrimitives.GenericOne<'T> / p)
102 | | ZeroVector z -> z
103 | /// Gets the minimum element of this vector
104 | member inline v.GetMin() =
105 | match v with
106 | | Vector v -> Array.min v
107 | | ZeroVector z -> z
108 | /// Gets the minimum element of this vector, compared by using Operators.min on the result of function `f`
109 | member inline v.GetMinBy(f) =
110 | match v with
111 | | Vector v -> Array.minBy f v
112 | | ZeroVector z -> z
113 | /// Gets the maximum element of this vector
114 | member inline v.GetMax() =
115 | match v with
116 | | Vector v -> Array.max v
117 | | ZeroVector z -> z
118 | /// Gets the maximum element of this vector, compared by using Operators.max on the result of function `f`
119 | member inline v.GetMaxBy(f) =
120 | match v with
121 | | Vector v -> Array.maxBy f v
122 | | ZeroVector z -> z
123 | /// Gets the unit vector codirectional with this vector
124 | member inline v.GetUnitVector() =
125 | match v with
126 | | Vector vv -> let n = v.GetL2Norm() in Vector (Array.map (fun x -> x / n) vv)
127 | | ZeroVector z -> ZeroVector z
128 | /// Returns a sequence of vectors that are obtained by splitting this vector into subvectors whose lengths are given in sequence `n`
129 | member inline v.Split(n:seq) =
130 | match v with
131 | | Vector v ->
132 | let i = ref 0
133 | seq {for j in n do
134 | yield Array.sub v !i j |> Vector
135 | i := !i + j}
136 | | ZeroVector _ -> seq {yield v}
137 | /// Returns a sequence of vectors that are obtained by splitting this vector into `n` subvectors of equal length. The length of this vector must be an integer multiple of `n`, otherwise ArgumentException is raised.
138 | member inline v.SplitEqual(n:int) =
139 | match v with
140 | | Vector v ->
141 | if n <= 0 then invalidArg "" "For splitting this vector, n should be a positive integer."
142 | let l = (float v.Length) / (float n)
143 | if not (isInteger l) then invalidArg "" "Cannot split vector into n equal pieces when length of vector is not an integer multiple of n."
144 | seq {for i in 0 .. (int l) .. (v.Length - 1) do yield Vector (Array.sub v i (int l))}
145 | | ZeroVector _ -> seq {yield v}
146 | /// Gets a string representation of this vector that can be pasted into a Mathematica notebook
147 | member inline v.ToMathematicaString() =
148 | let sb = System.Text.StringBuilder()
149 | sb.Append("{") |> ignore
150 | for i = 0 to v.Length - 1 do
151 | sb.Append(sprintf "%.2f" (float v.[i])) |> ignore
152 | if i < v.Length - 1 then sb.Append(", ") |> ignore
153 | sb.Append("}") |> ignore
154 | sb.ToString()
155 | /// Gets a string representation of this vector that can be pasted into MATLAB
156 | member inline v.ToMatlabString() =
157 | let sb = System.Text.StringBuilder()
158 | sb.Append("[") |> ignore
159 | for i = 0 to v.Length - 1 do
160 | sb.Append(sprintf "%.2f" (float v.[i])) |> ignore
161 | if i < v.Length - 1 then sb.Append(" ") |> ignore
162 | sb.Append("]") |> ignore
163 | sb.ToString()
164 | /// Converts the elements of this vector to another type, using the given conversion function `f`
165 | member inline v.Convert(f:'T->'a):Vector<'a> =
166 | match v with
167 | | Vector v -> Vector (Array.map f v)
168 | | ZeroVector _ -> ZeroVector LanguagePrimitives.GenericZero<'a>
169 | /// Creates a copy of this vector
170 | member inline v.Copy() =
171 | match v with
172 | | Vector v -> Vector (Array.copy v)
173 | | ZeroVector z -> ZeroVector z
174 | /// Converts this vector to an array
175 | member inline v.ToArray() =
176 | match v with
177 | | Vector v -> v
178 | | ZeroVector _ -> [||]
179 | /// Converts this vector to a sequence
180 | member inline v.ToSeq() =
181 | match v with
182 | | Vector v -> Array.toSeq v
183 | | ZeroVector _ -> Seq.empty
184 | /// Creates a new vector that contains the given subrange of elements, specified by start index `s` and count `c`
185 | member inline v.GetSubVector(s, c) =
186 | match v with
187 | | Vector v -> Vector (Array.sub v s c)
188 | | ZeroVector _ -> Vector.Zero
189 | /// Returns an enumerator that iterates through the elements of this vector
190 | member inline v.GetEnumerator() =
191 | v.ToSeq().GetEnumerator()
192 | /// Adds vector `a` to vector `b`
193 | static member inline (+) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
194 | match a, b with
195 | | Vector a, Vector b -> try Vector (Array.map2 (+) a b) with | _ -> invalidArg "" "Cannot add two vectors of different dimensions."
196 | | Vector _, ZeroVector _ -> a
197 | | ZeroVector _, Vector _ -> b
198 | | ZeroVector _, ZeroVector _ -> Vector.Zero
199 | /// Subtracts vector `b` from vector `a`
200 | static member inline (-) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
201 | match a, b with
202 | | Vector a, Vector b -> try Vector (Array.map2 (-) a b) with | _ -> invalidArg "" "Cannot subtract two vectors of different dimensions."
203 | | Vector _, ZeroVector _ -> a
204 | | ZeroVector _, Vector b -> Vector (Array.map (~-) b)
205 | | ZeroVector _, ZeroVector _ -> Vector.Zero
206 | /// Computes the inner product (dot / scalar product) of vector `a` and vector `b`
207 | static member inline (*) (a:Vector<'T>, b:Vector<'T>):'T =
208 | match a, b with
209 | | Vector a, Vector b -> try Array.map2 (*) a b |> Array.sum with | _ -> invalidArg "" "Cannot multiply two vectors of different dimensions."
210 | | Vector _, ZeroVector _ -> LanguagePrimitives.GenericZero<'T>
211 | | ZeroVector _, Vector _ -> LanguagePrimitives.GenericZero<'T>
212 | | ZeroVector _, ZeroVector _ -> LanguagePrimitives.GenericZero<'T>
213 | /// Computes the cross product of vector `a` and vector `b` (three-dimensional)
214 | static member inline (%*) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
215 | match a, b with
216 | | Vector va, Vector vb ->
217 | if (a.Length <> 3) || (b.Length <> 3) then invalidArg "" "The cross product is only defined for three-dimensional vectors."
218 | Vector [|va.[1] * vb.[2] - va.[2] * vb.[1]; va.[2] * vb.[0] - va.[0] * vb.[2]; va.[0] * vb.[1] - va.[1] * vb.[0]|]
219 | | Vector _, ZeroVector _ -> Vector.Zero
220 | | ZeroVector _, Vector _ -> Vector.Zero
221 | | ZeroVector _, ZeroVector _ -> Vector.Zero
222 | /// Multiplies vector `a` and vector `b` element-wise (Hadamard product)
223 | static member inline (.*) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
224 | match a, b with
225 | | Vector a, Vector b -> try Vector (Array.map2 (*) a b) with | _ -> invalidArg "" "Cannot multiply two vectors of different dimensions."
226 | | Vector _, ZeroVector _ -> Vector.Zero
227 | | ZeroVector _, Vector _ -> Vector.Zero
228 | | ZeroVector _, ZeroVector _ -> Vector.Zero
229 | /// Divides vector `a` by vector `b` element-wise (Hadamard division)
230 | static member inline (./) (a:Vector<'T>, b:Vector<'T>):Vector<'T> =
231 | match a, b with
232 | | Vector a, Vector b -> try Vector (Array.map2 (/) a b) with | _ -> invalidArg "" "Cannot divide two vectors of different dimensions."
233 | | Vector _, ZeroVector _-> raise (new System.DivideByZeroException("Attempted to divide a Vector by a ZeroVector."))
234 | | ZeroVector _, Vector _ -> Vector.Zero
235 | | ZeroVector _, ZeroVector _ -> raise (new System.DivideByZeroException("Attempted to divide a ZeroVector by a ZeroVector."))
236 | /// Adds scalar `b` to each element of vector `a`
237 | static member inline (+) (a:Vector<'T>, b:'T):Vector<'T> =
238 | match a with
239 | | Vector a -> Vector (Array.map ((+) b) a)
240 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroVector."
241 | /// Adds scalar `a` to each element of vector `b`
242 | static member inline (+) (a:'T, b:Vector<'T>):Vector<'T> =
243 | match b with
244 | | Vector b -> Vector (Array.map ((+) a) b)
245 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroVector."
246 | /// Subtracts scalar `b` from each element of vector `a`
247 | static member inline (-) (a:Vector<'T>, b:'T):Vector<'T> =
248 | match a with
249 | | Vector a -> Vector (Array.map (fun x -> x - b) a)
250 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot subtract a scalar from a ZeroVector."
251 | /// Subtracts each element of vector `b` from scalar `a`
252 | static member inline (-) (a:'T, b:Vector<'T>):Vector<'T> =
253 | match b with
254 | | Vector b -> Vector (Array.map ((-) a) b)
255 | | ZeroVector _ -> invalidArg "" "Unsupported operation. Cannot add subtract a ZeroVector from a scalar."
256 | /// Multiplies each element of vector `a` by scalar `b`
257 | static member inline (*) (a:Vector<'T>, b:'T):Vector<'T> =
258 | match a with
259 | | Vector a -> Vector (Array.map ((*) b) a)
260 | | ZeroVector _ -> Vector.Zero
261 | /// Multiplies each element of vector `b` by scalar `a`
262 | static member inline (*) (a:'T, b:Vector<'T>):Vector<'T> =
263 | match b with
264 | | Vector b -> Vector (Array.map ((*) a) b)
265 | | ZeroVector _ -> Vector.Zero
266 | /// Divides each element of vector `a` by scalar `b`
267 | static member inline (/) (a:Vector<'T>, b:'T):Vector<'T> =
268 | match a with
269 | | Vector a -> Vector (Array.map (fun x -> x / b) a)
270 | | ZeroVector _ -> Vector.Zero
271 | /// Divides scalar `a` by each element of vector `b`
272 | static member inline (/) (a:'T, b:Vector<'T>):Vector<'T> =
273 | match b with
274 | | Vector b -> Vector (Array.map ((/) a) b)
275 | | ZeroVector _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroVector."))
276 | /// Gets the negative of Vector `a`
277 | static member inline (~-) (a:Vector<'T>):Vector<'T> =
278 | match a with
279 | | Vector a -> Vector (Array.map (~-) a)
280 | | ZeroVector _ -> Vector.Zero
281 |
282 |
283 | /// Operations on Vector type. (Implementing functionality similar to Microsoft.FSharp.Collections.Array)
284 | []
285 | module Vector =
286 | /// Createsa vector from array `v`
287 | let inline ofArray (v:'T[]) = Vector v
288 | /// Converts vector `v` to an array
289 | let inline toArray (v:Vector<'T>):'T[] = v.ToArray()
290 | /// Creates a vector from sequence `s`
291 | let inline ofSeq (s:seq<'T>):Vector<'T> = Vector (Array.ofSeq s)
292 | /// Returns vector `v` as a sequence
293 | let inline toSeq (v:Vector<'T>):seq<'T> = v.ToSeq()
294 | /// Creates a vector that contains the elements of vector `v1` followed by the elements of vector `v2`
295 | let inline append (v1:Vector<'T>) (v2:Vector<'T>):Vector<'T> = Array.append (v1 |> toArray) (v2 |> toArray) |> Vector
296 | /// Builds a new vector that contains the elements of each of the given sequence of vectors `v`
297 | let inline concat (v:seq>):Vector<'T> = Seq.map toArray v |> Array.concat |> ofSeq
298 | /// Creates a copy of vector `v`
299 | let inline copy (v:Vector<'T>):Vector<'T> = v.Copy()
300 | /// Creates a vector with `n` elements, all having value `v`
301 | let inline create (n:int) (v:'T):Vector<'T> = Vector (Array.create n v)
302 | /// Creates a vector with `n` elements, where the element with index `i` has value `v` and the rest of the elements have value 0
303 | let inline createBasis (n:int) (i:int) (v:'T):Vector<'T> = Vector (Array.init n (fun j -> if j = i then v else LanguagePrimitives.GenericZero))
304 | /// Tests if any element of vector `v` satisfies predicate `p`
305 | let inline exists (p:'T->bool) (v:Vector<'T>):bool = v |> toArray |> Array.exists p
306 | /// Tests if any pair of corresponding elements of vectors `v1` and `v2` satisfies predicate `p`
307 | let inline exists2 (p:'T1->'T2->bool) (v1:Vector<'T1>) (v2:Vector<'T2>):bool = Array.exists2 p (v1 |> toArray) (v2 |> toArray)
308 | /// Fills a range of elements of vector `v` with value `a` starting with index `s` and counting `c` elements
309 | let inline fill (v:Vector<'T>) (s:int) (c:int) (a:'T):unit = Array.fill (v |> toArray) s c a
310 | /// Returns the first element of vector `v` for which predicate `p` is true
311 | let inline find (p:'T->bool) (v:Vector<'T>):'T = v |> toArray |> Array.find p
312 | /// Returns the index of the first element of vector `v` for which predicate `p` is true
313 | let inline findIndex (p:'T->bool) (v:Vector<'T>):int = v |> toArray |> Array.findIndex p
314 | /// Applies function `f` to each element of vector `v`, threading an accumulator (with initial state `s`) through the computation. If the input function is f and the elements are i0...iN then computes f (... (f s i0)...) iN.
315 | let inline fold (f:'S->'T->'S) (s:'S) (v:Vector<'T>):'S = v |> toArray |> Array.fold f s
316 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2`, threading an accumulator (with initial state `s`) through the computation. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
317 | let inline fold2 (f:'S->'T1->'T2->'S) (s:'S) (v1:Vector<'T1>) (v2:Vector<'T2>):'S = Array.fold2 f s (v1 |> toArray) (v2 |> toArray)
318 | /// Applies function `f` to each element of vector `v`, threading an accumulator (with initial state `s`) through the computation. If the input function is f and the elements are i0...iN then computes f i0 (...(f iN s)).
319 | let inline foldBack (f:'T->'S->'S) (v:Vector<'T>) (s:'S):'S = Array.foldBack f (v |> toArray) s
320 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2`, threading an accumulator (with initial state `s`) through the computation. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
321 | let inline foldBack2 (f:'T1->'T2->'S->'S) (v1:Vector<'T1>) (v2:Vector<'T2>) (s:'S):'S = Array.foldBack2 f (v1 |> toArray) (v2 |> toArray) s
322 | /// Tests if all elements of vector `v` satisfy predicate `p`
323 | let inline forall (p:'T->bool) (v:Vector<'T>):bool = v |> toArray |> Array.forall p
324 | /// Tests if all corresponding elements of vectors `v1` and `v2` satisfy predicate `p` pairwise
325 | let inline forall2 (p:'T1->'T2->bool) (v1:Vector<'T1>) (v2:Vector<'T2>):bool = Array.forall2 p (v1 |> toArray) (v2 |> toArray)
326 | /// Gets the element of vector `v` with index `i`
327 | let inline get (v:Vector<'T>) (i:int):'T = v.[i]
328 | /// Creates a vector with dimension `n` and a generator function `f` to compute the elements
329 | let inline init (n:int) (f:int->'T):Vector<'T> = Vector (Array.init n f)
330 | /// Applies function `f` to each element of vector `v`
331 | let inline iter (f:'T->unit) (v:Vector<'T>):unit = v |> toArray |> Array.iter f
332 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
333 | let inline iter2 (f:'T1->'T2->unit) (v1:Vector<'T1>) (v2:Vector<'T2>):unit = Array.iter2 f (v1 |> toArray) (v2 |> toArray)
334 | /// Applies function `f` to each element of vector `v`. The integer passed to function `f` indicates the index of element.
335 | let inline iteri (f:int->'T->unit) (v:Vector<'T>):unit = v |> toArray |> Array.iteri f
336 | /// Applies function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The integer passed to function `f` indicates the index of element. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
337 | let inline iteri2 (f:int->'T1->'T2->unit) (v1:Vector<'T1>) (v2:Vector<'T2>):unit = Array.iteri2 f (v1 |> toArray) (v2 |> toArray)
338 | /// Gets the L1 (Manhattan) norm of vector `v`
339 | let inline l1norm (v:Vector<'T>):'T = v.GetL1Norm()
340 | /// Gets the L2 (Euclidean) norm of vector `v`. This is the same with `Vector.norm`.
341 | let inline l2norm (v:Vector<'T>):'T = v.GetL2Norm()
342 | /// Gets the squared L2 (Euclidean) norm of vector `v`. This is the same with `Vector.normSq`.
343 | let inline l2normSq (v:Vector<'T>):'T = v.GetL2NormSq()
344 | /// Returns the length of vector `v`
345 | let inline length (v:Vector<'T>):int = v.Length
346 | /// Gets the Lp norm (or p-norm) of vector `v`, with the given `p`
347 | let inline lpnorm (p:'T) (v:Vector<'T>):'T = v.GetLPNorm(p)
348 | /// Creates a vector whose elements are the results of applying function `f` to each element of vector `v`
349 | let inline map (f:'T->'U) (v:Vector<'T>):Vector<'U> = v |> toArray |> Array.map f |> Vector
350 | /// Creates a vector whose elements are the results of applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
351 | let inline map2 (f:'T1->'T2->'U) (v1:Vector<'T1>) (v2:Vector<'T2>):Vector<'U> = Array.map2 f (v1 |> toArray) (v2 |> toArray) |> Vector
352 | /// Creates a vector whose elements are the results of applying function `f` to each element of vector `v`. An element index is also supplied to function `f`.
353 | let inline mapi (f:int->'T->'U) (v:Vector<'T>):Vector<'U> = v |> toArray |> Array.mapi f |> Vector
354 | /// Creates a vector whose elements are the results of applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The integer passed to function `f` indicates the index of element. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
355 | let inline mapi2 (f:int->'T1->'T2->'U) (v1:Vector<'T1>) (v2:Vector<'T2>):Vector<'U> = Array.mapi2 f (v1 |> toArray) (v2 |> toArray) |> Vector
356 | /// Returns the maximum of all elements of vector `v`
357 | let inline max (v:Vector<'T>):'T = v.GetMax()
358 | /// Returns the maximum of all elements of vector `v`, compared by using Operators.max on the result of function `f`
359 | let inline maxBy (f:'T->'U) (v:Vector<'T>):'T = v.GetMaxBy(f)
360 | /// Returns the minimum of all elements of vector `v`
361 | let inline min (v:Vector<'T>):'T = v.GetMin()
362 | /// Returns the minimum of all elements of vector `v`, compared by using Operators.min on the result of function `f`
363 | let inline minBy (f:'T->'U) (v:Vector<'T>):'T = v.GetMinBy(f)
364 | /// Gets the L2 (Euclidean) norm of vector `v`. This is the same with `Vector.l2norm`.
365 | let inline norm (v:Vector<'T>):'T = l2norm v
366 | /// Gets the squared L2 (Euclidean) norm of vector `v`. This is the same with `Vector.l2normSq`.
367 | let inline normSq (v:Vector<'T>):'T = l2normSq v
368 | /// Applies function `f` to each element of vector `v`, threading an accumulator argument through the computation. If the input function is f and the elements are i0...iN, then computes f (... (f i0 i1)...) iN.
369 | let inline reduce (f:'T->'T->'T) (v:Vector<'T>):'T = v |> toArray |> Array.reduce f
370 | /// Applies function `f` to each element of vector `v`, threading an accumulator argument through the computation. If the input function is f and the elements are i0...iN then computes f i0 (...(f iN-1 iN)).
371 | let inline reduceBack (f:'T->'T->'T) (v:Vector<'T>):'T = v |> toArray |> Array.reduceBack f
372 | /// Replaces the elements of vector `v` by mutating them in place, passing them through function `f`
373 | let inline replace (f:'T->'T) (v:Vector<'T>):unit = for i in 0..(v.Length - 1) do v.[i] <- f v.[i]
374 | /// Replaces the elements of vector `v1` by mutating them in place. The new values are computed by applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. The two input vectors must have the same lengths, otherwie ArgumentException is raised.
375 | let inline replace2 (f:'T1->'T2->'T1) (v1:Vector<'T1>) (v2:Vector<'T2>):unit =
376 | if v1.Length <> v2.Length then invalidArg "" "The vectors should have the same length."
377 | for i in 0..(v1.Length - 1) do v1.[i] <- f v1.[i] v2.[i]
378 | /// Replaces the elements of vector `v` by mutating them in place, passing them through function `f`. An element index is also supplied to function `f`.
379 | let inline replacei (f:int->'T->'T) (v:Vector<'T>):unit = for i in 0..(v.Length - 1) do v.[i] <- f i v.[i]
380 | /// Replaces the elements of vector `v1` by mutating them in place. The new values are computed by applying function `f` to corresponding elements of vectors `v1` and `v2` pairwise. An element index is also supplied to function `f`. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
381 | let inline replacei2 (f:int->'T1->'T2->'T1) (v1:Vector<'T1>) (v2:Vector<'T2>):unit =
382 | if v1.Length <> v2.Length then invalidArg "" "The vectors should have the same length."
383 | for i in 0..(v1.Length - 1) do v1.[i] <- f i v1.[i] v2.[i]
384 | /// Replaces the elements of vector `v1` with the elements of vector `v2`, by mutating them in place. The two input vectors must have the same lengths, otherwise ArgumentException is raised.
385 | let inline replaceWith (v1:Vector<'T>) (v2:Vector<'T>):unit =
386 | if v1.Length <> v2.Length then invalidArg "" "The vectors should have the same length."
387 | Array.blit (v2 |> toArray) 0 (v1 |> toArray) 0 v1.Length
388 | /// Like Vector.fold, but returns the intermediate and final results
389 | let inline scan (f:'S->'T->'S) (s:'S) (v:Vector<'T>):Vector<'S> = v |> toArray |> Array.scan f s |> ofSeq
390 | /// Like Vector.foldBack, but returns both the intermediate and final results
391 | let inline scanBack (f:'T->'S->'S) (s:'S) (v:Vector<'T>):Vector<'S> = Array.scanBack f (v |> toArray) s |> ofSeq
392 | /// Sets the element of vector`v` with index `i` to value `a`
393 | let inline set (v:Vector<'T>) (i:int) (a:'T):unit = v.[i] <- a
394 | /// Returns a sequence of vectors that are obtained by splitting vector `v` into subvectors whose lengths are given in the sequence `n`
395 | let inline split (n:seq) (v:Vector<'T>):seq> = v.Split(n)
396 | /// Returns a sequence of vectors that are obtained by splitting vector `v` into `n` subvectors of equal length. The length of vector `v` must be an integer multiple of `n`, otherwise ArgumentException is raised.
397 | let inline splitEqual (n:int) (v:Vector<'T>):seq> = v.SplitEqual(n)
398 | /// Creates a vector with `n` elements, where the `i`-th element is 1 and the rest of the elements are 0
399 | let inline standardBasis (n:int) (i:int):Vector<'T> = createBasis n i LanguagePrimitives.GenericOne
400 | /// Creates a new vector that contains the given subrange of vector `v`, specified by start index `s` and count `c`
401 | let inline sub (v:Vector<'T>) (s:int) (c:int):Vector<'T> = v.GetSubVector(s, c)
402 | /// Returns the sum of all the elements in vector `v`
403 | let inline sum (v:Vector<'T>):'T = v |> toArray |> Array.sum
404 | /// Returns the sum of the results generated by applying function `f` to each element of vector `v`
405 | let inline sumBy (f:'T->'U) (v:Vector<'T>):'U = v |> toArray |> Array.sumBy f
406 | /// Gets the unit vector codirectional with vector `v`
407 | let inline unitVector (v:Vector<'T>) = v.GetUnitVector()
--------------------------------------------------------------------------------
/src/FsAlg/Matrix.fs:
--------------------------------------------------------------------------------
1 | //
2 | // This file is part of
3 | // FsAlg: Generic Linear Algebra Library
4 | //
5 | // Copyright (c) 2015, National University of Ireland Maynooth (Atilim Gunes Baydin, Barak A. Pearlmutter)
6 | //
7 | // FsAlg is released under the BSD license.
8 | // (See accompanying LICENSE file.)
9 | //
10 | // Written by:
11 | //
12 | // Atilim Gunes Baydin
13 | // atilimgunes.baydin@nuim.ie
14 | //
15 | // Barak A. Pearlmutter
16 | // barak@cs.nuim.ie
17 | //
18 | // Brain and Computation Lab
19 | // Hamilton Institute & Department of Computer Science
20 | // National University of Ireland Maynooth
21 | // Maynooth, Co. Kildare
22 | // Ireland
23 | //
24 | // www.bcl.hamilton.ie
25 | //
26 |
27 | namespace FsAlg.Generic
28 |
29 | open FsAlg.Generic.Util
30 |
31 | /// Generic matrix type
32 | []
33 | type Matrix<'T when 'T : (static member Zero : 'T)
34 | and 'T : (static member One : 'T)
35 | and 'T : (static member (+) : 'T * 'T -> 'T)
36 | and 'T : (static member (-) : 'T * 'T -> 'T)
37 | and 'T : (static member (*) : 'T * 'T -> 'T)
38 | and 'T : (static member (/) : 'T * 'T -> 'T)
39 | and 'T : (static member (~-) : 'T -> 'T)
40 | and 'T : (static member Abs : 'T -> 'T)
41 | and 'T : (static member Pow : 'T * 'T -> 'T)
42 | and 'T : (static member Sqrt : 'T -> 'T)
43 | and 'T : (static member op_Explicit : 'T -> float)
44 | and 'T : comparison> =
45 | | ZeroMatrix of 'T
46 | | Matrix of 'T[,]
47 | /// ZeroMatrix
48 | static member inline Zero = ZeroMatrix LanguagePrimitives.GenericZero<'T>
49 | /// Converts Matrix `m` to float[,]
50 | static member inline op_Explicit(m:Matrix<'T>) =
51 | match m with
52 | | Matrix m -> Array2D.map float m
53 | | ZeroMatrix _ -> Array2D.zeroCreate 0 0
54 | /// Gets the number of rows of this matrix
55 | member inline m.Rows =
56 | match m with
57 | | Matrix m -> m.GetLength 0
58 | | ZeroMatrix _ -> 0
59 | /// Gets the number of columns of this matrix
60 | member inline m.Cols =
61 | match m with
62 | | Matrix m -> m.GetLength 1
63 | | ZeroMatrix _ -> 0
64 | /// The entry of this matrix at row `i` and column `j`
65 | member inline m.Item
66 | with get (i, j) =
67 | match m with
68 | | Matrix m -> m.[i, j]
69 | | ZeroMatrix z -> z
70 | and set (i, j) v =
71 | match m with
72 | | Matrix m -> m.[i, j] <- v
73 | | ZeroMatrix _ -> ()
74 | /// Gets a submatrix of this matrix with the bounds given in `rowStart`, `rowFinish`, `colStart`, `colFinish`
75 | member inline m.GetSlice(rowStart, rowFinish, colStart, colFinish) =
76 | match m with
77 | | Matrix mm ->
78 | let rowStart = defaultArg rowStart 0
79 | let rowFinish = defaultArg rowFinish (m.Rows - 1)
80 | if rowStart > rowFinish then invalidArg "" "Given row slice bounds are invalid."
81 | let colStart = defaultArg colStart 0
82 | let colFinish = defaultArg colFinish (m.Cols - 1)
83 | if colStart > colFinish then invalidArg "" "Given column slice bounds are invalid."
84 | Matrix mm.[rowStart..rowFinish, colStart..colFinish]
85 | | ZeroMatrix _ -> invalidArg "" "Cannot get slice of a ZeroMatrix."
86 | /// Gets a row of this matrix with the given row index `row` and column bounds `colStart` and `colFinish`
87 | member inline m.GetSlice(row, colStart, colFinish) =
88 | match m with
89 | | Matrix mm ->
90 | let colStart = defaultArg colStart 0
91 | let colFinish = defaultArg colFinish (m.Cols - 1)
92 | if colStart > colFinish then invalidArg "" "Given column slice bounds are invalid."
93 | [mm.[row, colStart..colFinish]] |> array2D |> Matrix
94 | | ZeroMatrix _ -> invalidArg "" "Cannot get slice of a ZeroMatrix."
95 | /// Gets a column of this matrix with the given column index `col` and row bounds `rowStart` and `rowFinish`
96 | member inline m.GetSlice(rowStart, rowFinish, col) =
97 | match m with
98 | | Matrix mm ->
99 | let rowStart = defaultArg rowStart 0
100 | let rowFinish = defaultArg rowFinish (m.Rows - 1)
101 | if rowStart > rowFinish then invalidArg "" "Given row slice bounds are invalid."
102 | [mm.[rowStart..rowFinish, col]] |> array2D |> transpose |> Matrix
103 | | ZeroMatrix _ -> invalidArg "" "Cannot get slice of a ZeroMatrix."
104 | /// Gets a string representation of this matrix that can be pasted into a Mathematica notebook
105 | member inline m.ToMathematicaString() =
106 | let sb = System.Text.StringBuilder()
107 | sb.Append("{") |> ignore
108 | for i = 0 to m.Rows - 1 do
109 | sb.Append("{") |> ignore
110 | for j = 0 to m.Cols - 1 do
111 | sb.Append(sprintf "%.2f" (float m.[i, j])) |> ignore
112 | if j <> m.Cols - 1 then sb.Append(", ") |> ignore
113 | sb.Append("}") |> ignore
114 | if i <> m.Rows - 1 then sb.Append(", ") |> ignore
115 | sb.Append("}") |> ignore
116 | sb.ToString()
117 | /// Gets a string representation of this matrix that can be pasted into MATLAB
118 | member inline m.ToMatlabString() =
119 | let sb = System.Text.StringBuilder()
120 | sb.Append("[") |> ignore
121 | for i = 0 to m.Rows - 1 do
122 | for j = 0 to m.Cols - 1 do
123 | sb.Append(sprintf "%.2f" (float m.[i, j])) |> ignore
124 | if j < m.Cols - 1 then sb.Append(" ") |> ignore
125 | if i < m.Rows - 1 then sb.Append("; ") |> ignore
126 | sb.Append("]") |> ignore
127 | sb.ToString()
128 | /// Converts this matrix into a 2d array
129 | member inline m.ToArray2D() =
130 | match m with
131 | | Matrix m -> m
132 | | ZeroMatrix _ -> Array2D.zeroCreate 0 0
133 | /// Converts this matrix into a jagged array, e.g. from Matrix to float[][]
134 | member inline m.ToArray() =
135 | let a = m.ToArray2D()
136 | [|for i = 0 to m.Rows - 1 do yield [|for j = 0 to m.Cols - 1 do yield a.[i, j]|]|]
137 | /// Converts this matrix into a one dimensional sequence, scanning columns from left to right and rows from top to bottom
138 | member inline m.ToSeq() =
139 | let a = m.ToArray()
140 | seq {for aa in a do yield! Seq.ofArray aa}
141 | /// Returns an enumerator that iterates through this matrix
142 | member inline m.GetEnumerator() =
143 | m.ToSeq().GetEnumerator()
144 | /// Creates a copy of this matrix
145 | member inline m.Copy() =
146 | match m with
147 | | Matrix m -> Matrix (Array2D.copy m)
148 | | ZeroMatrix z -> ZeroMatrix z
149 | /// Gets the trace of this matrix
150 | member inline m.GetTrace() =
151 | match m with
152 | | Matrix m -> trace m
153 | | ZeroMatrix z -> z
154 | /// Gets the transpose of this matrix
155 | member inline m.GetTranspose() =
156 | match m with
157 | | Matrix m -> Matrix (transpose m)
158 | | ZeroMatrix z -> ZeroMatrix z
159 | /// Gets a vector of the diagonal elements of this matrix
160 | member inline m.GetDiagonal() =
161 | match m with
162 | | Matrix mm ->
163 | if m.Rows <> m.Cols then invalidArg "" "Cannot get the diagonal entries of a nonsquare matrix."
164 | Array.init m.Rows (fun i -> mm.[i, i])
165 | | ZeroMatrix z -> [||]
166 | /// Returns the LU decomposition of this matrix. The return values are the LU matrix, pivot indices, and a toggle value indicating the number of row exchanges during the decomposition, which is +1 if the number of exchanges were even, -1 if odd.
167 | member inline m.GetLUDecomposition() =
168 | match m with
169 | | Matrix mm ->
170 | if (m.Rows <> m.Cols) then invalidArg "" "Cannot compute the LU decomposition of a nonsquare matrix."
171 | let res = Array2D.copy mm
172 | let perm = Array.init m.Rows (fun i -> i)
173 | let mutable toggle = LanguagePrimitives.GenericOne<'T>
174 | for j = 0 to m.Rows - 2 do
175 | let mutable colmax:'T = abs res.[j, j]
176 | let mutable prow = j
177 | for i = j + 1 to m.Rows - 1 do
178 | let absresij = abs res.[i, j]
179 | if absresij > colmax then
180 | colmax <- absresij
181 | prow <- i
182 | if prow <> j then
183 | let tmprow = res.[prow, 0..]
184 | res.[prow, 0..] <- res.[j, 0..]
185 | res.[j, 0..] <- tmprow
186 | let tmp = perm.[prow]
187 | perm.[prow] <- perm.[j]
188 | perm.[j] <- tmp
189 | toggle <- -toggle
190 | for i = j + 1 to m.Rows - 1 do
191 | res.[i, j] <- res.[i, j] / res.[j, j]
192 | for k = j + 1 to m.Rows - 1 do
193 | res.[i, k] <- res.[i, k] - res.[i, j] * res.[j, k]
194 | Matrix res, perm, toggle
195 | | ZeroMatrix z -> ZeroMatrix z, [||], LanguagePrimitives.GenericZero<'T>
196 | /// Gets the determinant of this matrix
197 | member inline m.GetDeterminant() =
198 | match m with
199 | | Matrix _ ->
200 | if (m.Rows <> m.Cols) then invalidArg "" "Cannot compute the determinant of a nonsquare matrix."
201 | let lu, _, toggle = m.GetLUDecomposition()
202 | toggle * Array.fold (fun s x -> s * x) LanguagePrimitives.GenericOne<'T> (lu.GetDiagonal())
203 | | ZeroMatrix z -> z
204 | /// Gets the inverse of this matrix
205 | member inline m.GetInverse() =
206 | match m with
207 | | Matrix mm ->
208 | if (m.Rows <> m.Cols) then invalidArg "" "Cannot compute the inverse of a nonsquare matrix."
209 | let res = Array2D.copy mm
210 | let lu, perm, _ = m.GetLUDecomposition()
211 | let b:'T[] = Array.zeroCreate m.Rows
212 | for i = 0 to m.Rows - 1 do
213 | for j = 0 to m.Rows - 1 do
214 | if i = perm.[j] then
215 | b.[j] <- LanguagePrimitives.GenericOne<'T>
216 | else
217 | b.[j] <- LanguagePrimitives.GenericZero<'T>
218 | let x = matrixSolveHelper (lu.ToArray2D()) b
219 | res.[0.., i] <- x
220 | Matrix res
221 | | ZeroMatrix z -> ZeroMatrix z
222 | /// Adds matrix `a` to matrix `b`
223 | static member inline (+) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
224 | match a, b with
225 | | Matrix ma, Matrix mb ->
226 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot add matrices of different sizes."
227 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] + mb.[i, j]))
228 | | Matrix _, ZeroMatrix _ -> a
229 | | ZeroMatrix _, Matrix _ -> b
230 | | ZeroMatrix z, ZeroMatrix _ -> ZeroMatrix z
231 | /// Subtracts matrix `b` from matrix `a`
232 | static member inline (-) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
233 | match a, b with
234 | | Matrix ma, Matrix mb ->
235 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot subtract matrices of different sizes."
236 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] - mb.[i, j]))
237 | | Matrix _, ZeroMatrix _ -> a
238 | | ZeroMatrix _, Matrix b -> Matrix (Array2D.map (~-) b)
239 | | ZeroMatrix _, ZeroMatrix _ -> Matrix.Zero
240 | /// Multiplies matrix `a` and matrix `b` (matrix product)
241 | static member inline (*) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
242 | match a, b with
243 | | Matrix ma, Matrix mb ->
244 | if (a.Cols <> b.Rows) then invalidArg "" "Cannot multiply two matrices of incompatible sizes."
245 | Matrix (Array2D.init a.Rows b.Cols (fun i j -> Array.sumBy (fun k -> ma.[i, k] * mb.[k, j]) [|0..(b.Rows - 1)|] ))
246 | | Matrix _, ZeroMatrix _ -> Matrix.Zero
247 | | ZeroMatrix _, Matrix _ -> Matrix.Zero
248 | | ZeroMatrix _, ZeroMatrix _ -> Matrix.Zero
249 | /// Multiplies matrix `a` and matrix `b` element-wise (Hadamard product)
250 | static member inline (.*) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
251 | match a, b with
252 | | Matrix ma, Matrix mb ->
253 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot multiply matrices of different sizes."
254 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] * mb.[i, j]))
255 | | Matrix _, ZeroMatrix _ -> Matrix.Zero
256 | | ZeroMatrix _, Matrix _ -> Matrix.Zero
257 | | ZeroMatrix _, ZeroMatrix _ -> Matrix.Zero
258 | /// Divides matrix `a` by matrix `b` element-wise (Hadamard division)
259 | static member inline (./) (a:Matrix<'T>, b:Matrix<'T>):Matrix<'T> =
260 | match a, b with
261 | | Matrix ma, Matrix mb ->
262 | if (a.Rows <> b.Rows) || (a.Cols <> b.Cols) then invalidArg "" "Cannot divide matrices of different sizes."
263 | Matrix (Array2D.init a.Rows a.Cols (fun i j -> ma.[i, j] / mb.[i, j]))
264 | | Matrix _, ZeroMatrix _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroMatrix."))
265 | | ZeroMatrix _, Matrix _ -> Matrix.Zero
266 | | ZeroMatrix _, ZeroMatrix _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroMatrix."))
267 | /// Computes the matrix-vector product of matrix `a` and vector `b`
268 | static member inline (*) (a:Matrix<'T>, b:Vector<'T>):Vector<'T> =
269 | match a, b with
270 | | Matrix ma, Vector vb ->
271 | if (a.Cols <> b.Length) then invalidArg "" "Cannot compute the matrix-vector product of a matrix and a vector of incompatible sizes."
272 | Vector (Array.init a.Rows (fun i -> Array.sumBy (fun j -> ma.[i, j] * vb.[j]) [|0..(b.Length - 1)|] ))
273 | | Matrix _, ZeroVector _ -> Vector.Zero
274 | | ZeroMatrix _, Vector _ -> Vector.Zero
275 | | ZeroMatrix _, ZeroVector _ -> Vector.Zero
276 | /// Computes the vector-matrix product of vector `a` and matrix `b`
277 | static member inline (*) (a:Vector<'T>, b:Matrix<'T>):Vector<'T> =
278 | match a, b with
279 | | Vector va, Matrix mb ->
280 | if (a.Length <> b.Rows) then invalidArg "" "Cannot compute the vector-matrix product of a vector and matrix of incompatible sizes."
281 | Vector (Array.init b.Cols (fun i -> Array.sumBy (fun j -> va.[j] * mb.[j, i]) [|0..(a.Length - 1)|]))
282 | | Vector _, ZeroMatrix _ -> Vector.Zero
283 | | ZeroVector _, Matrix _ -> Vector.Zero
284 | | ZeroVector _, ZeroMatrix _ -> Vector.Zero
285 | /// Adds scalar `b` to each element of matrix `a`
286 | static member inline (+) (a:Matrix<'T>, b:'T):Matrix<'T> =
287 | match a with
288 | | Matrix a -> Matrix (Array2D.map ((+) b) a)
289 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroMatrix."
290 | /// Adds scalar `a` to each element of matrix `b`
291 | static member inline (+) (a:'T, b:Matrix<'T>):Matrix<'T> =
292 | match b with
293 | | Matrix b -> Matrix (Array2D.map ((+) a) b)
294 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot add a scalar to a ZeroMatrix."
295 | /// Subtracts scalar `b` from each element of matrix `a`
296 | static member inline (-) (a:Matrix<'T>, b:'T):Matrix<'T> =
297 | match a with
298 | | Matrix a -> Matrix (Array2D.map (fun x -> x - b) a)
299 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot subtract a scalar from a ZeroMatrix."
300 | /// Subtracts each element of of matrix `b` from scalar `a`
301 | static member inline (-) (a:'T, b:Matrix<'T>):Matrix<'T> =
302 | match b with
303 | | Matrix b -> Matrix (Array2D.map ((-) a) b)
304 | | ZeroMatrix z -> invalidArg "" "Unsupported operation. Cannot subtract a ZeroMatrix from a scalar."
305 | /// Multiplies each element of matrix `a` by scalar `b`
306 | static member inline (*) (a:Matrix<'T>, b:'T):Matrix<'T> =
307 | match a with
308 | | Matrix a -> Matrix (Array2D.map ((*) b) a)
309 | | ZeroMatrix _ -> Matrix.Zero
310 | /// Multiplies each element of matrix `b` by scalar `a`
311 | static member inline (*) (a:'T, b:Matrix<'T>):Matrix<'T> =
312 | match b with
313 | | Matrix b -> Matrix (Array2D.map ((*) a) b)
314 | | ZeroMatrix _ -> Matrix.Zero
315 | /// Divides each element of matrix `a` by scalar `b`
316 | static member inline (/) (a:Matrix<'T>, b:'T):Matrix<'T> =
317 | match a with
318 | | Matrix a -> Matrix (Array2D.map (fun x -> x / b) a)
319 | | ZeroMatrix _ -> Matrix.Zero
320 | /// Creates a matrix whose elements are scalar `a` divided by each element of matrix `b`
321 | static member inline (/) (a:'T, b:Matrix<'T>):Matrix<'T> =
322 | match b with
323 | | Matrix b -> Matrix (Array2D.map ((/) a) b)
324 | | ZeroMatrix _ -> raise (new System.DivideByZeroException("Attempted division by a ZeroMatrix."))
325 | /// Gets the negative of matrix `a`
326 | static member inline (~-) (a:Matrix<'T>) =
327 | match a with
328 | | Matrix a -> Matrix (Array2D.map (~-) a)
329 | | ZeroMatrix _ -> Matrix.Zero
330 | /// Returns the QR decomposition of this matrix
331 | member inline m.GetQRDecomposition() =
332 | match m with
333 | | ZeroMatrix z -> failwith "Cannot compute the QR decomposition of ZeroMatrix."
334 | | Matrix mm ->
335 | let minor (m:_[,]) (d) =
336 | let rows = Array2D.length1 m
337 | let cols = Array2D.length2 m
338 | let ret = Array2D.zeroCreate rows cols
339 | for i = 0 to d - 1 do
340 | ret.[i, i] <- LanguagePrimitives.GenericOne
341 | Array2D.blit m d d ret d d (rows - d) (cols - d)
342 | ret
343 | let identity d = Array2D.init d d (fun i j -> if i = j then LanguagePrimitives.GenericOne else LanguagePrimitives.GenericZero)
344 | // Householder
345 | let kmax = -1 + min (m.Rows - 1) m.Cols
346 | let mutable z = m.Copy()
347 | let q = Array.create m.Rows Matrix.Zero
348 | for k = 0 to kmax do
349 | z <- Matrix (minor (z.ToArray2D()) k)
350 | let x = z.[*, k].ToSeq() |> Array.ofSeq |> Vector
351 | let mutable a = x.GetL2Norm()
352 | if mm.[k, k] > LanguagePrimitives.GenericZero then a <- -a
353 | let e = (x + Vector.createBasis m.Rows k a).GetUnitVector()
354 | q.[k] <- Matrix (identity m.Rows) + Matrix (Array2D.init m.Rows m.Rows (fun i j -> -(e.[i] * e.[j] + e.[i] * e.[j])))
355 | z <- q.[k] * z
356 | let mutable q' = q.[0]
357 | for i = 1 to kmax do
358 | q' <- q.[i] * q'
359 | q'.GetTranspose(), q' * m
360 | /// Returns the eigenvalues of this matrix. (Experimental code, complex eigenvalues are not supported.)
361 | member inline m.GetEigenvalues() =
362 | let mutable m' = m.Copy()
363 | for i = 0 to 20 do
364 | let q, r = m'.GetQRDecomposition()
365 | m' <- r * q
366 | m'.GetDiagonal()
367 |
368 | /// Operations on Matrix type. (Implementing functionality similar to Microsoft.FSharp.Collections.Array2D)
369 | []
370 | module Matrix =
371 | /// Creates a matrix from 2d array `m`
372 | let inline ofArray2D (m:'T[,]):Matrix<'T> = Matrix m
373 | /// Converts matrix `m` to a 2d array, e.g. from Matrix to float[,]
374 | let inline toArray2D (m:Matrix<'T>):'T[,] = m.ToArray2D()
375 | /// Creates a matrix from a jagged array, e.g. from float[][] to Matrix
376 | let inline ofArrayArray (m:'T[][]):Matrix<'T> = m |> array2D |> Matrix
377 | /// Converts matrix `m` to a jagged array, e.g. from Matrix to float[][]
378 | let inline toArrayArray (m:Matrix<'T>):'T[][] = m.ToArray()
379 | /// Creates a matrix from sequence of sequences `s`
380 | let inline ofSeqSeq (s:seq>):Matrix<'T> = s |> array2D |> Matrix
381 | /// Converts matrix `m` to a sequence of sequences
382 | let inline toSeqSeq (m:Matrix<'T>):seq> = m.ToArray() |> Array.map Seq.ofArray |> Seq.ofArray
383 | /// Creates a matrix with `m` rows from the one dimensional sequence `s`, filling columns from left to right and rows from top to bottom. The number of columns will be deduced from `m` and the length of the sequence `s`. The length of `s` must be an integer multiple of `m`.
384 | let inline ofSeq (m:int) (s:seq<'T>):Matrix<'T> =
385 | let n = Seq.length s / m
386 | Array2D.init m n (fun i j -> Seq.nth (i * n + j) s) |> Matrix
387 | /// Converts matrix `m` to a one dimensional sequence, scanning columns from left to right and rows from top to bottom
388 | let inline toSeq (m:Matrix<'T>):seq<'T> = m.ToSeq()
389 | /// Creates a matrix with `m` rows from the one dimensional array `a`, filling columns from left to right and rows from top to bottom. The number of columns will be deduced from `m` and the length of the array `a`. The length of `a` must be an integer multiple of `m`.
390 | let inline ofArray (m:int) (a:'T[]):Matrix<'T> = ofSeq m a
391 | /// Converts matrix `m` to a one dimensional array, scanning columns from left to right and rows from top to bottom
392 | let inline toArray (m:Matrix<'T>):'T[] = m |> toSeq |> Array.ofSeq
393 | /// Creates a matrix with `m` rows from the vector `v`, filling columns from left to right and rows from top to bottom. The number of columns will be deduced from `m` and the length of the vector `v`. The length of `v` must be an integer multiple of `m`.
394 | let inline ofVector (m:int) (v:Vector<'T>):Matrix<'T> = v |> Vector.toSeq |> ofSeq m
395 | /// Converts matrix `m` to a vector, scanning columns from left to right and rows from top to bottom
396 | let inline toVector (m:Matrix<'T>):Vector<'T> = m |> toSeq |> Vector.ofSeq
397 | /// Returns the j-th column of matrix `m`
398 | let inline col (j:int) (m:Matrix<'T>):Matrix<'T> = m.[*,j]
399 | /// Returns the number of columns in matrix `m`. This is the same with `Matrix.length2`.
400 | let inline cols (m:Matrix<'T>):int = m.Cols
401 | /// Creates a copy of Matrix `m`
402 | let inline copy (m:Matrix<'T>):Matrix<'T> = m.Copy()
403 | /// Creates a matrix with `m` rows, `n` columns, and all entries having value `v`
404 | let inline create (m:int) (n:int) (v:'T):Matrix<'T> = Matrix (Array2D.create m n v)
405 | /// Creates a matrix with `m` rows and all rows equal to array `v`
406 | let inline createRows (m:int) (v:'T[]):Matrix<'T> = Matrix (array2D (Array.init m (fun _ -> v)))
407 | /// Gets the LU decomposition of matrix `m`. The return values are the LU matrix, pivot indices, and a toggle value indicating the number of row exchanges during the decomposition, which is +1 if the number of exchanges were even, -1 if odd.
408 | let inline decomposeLU (m:Matrix<'T>):(Matrix<'T>*int[]*'T) = m.GetLUDecomposition()
409 | /// Gets the QR decomposition of matrix `m`
410 | let inline decomposeQR (m:Matrix<'T>):(Matrix<'T>*Matrix<'T>) = m.GetQRDecomposition()
411 | /// Gets the determinant of matrix `m`
412 | let inline det (m:Matrix<'T>):'T = m.GetDeterminant()
413 | /// Gets the diagonal elements of matrix `m`
414 | let inline diagonal (m:Matrix<'T>):Vector<'T> = m.GetDiagonal() |> Vector.ofSeq
415 | /// Creates the identity matrix with `m` rows and columns
416 | let inline identity (m:int):Matrix<'T> =
417 | Matrix (Array2D.init m m (fun i j -> if i = j then LanguagePrimitives.GenericOne<'T> else LanguagePrimitives.GenericZero<'T>))
418 | /// Gets the eigenvalues of matrix `m`
419 | let inline eigenvalues (m:Matrix<'T>):Vector<'T> = m.GetEigenvalues() |> Vector.ofSeq
420 | /// Gets the entry of matrix `m` with indices `i` and `j`
421 | let inline get (m:Matrix<'T>) (i:int) (j:int):'T = m.[i, j]
422 | /// Creates a matrix with `m` rows, `n` columns and a generator function `f` to compute the entries
423 | let inline init (m:int) (n:int) (f:int->int->'T):Matrix<'T> = Matrix (Array2D.init m n f)
424 | /// Creates a square matrix with `m` rows and columns and a generator function `f` to compute the elements. Function `f` is used only for populating the diagonal and the upper triangular part of the matrix, the lower triangular part will be the reflection.
425 | let inline initSymmetric (m:int) (f:int->int->'T):Matrix<'T> =
426 | if m = 0 then
427 | Matrix.Zero
428 | else
429 | let s = Array2D.zeroCreate<'T> m m
430 | for i = 0 to m - 1 do
431 | for j = i to m - 1 do
432 | s.[i, j] <- f i j
433 | Matrix (copyUpperToLower s)
434 | /// Gets the inverse of matrix `m`
435 | let inline inverse (m:Matrix<'T>):Matrix<'T> = m.GetInverse()
436 | /// Applies function `f` to each element of matrix `m`
437 | let inline iter (f:'T->unit) (m:Matrix<'T>):unit = m |> toArray2D |> Array2D.iter f
438 | /// Applies function `f` to each element of matrix `m`. Element indices are also supplied to function `f`.
439 | let inline iteri (f:int->int->'T->unit) (m:Matrix<'T>):unit = m |> toArray2D |> Array2D.iteri f
440 | /// Returns the number of rows in matrix `m`. This is the same with `Matrix.rows`.
441 | let inline length1 (m:Matrix<'T>):int = m.Rows
442 | /// Returns the number of columns in matrix `m`. This is the same with `Matrix.cols`.
443 | let inline length2 (m:Matrix<'T>):int = m.Cols
444 | /// Creates a matrix whose entries are the results of applying function `f` to each entry of matrix `m`
445 | let inline map (f:'T->'U) (m:Matrix<'T>):Matrix<'U> = m |> toArray2D |> Array2D.map f |> Matrix
446 | /// Creates a matrix whose entries are the results of applying function `f` to each entry of matrix `m`. Element indices are also supplied to function `f`.
447 | let inline mapi (f:int->int->'T->'U) (m:Matrix<'T>):Matrix<'U> = m |> toArray2D |> Array2D.mapi f |> Matrix
448 | /// Replaces the elements of matrix `m` by mutating them in place, passing them through function `f`
449 | let inline replace (f:'T->'T) (m:Matrix<'T>):unit =
450 | for i in 0..(m.Rows - 1) do
451 | for j in 0..(m.Cols - 1) do
452 | m.[i,j] <- f m.[i, j]
453 | /// Replaces the elements of matrix `m1` by mutating them in place. The new values are computed by applying function `f` to the corresponding elements of matrices `m1` and `m2` pairwise. The two input matrices should have the same dimensions, otherwise ArgumentException is raised.
454 | let inline replace2 (f:'T1->'T2->'T1) (m1:Matrix<'T1>) (m2:Matrix<'T2>):unit =
455 | if (m1.Rows <> m2.Rows) || (m1.Cols <> m2.Cols) then invalidArg "" "The matrices should have the same dimensions."
456 | for i in 0..(m1.Rows - 1) do
457 | for j in 0..(m1.Cols - 1) do
458 | m1.[i, j] <- f m1.[i, j] m2.[i, j]
459 | /// Replaces the elements of matrix `m` by mutating them in place, passing them through function` f`. Element indices are also supplied to function `f`.
460 | let inline replacei (f:int->int->'T->'T) (m:Matrix<'T>):unit =
461 | for i in 0..(m.Rows - 1) do
462 | for j in 0..(m.Cols - 1) do
463 | m.[i,j] <- f i j m.[i, j]
464 | /// Replaces the elements of matrix `m1` by mutating them in place. The new values are computed by applying function `f` to the corresponding elements of matrices `m1` and `m2` pairwise. Element indices are also supplied to function `f`. The two input matrices should have the same dimensions, otherwise ArgumentException is raised.
465 | let inline replacei2 (f:int->int->'T1->'T2->'T1) (m1:Matrix<'T1>) (m2:Matrix<'T2>):unit =
466 | if (m1.Rows <> m2.Rows) || (m1.Cols <> m2.Cols) then invalidArg "" "The matrices should have the same dimensions."
467 | for i in 0..(m1.Rows - 1) do
468 | for j in 0..(m1.Cols - 1) do
469 | m1.[i, j] <- f i j m1.[i, j] m2.[i, j]
470 | /// Replaces the elements of matrix `m1` with the elements of matrix `m2`, by mutating them in place. The two input matrices should have the same dimensions, otherwise ArgumentException is raised.
471 | let inline replaceWith (m1:Matrix<'T>) (m2:Matrix<'T>) =
472 | if (m1.Rows <> m2.Rows) || (m1.Cols <> m2.Cols) then invalidArg "" "The matrices should have the same dimensions."
473 | Array2D.blit (m2 |> toArray2D) 0 0 (m1 |> toArray2D) 0 0 m1.Rows m1.Cols
474 | /// Returns the i-th row of matrix `m`
475 | let inline row (i:int) (m:Matrix<'T>):Matrix<'T> = m.[i,*]
476 | /// Returns the number of rows in matrix `m`. This is the same with `Matrix.length1`.
477 | let inline rows (m:Matrix<'T>):int = m.Rows
478 | /// Sets the entry of matrix `m` with indices `i` and `j` to value `a`
479 | let inline set (m:Matrix<'T>) (i:int) (j:int) (a:'T) = m.[i, j] <- a
480 | /// Solves a system of linear equations ax = b, where the coefficients are given in matrix `a` and the result vector is vector `b`. The returned vector will correspond to x.
481 | let inline solve (a:Matrix<'T>) (b:Vector<'T>):Vector<'T> =
482 | if a.Cols <> b.Length then invalidArg "" "Cannot solve the system of equations using a matrix and a vector of incompatible sizes."
483 | let lu, perm, _ = a.GetLUDecomposition()
484 | let bp = Array.init a.Rows (fun i -> b.[perm.[i]])
485 | Vector (matrixSolveHelper (lu.ToArray2D()) bp)
486 | /// Gets the trace of matrix `m`
487 | let inline trace (m:Matrix<'T>):'T = m.GetTrace()
488 | /// Gets the transpose of matrix `m`
489 | let inline transpose (m:Matrix<'T>):Matrix<'T> = m.GetTranspose()
490 | /// Constructs a matrix out of a sequence of row vectors `v`. The row vectors should be of equal length.
491 | let inline ofRows (v:seq>):Matrix<'T> = v |> Seq.map Vector.toSeq |> ofSeqSeq
492 | /// Returns the rows of matrix `m` as a sequence of matrices
493 | let inline toRows (m:Matrix<'T>):seq> = Seq.init m.Rows (fun i -> m.[i,*])
494 | /// Constructs a matrix out of a sequence of column vectors `v`. The column vectors should be of equal length.
495 | let inline ofCols (v:seq>):Matrix<'T> = v |> ofRows |> transpose
496 | /// Returns the columns of matrix `m` as a sequence of matrices
497 | let inline toCols (m:Matrix<'T>):seq> = m |> transpose |> toRows
498 | /// Returns a matrix where vector `v` is appended as a new row to matrix `m`
499 | let inline appendRow (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
500 | if m.Cols <> v.Length then invalidArg "" "The length of the appended row should be the same with the number of the columns in the matrix."
501 | let mm = m |> toArray2D
502 | let ret = Array2D.zeroCreate (m.Rows + 1) m.Cols
503 | Array2D.blit mm 0 0 ret 0 0 m.Rows m.Cols
504 | for i = 0 to m.Cols - 1 do
505 | ret.[m.Rows, i] <- v.[i]
506 | Matrix ret
507 | /// Returns a matrix where vector `v` is appended as a new column to matrix `m`
508 | let inline appendCol (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
509 | if m.Rows <> v.Length then invalidArg "" "The length of the appended column should be the same with the number of rows in the matrix."
510 | let mm = m |> toArray2D
511 | let ret = Array2D.zeroCreate m.Rows (m.Cols + 1)
512 | Array2D.blit mm 0 0 ret 0 0 m.Rows m.Cols
513 | for i = 0 to m.Rows - 1 do
514 | ret.[i, m.Cols] <- v.[i]
515 | Matrix ret
516 | /// Returns a matrix where vector `v` is appended as a new row to matrix `m`
517 | let inline prependRow (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
518 | if m.Cols <> v.Length then invalidArg "" "The length of the prepended row should be the same with the number of the columns in the matrix."
519 | let mm = m |> toArray2D
520 | let ret = Array2D.zeroCreate (m.Rows + 1) m.Cols
521 | Array2D.blit mm 0 0 ret 1 0 m.Rows m.Cols
522 | for i = 0 to m.Cols - 1 do
523 | ret.[0, i] <- v.[i]
524 | Matrix ret
525 | /// Returns a matrix where vector `v` is prepended as a new column to matrix `m`
526 | let inline prependCol (v:Vector<'T>) (m:Matrix<'T>):Matrix<'T> =
527 | if m.Rows <> v.Length then invalidArg "" "The length of the prepended column should be the same with the number of rows in the matrix."
528 | let mm = m |> toArray2D
529 | let ret = Array2D.zeroCreate m.Rows (m.Cols + 1)
530 | Array2D.blit mm 0 0 ret 0 1 m.Rows m.Cols
531 | for i = 0 to m.Rows - 1 do
532 | ret.[i, 0] <- v.[i]
533 | Matrix ret
534 | /// Creates a matrix with `n` columns and a generator function `f` that gives each column as a vector
535 | let inline initCols (n:int) (f:int->Vector<'T>):Matrix<'T> = Matrix (array2D (Array.init n (f >> Vector.toArray))) |> transpose
536 | /// Creates a matrix with `m` rows and a generator function `f` that gives each row as a a vector
537 | let inline initRows (m:int) (f:int->Vector<'T>):Matrix<'T> = Matrix (array2D (Array.init m (f >> Vector.toArray)))
538 |
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