├── README.md ├── upload.sh ├── newton-krawczyk-time.jl ├── atahualpa.jl ├── krawczyk2d-tests.jl ├── newton-krawczyk-time.txt ├── logistic-map.jl ├── newton2d.jl ├── modules ├── NewtonMethod.jl ├── tests-krawczyk.jl ├── tests-newton.jl ├── ad-benet.jl ├── AutoDiff.jl ├── KrawczykMethod.jl └── IntervalArithmetic.jl ├── krawczyk2d.jl ├── 2D bisection.ipynb ├── krawczyk2d-tests-solutions.nb ├── newton-sin.ipynb ├── .ipynb_checkpoints ├── newton-sin-checkpoint.ipynb └── krawczyk2d-examples-checkpoint.ipynb ├── logistic-arbitrary.jl ├── krawczyk2d-examples.ipynb ├── LICENSE └── newton2d-examples.ipynb /README.md: -------------------------------------------------------------------------------- 1 | interval-methods 2 | =========== 3 | -------------------------------------------------------------------------------- /upload.sh: -------------------------------------------------------------------------------- 1 | git push https://kriukov@github.com/kriukov/interval-methods 2 | -------------------------------------------------------------------------------- /newton-krawczyk-time.jl: -------------------------------------------------------------------------------- 1 | using NewtonMethod 2 | using KrawczykMethod 3 | 4 | f(x) = 5040 - 3828x - 2356x^2 + 1009x^3 + 200x^4 - 62x^5 - 4x^6 + x^7 5 | a = Interval(-20, 20) 6 | @time newton(f, a, 128) 7 | @time krawczyk(f, a, 128) 8 | 9 | f(x) = x^2 - 4 10 | @time newton(f, a, 128) 11 | @time krawczyk(f, a, 128) 12 | 13 | f(x) = exp(x^2) - 4 14 | a = Interval(-4, 4) 15 | @time newton(f, a, 128) 16 | @time krawczyk(f, a, 128) 17 | 18 | f(x) = log(x^2 - 1) + x + 1 19 | a = Interval(-10, -1.1) 20 | @time newton(f, a, 128) 21 | @time krawczyk(f, a, 128) 22 | 23 | a = Interval(1.1, 10) 24 | @time newton(f, a, 128) 25 | @time krawczyk(f, a, 128) 26 | -------------------------------------------------------------------------------- /atahualpa.jl: -------------------------------------------------------------------------------- 1 | 2 | function first_collision(irr, delta) 3 | 4 | irr0 = irr 5 | 6 | arr_a = Integer[] 7 | 8 | n = 0 9 | while n <= 38 10 | a = floor(irr) 11 | #println(a) 12 | @show push!(arr_a, a) 13 | @show irr = 1/(irr - a) 14 | n += 1 15 | end 16 | 17 | 18 | h = Integer[] 19 | push!(h, 0, 1) 20 | for i = 3:40 21 | push!(h, arr_a[i-2]*h[i-1] + h[i-2]) 22 | end 23 | shift!(h) 24 | shift!(h) 25 | 26 | 27 | k = Integer[] 28 | push!(k, 1, 0) 29 | for i = 3:40 30 | push!(k, arr_a[i-2]*k[i-1] + k[i-2]) 31 | end 32 | shift!(k) 33 | shift!(k) 34 | 35 | #for i = 1:length(h) 36 | # println("$(abs(irr0 - h[i]/k[i])), $(1/(sqrt(5)*(k[i])^2))") 37 | # @show abs(irr0 - h[i]/k[i]) <= 1/(sqrt(5)*(k[i])^2) 38 | #end 39 | 40 | 41 | function first_solution(alpha, delta) 42 | n = 1 43 | while k[n] < 1/(sqrt(1 + alpha^2)*sqrt(5)*delta) 44 | n +=1 45 | end 46 | [k[n], h[n]] 47 | end 48 | 49 | first_solution(irr0, delta) 50 | end 51 | -------------------------------------------------------------------------------- /krawczyk2d-tests.jl: -------------------------------------------------------------------------------- 1 | # Exercises from the Ramon Moore book 2 | 3 | f(x) = [x[1]^2 + x[2]^2 - 1, x[1] - x[2]^2] 4 | a = [Interval(0.5, 0.8), Interval(0.6, 0.9)] 5 | krawczyk2d(f, a, 64) 6 | trueroots = [(sqrt(5) - 1)/2, sqrt((sqrt(5) - 1)/2)] 7 | 8 | 9 | f(x) = [20 - 20x[1] - x[2], x[1] - x[2]/20 - 1e-9*exp(x[2]/0.052)] 10 | a = [Interval(0.5, 1.2), Interval(0.6, 1.2)] 11 | krawczyk2d(f, a, 64) 12 | trueroots = [0.9464142468335176, 1.0717150633296477] 13 | 14 | 15 | f(x) = [0.5*(-(17.76x[1] - 103.79x[1]^2 + 229.62x[1]^3 - 226.31x[1]^4 + 83.72x[1]^5) + x[2]), 0.2*(-x[1] - 1.5x[2] + 1.2)] 16 | 17 | a = [Interval(0.01, 0.1), Interval(0.7, 0.9)] 18 | a = [Interval(0.7, 0.9), Interval(0.1, 0.3)] 19 | a = [Interval(0.11, 0.3), Interval(0.5, 0.69)] 20 | 21 | a = [Interval(0.01, 1), Interval(0.01, 1)] 22 | 23 | trueroots = [0.06263595920972119, 0.758242693860186] 24 | trueroots = [0.8844295888702942, 0.21038027408647064] 25 | trueroots = [0.2853687241300338, 0.6097541839133109] 26 | -------------------------------------------------------------------------------- /newton-krawczyk-time.txt: -------------------------------------------------------------------------------- 1 | Comparison between Newton and Krawczyk methods made by running file newton-krawczyk-time.jl. It is seen that Newton is faster than Krawczyk (after "Function calls", the first time value is of Newton, the second is of Krawczyk) 2 | 3 | Function calls: 16 4 | elapsed time: 1.565580033 seconds (252322430 bytes allocated, 4.00% gc time) 5 | elapsed time: 9.915345432 seconds (3577530829 bytes allocated, 15.90% gc time) 6 | 7 | Function calls: 10 8 | elapsed time: 0.017599943 seconds (3370058 bytes allocated) 9 | elapsed time: 0.258211858 seconds (113664575 bytes allocated) 10 | 11 | Function calls: 11 12 | elapsed time: 0.024908051 seconds (5353331 bytes allocated) 13 | elapsed time: 0.475278221 seconds (156370354 bytes allocated, 18.42% gc time) 14 | 15 | Function calls: 11 16 | elapsed time: 0.039670328 seconds (6722259 bytes allocated) 17 | elapsed time: 0.585135394 seconds (206057476 bytes allocated, 13.86% gc time) 18 | 19 | Function calls: 6 20 | elapsed time: 0.00397811 seconds (1490223 bytes allocated) 21 | elapsed time: 0.022749604 seconds (9578812 bytes allocated) 22 | -------------------------------------------------------------------------------- /logistic-map.jl: -------------------------------------------------------------------------------- 1 | using NewtonMethod 2 | 3 | # Function composition 4 | compose(f::Function, g::Function) = x -> f(g(x)) 5 | 6 | # Logistic function T(x) of the first order 7 | T1(x) = Interval(r)*(Interval(1/4)-(x-Interval(1/2))^2) 8 | 9 | # T^p (x) function - T(T(T(...T(x)))) p times 10 | function T(x, p) 11 | F = T1 12 | for i = 1:p-1 13 | F = compose(F, T1) 14 | end 15 | return F(x) 16 | end 17 | 18 | # Equation to solve for periodic points 19 | 20 | f(x, p) = T(x, p) - x 21 | 22 | point(f) = newton(f, Interval(0.0001, 1.), 50) 23 | 24 | r = 1.5 25 | f(x) = T(x, 1) - x 26 | 27 | println("$r $(point(f))") 28 | 29 | r = 2 30 | f(x) = T(x, 1) - x 31 | println("$r $(point(f))") 32 | 33 | r = 2.5 34 | f(x) = T(x, 1) - x 35 | println("$r $(point(f))") 36 | 37 | r = 3 38 | f(x) = T(x, 1) - x 39 | println("$r $(point(f))") 40 | 41 | r = 3.2 42 | f(x) = (T(x, 2) - x)/(T(x, 1) - x) 43 | println("$r $(point(f))") 44 | 45 | r = 3.5 46 | f(x) = (T(x, 4) - x)/(T(x, 2) - x) 47 | println("$r $(point(f))") 48 | 49 | r = 3.75 50 | for i = 1:8 51 | f(x) = T(x, i) - x 52 | println("$r $(point(f))") 53 | end 54 | -------------------------------------------------------------------------------- /newton2d.jl: -------------------------------------------------------------------------------- 1 | using IntervalArithmetic 2 | using AutoDiff 3 | 4 | 5 | function newton2d(f, a::Array{Interval, 1}, bigprec::Integer=64) 6 | 7 | set_bigfloat_precision(bigprec) 8 | 9 | # center() makes degenerate interval midpoints of the intervals in an array 10 | center(x) = make_intervals(mid(x)) 11 | N(x) = center(x) - inv(jacobian(f, x))*f(center(x)) 12 | 13 | # If a is symmetric, i.e., mid(a) = 0, the process may stall. The initial interval should be slightly asymmetrized then 14 | #if mid(a) == 0 15 | # a = Interval(a.lo, a.hi + 0.0001*mag(a)) 16 | #end 17 | 18 | roots_array = Array{Interval, 1}[] 19 | 20 | push!(roots_array, a) 21 | 22 | k = 0 23 | while true 24 | 25 | roots_array_new = Array{Interval, 1}[] 26 | 27 | for i = 1:length(roots_array) 28 | if isectext(roots_array[i], N(roots_array[i])) != false 29 | @show roots_array_new = push!(roots_array_new, isectext(roots_array[i], N(roots_array[i]))) 30 | end 31 | end 32 | 33 | # Exit criterion 34 | if roots_array_new == roots_array 35 | break 36 | end 37 | 38 | @show roots_array = roots_array_new 39 | k += 1 40 | end 41 | 42 | println("Function calls: ", k) 43 | return roots_array 44 | end 45 | -------------------------------------------------------------------------------- /modules/NewtonMethod.jl: -------------------------------------------------------------------------------- 1 | ## Newton's method (interval) code 2 | 3 | module NewtonMethod 4 | export newton, differentiate, Interval, rad, diam, mid, mig, mag, belong, hd, hull, isect, isectext 5 | 6 | using AutoDiff 7 | 8 | println("Syntax: newton(function, Interval(lo, hi), precision [default is 64])") 9 | 10 | function newton(f::Function, a::Interval, bigprec::Integer=64) 11 | 12 | set_bigfloat_precision(bigprec) 13 | 14 | center(x) = Interval(mid(x)) 15 | N(x) = center(x) - f(center(x))//differentiate(f, x) 16 | 17 | # If a is symmetric, i.e., mid(a) = 0, the process may stall. The initial interval should be slightly asymmetrized then 18 | if mid(a) == 0 19 | a = Interval(a.lo, a.hi + 0.0001*mag(a)) 20 | end 21 | 22 | roots_array = Interval[] 23 | 24 | push!(roots_array, a) 25 | 26 | k = 0 27 | while true 28 | 29 | roots_array_new = Interval[] 30 | 31 | for i = 1:length(roots_array) 32 | if isectext(roots_array[i], N(roots_array[i])) != false 33 | roots_array_new = vcat(roots_array_new, isectext(roots_array[i], N(roots_array[i]))) 34 | end 35 | end 36 | 37 | # Exit criterion 38 | if roots_array_new == roots_array 39 | break 40 | end 41 | 42 | roots_array = roots_array_new 43 | k += 1 44 | end 45 | 46 | println("Function calls: ", k) 47 | return roots_array 48 | end 49 | 50 | # end of module 51 | end 52 | -------------------------------------------------------------------------------- /modules/tests-krawczyk.jl: -------------------------------------------------------------------------------- 1 | using Base.Test 2 | using KrawczykMethod 3 | 4 | # Transcendental true roots were found using Wolfram Mathematica 5 | 6 | a = Interval(-20, 20) 7 | 8 | enclosure(x) = Interval(x - 5000*eps(BigFloat), x + 5000*eps(BigFloat)) 9 | 10 | function checkroots(f, trueroots) 11 | roots = krawczyk(f, a, 64) 12 | # Check if both ends of the found interval lie within the 10eps-enclosure of the true root 13 | n = 0 14 | for i = 1:length(roots) 15 | if belong(roots[i][1].lo, enclosure(trueroots[i])) == false && belong(roots[i][1].hi, enclosure(trueroots[i])) == false 16 | n += 1 17 | end 18 | end 19 | n == 0 20 | end 21 | 22 | f(x) = x^2 - 4 23 | trueroots = [-2, 2] 24 | @test checkroots(f, trueroots) 25 | 26 | f(x) = 5040 - 3828x - 2356x^2 + 1009x^3 + 200x^4 - 62x^5 - 4x^6 + x^7 27 | trueroots = [-6, -4, -2, 1, 3, 5, 7] 28 | @test checkroots(f, trueroots) 29 | 30 | f(x) = exp(x^2) - 4 31 | a = Interval(-4, 4) 32 | trueroots = [-sqrt(log(4)), sqrt(log(4))] 33 | @test checkroots(f, trueroots) 34 | 35 | # The next function is undefined in [-1, 1], so we split the interval manually 36 | f(x) = log(x^2 - 1) + x + 1 37 | a = Interval(-10, -1.01) 38 | trueroots = [-3.274277505524962, -1.7894889253208424] 39 | @test checkroots(f, trueroots) 40 | 41 | f(x) = log(x^2 - 1) + x + 1 42 | a = Interval(1.01, 10) 43 | trueroots = [1.0617135926109396] 44 | @test checkroots(f, trueroots) 45 | 46 | 47 | -------------------------------------------------------------------------------- /modules/tests-newton.jl: -------------------------------------------------------------------------------- 1 | using Base.Test 2 | using NewtonMethod 3 | 4 | # Transcendental true roots were found using Wolfram Mathematica 5 | 6 | # Choose interval [-20, 20] 7 | a = Interval(-20, 20) 8 | 9 | enclosure(x) = Interval(x - 10*eps(), x + 10*eps()) 10 | 11 | function checkroots(f, trueroots) 12 | roots = newton(f, a, 128) 13 | # Check if both ends of the found interval lie within the 10eps-enclosure of the true root 14 | n = 0 15 | for i = 1:length(roots) 16 | if belong(roots[i].lo, enclosure(trueroots[i])) == false && belong(roots[i].hi, enclosure(trueroots[i])) == false 17 | n += 1 18 | end 19 | end 20 | n == 0 21 | end 22 | 23 | f(x) = x^2 - 4 24 | trueroots = [-2, 2] 25 | @test checkroots(f, trueroots) 26 | 27 | f(x) = 5040 - 3828x - 2356x^2 + 1009x^3 + 200x^4 - 62x^5 - 4x^6 + x^7 28 | trueroots = [-6, -4, -2, 1, 3, 5, 7] 29 | @test checkroots(f, trueroots) 30 | 31 | f(x) = exp(x^2) - 4 32 | a = Interval(-4, 4) 33 | trueroots = [-sqrt(log(4)), sqrt(log(4))] 34 | @test checkroots(f, trueroots) 35 | 36 | # The next function is undefined in [-1, 1], so we split the interval manually 37 | f(x) = log(x^2 - 1) + x + 1 38 | a = Interval(-10, -1.11) 39 | trueroots = [-3.274277505524962, -1.7894889253208424] 40 | @test checkroots(f, trueroots) 41 | 42 | f(x) = log(x^2 - 1) + x + 1 43 | a = Interval(1.01, 10) 44 | trueroots = [1.0617135926109396] 45 | @test checkroots(f, trueroots) 46 | 47 | 48 | -------------------------------------------------------------------------------- /modules/ad-benet.jl: -------------------------------------------------------------------------------- 1 | ## Automatic differentiation 2 | 3 | #= 4 | type Dual{T <: Real} 5 | fun::T 6 | der::T 7 | end 8 | =# 9 | 10 | type Dual 11 | fun 12 | der 13 | end 14 | 15 | 16 | # Basic arithmetic 17 | +(u::Dual, w::Dual) = Dual(u.fun + w.fun, u.der + w.der) 18 | -(u::Dual, w::Dual) = Dual(u.fun - w.fun, u.der - w.der) 19 | *(u::Dual, w::Dual) = Dual(u.fun*w.fun, u.der*w.fun + u.fun*w.der) 20 | function /(u::Dual, w::Dual) 21 | ff = u.fun/w.fun 22 | dd = (u.der - ff*w.der)/w.fun 23 | Dual(ff, dd) 24 | end 25 | 26 | # Power; two functions for power because it is less costly 27 | ^(u::Dual, a::Integer)= Dual(u.fun^a, a*u.fun^(a-1)*u.der) 28 | 29 | function ^(u::Dual, a::Real) 30 | ff = u.fun^(a-1) 31 | Dual(ff*u.fun, a*ff*u.der) 32 | end 33 | 34 | # Enable dual-real operations as normal 35 | for op in (:+, :-, :*, :/) 36 | @eval begin 37 | $op(u::Dual, c::Real) = $op(u, Dual(c)) 38 | $op(c::Real, u::Dual) = $op(Dual(c), u) 39 | end 40 | end 41 | +(u::Dual) = u 42 | -(u::Dual) = Dual(-u.fun, -u.der) 43 | 44 | 45 | #= Trying to add intervals - unnecessary??? 46 | for op in (:+, :-, :*, :/) 47 | @eval begin 48 | $op(u::Dual, c::Interval) = $op(u, Dual(c)) 49 | $op(c::Interval, u::Dual) = $op(Dual(c), u) 50 | end 51 | end 52 | =# 53 | 54 | # Dual(constant) 55 | Dual{T<:Real}(c::T) = Dual{T}(c, zero(T)) 56 | 57 | # Elementary functions and their derivatives 58 | for (ff, dd) in ((:(Base.exp), :(Base.exp)), (:(Base.log), :(x->1/x)), (:(Base.sin), :(Base.cos)), (:(Base.cos), :(x->-sin(x))), (:(Base.tan), :(x->(sec(x))^2))) 59 | @eval begin 60 | function $ff(u::Dual) 61 | Dual($ff(u.fun), u.der*$dd(u.fun)) 62 | end 63 | end 64 | end 65 | 66 | # Differentiating function f at point a 67 | 68 | #= 69 | function differentiate(f, a) 70 | deriv(a) = f(Dual(a, one(a))).der 71 | if typeof(a) != Interval 72 | return deriv(a) 73 | else return Interval(min(deriv(a.lo), deriv(a.hi)), max(deriv(a.lo), deriv(a.hi))) 74 | end 75 | end 76 | =# 77 | 78 | differentiate(f, a) = f(Dual(a, one(a))).der 79 | 80 | 81 | #diff(f, a) = 82 | 83 | #diff(f, a::Interval) = Interval(diff(f, a.lo), diff(a.hi)) 84 | 85 | 86 | 87 | 88 | 89 | -------------------------------------------------------------------------------- /modules/AutoDiff.jl: -------------------------------------------------------------------------------- 1 | ## Automatic differentiation (interval version) 2 | 3 | module AutoDiff 4 | export differentiate, Ad, Interval, rad, diam, mid, mig, mag, belong, hd, hull, isect, isectext, lo, hi, make_intervals, det2 5 | 6 | using IntervalArithmetic 7 | 8 | type Ad 9 | u 10 | up 11 | end 12 | 13 | Ad(c) = Ad(c, Interval(0)) 14 | 15 | # Arithmetic between two Ad 16 | +(x::Ad, y::Ad) = Ad(x.u + y.u, x.up + y.up) 17 | -(x::Ad, y::Ad) = Ad(x.u - y.u, x.up - y.up) 18 | *(x::Ad, y::Ad) = Ad(x.u*y.u, x.u*y.up + y.u*x.up) 19 | function /(x::Ad, y::Ad) 20 | ff = x.u/y.u 21 | dd = (x.up - ff*y.up)/y.u 22 | Ad(ff, dd) 23 | end 24 | 25 | # Power; two functions for power because it is less costly 26 | # Commented out because it gave errors 27 | ^(x::Ad, a::Integer)= Ad(x.u^a, a*x.u^(a-1)*x.up) 28 | 29 | function ^(x::Ad, a::Real) 30 | ff = x.u^(a-1) 31 | Ad(ff*x.u, a*ff*x.up) 32 | end 33 | 34 | 35 | # Arithmetic operations between Ad and intervals/numbers 36 | 37 | for op in (:+, :-, :*, :/) 38 | @eval begin 39 | $op(u::Ad, c::Interval) = $op(u, Ad(c)) 40 | $op(c::Interval, u::Ad) = $op(Ad(c), u) 41 | $op(u::Ad, c::Real) = $op(u, Ad(c)) 42 | $op(c::Real, u::Ad) = $op(Ad(c), u) 43 | end 44 | end 45 | 46 | +(x::Ad) = x 47 | -(x::Ad) = Ad(-x.u, -x.up) 48 | 49 | # Elementary functions 50 | 51 | import Base.sin 52 | sin(x::Ad) = Ad(sin(x.u), x.up*cos(x.u)) 53 | 54 | import Base.cos 55 | cos(x::Ad) = Ad(cos(x.u), -x.up*sin(x.u)) 56 | 57 | e^(x::Ad) = Ad(e^x.u, x.up*e^x.u) 58 | 59 | import Base.exp 60 | exp(x::Ad) = Ad(exp(x.u), x.up*exp(x.u)) 61 | 62 | import Base.log 63 | log(x::Ad) = Ad(log(x.u), x.up/x.u) 64 | 65 | import Base.abs 66 | abs(x::Ad) = Ad(abs(x.u), x.up*sign(x.u)) 67 | 68 | #(x::Ad)^y::Interval = Ad(x.u^y, x.up*y*x.u^(y-1)) 69 | 70 | 71 | 72 | 73 | differentiate(f, a) = f(Ad(a, 1.)).up 74 | 75 | # End of module 76 | end 77 | 78 | function jacobian(f, a) 79 | 80 | f1(x) = f(x)[1] 81 | f2(x) = f(x)[2] 82 | 83 | f11(x1) = f1([x1, a[2]]) 84 | J11 = differentiate(f11, a[1]) 85 | 86 | f12(x2) = f1([a[1], x2]) 87 | J12 = differentiate(f12, a[2]) 88 | 89 | f21(x1) = f2([x1, a[2]]) 90 | J21 = differentiate(f21, a[1]) 91 | 92 | f22(x2) = f2([a[1], x2]) 93 | J22 = differentiate(f22, a[2]) 94 | 95 | [J11 J12; J21 J22] 96 | 97 | end 98 | -------------------------------------------------------------------------------- /modules/KrawczykMethod.jl: -------------------------------------------------------------------------------- 1 | ## Krawczyk method, based on Tucker p. 86 2 | 3 | module KrawczykMethod 4 | export krawczyk, differentiate, Interval, rad, diam, mid, mig, mag, belong, hd, hull, isect, isectext, K 5 | 6 | using IntervalArithmetic 7 | using AutoDiff 8 | 9 | 10 | println("Syntax: krawczyk(function, Interval(lo, hi), precision [default is 64])") 11 | 12 | # If the derivative is 0, the constant C will return a "division by thin zero" error. We slightly change the denominator of the constant if this is the case. 13 | function C(f, x) 14 | if differentiate(f, mid(x)) == Interval(0) 15 | return Interval(1)//Interval(differentiate(f, mid(x)) + 0.0001*rad(x)) 16 | else 17 | return Interval(1)//Interval(differentiate(f, mid(x))) 18 | end 19 | end 20 | 21 | K(f, x) = mid(x) - C(f, x)*f(mid(x)) + (1 - C(f, x)*differentiate(f, x))*(x - mid(x)) 22 | 23 | # Outside wrapping function was made in order for it to clean up the array of roots every time 24 | function krawczyk(f::Function, a::Interval, bigprec::Integer=64) 25 | 26 | arr_a = Any[] # instead of Interval[] to put symbols at each encountered root 27 | 28 | function krawczyk_internal(f::Function, a::Interval, bigprec::Integer) 29 | 30 | set_bigfloat_precision(bigprec) 31 | tol = 1e-15 #100eps(BigFloat) 32 | 33 | # If a is symmetric, i.e., mid(a) = 0, the process may stall. The initial interval should be slightly asymmetrized then. This fix also removes the "possible" roots which correctly merge into "unique". 34 | if mid(a) == 0 35 | a = Interval(a.lo, a.hi + 0.0001*mag(a)) 36 | end 37 | 38 | Ka = isect(a, K(f, a)) 39 | #@show Ka 40 | 41 | if Ka != false 42 | #@show diam(a) 43 | #@show diam(K(a)) 44 | 45 | if diam(Ka) < tol 46 | if inside(Ka, a) # inside 47 | println("Unique zero in $(Ka)") 48 | push!(arr_a, [Ka, :unique]) 49 | else 50 | println("Maybe a zero in $(Ka)") 51 | push!(arr_a, [Ka, :possible]) 52 | end 53 | 54 | else 55 | krawczyk_internal(f, Interval(Ka.lo, mid(Ka)), bigprec) 56 | krawczyk_internal(f, Interval(mid(Ka), Ka.hi), bigprec) 57 | end 58 | 59 | end 60 | 61 | return arr_a 62 | end 63 | 64 | 65 | return krawczyk_internal(f, a, bigprec) 66 | end 67 | 68 | #end of module 69 | end 70 | -------------------------------------------------------------------------------- /krawczyk2d.jl: -------------------------------------------------------------------------------- 1 | using IntervalArithmetic 2 | using AutoDiff 3 | 4 | function all_inside(x::Array{Interval, 1}, y::Array{Interval, 1}) 5 | k = 0 6 | for i = 1:length(x) 7 | if !inside(x[i], y[i]) 8 | k += 1 9 | end 10 | end 11 | if k > 0 12 | return false 13 | else 14 | return true 15 | end 16 | end 17 | 18 | # Functions for bisection 19 | 20 | function lower(x::Array{Interval, 1}) 21 | z = Interval[] 22 | for i = 1:length(x) 23 | push!(z, Interval(x[i].lo, mid(x[i]))) 24 | end 25 | return z 26 | end 27 | 28 | function upper(x::Array{Interval, 1}) 29 | z = Interval[] 30 | for i = 1:length(x) 31 | push!(z, Interval(mid(x[i]), x[i].hi)) 32 | end 33 | return z 34 | end 35 | 36 | function krawczyk2d(f, a::Array{Interval, 1}, bigprec::Integer=64) 37 | 38 | arr_a = Array{Interval, 1}[] 39 | 40 | set_bigfloat_precision(bigprec) 41 | #tol = (1e-10)*eps(BigFloat) 42 | tol = 1e-10 43 | 44 | I = [Interval(1) Interval(0); Interval(0) Interval(1)] 45 | Y(x) = make_intervals(inv(mid(jacobian(f, mid(x))))) 46 | M(x) = I - Y(x)*jacobian(f, x) 47 | K(x) = make_intervals(mid(x)) - Y(x)*make_intervals(f(mid(x))) + M(x)*(x - make_intervals(mid(x))) 48 | 49 | Y1 = Array{Interval, 2}[] 50 | push!(Y1, Y(a)) 51 | 52 | Kprev(x) = make_intervals(mid(x)) - Y1[k-1]*make_intervals(f(mid(x))) + (I - Y1[k-1]*jacobian(f, x))*(x - make_intervals(mid(x))) 53 | 54 | 55 | k = 1 56 | 57 | function krawczyk2d_internal(f, a::Array{Interval, 1}, bigprec::Integer) 58 | 59 | # If a is symmetric, i.e., mid(a) = 0, the process may stall. The initial interval should be slightly asymmetrized then 60 | #if mid(a) == 0 61 | # a = Interval(a.lo, a.hi + 0.0001*mag(a)) 62 | #end 63 | 64 | Ka = isectext(a, K(a)) 65 | if Ka != false 66 | @show a 67 | @show diam(a) 68 | 69 | d = diam(a) 70 | dK = diam(Ka) 71 | 72 | if dK[1] < tol && dK[2] < tol #d == dK 73 | if all_inside(Ka, a) 74 | println("Unique zero in $Ka") 75 | push!(arr_a, Ka) 76 | else 77 | println("Maybe a zero in $Ka") 78 | end 79 | k += 1 80 | else 81 | 82 | 83 | if k == 1 84 | 85 | if true #@show mid(det2(I - Y1[k]*jacobian(f, a))) <= mid(det2(I - Y(a)*jacobian(f, a))) 86 | k += 1 87 | push!(Y1, Y(a)) 88 | @show krawczyk2d_internal(f, lower(Ka), bigprec) 89 | @show krawczyk2d_internal(f, upper(Ka), bigprec) 90 | 91 | else 92 | k += 1 93 | push!(Y1, Y(a)) 94 | @show krawczyk2d_internal(f, lower(Kprev(a)), bigprec) 95 | @show krawczyk2d_internal(f, upper(Kprev(a)), bigprec) 96 | 97 | end 98 | 99 | else 100 | 101 | 102 | 103 | if mid(det2(I - Y1[k]*jacobian(f, a))) <= mid(det2(I - Y1[k-1]*jacobian(f, a))) 104 | @show k += 1 105 | push!(Y1, Y(a)) 106 | krawczyk2d_internal(f, lower(Ka), bigprec) 107 | krawczyk2d_internal(f, upper(Ka), bigprec) 108 | 109 | else 110 | @show k += 1 111 | push!(Y1, Y(a)) 112 | krawczyk2d_internal(f, lower(Kprev(a)), bigprec) 113 | krawczyk2d_internal(f, upper(Kprev(a)), bigprec) 114 | 115 | end 116 | 117 | 118 | end 119 | 120 | 121 | end 122 | 123 | end 124 | 125 | return arr_a 126 | end 127 | 128 | return krawczyk2d_internal(f, a, bigprec) 129 | end 130 | 131 | -------------------------------------------------------------------------------- /2D bisection.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "metadata": { 3 | "language": "Julia", 4 | "name": "", 5 | "signature": "sha256:3d0b2c3cb6a8915ae17995f871bc029041c6e8c0706ecc6bcb4154952aac23ca" 6 | }, 7 | "nbformat": 3, 8 | "nbformat_minor": 0, 9 | "worksheets": [ 10 | { 11 | "cells": [ 12 | { 13 | "cell_type": "code", 14 | "collapsed": false, 15 | "input": [ 16 | "using IntervalArithmetic" 17 | ], 18 | "language": "python", 19 | "metadata": {}, 20 | "outputs": [], 21 | "prompt_number": 1 22 | }, 23 | { 24 | "cell_type": "code", 25 | "collapsed": false, 26 | "input": [ 27 | "left(x::Interval) = Interval(x.lo, mid(x))\n", 28 | "right(x::Interval) = Interval(mid(x), x.hi)\n", 29 | "\n", 30 | "\n", 31 | "function bisect(xx::Vector{Interval})\n", 32 | " \n", 33 | " if length(xx) != 2\n", 34 | " error(\"Only works for 2 at the moment\")\n", 35 | " end\n", 36 | " \n", 37 | " x, y = xx\n", 38 | " \n", 39 | " @show x\n", 40 | " \n", 41 | " intervals = Vector{Interval}[]\n", 42 | " \n", 43 | " push!(intervals, [left(x), left(y)])\n", 44 | " push!(intervals, [left(x), right(y)])\n", 45 | " push!(intervals, [right(x), left(y)])\n", 46 | " push!(intervals, [right(x), right(y)])\n", 47 | " \n", 48 | " intervals\n", 49 | "end" 50 | ], 51 | "language": "python", 52 | "metadata": {}, 53 | "outputs": [ 54 | { 55 | "metadata": {}, 56 | "output_type": "pyout", 57 | "prompt_number": 10, 58 | "text": [ 59 | "bisect (generic function with 1 method)" 60 | ] 61 | } 62 | ], 63 | "prompt_number": 10 64 | }, 65 | { 66 | "cell_type": "code", 67 | "collapsed": false, 68 | "input": [ 69 | "a = Interval(0, 1)\n", 70 | "b = Interval(0, 1)\n", 71 | "\n", 72 | "bisect([a,b])" 73 | ], 74 | "language": "python", 75 | "metadata": {}, 76 | "outputs": [ 77 | { 78 | "output_type": "stream", 79 | "stream": "stdout", 80 | "text": [ 81 | "x => Interval(0e+00 with 256 bits of precision,1e+00 with 256 bits of precision)" 82 | ] 83 | }, 84 | { 85 | "output_type": "stream", 86 | "stream": "stdout", 87 | "text": [ 88 | "\n" 89 | ] 90 | }, 91 | { 92 | "metadata": {}, 93 | "output_type": "pyout", 94 | "prompt_number": 11, 95 | "text": [ 96 | "4-element Array{Array{Interval,1},1}:\n", 97 | " [Interval(0e+00 with 256 bits of precision,5e-01 with 256 bits of precision),Interval(0e+00 with 256 bits of precision,5e-01 with 256 bits 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(*CellTagsIndex 266 | CellTagsIndex->{} 267 | *) 268 | (*NotebookFileOutline 269 | Notebook[{ 270 | Cell[CellGroupData[{ 271 | Cell[567, 22, 490, 14, 32, "Input"], 272 | Cell[1060, 38, 1872, 69, 121, "Output"] 273 | }, Open ]], 274 | Cell[CellGroupData[{ 275 | Cell[2969, 112, 116, 2, 32, "Input"], 276 | Cell[3088, 116, 984, 27, 52, "Output"] 277 | }, Open ]], 278 | Cell[CellGroupData[{ 279 | Cell[4109, 148, 834, 24, 32, "Input"], 280 | Cell[4946, 174, 213, 5, 31, "Output"] 281 | }, Open ]], 282 | Cell[CellGroupData[{ 283 | Cell[5196, 184, 988, 31, 55, "Input"], 284 | Cell[6187, 217, 1362, 33, 52, "Output"] 285 | }, Open ]] 286 | } 287 | ] 288 | *) 289 | 290 | (* End of internal cache information *) 291 | -------------------------------------------------------------------------------- /newton-sin.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "metadata": { 3 | "language": "Julia", 4 | "name": "" 5 | }, 6 | "nbformat": 3, 7 | "nbformat_minor": 0, 8 | "worksheets": [ 9 | { 10 | "cells": [ 11 | { 12 | "cell_type": "code", 13 | "collapsed": false, 14 | "input": [ 15 | "using NewtonMethod" 16 | ], 17 | "language": "python", 18 | "metadata": {}, 19 | "outputs": [ 20 | { 21 | "output_type": "stream", 22 | "stream": "stdout", 23 | "text": [ 24 | "Syntax: newton(function, Interval(lo, hi), calls)\n" 25 | ] 26 | }, 27 | { 28 | "output_type": "stream", 29 | "stream": "stderr", 30 | "text": [ 31 | "Warning: requiring \"NewtonMethod\" did not define a corresponding module.\n" 32 | ] 33 | } 34 | ], 35 | "prompt_number": 1 36 | }, 37 | { 38 | "cell_type": "code", 39 | "collapsed": false, 40 | "input": [ 41 | "f(x) = sin(x)" 42 | ], 43 | "language": "python", 44 | "metadata": {}, 45 | "outputs": [ 46 | { 47 | "metadata": {}, 48 | "output_type": "pyout", 49 | "prompt_number": 2, 50 | "text": [ 51 | "f (generic function with 1 method)" 52 | ] 53 | } 54 | ], 55 | "prompt_number": 2 56 | }, 57 | { 58 | "cell_type": "code", 59 | "collapsed": false, 60 | "input": [ 61 | "newton(f, Interval(-20., 20.), 40)" 62 | ], 63 | "language": "python", 64 | "metadata": {}, 65 | "outputs": [ 66 | { 67 | "metadata": {}, 68 | "output_type": "pyout", 69 | "prompt_number": 3, 70 | "text": [ 71 | "7-element Array{Interval,1}:\n", 72 | " Interval(-18.849555921516185,-18.84955592151618) \n", 73 | " Interval(-15.707963267926392,-15.707963267926392)\n", 74 | " Interval(-9.424777960746805,-9.424777960746804) \n", 75 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 76 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 77 | " Interval(-0.0,5.0e-324) \n", 78 | " Interval(12.566370614381748,12.56637061438175) " 79 | ] 80 | } 81 | ], 82 | "prompt_number": 3 83 | }, 84 | { 85 | "cell_type": "code", 86 | "collapsed": false, 87 | "input": [ 88 | "newton(f, Interval(-21., 21.), 40)" 89 | ], 90 | "language": "python", 91 | "metadata": {}, 92 | "outputs": [ 93 | { 94 | "metadata": {}, 95 | "output_type": "pyout", 96 | "prompt_number": 4, 97 | "text": [ 98 | "8-element Array{Interval,1}:\n", 99 | " Interval(-15.707963267948967,-15.707963267948966)\n", 100 | " Interval(-12.566370614359174,-12.566370614359172)\n", 101 | " Interval(-9.424777960746805,-9.424777960746804) \n", 102 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 103 | " Interval(-0.0,5.0e-324) \n", 104 | " Interval(12.566370614381748,12.56637061438175) \n", 105 | " Interval(15.707963267971541,15.707963267971541) \n", 106 | " Interval(18.849555921561333,18.849555921561336) " 107 | ] 108 | } 109 | ], 110 | "prompt_number": 4 111 | }, 112 | { 113 | "cell_type": "code", 114 | "collapsed": false, 115 | "input": [ 116 | "newton(f, Interval(-23., 23.), 40)" 117 | ], 118 | "language": "python", 119 | "metadata": {}, 120 | "outputs": [ 121 | { 122 | "metadata": {}, 123 | "output_type": "pyout", 124 | "prompt_number": 5, 125 | "text": [ 126 | "7-element Array{Interval,1}:\n", 127 | " Interval(-21.991148575105978,-21.991148575105974)\n", 128 | " Interval(-18.849555921516185,-18.84955592151618) \n", 129 | " Interval(-15.707963267926392,-15.70796326792639) \n", 130 | " Interval(-12.566370614336599,-12.566370614336597)\n", 131 | " Interval(-9.424777960746805,-9.424777960746804) \n", 132 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 133 | " Interval(-0.0,5.0e-324) " 134 | ] 135 | } 136 | ], 137 | "prompt_number": 5 138 | }, 139 | { 140 | "cell_type": "code", 141 | "collapsed": false, 142 | "input": [ 143 | "newton(f, Interval(-21., 22.), 40)" 144 | ], 145 | "language": "python", 146 | "metadata": {}, 147 | "outputs": [ 148 | { 149 | "metadata": {}, 150 | "output_type": "pyout", 151 | "prompt_number": 6, 152 | "text": [ 153 | "8-element Array{Interval,1}:\n", 154 | " Interval(-15.707963267948967,-15.707963267948966)\n", 155 | " Interval(-12.566370614359174,-12.566370614359172)\n", 156 | " Interval(-9.424777960746805,-9.424777960746804) \n", 157 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 158 | " Interval(-0.0,5.0e-324) \n", 159 | " Interval(12.566370614381748,12.56637061438175) \n", 160 | " Interval(15.707963267971541,15.707963267971543) \n", 161 | " Interval(21.991148575151126,21.99114857515113) " 162 | ] 163 | } 164 | ], 165 | "prompt_number": 6 166 | }, 167 | { 168 | "cell_type": "code", 169 | "collapsed": false, 170 | "input": [ 171 | "newton(f, Interval(-22., 21.), 40)" 172 | ], 173 | "language": "python", 174 | "metadata": {}, 175 | "outputs": [ 176 | { 177 | "metadata": {}, 178 | "output_type": "pyout", 179 | "prompt_number": 7, 180 | "text": [ 181 | "9-element Array{Interval,1}:\n", 182 | " Interval(-21.991148575128555,-21.991148575128552)\n", 183 | " Interval(-18.849555921516185,-18.849555921516185)\n", 184 | " Interval(-15.707963267948967,-15.707963267948966)\n", 185 | " Interval(-9.424777960746805,-9.424777960746804) \n", 186 | " Interval(-6.283185307179587,-6.283185307179586) \n", 187 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 188 | " Interval(12.566370614381748,12.56637061438175) \n", 189 | " Interval(15.707963267971541,15.707963267971541) \n", 190 | " Interval(18.849555921561333,18.849555921561336) " 191 | ] 192 | } 193 | ], 194 | "prompt_number": 7 195 | }, 196 | { 197 | "cell_type": "code", 198 | "collapsed": false, 199 | "input": [ 200 | "newton(f, Interval(-20., 20.), 40)" 201 | ], 202 | "language": "python", 203 | "metadata": {}, 204 | "outputs": [ 205 | { 206 | "metadata": {}, 207 | "output_type": "pyout", 208 | "prompt_number": 8, 209 | "text": [ 210 | "7-element Array{Interval,1}:\n", 211 | " Interval(-18.849555921516185,-18.84955592151618) \n", 212 | " Interval(-15.707963267926392,-15.707963267926392)\n", 213 | " Interval(-9.424777960746805,-9.424777960746804) \n", 214 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 215 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 216 | " Interval(-0.0,5.0e-324) \n", 217 | " Interval(12.566370614381748,12.56637061438175) " 218 | ] 219 | } 220 | ], 221 | "prompt_number": 8 222 | }, 223 | { 224 | "cell_type": "code", 225 | "collapsed": false, 226 | "input": [ 227 | "newton(f, Interval(-20., 20.), 50)" 228 | ], 229 | "language": "python", 230 | "metadata": {}, 231 | "outputs": [ 232 | { 233 | "metadata": {}, 234 | "output_type": "pyout", 235 | "prompt_number": 9, 236 | "text": [ 237 | "7-element Array{Interval,1}:\n", 238 | " Interval(-18.849555921516185,-18.84955592151618) \n", 239 | " Interval(-15.707963267926392,-15.707963267926392)\n", 240 | " Interval(-9.424777960746805,-9.424777960746804) \n", 241 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 242 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 243 | " Interval(-0.0,5.0e-324) \n", 244 | " Interval(12.566370614381748,12.56637061438175) " 245 | ] 246 | } 247 | ], 248 | "prompt_number": 9 249 | }, 250 | { 251 | "cell_type": "code", 252 | "collapsed": false, 253 | "input": [ 254 | "newton(f, Interval(-20., 20.), 30)" 255 | ], 256 | "language": "python", 257 | "metadata": {}, 258 | "outputs": [ 259 | { 260 | "metadata": {}, 261 | "output_type": "pyout", 262 | "prompt_number": 10, 263 | "text": [ 264 | "7-element Array{Interval,1}:\n", 265 | " Interval(-18.849555921516185,-18.84955592151618) \n", 266 | " Interval(-15.707963267926392,-15.707963267926392)\n", 267 | " Interval(-9.424777960746805,-9.424777960746804) \n", 268 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 269 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 270 | " Interval(-0.0,2.2538265914667098e-260) \n", 271 | " Interval(12.566370614381748,12.56637061438175) " 272 | ] 273 | } 274 | ], 275 | "prompt_number": 10 276 | }, 277 | { 278 | "cell_type": "code", 279 | "collapsed": false, 280 | "input": [ 281 | "newton(f, Interval(-24., 22.), 40)" 282 | ], 283 | "language": "python", 284 | "metadata": {}, 285 | "outputs": [ 286 | { 287 | "metadata": {}, 288 | "output_type": "pyout", 289 | "prompt_number": 11, 290 | "text": [ 291 | "9-element Array{Interval,1}:\n", 292 | " Interval(-21.991148575128555,-21.991148575128552)\n", 293 | " Interval(-18.849555921538762,-18.84955592153876) \n", 294 | " Interval(-9.424777960746805,-9.424777960746804) \n", 295 | " Interval(-6.283185307179587,-6.283185307179586) \n", 296 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 297 | " Interval(-0.0,0.0) \n", 298 | " Interval(6.283185307202162,6.283185307202162) \n", 299 | " Interval(18.849555921561333,18.849555921561336) \n", 300 | " Interval(21.991148575151126,21.99114857515113) " 301 | ] 302 | } 303 | ], 304 | "prompt_number": 11 305 | }, 306 | { 307 | "cell_type": "markdown", 308 | "metadata": {}, 309 | "source": [ 310 | "The roots that are lost depend on the points of the interval. No other function (except sin and based on it cos) does this. Interestingly, changing the procedure of sin into an alternative one left the problem unchanged." 311 | ] 312 | }, 313 | { 314 | "cell_type": "code", 315 | "collapsed": false, 316 | "input": [], 317 | "language": "python", 318 | "metadata": {}, 319 | "outputs": [] 320 | } 321 | ], 322 | "metadata": {} 323 | } 324 | ] 325 | } -------------------------------------------------------------------------------- /.ipynb_checkpoints/newton-sin-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "metadata": { 3 | "language": "Julia", 4 | "name": "" 5 | }, 6 | "nbformat": 3, 7 | "nbformat_minor": 0, 8 | "worksheets": [ 9 | { 10 | "cells": [ 11 | { 12 | "cell_type": "code", 13 | "collapsed": false, 14 | "input": [ 15 | "using NewtonMethod" 16 | ], 17 | "language": "python", 18 | "metadata": {}, 19 | "outputs": [ 20 | { 21 | "output_type": "stream", 22 | "stream": "stdout", 23 | "text": [ 24 | "Syntax: newton(function, Interval(lo, hi), calls)\n" 25 | ] 26 | }, 27 | { 28 | "output_type": "stream", 29 | "stream": "stderr", 30 | "text": [ 31 | "Warning: requiring \"NewtonMethod\" did not define a corresponding module.\n" 32 | ] 33 | } 34 | ], 35 | "prompt_number": 1 36 | }, 37 | { 38 | "cell_type": "code", 39 | "collapsed": false, 40 | "input": [ 41 | "f(x) = sin(x)" 42 | ], 43 | "language": "python", 44 | "metadata": {}, 45 | "outputs": [ 46 | { 47 | "metadata": {}, 48 | "output_type": "pyout", 49 | "prompt_number": 2, 50 | "text": [ 51 | "f (generic function with 1 method)" 52 | ] 53 | } 54 | ], 55 | "prompt_number": 2 56 | }, 57 | { 58 | "cell_type": "code", 59 | "collapsed": false, 60 | "input": [ 61 | "newton(f, Interval(-20., 20.), 40)" 62 | ], 63 | "language": "python", 64 | "metadata": {}, 65 | "outputs": [ 66 | { 67 | "metadata": {}, 68 | "output_type": "pyout", 69 | "prompt_number": 3, 70 | "text": [ 71 | "7-element Array{Interval,1}:\n", 72 | " Interval(-18.849555921516185,-18.84955592151618) \n", 73 | " Interval(-15.707963267926392,-15.707963267926392)\n", 74 | " Interval(-9.424777960746805,-9.424777960746804) \n", 75 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 76 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 77 | " Interval(-0.0,5.0e-324) \n", 78 | " Interval(12.566370614381748,12.56637061438175) " 79 | ] 80 | } 81 | ], 82 | "prompt_number": 3 83 | }, 84 | { 85 | "cell_type": "code", 86 | "collapsed": false, 87 | "input": [ 88 | "newton(f, Interval(-21., 21.), 40)" 89 | ], 90 | "language": "python", 91 | "metadata": {}, 92 | "outputs": [ 93 | { 94 | "metadata": {}, 95 | "output_type": "pyout", 96 | "prompt_number": 4, 97 | "text": [ 98 | "8-element Array{Interval,1}:\n", 99 | " Interval(-15.707963267948967,-15.707963267948966)\n", 100 | " Interval(-12.566370614359174,-12.566370614359172)\n", 101 | " Interval(-9.424777960746805,-9.424777960746804) \n", 102 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 103 | " Interval(-0.0,5.0e-324) \n", 104 | " Interval(12.566370614381748,12.56637061438175) \n", 105 | " Interval(15.707963267971541,15.707963267971541) \n", 106 | " Interval(18.849555921561333,18.849555921561336) " 107 | ] 108 | } 109 | ], 110 | "prompt_number": 4 111 | }, 112 | { 113 | "cell_type": "code", 114 | "collapsed": false, 115 | "input": [ 116 | "newton(f, Interval(-23., 23.), 40)" 117 | ], 118 | "language": "python", 119 | "metadata": {}, 120 | "outputs": [ 121 | { 122 | "metadata": {}, 123 | "output_type": "pyout", 124 | "prompt_number": 5, 125 | "text": [ 126 | "7-element Array{Interval,1}:\n", 127 | " Interval(-21.991148575105978,-21.991148575105974)\n", 128 | " Interval(-18.849555921516185,-18.84955592151618) \n", 129 | " Interval(-15.707963267926392,-15.70796326792639) \n", 130 | " Interval(-12.566370614336599,-12.566370614336597)\n", 131 | " Interval(-9.424777960746805,-9.424777960746804) \n", 132 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 133 | " Interval(-0.0,5.0e-324) " 134 | ] 135 | } 136 | ], 137 | "prompt_number": 5 138 | }, 139 | { 140 | "cell_type": "code", 141 | "collapsed": false, 142 | "input": [ 143 | "newton(f, Interval(-21., 22.), 40)" 144 | ], 145 | "language": "python", 146 | "metadata": {}, 147 | "outputs": [ 148 | { 149 | "metadata": {}, 150 | "output_type": "pyout", 151 | "prompt_number": 6, 152 | "text": [ 153 | "8-element Array{Interval,1}:\n", 154 | " Interval(-15.707963267948967,-15.707963267948966)\n", 155 | " Interval(-12.566370614359174,-12.566370614359172)\n", 156 | " Interval(-9.424777960746805,-9.424777960746804) \n", 157 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 158 | " Interval(-0.0,5.0e-324) \n", 159 | " Interval(12.566370614381748,12.56637061438175) \n", 160 | " Interval(15.707963267971541,15.707963267971543) \n", 161 | " Interval(21.991148575151126,21.99114857515113) " 162 | ] 163 | } 164 | ], 165 | "prompt_number": 6 166 | }, 167 | { 168 | "cell_type": "code", 169 | "collapsed": false, 170 | "input": [ 171 | "newton(f, Interval(-22., 21.), 40)" 172 | ], 173 | "language": "python", 174 | "metadata": {}, 175 | "outputs": [ 176 | { 177 | "metadata": {}, 178 | "output_type": "pyout", 179 | "prompt_number": 7, 180 | "text": [ 181 | "9-element Array{Interval,1}:\n", 182 | " Interval(-21.991148575128555,-21.991148575128552)\n", 183 | " Interval(-18.849555921516185,-18.849555921516185)\n", 184 | " Interval(-15.707963267948967,-15.707963267948966)\n", 185 | " Interval(-9.424777960746805,-9.424777960746804) \n", 186 | " Interval(-6.283185307179587,-6.283185307179586) \n", 187 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 188 | " Interval(12.566370614381748,12.56637061438175) \n", 189 | " Interval(15.707963267971541,15.707963267971541) \n", 190 | " Interval(18.849555921561333,18.849555921561336) " 191 | ] 192 | } 193 | ], 194 | "prompt_number": 7 195 | }, 196 | { 197 | "cell_type": "code", 198 | "collapsed": false, 199 | "input": [ 200 | "newton(f, Interval(-20., 20.), 40)" 201 | ], 202 | "language": "python", 203 | "metadata": {}, 204 | "outputs": [ 205 | { 206 | "metadata": {}, 207 | "output_type": "pyout", 208 | "prompt_number": 8, 209 | "text": [ 210 | "7-element Array{Interval,1}:\n", 211 | " Interval(-18.849555921516185,-18.84955592151618) \n", 212 | " Interval(-15.707963267926392,-15.707963267926392)\n", 213 | " Interval(-9.424777960746805,-9.424777960746804) \n", 214 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 215 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 216 | " Interval(-0.0,5.0e-324) \n", 217 | " Interval(12.566370614381748,12.56637061438175) " 218 | ] 219 | } 220 | ], 221 | "prompt_number": 8 222 | }, 223 | { 224 | "cell_type": "code", 225 | "collapsed": false, 226 | "input": [ 227 | "newton(f, Interval(-20., 20.), 50)" 228 | ], 229 | "language": "python", 230 | "metadata": {}, 231 | "outputs": [ 232 | { 233 | "metadata": {}, 234 | "output_type": "pyout", 235 | "prompt_number": 9, 236 | "text": [ 237 | "7-element Array{Interval,1}:\n", 238 | " Interval(-18.849555921516185,-18.84955592151618) \n", 239 | " Interval(-15.707963267926392,-15.707963267926392)\n", 240 | " Interval(-9.424777960746805,-9.424777960746804) \n", 241 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 242 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 243 | " Interval(-0.0,5.0e-324) \n", 244 | " Interval(12.566370614381748,12.56637061438175) " 245 | ] 246 | } 247 | ], 248 | "prompt_number": 9 249 | }, 250 | { 251 | "cell_type": "code", 252 | "collapsed": false, 253 | "input": [ 254 | "newton(f, Interval(-20., 20.), 30)" 255 | ], 256 | "language": "python", 257 | "metadata": {}, 258 | "outputs": [ 259 | { 260 | "metadata": {}, 261 | "output_type": "pyout", 262 | "prompt_number": 10, 263 | "text": [ 264 | "7-element Array{Interval,1}:\n", 265 | " Interval(-18.849555921516185,-18.84955592151618) \n", 266 | " Interval(-15.707963267926392,-15.707963267926392)\n", 267 | " Interval(-9.424777960746805,-9.424777960746804) \n", 268 | " Interval(-6.283185307157011,-6.2831853071570105) \n", 269 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 270 | " Interval(-0.0,2.2538265914667098e-260) \n", 271 | " Interval(12.566370614381748,12.56637061438175) " 272 | ] 273 | } 274 | ], 275 | "prompt_number": 10 276 | }, 277 | { 278 | "cell_type": "code", 279 | "collapsed": false, 280 | "input": [ 281 | "newton(f, Interval(-24., 22.), 40)" 282 | ], 283 | "language": "python", 284 | "metadata": {}, 285 | "outputs": [ 286 | { 287 | "metadata": {}, 288 | "output_type": "pyout", 289 | "prompt_number": 11, 290 | "text": [ 291 | "9-element Array{Interval,1}:\n", 292 | " Interval(-21.991148575128555,-21.991148575128552)\n", 293 | " Interval(-18.849555921538762,-18.84955592153876) \n", 294 | " Interval(-9.424777960746805,-9.424777960746804) \n", 295 | " Interval(-6.283185307179587,-6.283185307179586) \n", 296 | " Interval(-3.1415926535897936,-3.141592653589793) \n", 297 | " Interval(-0.0,0.0) \n", 298 | " Interval(6.283185307202162,6.283185307202162) \n", 299 | " Interval(18.849555921561333,18.849555921561336) \n", 300 | " Interval(21.991148575151126,21.99114857515113) " 301 | ] 302 | } 303 | ], 304 | "prompt_number": 11 305 | }, 306 | { 307 | "cell_type": "markdown", 308 | "metadata": {}, 309 | "source": [ 310 | "The roots that are lost depend on the points of the interval. No other function (except sin and based on it cos) does this. Interestingly, changing the procedure of sin into an alternative one left the problem unchanged." 311 | ] 312 | }, 313 | { 314 | "cell_type": "code", 315 | "collapsed": false, 316 | "input": [], 317 | "language": "python", 318 | "metadata": {}, 319 | "outputs": [] 320 | } 321 | ], 322 | "metadata": {} 323 | } 324 | ] 325 | } -------------------------------------------------------------------------------- /logistic-arbitrary.jl: -------------------------------------------------------------------------------- 1 | using NewtonMethod 2 | 3 | # Function composition 4 | compose(f::Function, g::Function) = x -> f(g(x)) 5 | 6 | # Logistic function T(x) of the first order 7 | T1(x) = Interval(r)*(Interval(1/4)-(x-Interval(1/2))^2) 8 | 9 | # T^p (x) function - T(T(T(...T(x)))) p times 10 | function T(x, p) 11 | F = T1 12 | for i = 1:p-1 13 | F = compose(F, T1) 14 | end 15 | return F(x) 16 | end 17 | 18 | # Equation to solve for periodic points 19 | 20 | f(x, p) = T(x, p) - x 21 | 22 | r = 3.75 23 | 24 | f(x) = T(x, 8) - x 25 | newton(f, Interval(0.0000000001, 1), 512) 26 | 27 | #= Example output 28 | Function calls: 19 29 | 31-element Array{Interval,1}: 30 | Interval(2.70141733997309131900935211636288574483901429482501059532154506208297183996807957847829633116905071757806038469894488059344025411607098298341427286088463345e-01 with 512 bits of precision,2.70141733997309131900935211636288574483901429482501059532154506208297183996807957847829633116905071757806038469894488059344025411607098298341427286088463792e-01 with 512 bits of precision) 31 | Interval(2.74647636760931918840046974378059197911420565436949728776636765594938388294429628138380719979999854026710482365607036692728959040187437603028573367452714287e-01 with 512 bits of precision,2.74647636760931918840046974378059197911420565436949728776636765594938388294429628138380719979999854026710482365607036692728959040187437603028573367452714995e-01 with 512 bits of precision) 32 | Interval(3.0138171181726633936334181339465856447034349337873780523398986372086669732463733938459142325062332980240687180757639378136694887559077626546201191151561438e-01 with 512 bits of precision,3.01381711817266339363341813394658564470343493378737805233989863720866697324637339384591423250623329802406871807576393781366948875590776265462011911515615051e-01 with 512 bits of precision) 33 | Interval(3.08712576277511324461829601788358753550049139843400090231968259503551955534161812097844577167792000843709441476587235252683055900941920341118109072443522405e-01 with 512 bits of precision,3.08712576277511324461829601788358753550049139843400090231968259503551955534161812097844577167792000843709441476587235252683055900941920341118109072443523113e-01 with 512 bits of precision) 34 | Interval(3.66798186806905272073379444297018530917438548740483128515808130875022090521082841006090892219720057598088967935897122514296512155755215449527547847036711597e-01 with 512 bits of precision,3.66798186806905272073379444297018530917438548740483128515808130875022090521082841006090892219720057598088967935897122514296512155755215449527547847036712529e-01 with 512 bits of precision) 35 | Interval(3.78450775803430692686135645677894530453544376939824698561012796130981078865771575523406853340168442671144391465313379814252691099831785483761418991000291029e-01 with 512 bits of precision,3.78450775803430692686135645677894530453544376939824698561012796130981078865771575523406853340168442671144391465313379814252691099831785483761418991000291812e-01 with 512 bits of precision) 36 | Interval(4.10617758685194158020084074644597303577274924175521424154067697587735987938351697225502193609619674386681578126905119369812609634056998083116702622580651832e-01 with 512 bits of precision,4.10617758685194158020084074644597303577274924175521424154067697587735987938351697225502193609619674386681578126905119369812609634056998083116702622580652652e-01 with 512 bits of precision) 37 | Interval(4.2429101104593384973864956550002890959249659108940410455091507925280570816140225726198904774417796096204969838775219057848959796394078355287684139009193047e-01 with 512 bits of precision,4.24291011045933849738649565500028909592496591089404104550915079252805708161402257261989047744177960962049698387752190578489597963940783552876841390091931402e-01 with 512 bits of precision) 38 | Interval(5.7318379212630204408898661168704302350504723301380957392606699926998944631536054103660052370201266124561100075600772333537134467476609640409853155968299533e-01 with 512 bits of precision,5.73183792126302044088986611687043023505047233013809573926066999269989446315360541036600523702012661245611000756007723335371344674766096404098531559682996374e-01 with 512 bits of precision) 39 | Interval(5.92824367624351407929292833268682100619705230441170169638687726104889731985372093688432443846419608465710504370104633194043820729091546982633026028294981817e-01 with 512 bits of precision,5.92824367624351407929292833268682100619705230441170169638687726104889731985372093688432443846419608465710504370104633194043820729091546982633026028294982786e-01 with 512 bits of precision) 40 | Interval(6.17131705864440229122211127440407914709695646062694298168624551999972722541206527139384023687813497244366575775460548194871117747058392337549648803810393639e-01 with 512 bits of precision,6.17131705864440229122211127440407914709695646062694298168624551999972722541206527139384023687813497244366575775460548194871117747058392337549648803810394608e-01 with 512 bits of precision) 41 | Interval(6.37955377457103435004486384384682116483545332037301218881664118057901620563418872050203700796025592418384691816665978556041724706992728489161747287751938249e-01 with 512 bits of precision,6.37955377457103435004486384384682116483545332037301218881664118057901620563418872050203700796025592418384691816665978556041724706992728489161747287751939218e-01 with 512 bits of precision) 42 | Interval(6.85346050849424318856032513746113757716051637920528737020625551877192105901703809140235069074479417872458362294140322896647053262565609043898557787867166097e-01 with 512 bits of precision,6.85346050849424318856032513746113757716051637920528737020625551877192105901703809140235069074479417872458362294140322896647053262565609043898557787867167066e-01 with 512 bits of precision) 43 | Interval(7.04900116012334156176621269851610244955462747230100481462411647607712773260270648819212638188221137376596713557034120853859129627280751971519795473931089921e-01 with 512 bits of precision,7.04900116012334156176621269851610244955462747230100481462411647607712773260270648819212638188221137376596713557034120853859129627280751971519795473931090517e-01 with 512 bits of precision) 44 | Interval(7.19311930645055173976491685390702388906529225268829737848845774664565388772166593131585183054951655390981694560401685618093713472304695228549149382738651329e-01 with 512 bits of precision,7.19311930645055173976491685390702388906529225268829737848845774664565388772166593131585183054951655390981694560401685618093713472304695228549149382738652074e-01 with 512 bits of precision) 45 | Interval(7.34748010610079575596816976127320954907161803713527851458885941644562334217506631299734748010610079575596816976127320954907161803713527851458885941644562029e-01 with 512 bits of precision,7.34748010610079575596816976127320954907161803713527851458885941644562334217506631299734748010610079575596816976127320954907161803713527851458885941644562625e-01 with 512 bits of precision) 46 | Interval(7.43312719364390502084318158791807622280104302462812472248981692407736048176209257050948176267230584261223661807748755741792552548734499892986579584486721483e-01 with 512 bits of precision,7.43312719364390502084318158791807622280104302462812472248981692407736048176209257050948176267230584261223661807748755741792552548734499892986579584486722154e-01 with 512 bits of precision) 47 | Interval(7.5104549768227803081970631368228244762456586980133639156814468635103084995838555391967209341105529560318669953280033419090968945666029170278234873308438276e-01 with 512 bits of precision,7.5104549768227803081970631368228244762456586980133639156814468635103084995838555391967209341105529560318669953280033419090968945666029170278234873308438358e-01 with 512 bits of precision) 48 | Interval(7.61171584579304179915340358137628228608421111014895791246861220990877962339249435867362697679562840186647952273869267430436233586348096062296668790335239458e-01 with 512 bits of precision,7.61171584579304179915340358137628228608421111014895791246861220990877962339249435867362697679562840186647952273869267430436233586348096062296668790335240203e-01 with 512 bits of precision) 49 | Interval(7.84220103067157654843033781897425092460727931199554730113279865045357185986144239323802489621119072480368541527394187279839522116755503756071408934006958035e-01 with 512 bits of precision,7.84220103067157654843033781897425092460727931199554730113279865045357185986144239323802489621119072480368541527394187279839522116755503756071408934006958631e-01 with 512 bits of precision) 50 | Interval(7.93776424009589319853493368401011071936485127060845317825889083603345620789396876053449968867493328104043458354844221416456635412771784211577912853980185028e-01 with 512 bits of precision,7.93776424009589319853493368401011071936485127060845317825889083603345620789396876053449968867493328104043458354844221416456635412771784211577912853980185848e-01 with 512 bits of precision) 51 | Interval(8.04552388151561299148634786510741761220096592900707858241696023575283396912976239489208141454224715921253846782433428187669021737059581021323772518989637233e-01 with 512 bits of precision,8.04552388151561299148634786510741761220096592900707858241696023575283396912976239489208141454224715921253846782433428187669021737059581021323772518989637904e-01 with 512 bits of precision) 52 | Interval(8.12988592208150257534610957028715655742066381012222441901505883844702967193437423750979898452071608274906355087360433413277971761815499476183160699218476699e-01 with 512 bits of precision,8.12988592208150257534610957028715655742066381012222441901505883844702967193437423750979898452071608274906355087360433413277971761815499476183160699218477519e-01 with 512 bits of precision) 53 | Interval(8.70750543141618802533829389893130091823114804541495869218356161091015958734584878277488615636706326563347969541691136494586734588215821766796238021878908592e-01 with 512 bits of precision,8.70750543141618802533829389893130091823114804541495869218356161091015958734584878277488615636706326563347969541691136494586734588215821766796238021878909188e-01 with 512 bits of precision) 54 | Interval(8.75609934147011045211553765267387482681630049321819435887434300929107282839936909319388668379522958430347765968856231961243021526730775833671778687258206706e-01 with 512 bits of precision,8.75609934147011045211553765267387482681630049321819435887434300929107282839936909319388668379522958430347765968856231961243021526730775833671778687258207302e-01 with 512 bits of precision) 55 | Interval(8.86801213586489731717047378194784514639293819346647449980101262224456586916721793176858398649221477753258791558559299230840147096454686664779695067355145861e-01 with 512 bits of precision,8.86801213586489731717047378194784514639293819346647449980101262224456586916721793176858398649221477753258791558559299230840147096454686664779695067355146532e-01 with 512 bits of precision) 56 | Interval(8.907762163244491943089701201751873386534206694891951320407813813019362766965457330901057032391993843796882878715455859921959896022646389727939767144241214e-01 with 512 bits of precision,8.90776216324449194308970120175187338653420669489195132040781381301936276696545733090105703239199384379688287871545585992195989602264638972793976714424121996e-01 with 512 bits of precision) 57 | Interval(9.1001631064227501833186942863724909361476707601231319219434523224869473655128110812223870030608370751808905763650777553605756587484943294008014517567960139e-01 with 512 bits of precision,9.10016310642275018331869428637249093614767076012313192194345232248694736551281108122238700306083707518089057636507775536057565874849432940080145175679601837e-01 with 512 bits of precision) 58 | Interval(9.12380772314532644920514764099638296242933300313307492835808421346413940468740772156311160803982888374582174267039453718403687957934333648677822884590105313e-01 with 512 bits of precision,9.12380772314532644920514764099638296242933300313307492835808421346413940468740772156311160803982888374582174267039453718403687957934333648677822884590105984e-01 with 512 bits of precision) 59 | Interval(9.20890921698155147969462296192560975775690925359806528946589854389722444246393420707518885122706151221206601799887257757797093772863598413604419382369652742e-01 with 512 bits of precision,9.20890921698155147969462296192560975775690925359806528946589854389722444246393420707518885122706151221206601799887257757797093772863598413604419382369653189e-01 with 512 bits of precision) 60 | Interval(9.22308379788953575230534660463328165543626397524461957239433848283663754231803760000472109501536903221845725164535600873672161639508568015452506655300386598e-01 with 512 bits of precision,9.22308379788953575230534660463328165543626397524461957239433848283663754231803760000472109501536903221845725164535600873672161639508568015452506655300387045e-01 with 512 bits of precision) 61 | =# 62 | 63 | 64 | -------------------------------------------------------------------------------- /modules/IntervalArithmetic.jl: -------------------------------------------------------------------------------- 1 | ## Interval arithmetic 2 | 3 | module IntervalArithmetic 4 | export Interval, rad, diam, mid, mig, mag, belong, hd, hull, isect, isectext, lo, hi, make_intervals, det2, inside 5 | 6 | typealias prec BigFloat 7 | 8 | type Interval 9 | 10 | lo 11 | hi 12 | 13 | function Interval(a, b) 14 | set_rounding(prec, RoundDown) 15 | lo = BigFloat("$a") 16 | 17 | set_rounding(prec, RoundUp) 18 | hi = BigFloat("$b") 19 | 20 | new(lo, hi) 21 | end 22 | 23 | end 24 | 25 | #import MPFR.BigFloat 26 | #BigFloat(x::Interval) = Interval(BigFloat(x.lo), BigFloat(x.hi)) 27 | 28 | # Thin (degenerate) interval and functions zero() and one() 29 | 30 | Interval(x::Number) = Interval(x, x) 31 | 32 | # Prevent errors from embedded interval functions 33 | Interval(x::Interval) = x 34 | 35 | import Base.one 36 | one(x::Interval) = Interval(1.0) 37 | 38 | import Base.zero 39 | zero(x::Interval) = Interval(0.0) 40 | zero(Interval) = Interval(0) 41 | 42 | # The basic operations on intervals. Left end is rounded down, right end is rounded up. 43 | 44 | # Addition 45 | 46 | function +(x::Interval, y::Interval) 47 | z1 = with_rounding(prec, RoundDown) do 48 | x.lo + y.lo 49 | end 50 | z2 = with_rounding(prec, RoundUp) do 51 | x.hi + y.hi 52 | end 53 | Interval(z1, z2) 54 | end 55 | 56 | +(x::Interval, y::Real) = x + Interval(y) 57 | +(x::Real, y::Interval) = Interval(x) + y 58 | 59 | # Subtraction 60 | 61 | function -(x::Interval, y::Interval) 62 | z1 = with_rounding(prec, RoundDown) do 63 | x.lo - y.hi 64 | end 65 | z2 = with_rounding(prec, RoundUp) do 66 | x.hi - y.lo 67 | end 68 | Interval(z1, z2) 69 | end 70 | 71 | -(x::Interval) = Interval(-x.hi, -x.lo) 72 | 73 | -(x::Interval, y::Real) = x - Interval(y) 74 | -(x::Real, y::Interval) = Interval(x) - y 75 | 76 | # Multiplication 77 | 78 | function *(x::Interval, y::Interval) 79 | z1 = with_rounding(prec, RoundDown) do 80 | min(x.lo*y.lo, x.lo*y.hi, x.hi*y.lo, x.hi*y.hi) 81 | end 82 | z2 = with_rounding(prec, RoundUp) do 83 | max(x.lo*y.lo, x.lo*y.hi, x.hi*y.lo, x.hi*y.hi) 84 | end 85 | Interval(z1, z2) 86 | end 87 | 88 | *(x::Interval, y::Real) = x*Interval(y) 89 | *(x::Real, y::Interval) = Interval(x)*y 90 | 91 | 92 | # Division 93 | 94 | function /(x::Interval, y::Interval) 95 | z1 = with_rounding(prec, RoundDown) do 96 | 1/y.hi 97 | end 98 | z2 = with_rounding(prec, RoundUp) do 99 | 1/y.lo 100 | end 101 | x*Interval(z1, z2) 102 | end 103 | 104 | /(x::Interval, y::Real) = x/Interval(y) 105 | /(x::Real, y::Interval) = Interval(x)/y 106 | 107 | # Extended division 108 | 109 | function //(x::Interval, y::Interval) 110 | if belong(0., y) == false 111 | return x/y 112 | elseif belong(0., x) == true && belong(0, y) == true 113 | return Interval(-Inf, Inf) 114 | elseif x.hi < 0 && y.lo < y.hi == 0 115 | z1 = with_rounding(prec, RoundDown) do 116 | x.hi/y.lo 117 | end 118 | return Interval(z1, Inf) 119 | elseif x.hi < 0 && y.lo < 0 < y.hi 120 | z1 = with_rounding(prec, RoundDown) do 121 | x.hi/y.lo 122 | end 123 | z2 = with_rounding(prec, RoundUp) do 124 | x.hi/y.hi 125 | end 126 | return Interval(z1, z2) 127 | elseif x.hi < 0 && 0 == y.lo < y.hi 128 | z2 = with_rounding(prec, RoundUp) do 129 | x.hi/y.hi 130 | end 131 | return Interval(-Inf, z2) 132 | elseif 0 < x.lo && y.lo < y.hi == 0 133 | z2 = with_rounding(prec, RoundUp) do 134 | x.lo/y.lo 135 | end 136 | return Interval(-Inf, z2) 137 | elseif 0 < x.lo && y.lo < 0 < y.hi 138 | z1 = with_rounding(prec, RoundDown) do 139 | x.lo/y.hi 140 | end 141 | z2 = with_rounding(prec, RoundUp) do 142 | x.lo/y.lo 143 | end 144 | return Interval(z1, z2) 145 | elseif 0 < x.lo && 0 == y.lo < y.hi 146 | z1 = with_rounding(prec, RoundDown) do 147 | x.lo/y.hi 148 | end 149 | return Interval(z1, Inf) 150 | elseif belong(0, x) == false && y.lo == 0 && y.hi == 0 151 | return error("\nEmpty set: extended division by thin zero") 152 | end 153 | end 154 | 155 | 156 | ## Interval properties 157 | 158 | # Whether the point belongs to the interval: belong(point, interval) 159 | 160 | function belong(p::Real, x::Interval) 161 | if p >= x.lo && p <= x.hi 162 | return true 163 | else 164 | return false 165 | end 166 | end 167 | 168 | # Whether one interval is inside of the other 169 | 170 | function inside(x::Interval, y::Interval) 171 | if x.lo > y.lo && x.hi < y.hi 172 | return true 173 | else 174 | return false 175 | end 176 | end 177 | 178 | # Lower end, higher end, bottom, top, radius, diameter, midpoint, mignitude, magnitude, absolute value 179 | 180 | lo(x::Interval) = x.lo 181 | hi(x::Interval) = x.hi 182 | rad(x::Interval) = (x.hi - x.lo)/2 183 | diam(x::Interval) = x.hi - x.lo 184 | mid(x::Interval) = (x.hi + x.lo)/2 185 | mid(x::prec) = x 186 | 187 | function mig(x::Interval) 188 | if belong(0.0, x) == true 189 | return 0 190 | else return min(abs(x.lo), abs(x.hi)) 191 | end 192 | end 193 | 194 | mag(x::Interval) = max(abs(x.lo), abs(x.hi)) 195 | 196 | import Base.abs 197 | abs(x::Interval) = Interval(mig(x), mag(x)) 198 | 199 | 200 | # Hausdorff distance 201 | 202 | hd(x::Interval, y::Interval) = max(abs(x.lo - y.lo), abs(x.hi - y.hi)) 203 | 204 | 205 | # "Union" (hull) and intersection 206 | 207 | hull(x::Interval, y::Interval) = Interval(min(x.lo, y.lo), max(x.hi, y.hi)) 208 | 209 | function isect(x::Interval, y::Interval) 210 | if x.hi < y.lo || y.hi < x.lo 211 | return false 212 | else 213 | z1 = with_rounding(prec, RoundDown) do 214 | max(x.lo, y.lo) 215 | end 216 | z2 = with_rounding(prec, RoundUp) do 217 | min(x.hi, y.hi) 218 | end 219 | return Interval(z1, z2) 220 | end 221 | end 222 | 223 | # Extended intersection (involving extended intervals [a, b] with a > b) 224 | 225 | function isectext(x::Interval, y::Interval) 226 | if x.hi < x.lo && y.hi >= y.lo # x is an extended interval and y is a normal one 227 | if y.lo <= x.hi && x.lo <= y.hi 228 | return [Interval(y.lo, x.hi), Interval(x.lo, y.hi)] 229 | elseif y.lo > x.hi && x.lo <= y.hi && y.lo <= x.lo 230 | return Interval(x.lo, y.hi) 231 | elseif y.lo <= x.hi && x.lo > y.hi && y.hi >= x.hi 232 | return Interval(y.lo, x.hi) 233 | elseif y.lo > x.hi && x.lo > y.hi 234 | return false 235 | elseif y.lo <= x.hi && x.lo > y.hi && y.hi < x.hi 236 | return y 237 | elseif y.lo > x.hi && x.lo <= y.hi && y.lo > x.lo 238 | return y 239 | end 240 | elseif x.hi < x.lo && y.hi < y.lo # both intervals are extended 241 | return Interval(max(x.lo, y.lo), min(x.hi, y.hi)) # Returns also an extended interval 242 | elseif x.hi >= x.lo && y.hi < y.lo # x normal, y extended 243 | return isectext(y, x) 244 | elseif x.hi >= x.lo && y.hi >= y.lo # both intervals are normal 245 | return isect(x, y) 246 | end 247 | end 248 | 249 | 250 | # Integer power 251 | 252 | function ^(x::Interval, n::Integer) 253 | if n > 0 && n % 2 == 1 254 | return Interval(x.lo^n, x.hi^n) 255 | elseif n > 0 && n % 2 == 0 256 | return Interval((mig(x))^n, (mag(x))^n) 257 | elseif n == 0 258 | return Interval(1, 1) 259 | elseif n < 0 && belong(0, x) == false 260 | return Interval(1/x.hi, 1/x.lo)^(-n) 261 | # elseif return println("Error") 262 | end 263 | end 264 | 265 | 266 | # Real power function - taken from https://github.com/dpsanders/ValidatedNumerics.jl 267 | 268 | 269 | 270 | macro round_down(expr) 271 | quote 272 | with_rounding(BigFloat, RoundDown) do 273 | $expr 274 | end 275 | end 276 | end 277 | 278 | macro round_up(expr) 279 | quote 280 | with_rounding(BigFloat, RoundUp) do 281 | $expr 282 | end 283 | end 284 | end 285 | 286 | macro round(expr1, expr2) 287 | quote 288 | Interval(@round_down($expr1), @round_up($expr2)) 289 | end 290 | end 291 | 292 | function reciprocal(a::Interval) 293 | uno = one(BigFloat) 294 | z = zero(BigFloat) 295 | if belong(z, a) 296 | #if z in a 297 | warn("\nInterval in denominator contains 0.") 298 | return Interval(-inf(z),inf(z)) # inf(z) returns inf of type of z 299 | end 300 | @round(uno/a.hi, uno/a.lo) 301 | end 302 | 303 | function ^(a::Interval, x::Real) 304 | x == int(x) && return a^(int(x)) 305 | x < zero(x) && return reciprocal( a^(-x) ) 306 | x == 0.5*one(x) && return sqrt(a) 307 | # 308 | z = zero(BigFloat) 309 | z > a.hi && error("Undefined operation; Interval is strictly negative and power is not an integer") 310 | # 311 | xInterv = Interval( x ) 312 | diam( xInterv ) >= eps(x) && return a^xInterv 313 | # xInterv is a thin interval 314 | domainPow = Interval(z, big(Inf)) 315 | aRestricted = isect(a, domainPow) 316 | @round(aRestricted.lo^x, aRestricted.hi^x) 317 | end 318 | 319 | 320 | 321 | 322 | 323 | # Interval power - exercise 3.5 from Tucker "Validated Numerics" 324 | 325 | function ^(x::Interval, n::Interval) 326 | if x.lo > 0 327 | z1 = min(x.lo^n.lo, x.lo^n.hi, x.hi^n.lo, x.hi^n.hi) 328 | z2 = max(x.lo^n.lo, x.lo^n.hi, x.hi^n.lo, x.hi^n.hi) 329 | return Interval(z1, z2) 330 | end 331 | end 332 | 333 | 334 | # Trigonometry 335 | 336 | #= COMMENTED OUT: old Tucker definition of sin 337 | import Base.sin 338 | function sin(x::Interval) 339 | 340 | k = 0 341 | pcount = 0 342 | for k = -1000:1000 343 | p = pi/2 + 2pi*k 344 | if belong(p, x) == true 345 | pcount = pcount + 1 346 | end 347 | end 348 | 349 | k = 0 350 | qcount = 0 351 | for k = -1000:1000 352 | q = - pi/2 + 2pi*k 353 | if belong(q, x) == true 354 | qcount = qcount + 1 355 | end 356 | end 357 | 358 | if qcount != 0 && pcount != 0 359 | return Interval(-1., 1.) 360 | elseif qcount != 0 && pcount == 0 361 | return Interval(-1., max(sin(x.lo), sin(x.hi))) 362 | elseif qcount == 0 && pcount != 0 363 | return Interval(min(sin(x.lo), sin(x.hi)), 1.) 364 | elseif qcount == 0 && pcount == 0 365 | return Interval(min(sin(x.lo), sin(x.hi)), max(sin(x.lo), sin(x.hi))) 366 | end 367 | end 368 | =# 369 | 370 | 371 | # Taken from http://jenchienjackchang.com/sample-page/implicit-solid-modeling-using-interval-methods/interval-arithmetic/ 372 | 373 | #= Deprecated in favour of the next 374 | import Base.sin 375 | function sin(x::Interval) 376 | if x.lo%2pi >= 0 377 | low = x.lo%2pi 378 | else low = x.lo%2pi + 2pi 379 | end 380 | x1 = Interval(low, low + diam(x)) 381 | # If the interval has a diameter equal or greater than 2pi or it is an extended interval, return [-1, 1] 382 | if diam(x) >= 2pi || x.lo > x.hi 383 | return Interval(-1, 1) 384 | elseif 0 <= x1.lo <= x1.hi <= pi/2 || 3pi/2 <= x1.lo <= x1.hi <= 5pi/2 || 7pi/2 <= x1.lo <= x1.hi <= 4pi 385 | return Interval(sin(x1.lo), sin(x1.hi)) 386 | elseif pi/2 <= x1.lo <= x1.hi <= 3pi/2 || 5pi/2 <= x1.lo <= x1.hi <= 7pi/2 387 | return Interval(sin(x1.hi), sin(x1.lo)) 388 | elseif (0 <= x1.lo <= pi/2 && pi/2 <= x1.hi <= 3pi/2) || (3pi/2 <= x1.lo <= 5pi/2 && 5pi/2 <= x1.hi <= 7pi/2) 389 | return Interval(min(sin(x1.lo), sin(x1.hi)), 1) 390 | elseif (pi/2 <= x1.lo <= 3pi/2 && 3pi/2 <= x1.hi <= 5pi/2) || (5pi/2 <= x1.lo <= 7pi/2 && 7pi/2 <= x1.hi <= 4pi) 391 | return Interval(-1, max(sin(x1.lo), sin(x1.hi))) 392 | elseif (0 <= x1.lo <= pi/2 && 3pi/2 <= x1.hi <= 5pi/2) || (pi/2 <= x1.lo <= 3pi/2 && 5pi/2 <= x1.hi <= 7pi/2) || (3pi/2 <= x1.lo <= 5pi/2 && 7pi/2 <= x1.hi <= 4pi) 393 | return Interval(-1, 1) 394 | end 395 | 396 | end 397 | =# 398 | 399 | # Using Sanders/Benet sin() from https://github.com/dpsanders/ValidatedNumerics.jl 400 | import Base.sin 401 | function sin(a::Interval) 402 | piHalf = pi*BigFloat("0.5") 403 | twoPi = pi*BigFloat("2.0") 404 | domainSin = Interval( BigFloat(-1.0), BigFloat(1.0) ) 405 | 406 | # Checking the specific case 407 | diam(a) >= twoPi && return domainSin 408 | 409 | # Limits within 1 full period of sin(x) 410 | # Abbreviations 411 | loMod2pi = mod(a.lo, twoPi) 412 | hiMod2pi = mod(a.hi, twoPi) 413 | loQuartile = floor( loMod2pi / piHalf ) 414 | hiQuartile = floor( hiMod2pi / piHalf ) 415 | 416 | # 20 different cases 417 | if loQuartile == hiQuartile # Interval limits in the same quartile 418 | loMod2pi > hiMod2pi && return domainSin 419 | set_rounding(BigFloat, RoundDown) 420 | lo = sin( a.lo ) 421 | set_rounding(BigFloat, RoundUp) 422 | hi = sin( a.hi ) 423 | set_rounding(BigFloat, RoundNearest) 424 | return Interval( lo, hi ) 425 | elseif loQuartile == 3 && hiQuartile==0 426 | set_rounding(BigFloat, RoundDown) 427 | lo = sin( a.lo ) 428 | set_rounding(BigFloat, RoundUp) 429 | hi = sin( a.hi ) 430 | set_rounding(BigFloat, RoundNearest) 431 | return Interval( lo, hi ) 432 | elseif loQuartile == 1 && hiQuartile==2 433 | set_rounding(BigFloat, RoundDown) 434 | lo = sin( a.hi ) 435 | set_rounding(BigFloat, RoundUp) 436 | hi = sin( a.lo ) 437 | set_rounding(BigFloat, RoundNearest) 438 | return Interval( lo, hi ) 439 | elseif ( loQuartile == 0 || loQuartile==3 ) && ( hiQuartile==1 || hiQuartile==2 ) 440 | set_rounding(BigFloat, RoundDown) 441 | slo = sin( a.lo ) 442 | shi = sin( a.hi ) 443 | set_rounding(BigFloat, RoundNearest) 444 | lo = min( slo, shi ) 445 | return Interval( lo, BigFloat(1.0) ) 446 | elseif ( loQuartile == 1 || loQuartile==2 ) && ( hiQuartile==3 || hiQuartile==0 ) 447 | set_rounding(BigFloat, RoundUp) 448 | slo = sin( a.lo ) 449 | shi = sin( a.hi ) 450 | set_rounding(BigFloat, RoundNearest) 451 | hi = max( slo, shi ) 452 | return Interval( BigFloat(-1.0), hi ) 453 | elseif ( loQuartile == 0 && hiQuartile==3 ) || ( loQuartile == 2 && hiQuartile==1 ) 454 | return domainSin 455 | else 456 | # This should be never reached! 457 | error(string("SOMETHING WENT WRONG in sin.\nThis should have never been reached") ) 458 | end 459 | end 460 | 461 | 462 | import Base.cos 463 | cos(x::Interval) = sin(Interval(x.lo + pi/2, x.hi + pi/2)) 464 | 465 | # Miscellaneous 466 | 467 | import Base.complex 468 | complex(x::Interval) = Interval(complex(x.lo), complex(x.hi)) 469 | 470 | function *(x::Array{Interval, 1}, y::Interval) 471 | [x[1]*y, x[2]*y] 472 | end 473 | 474 | 475 | function /(x::Array{Any, 1}, y::Interval) 476 | [x[1]/y, x[2]/y] 477 | end 478 | 479 | # Equality of two intervals 480 | ==(a::Interval, b::Interval) = a.lo == b.lo && a.hi == b.hi 481 | 482 | # Monotonic functions 483 | 484 | import Base.exp 485 | exp(x::Interval) = Interval(exp(x.lo), exp(x.hi)) 486 | 487 | import Base.sqrt 488 | sqrt(x::Interval) = Interval(sqrt(x.lo), sqrt(x.hi)) 489 | 490 | import Base.log 491 | log(x::Interval) = Interval(log(x.lo), log(x.hi)) 492 | 493 | import Base.asin 494 | asin(x::Interval) = Interval(asin(x.lo), asin(x.hi)) 495 | 496 | import Base.acos 497 | acos(x::Interval) = Interval(acos(x.hi), acos(x.lo)) 498 | 499 | 500 | #------------------------------------------------------- 501 | 502 | ## Interval arithmetic for 2D objects 503 | 504 | function lo(x::Array{Interval}) 505 | y = prec[] 506 | for i = 1:length(x) 507 | push!(y, lo(x[i])) 508 | end 509 | y 510 | end 511 | 512 | function hi(x::Array{Interval}) 513 | y = prec[] 514 | for i = 1:length(x) 515 | push!(y, hi(x[i])) 516 | end 517 | y 518 | end 519 | 520 | # Making mid() process 1-D and 2-D interval arrays into arrays of midpoints 521 | function mid(array::Array{Interval, 1}) 522 | array1 = prec[] 523 | for i = 1:length(array) 524 | push!(array1, mid(array[i])) 525 | end 526 | array1 527 | end 528 | 529 | function mid(array::Array{Interval, 2}) 530 | array1 = prec[] 531 | for i = 1:length(array) 532 | push!(array1, mid(array[i])) 533 | end 534 | reshape(array1, (2, 2)) 535 | end 536 | 537 | function mid(array::Array{Array{Interval, 1}, 1}) 538 | array1 = Array{prec, 1}[] 539 | for i = 1:length(array) 540 | push!(array1, mid(array[i])) 541 | end 542 | array1 543 | end 544 | 545 | function diam(x::Array{Interval, 1}) 546 | y = prec[] 547 | for i = 1:length(x) 548 | push!(y, diam(x[i])) 549 | end 550 | y 551 | end 552 | 553 | 554 | # Function that makes numbers into thin intervals in arrays 555 | function make_intervals(x::Array{prec, 1}) 556 | y = Interval[] 557 | for i = 1:length(x) 558 | push!(y, Interval(x[i])) 559 | end 560 | return y 561 | end 562 | 563 | 564 | function make_intervals(x::Array{prec, 2}) 565 | y = Interval[] 566 | for i = 1:length(x) 567 | push!(y, Interval(x[i])) 568 | end 569 | return reshape(y, (2, 2)) 570 | end 571 | 572 | # Intersection of 2-D vectors 573 | function isect(x::Array{Interval, 1}, y::Array{Interval, 1}) 574 | z = Interval[] 575 | if length(x) == length(y) 576 | for i = 1:length(x) 577 | if isect(x[i], y[i]) == false 578 | return false 579 | end 580 | end 581 | for i = 1:length(x) 582 | push!(z, isect(x[i], y[i])) 583 | end 584 | else return false 585 | end 586 | return z 587 | end 588 | 589 | 590 | function isectext(x::Array{Interval, 1}, y::Array{Interval, 1}) 591 | z = Interval[] 592 | if length(x) == length(y) 593 | for i = 1:length(x) 594 | if isectext(x[i], y[i]) == false 595 | return false 596 | end 597 | end 598 | for i = 1:length(x) 599 | push!(z, isectext(x[i], y[i])) 600 | end 601 | else return false 602 | end 603 | return z 604 | end 605 | 606 | # Determinant of a 2x2 interval matrix 607 | 608 | det2(x::Array{Interval, 2}) = x[1]*x[4] - x[3]*x[2] 609 | 610 | # Definitions for the inverse of an interval matrix 611 | 612 | import Base.one; one(::Type{Interval}) = Interval(1) 613 | import Base.real; real(x::Interval) = x 614 | import Base.inv; inv(x::Interval) = Interval(1)/x 615 | import Base.isless; isless(a::Interval, b::Interval) = a.hi < b.lo 616 | 617 | 618 | # End of module 619 | end 620 | -------------------------------------------------------------------------------- /krawczyk2d-examples.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "metadata": { 3 | "language": "Julia", 4 | "name": "" 5 | }, 6 | "nbformat": 3, 7 | "nbformat_minor": 0, 8 | "worksheets": [ 9 | { 10 | "cells": [ 11 | { 12 | "cell_type": "code", 13 | "collapsed": false, 14 | "input": [ 15 | "# Krawczyk 2D actually works, but when the intervals are large, it stalls" 16 | ], 17 | "language": "python", 18 | "metadata": {}, 19 | "outputs": [], 20 | "prompt_number": 1 21 | }, 22 | { 23 | "cell_type": "code", 24 | "collapsed": false, 25 | "input": [ 26 | "include(\"krawczyk2d.jl\")" 27 | ], 28 | "language": "python", 29 | "metadata": {}, 30 | "outputs": [ 31 | { 32 | "metadata": {}, 33 | "output_type": "pyout", 34 | "prompt_number": 6, 35 | "text": [ 36 | "krawczyk2d (generic function with 2 methods)" 37 | ] 38 | } 39 | ], 40 | "prompt_number": 6 41 | }, 42 | { 43 | "cell_type": "code", 44 | "collapsed": false, 45 | "input": [ 46 | "f(x) = [x[1]^2 + x[2]^2 - 1, x[1] - x[2]^2]" 47 | ], 48 | "language": "python", 49 | "metadata": {}, 50 | "outputs": [ 51 | { 52 | "metadata": {}, 53 | "output_type": "pyout", 54 | "prompt_number": 3, 55 | "text": [ 56 | "f (generic function with 1 method)" 57 | ] 58 | } 59 | ], 60 | "prompt_number": 3 61 | }, 62 | { 63 | "cell_type": "code", 64 | "collapsed": false, 65 | "input": [ 66 | "a = [Interval(0.5, 0.8), Interval(0.6, 0.9)]" 67 | ], 68 | "language": "python", 69 | "metadata": {}, 70 | "outputs": [ 71 | { 72 | "metadata": {}, 73 | "output_type": "pyout", 74 | "prompt_number": 4, 75 | "text": [ 76 | "2-element Array{Interval,1}:\n", 77 | " Interval(5e-01 with 256 bits of precision,8.000000000000000000000000000000000000000000000000000000000000000000000000000017e-01 with 256 bits of precision) \n", 78 | " Interval(5.999999999999999999999999999999999999999999999999999999999999999999999999999948e-01 with 256 bits of precision,9.000000000000000000000000000000000000000000000000000000000000000000000000000052e-01 with 256 bits of precision)" 79 | ] 80 | } 81 | ], 82 | "prompt_number": 4 83 | }, 84 | { 85 | "cell_type": "code", 86 | "collapsed": false, 87 | "input": [ 88 | "krawczyk2d(f, a)" 89 | ], 90 | "language": "python", 91 | "metadata": {}, 92 | "outputs": [ 93 | { 94 | "output_type": "stream", 95 | "stream": "stdout", 96 | "text": [ 97 | "a => [Interval(5e-01 with 256 bits of precision,8.000000000000000000000000000000000000000000000000000000000000000000000000000017e-01 with 256 bits of precision),Interval(5.999999999999999999999999999999999999999999999999999999999999999999999999999948e-01 with 256 bits of precision,9.000000000000000000000000000000000000000000000000000000000000000000000000000052e-01 with 256 bits of precision)]" 98 | ] 99 | }, 100 | { 101 | "output_type": "stream", 102 | "stream": "stdout", 103 | "text": [ 104 | "\n", 105 | "diam(a) => BigFloat[3.00000000000000000011e-01,3.00000000000000000011e-01]\n", 106 | "a => [Interval(5.59782608695652173609e-01 with 64 bits of precision,6.77173913043478261232e-01 with 64 bits of precision),Interval(7.44275362318840579489e-01 with 64 bits of precision,8.30362318840579710351e-01 with 64 bits of precision)]\n", 107 | "diam(a) => BigFloat[1.17391304347826087623e-01,8.60869565217391308621e-02]\n", 108 | "k += 1 => 3\n", 109 | "a => [Interval(6.11640871255334431689e-01 with 64 bits of precision,6.2442728271432796749e-01 with 64 bits of precision),Interval(7.81842914150279365824e-01 with 64 bits of precision,7.90461684587798984547e-01 with 64 bits of precision)]\n", 110 | "diam(a) => BigFloat[1.27864114589935358012e-02,8.61877043751961872346e-03]\n", 111 | "k += 1 => 4\n", 112 | "a => [Interval(6.17961699100631085921e-01 with 64 bits of precision,6.18106278469247463888e-01 with 64 bits of precision),Interval(7.86096586748077572548e-01 with 64 bits of precision,7.86206171543394608436e-01 with 64 bits of precision)]\n", 113 | "diam(a) => BigFloat[1.44579368616377967102e-04,1.09584795317035888461e-04]\n", 114 | "k += 1 => 5\n", 115 | "a => [Interval(6.18033978705287749924e-01 with 64 bits of precision,6.18033998794501946436e-01 with 64 bits of precision),Interval(7.8615137096578607801e-01 with 64 bits of precision,7.86151384549060496562e-01 with 64 bits of precision)]\n", 116 | "diam(a) => BigFloat[2.00892141965125677061e-08,1.35832744185522893976e-08]\n", 117 | "Unique zero in [Interval(6.18033988749894675277e-01 with 64 bits of precision,6.18033988749895021029e-01 with 64 bits of precision),Interval(7.86151377757423169867e-01 with 64 bits of precision,7.86151377757423402211e-01 with 64 bits of precision)]" 118 | ] 119 | }, 120 | { 121 | "output_type": "stream", 122 | "stream": "stdout", 123 | "text": [ 124 | "\n" 125 | ] 126 | }, 127 | { 128 | "metadata": {}, 129 | "output_type": "pyout", 130 | "prompt_number": 6, 131 | "text": [ 132 | "1-element Array{Array{Interval,1},1}:\n", 133 | " [Interval(6.18033988749894675277e-01 with 64 bits of precision,6.18033988749895021029e-01 with 64 bits of precision),Interval(7.86151377757423169867e-01 with 64 bits of precision,7.86151377757423402211e-01 with 64 bits of precision)]" 134 | ] 135 | } 136 | ], 137 | "prompt_number": 6 138 | }, 139 | { 140 | "cell_type": "code", 141 | "collapsed": false, 142 | "input": [ 143 | "# Works. Now let's increase starting intervals" 144 | ], 145 | "language": "python", 146 | "metadata": {}, 147 | "outputs": [], 148 | "prompt_number": 7 149 | }, 150 | { 151 | "cell_type": "code", 152 | "collapsed": false, 153 | "input": [ 154 | "a = [Interval(0, 2), Interval(0, 2)]" 155 | ], 156 | "language": "python", 157 | "metadata": {}, 158 | "outputs": [ 159 | { 160 | "metadata": {}, 161 | "output_type": "pyout", 162 | "prompt_number": 8, 163 | "text": [ 164 | "2-element Array{Interval,1}:\n", 165 | " Interval(0e+00 with 64 bits of precision,2e+00 with 64 bits of precision)\n", 166 | " Interval(0e+00 with 64 bits of precision,2e+00 with 64 bits of precision)" 167 | ] 168 | } 169 | ], 170 | "prompt_number": 8 171 | }, 172 | { 173 | "cell_type": "code", 174 | "collapsed": false, 175 | "input": [ 176 | "# This will set the algorithm into an infinite loop." 177 | ], 178 | "language": "python", 179 | "metadata": {}, 180 | "outputs": [], 181 | "prompt_number": 1 182 | }, 183 | { 184 | "cell_type": "code", 185 | "collapsed": false, 186 | "input": [ 187 | "# Let's see the next example: it actually does find the roots, but when the intervals are very small" 188 | ], 189 | "language": "python", 190 | "metadata": {}, 191 | "outputs": [], 192 | "prompt_number": 2 193 | }, 194 | { 195 | "cell_type": "code", 196 | "collapsed": false, 197 | "input": [ 198 | "f(x) = [20 - 20x[1] - x[2], x[1] - x[2]/20 - 1e-9*exp(x[2]/0.052)]" 199 | ], 200 | "language": "python", 201 | "metadata": {}, 202 | "outputs": [ 203 | { 204 | "metadata": {}, 205 | "output_type": "pyout", 206 | "prompt_number": 3, 207 | "text": [ 208 | "f (generic function with 1 method)" 209 | ] 210 | } 211 | ], 212 | "prompt_number": 3 213 | }, 214 | { 215 | "cell_type": "code", 216 | "collapsed": false, 217 | "input": [ 218 | "# trueroots = [0.9464142468335176, 1.0717150633296477]" 219 | ], 220 | "language": "python", 221 | "metadata": {}, 222 | "outputs": [], 223 | "prompt_number": 4 224 | }, 225 | { 226 | "cell_type": "code", 227 | "collapsed": false, 228 | "input": [ 229 | "a = [Interval(0.94, 0.95), Interval(1.05, 1.08)]" 230 | ], 231 | "language": "python", 232 | "metadata": {}, 233 | "outputs": [ 234 | { 235 | "metadata": {}, 236 | "output_type": "pyout", 237 | "prompt_number": 7, 238 | "text": [ 239 | "2-element Array{Interval,1}:\n", 240 | " Interval(9.399999999999999999999999999999999999999999999999999999999999999999999999999927e-01 with 256 bits of precision,9.500000000000000000000000000000000000000000000000000000000000000000000000000069e-01 with 256 bits of precision)\n", 241 | " Interval(1.049999999999999999999999999999999999999999999999999999999999999999999999999993e+00 with 256 bits of precision,1.08000000000000000000000000000000000000000000000000000000000000000000000000001e+00 with 256 bits of precision) " 242 | ] 243 | } 244 | ], 245 | "prompt_number": 7 246 | }, 247 | { 248 | "cell_type": "code", 249 | "collapsed": false, 250 | "input": [ 251 | "krawczyk2d(f, a)" 252 | ], 253 | "language": "python", 254 | "metadata": {}, 255 | "outputs": [ 256 | { 257 | "output_type": "stream", 258 | "stream": "stdout", 259 | "text": [ 260 | "a => [Interval(9.399999999999999999999999999999999999999999999999999999999999999999999999999927e-01 with 256 bits of precision,9.500000000000000000000000000000000000000000000000000000000000000000000000000069e-01 with 256 bits of precision),Interval(1.049999999999999999999999999999999999999999999999999999999999999999999999999993e+00 with 256 bits of precision,1.08000000000000000000000000000000000000000000000000000000000000000000000000001e+00 with 256 bits of precision)]" 261 | ] 262 | }, 263 | { 264 | "output_type": "stream", 265 | "stream": "stdout", 266 | "text": [ 267 | "\n", 268 | "diam(a) => BigFloat[1.00000000000000000006e-02,3.00000000000000000011e-02]\n", 269 | "a => [Interval(9.46142624135044351362e-01 with 64 bits of precision,9.4664088171185997489e-01 with 64 bits of precision),Interval(1.06718236576280050654e+00 with 64 bits of precision,1.07714751729911297124e+00 with 64 bits of precision)]\n", 270 | "diam(a) => BigFloat[4.98257576815623528122e-04,9.96515153631246470775e-03]\n", 271 | "k += 1 => 3\n", 272 | "a => [Interval(9.46389241901565050083e-01 with 64 bits of precision,9.46439058833313800773e-01 with 64 bits of precision),Interval(1.07121882333372398627e+00 with 64 bits of precision,1.07221516196869899313e+00 with 64 bits of precision)]\n", 273 | "diam(a) => BigFloat[4.98169317487506899816e-05,9.96338634975006860739e-04]\n", 274 | "k += 1 => 4\n", 275 | "a => [Interval(9.46413799486205053785e-01 with 64 bits of precision,9.46414692531945470576e-01 with 64 bits of precision),Interval(1.07170614936109058989e+00 with 64 bits of precision,1.07172401027589892007e+00 with 64 bits of precision)]\n", 276 | "diam(a) => BigFloat[8.93045740416791109656e-07,1.78609148083301843418e-05]\n", 277 | "k += 1 => 5\n", 278 | "a => [Interval(9.46414246757269344147e-01 with 64 bits of precision,9.46414246909765635783e-01 with 64 bits of precision),Interval(1.07171506180468728607e+00 with 64 bits of precision,1.07171506485461311684e+00 with 64 bits of precision)]\n", 279 | "diam(a) => BigFloat[1.5249629163605060711e-10,3.04992583076944823173e-09]\n", 280 | "Unique zero in [Interval(9.46414246833517617413e-01 with 64 bits of precision,9.46414246833517622346e-01 with 64 bits of precision),Interval(1.07171506332964755785e+00 with 64 bits of precision,1.0717150633296476474e+00 with 64 bits of precision)]" 281 | ] 282 | }, 283 | { 284 | "output_type": "stream", 285 | "stream": "stdout", 286 | "text": [ 287 | "\n" 288 | ] 289 | }, 290 | { 291 | "metadata": {}, 292 | "output_type": "pyout", 293 | "prompt_number": 8, 294 | "text": [ 295 | "1-element Array{Array{Interval,1},1}:\n", 296 | " [Interval(9.46414246833517617413e-01 with 64 bits of precision,9.46414246833517622346e-01 with 64 bits of precision),Interval(1.07171506332964755785e+00 with 64 bits of precision,1.0717150633296476474e+00 with 64 bits of precision)]" 297 | ] 298 | } 299 | ], 300 | "prompt_number": 8 301 | }, 302 | { 303 | "cell_type": "code", 304 | "collapsed": false, 305 | "input": [ 306 | "# In the next function it finds the root, but only when it is closely surrounded by a starting interval" 307 | ], 308 | "language": "python", 309 | "metadata": {}, 310 | "outputs": [], 311 | "prompt_number": 9 312 | }, 313 | { 314 | "cell_type": "code", 315 | "collapsed": false, 316 | "input": [ 317 | "f(x) = [0.5*(-(17.76x[1] - 103.79x[1]^2 + 229.62x[1]^3 - 226.31x[1]^4 + 83.72x[1]^5) + x[2]), 0.2*(-x[1] - 1.5x[2] + 1.2)]" 318 | ], 319 | "language": "python", 320 | "metadata": {}, 321 | "outputs": [ 322 | { 323 | "metadata": {}, 324 | "output_type": "pyout", 325 | "prompt_number": 10, 326 | "text": [ 327 | "f (generic function with 1 method)" 328 | ] 329 | } 330 | ], 331 | "prompt_number": 10 332 | }, 333 | { 334 | "cell_type": "code", 335 | "collapsed": false, 336 | "input": [ 337 | "# trueroots = [0.06263595920972119, 0.758242693860186]\n", 338 | "# trueroots = [0.8844295888702942, 0.21038027408647064]\n", 339 | "# trueroots = [0.2853687241300338, 0.6097541839133109]" 340 | ], 341 | "language": "python", 342 | "metadata": {}, 343 | "outputs": [], 344 | "prompt_number": 11 345 | }, 346 | { 347 | "cell_type": "code", 348 | "collapsed": false, 349 | "input": [ 350 | "a = [Interval(0.05, 0.1), Interval(-100, 100)]" 351 | ], 352 | "language": "python", 353 | "metadata": {}, 354 | "outputs": [ 355 | { 356 | "metadata": {}, 357 | "output_type": "pyout", 358 | "prompt_number": 12, 359 | "text": [ 360 | "2-element Array{Interval,1}:\n", 361 | " Interval(4.99999999999999999973e-02 with 64 bits of precision,1.00000000000000000001e-01 with 64 bits of precision)\n", 362 | " Interval(-1e+02 with 64 bits of precision,1e+02 with 64 bits of precision) " 363 | ] 364 | } 365 | ], 366 | "prompt_number": 12 367 | }, 368 | { 369 | "cell_type": "code", 370 | "collapsed": false, 371 | "input": [ 372 | "krawczyk2d(f, a)" 373 | ], 374 | "language": "python", 375 | "metadata": {}, 376 | "outputs": [ 377 | { 378 | "output_type": "stream", 379 | "stream": "stdout", 380 | "text": [ 381 | "a => [Interval(4.99999999999999999973e-02 with 64 bits of precision,1.00000000000000000001e-01 with 64 bits of precision),Interval(-1e+02 with 64 bits of precision,1e+02 with 64 bits of precision)]" 382 | ] 383 | }, 384 | { 385 | "output_type": "stream", 386 | "stream": "stdout", 387 | "text": [ 388 | "\n", 389 | "diam(a) => BigFloat[5.00000000000000000041e-02,2e+02]\n", 390 | "a => [Interval(4.99999999999999999973e-02 with 64 bits of precision,9.4550344742341260653e-02 with 64 bits of precision),Interval(7.36966436838439141379e-01 with 64 bits of precision,7.81488500003341015883e-01 with 64 bits of precision)]\n", 391 | "diam(a) => BigFloat[4.45503447423412606557e-02,4.4522063164901874504e-02]\n", 392 | "k += 1 => 3\n", 393 | "a => [Interval(4.99999999999999999973e-02 with 64 bits of precision,8.65160934538207583382e-02 with 64 bits of precision),Interval(7.42322604364119464221e-01 with 64 bits of precision,7.75318412680682777062e-01 with 64 bits of precision)]\n", 394 | "diam(a) => BigFloat[3.65160934538207583409e-02,3.29958083165633128415e-02]\n", 395 | "k += 1 => 4\n", 396 | "a => [Interval(4.43949064835282951901e-02 with 64 bits of precision,7.94399873513205070338e-02 with 64 bits of precision),Interval(7.47040008432452965479e-01 with 64 bits of precision,7.70403395677647773621e-01 with 64 bits of precision)]\n", 397 | "diam(a) => BigFloat[3.50450808677922118438e-02,2.33633872451948081421e-02]\n", 398 | "k += 1 => 5\n", 399 | "a => " 400 | ] 401 | }, 402 | { 403 | "output_type": "stream", 404 | "stream": "stdout", 405 | "text": [ 406 | "[Interval(5.06163960096364962984e-02 with 64 bits of precision,7.46470082780863032913e-02 with 64 bits of precision),Interval(7.5023532781460910133e-01 with 64 bits of precision,7.66255735993575639675e-01 with 64 bits of precision)]\n", 407 | "diam(a) => BigFloat[2.40306122684498069929e-02,1.60204081789665383454e-02]\n", 408 | "k += 1 => 6\n", 409 | "a => [Interval(5.69532489844786834348e-02 with 64 bits of precision,6.83186691336819782751e-02 with 64 bits of precision),Interval(7.54454220577545318048e-01 with 64 bits of precision,7.62031167343680848375e-01 with 64 bits of precision)]\n", 410 | "diam(a) => BigFloat[1.13654201492032948403e-02,7.57694676613553032719e-03]\n", 411 | "k += 1 => 7\n", 412 | "a => [Interval(6.13793095773444506483e-02 with 64 bits of precision,6.3892608842076667713e-02 with 64 bits of precision),Interval(7.57404927438615525144e-01 with 64 bits of precision,7.59080460281770336714e-01 with 64 bits of precision)]\n", 413 | "diam(a) => BigFloat[2.51329926473221706469e-03,1.67553284315481157071e-03]\n", 414 | "k += 1 => 8\n", 415 | "a => [Interval(6.25749570475021556183e-02 with 64 bits of precision,6.26969613719402846155e-02 with 64 bits of precision),Interval(7.58202025752039780542e-01 with 64 bits of precision,7.58283361968331866775e-01 with 64 bits of precision)]\n", 416 | "diam(a) => BigFloat[1.22004324438128997207e-04,8.13362162920862330484e-05]\n", 417 | "k += 1 => 9\n", 418 | "a => " 419 | ] 420 | }, 421 | { 422 | "output_type": "stream", 423 | "stream": "stdout", 424 | "text": [ 425 | "[Interval(6.26358157579486172535e-02 with 64 bits of precision,6.26361026614938229531e-02 with 64 bits of precision),Interval(7.58242598225670755047e-01 with 64 bits of precision,7.58242789494700892487e-01 with 64 bits of precision)]\n", 426 | "diam(a) => BigFloat[2.8690354520569962514e-07,1.91269030137440274042e-07]\n", 427 | "Unique zero in [Interval(6.26359592089280223205e-02 with 64 bits of precision,6.26359592105144178929e-02 with 64 bits of precision),Interval(7.58242693859657025137e-01 with 64 bits of precision,7.58242693860714622343e-01 with 64 bits of precision)]\n" 428 | ] 429 | }, 430 | { 431 | "metadata": {}, 432 | "output_type": "pyout", 433 | "prompt_number": 13, 434 | "text": [ 435 | "1-element Array{Array{Interval,1},1}:\n", 436 | " [Interval(6.26359592089280223205e-02 with 64 bits of precision,6.26359592105144178929e-02 with 64 bits of precision),Interval(7.58242693859657025137e-01 with 64 bits of precision,7.58242693860714622343e-01 with 64 bits of precision)]" 437 | ] 438 | } 439 | ], 440 | "prompt_number": 13 441 | }, 442 | { 443 | "cell_type": "markdown", 444 | "metadata": {}, 445 | "source": [ 446 | "Increasing the first interval destroys the root finding capability, and the second one in this example doesn't seem to affect it." 447 | ] 448 | } 449 | ], 450 | "metadata": {} 451 | } 452 | ] 453 | } -------------------------------------------------------------------------------- /.ipynb_checkpoints/krawczyk2d-examples-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "metadata": { 3 | "language": "Julia", 4 | "name": "" 5 | }, 6 | "nbformat": 3, 7 | "nbformat_minor": 0, 8 | "worksheets": [ 9 | { 10 | "cells": [ 11 | { 12 | "cell_type": "code", 13 | "collapsed": false, 14 | "input": [ 15 | "# Krawczyk 2D actually works, but when the intervals are large, it stalls" 16 | ], 17 | "language": "python", 18 | "metadata": {}, 19 | "outputs": [], 20 | "prompt_number": 1 21 | }, 22 | { 23 | "cell_type": "code", 24 | "collapsed": false, 25 | "input": [ 26 | "include(\"krawczyk2d.jl\")" 27 | ], 28 | "language": "python", 29 | "metadata": {}, 30 | "outputs": [ 31 | { 32 | "metadata": {}, 33 | "output_type": "pyout", 34 | "prompt_number": 6, 35 | "text": [ 36 | "krawczyk2d (generic function with 2 methods)" 37 | ] 38 | } 39 | ], 40 | "prompt_number": 6 41 | }, 42 | { 43 | "cell_type": "code", 44 | "collapsed": false, 45 | "input": [ 46 | "f(x) = [x[1]^2 + x[2]^2 - 1, x[1] - x[2]^2]" 47 | ], 48 | "language": "python", 49 | "metadata": {}, 50 | "outputs": [ 51 | { 52 | "metadata": {}, 53 | "output_type": "pyout", 54 | "prompt_number": 3, 55 | "text": [ 56 | "f (generic function with 1 method)" 57 | ] 58 | } 59 | ], 60 | "prompt_number": 3 61 | }, 62 | { 63 | "cell_type": "code", 64 | "collapsed": false, 65 | "input": [ 66 | "a = [Interval(0.5, 0.8), Interval(0.6, 0.9)]" 67 | ], 68 | "language": "python", 69 | "metadata": {}, 70 | "outputs": [ 71 | { 72 | "metadata": {}, 73 | "output_type": "pyout", 74 | "prompt_number": 4, 75 | "text": [ 76 | "2-element Array{Interval,1}:\n", 77 | " Interval(5e-01 with 256 bits of precision,8.000000000000000000000000000000000000000000000000000000000000000000000000000017e-01 with 256 bits of precision) \n", 78 | " Interval(5.999999999999999999999999999999999999999999999999999999999999999999999999999948e-01 with 256 bits of precision,9.000000000000000000000000000000000000000000000000000000000000000000000000000052e-01 with 256 bits of precision)" 79 | ] 80 | } 81 | ], 82 | "prompt_number": 4 83 | }, 84 | { 85 | "cell_type": "code", 86 | "collapsed": false, 87 | "input": [ 88 | "krawczyk2d(f, a)" 89 | ], 90 | "language": "python", 91 | "metadata": {}, 92 | "outputs": [ 93 | { 94 | "output_type": "stream", 95 | "stream": "stdout", 96 | "text": [ 97 | "a => [Interval(5e-01 with 256 bits of precision,8.000000000000000000000000000000000000000000000000000000000000000000000000000017e-01 with 256 bits of precision),Interval(5.999999999999999999999999999999999999999999999999999999999999999999999999999948e-01 with 256 bits of precision,9.000000000000000000000000000000000000000000000000000000000000000000000000000052e-01 with 256 bits of precision)]" 98 | ] 99 | }, 100 | { 101 | "output_type": "stream", 102 | "stream": "stdout", 103 | "text": [ 104 | "\n", 105 | "diam(a) => BigFloat[3.00000000000000000011e-01,3.00000000000000000011e-01]\n", 106 | "a => [Interval(5.59782608695652173609e-01 with 64 bits of precision,6.77173913043478261232e-01 with 64 bits of precision),Interval(7.44275362318840579489e-01 with 64 bits of precision,8.30362318840579710351e-01 with 64 bits of precision)]\n", 107 | "diam(a) => BigFloat[1.17391304347826087623e-01,8.60869565217391308621e-02]\n", 108 | "k += 1 => 3\n", 109 | "a => [Interval(6.11640871255334431689e-01 with 64 bits of precision,6.2442728271432796749e-01 with 64 bits of precision),Interval(7.81842914150279365824e-01 with 64 bits of precision,7.90461684587798984547e-01 with 64 bits of precision)]\n", 110 | "diam(a) => BigFloat[1.27864114589935358012e-02,8.61877043751961872346e-03]\n", 111 | "k += 1 => 4\n", 112 | "a => [Interval(6.17961699100631085921e-01 with 64 bits of precision,6.18106278469247463888e-01 with 64 bits of precision),Interval(7.86096586748077572548e-01 with 64 bits of precision,7.86206171543394608436e-01 with 64 bits of precision)]\n", 113 | "diam(a) => BigFloat[1.44579368616377967102e-04,1.09584795317035888461e-04]\n", 114 | "k += 1 => 5\n", 115 | "a => [Interval(6.18033978705287749924e-01 with 64 bits of precision,6.18033998794501946436e-01 with 64 bits of precision),Interval(7.8615137096578607801e-01 with 64 bits of precision,7.86151384549060496562e-01 with 64 bits of precision)]\n", 116 | "diam(a) => BigFloat[2.00892141965125677061e-08,1.35832744185522893976e-08]\n", 117 | "Unique zero in [Interval(6.18033988749894675277e-01 with 64 bits of precision,6.18033988749895021029e-01 with 64 bits of precision),Interval(7.86151377757423169867e-01 with 64 bits of precision,7.86151377757423402211e-01 with 64 bits of precision)]" 118 | ] 119 | }, 120 | { 121 | "output_type": "stream", 122 | "stream": "stdout", 123 | "text": [ 124 | "\n" 125 | ] 126 | }, 127 | { 128 | "metadata": {}, 129 | "output_type": "pyout", 130 | "prompt_number": 6, 131 | "text": [ 132 | "1-element Array{Array{Interval,1},1}:\n", 133 | " [Interval(6.18033988749894675277e-01 with 64 bits of precision,6.18033988749895021029e-01 with 64 bits of precision),Interval(7.86151377757423169867e-01 with 64 bits of precision,7.86151377757423402211e-01 with 64 bits of precision)]" 134 | ] 135 | } 136 | ], 137 | "prompt_number": 6 138 | }, 139 | { 140 | "cell_type": "code", 141 | "collapsed": false, 142 | "input": [ 143 | "# Works. Now let's increase starting intervals" 144 | ], 145 | "language": "python", 146 | "metadata": {}, 147 | "outputs": [], 148 | "prompt_number": 7 149 | }, 150 | { 151 | "cell_type": "code", 152 | "collapsed": false, 153 | "input": [ 154 | "a = [Interval(0, 2), Interval(0, 2)]" 155 | ], 156 | "language": "python", 157 | "metadata": {}, 158 | "outputs": [ 159 | { 160 | "metadata": {}, 161 | "output_type": "pyout", 162 | "prompt_number": 8, 163 | "text": [ 164 | "2-element Array{Interval,1}:\n", 165 | " Interval(0e+00 with 64 bits of precision,2e+00 with 64 bits of precision)\n", 166 | " Interval(0e+00 with 64 bits of precision,2e+00 with 64 bits of precision)" 167 | ] 168 | } 169 | ], 170 | "prompt_number": 8 171 | }, 172 | { 173 | "cell_type": "code", 174 | "collapsed": false, 175 | "input": [ 176 | "# This will set the algorithm into an infinite loop." 177 | ], 178 | "language": "python", 179 | "metadata": {}, 180 | "outputs": [], 181 | "prompt_number": 1 182 | }, 183 | { 184 | "cell_type": "code", 185 | "collapsed": false, 186 | "input": [ 187 | "# Let's see the next example: it actually does find the roots, but when the intervals are very small" 188 | ], 189 | "language": "python", 190 | "metadata": {}, 191 | "outputs": [], 192 | "prompt_number": 2 193 | }, 194 | { 195 | "cell_type": "code", 196 | "collapsed": false, 197 | "input": [ 198 | "f(x) = [20 - 20x[1] - x[2], x[1] - x[2]/20 - 1e-9*exp(x[2]/0.052)]" 199 | ], 200 | "language": "python", 201 | "metadata": {}, 202 | "outputs": [ 203 | { 204 | "metadata": {}, 205 | "output_type": "pyout", 206 | "prompt_number": 3, 207 | "text": [ 208 | "f (generic function with 1 method)" 209 | ] 210 | } 211 | ], 212 | "prompt_number": 3 213 | }, 214 | { 215 | "cell_type": "code", 216 | "collapsed": false, 217 | "input": [ 218 | "# trueroots = [0.9464142468335176, 1.0717150633296477]" 219 | ], 220 | "language": "python", 221 | "metadata": {}, 222 | "outputs": [], 223 | "prompt_number": 4 224 | }, 225 | { 226 | "cell_type": "code", 227 | "collapsed": false, 228 | "input": [ 229 | "a = [Interval(0.94, 0.95), Interval(1.05, 1.08)]" 230 | ], 231 | "language": "python", 232 | "metadata": {}, 233 | "outputs": [ 234 | { 235 | "metadata": {}, 236 | "output_type": "pyout", 237 | "prompt_number": 7, 238 | "text": [ 239 | "2-element Array{Interval,1}:\n", 240 | " Interval(9.399999999999999999999999999999999999999999999999999999999999999999999999999927e-01 with 256 bits of precision,9.500000000000000000000000000000000000000000000000000000000000000000000000000069e-01 with 256 bits of precision)\n", 241 | " Interval(1.049999999999999999999999999999999999999999999999999999999999999999999999999993e+00 with 256 bits of precision,1.08000000000000000000000000000000000000000000000000000000000000000000000000001e+00 with 256 bits of precision) " 242 | ] 243 | } 244 | ], 245 | "prompt_number": 7 246 | }, 247 | { 248 | "cell_type": "code", 249 | "collapsed": false, 250 | "input": [ 251 | "krawczyk2d(f, a)" 252 | ], 253 | "language": "python", 254 | "metadata": {}, 255 | "outputs": [ 256 | { 257 | "output_type": "stream", 258 | "stream": "stdout", 259 | "text": [ 260 | "a => [Interval(9.399999999999999999999999999999999999999999999999999999999999999999999999999927e-01 with 256 bits of precision,9.500000000000000000000000000000000000000000000000000000000000000000000000000069e-01 with 256 bits of precision),Interval(1.049999999999999999999999999999999999999999999999999999999999999999999999999993e+00 with 256 bits of precision,1.08000000000000000000000000000000000000000000000000000000000000000000000000001e+00 with 256 bits of precision)]" 261 | ] 262 | }, 263 | { 264 | "output_type": "stream", 265 | "stream": "stdout", 266 | "text": [ 267 | "\n", 268 | "diam(a) => BigFloat[1.00000000000000000006e-02,3.00000000000000000011e-02]\n", 269 | "a => [Interval(9.46142624135044351362e-01 with 64 bits of precision,9.4664088171185997489e-01 with 64 bits of precision),Interval(1.06718236576280050654e+00 with 64 bits of precision,1.07714751729911297124e+00 with 64 bits of precision)]\n", 270 | "diam(a) => BigFloat[4.98257576815623528122e-04,9.96515153631246470775e-03]\n", 271 | "k += 1 => 3\n", 272 | "a => [Interval(9.46389241901565050083e-01 with 64 bits of precision,9.46439058833313800773e-01 with 64 bits of precision),Interval(1.07121882333372398627e+00 with 64 bits of precision,1.07221516196869899313e+00 with 64 bits of precision)]\n", 273 | "diam(a) => BigFloat[4.98169317487506899816e-05,9.96338634975006860739e-04]\n", 274 | "k += 1 => 4\n", 275 | "a => [Interval(9.46413799486205053785e-01 with 64 bits of precision,9.46414692531945470576e-01 with 64 bits of precision),Interval(1.07170614936109058989e+00 with 64 bits of precision,1.07172401027589892007e+00 with 64 bits of precision)]\n", 276 | "diam(a) => BigFloat[8.93045740416791109656e-07,1.78609148083301843418e-05]\n", 277 | "k += 1 => 5\n", 278 | "a => [Interval(9.46414246757269344147e-01 with 64 bits of precision,9.46414246909765635783e-01 with 64 bits of precision),Interval(1.07171506180468728607e+00 with 64 bits of precision,1.07171506485461311684e+00 with 64 bits of precision)]\n", 279 | "diam(a) => BigFloat[1.5249629163605060711e-10,3.04992583076944823173e-09]\n", 280 | "Unique zero in [Interval(9.46414246833517617413e-01 with 64 bits of precision,9.46414246833517622346e-01 with 64 bits of precision),Interval(1.07171506332964755785e+00 with 64 bits of precision,1.0717150633296476474e+00 with 64 bits of precision)]" 281 | ] 282 | }, 283 | { 284 | "output_type": "stream", 285 | "stream": "stdout", 286 | "text": [ 287 | "\n" 288 | ] 289 | }, 290 | { 291 | "metadata": {}, 292 | "output_type": "pyout", 293 | "prompt_number": 8, 294 | "text": [ 295 | "1-element Array{Array{Interval,1},1}:\n", 296 | " [Interval(9.46414246833517617413e-01 with 64 bits of precision,9.46414246833517622346e-01 with 64 bits of precision),Interval(1.07171506332964755785e+00 with 64 bits of precision,1.0717150633296476474e+00 with 64 bits of precision)]" 297 | ] 298 | } 299 | ], 300 | "prompt_number": 8 301 | }, 302 | { 303 | "cell_type": "code", 304 | "collapsed": false, 305 | "input": [ 306 | "# In the next function it finds the root, but only when it is closely surrounded by a starting interval" 307 | ], 308 | "language": "python", 309 | "metadata": {}, 310 | "outputs": [], 311 | "prompt_number": 9 312 | }, 313 | { 314 | "cell_type": "code", 315 | "collapsed": false, 316 | "input": [ 317 | "f(x) = [0.5*(-(17.76x[1] - 103.79x[1]^2 + 229.62x[1]^3 - 226.31x[1]^4 + 83.72x[1]^5) + x[2]), 0.2*(-x[1] - 1.5x[2] + 1.2)]" 318 | ], 319 | "language": "python", 320 | "metadata": {}, 321 | "outputs": [ 322 | { 323 | "metadata": {}, 324 | "output_type": "pyout", 325 | "prompt_number": 10, 326 | "text": [ 327 | "f (generic function with 1 method)" 328 | ] 329 | } 330 | ], 331 | "prompt_number": 10 332 | }, 333 | { 334 | "cell_type": "code", 335 | "collapsed": false, 336 | "input": [ 337 | "# trueroots = [0.06263595920972119, 0.758242693860186]\n", 338 | "# trueroots = [0.8844295888702942, 0.21038027408647064]\n", 339 | "# trueroots = [0.2853687241300338, 0.6097541839133109]" 340 | ], 341 | "language": "python", 342 | "metadata": {}, 343 | "outputs": [], 344 | "prompt_number": 11 345 | }, 346 | { 347 | "cell_type": "code", 348 | "collapsed": false, 349 | "input": [ 350 | "a = [Interval(0.05, 0.1), Interval(-100, 100)]" 351 | ], 352 | "language": "python", 353 | "metadata": {}, 354 | "outputs": [ 355 | { 356 | "metadata": {}, 357 | "output_type": "pyout", 358 | "prompt_number": 12, 359 | "text": [ 360 | "2-element Array{Interval,1}:\n", 361 | " Interval(4.99999999999999999973e-02 with 64 bits of precision,1.00000000000000000001e-01 with 64 bits of precision)\n", 362 | " Interval(-1e+02 with 64 bits of precision,1e+02 with 64 bits of precision) " 363 | ] 364 | } 365 | ], 366 | "prompt_number": 12 367 | }, 368 | { 369 | "cell_type": "code", 370 | "collapsed": false, 371 | "input": [ 372 | "krawczyk2d(f, a)" 373 | ], 374 | "language": "python", 375 | "metadata": {}, 376 | "outputs": [ 377 | { 378 | "output_type": "stream", 379 | "stream": "stdout", 380 | "text": [ 381 | "a => [Interval(4.99999999999999999973e-02 with 64 bits of precision,1.00000000000000000001e-01 with 64 bits of precision),Interval(-1e+02 with 64 bits of precision,1e+02 with 64 bits of precision)]" 382 | ] 383 | }, 384 | { 385 | "output_type": "stream", 386 | "stream": "stdout", 387 | "text": [ 388 | "\n", 389 | "diam(a) => BigFloat[5.00000000000000000041e-02,2e+02]\n", 390 | "a => [Interval(4.99999999999999999973e-02 with 64 bits of precision,9.4550344742341260653e-02 with 64 bits of precision),Interval(7.36966436838439141379e-01 with 64 bits of precision,7.81488500003341015883e-01 with 64 bits of precision)]\n", 391 | "diam(a) => BigFloat[4.45503447423412606557e-02,4.4522063164901874504e-02]\n", 392 | "k += 1 => 3\n", 393 | "a => [Interval(4.99999999999999999973e-02 with 64 bits of precision,8.65160934538207583382e-02 with 64 bits of precision),Interval(7.42322604364119464221e-01 with 64 bits of precision,7.75318412680682777062e-01 with 64 bits of precision)]\n", 394 | "diam(a) => BigFloat[3.65160934538207583409e-02,3.29958083165633128415e-02]\n", 395 | "k += 1 => 4\n", 396 | "a => [Interval(4.43949064835282951901e-02 with 64 bits of precision,7.94399873513205070338e-02 with 64 bits of precision),Interval(7.47040008432452965479e-01 with 64 bits of precision,7.70403395677647773621e-01 with 64 bits of precision)]\n", 397 | "diam(a) => BigFloat[3.50450808677922118438e-02,2.33633872451948081421e-02]\n", 398 | "k += 1 => 5\n", 399 | "a => " 400 | ] 401 | }, 402 | { 403 | "output_type": "stream", 404 | "stream": "stdout", 405 | "text": [ 406 | "[Interval(5.06163960096364962984e-02 with 64 bits of precision,7.46470082780863032913e-02 with 64 bits of precision),Interval(7.5023532781460910133e-01 with 64 bits of precision,7.66255735993575639675e-01 with 64 bits of precision)]\n", 407 | "diam(a) => BigFloat[2.40306122684498069929e-02,1.60204081789665383454e-02]\n", 408 | "k += 1 => 6\n", 409 | "a => [Interval(5.69532489844786834348e-02 with 64 bits of precision,6.83186691336819782751e-02 with 64 bits of precision),Interval(7.54454220577545318048e-01 with 64 bits of precision,7.62031167343680848375e-01 with 64 bits of precision)]\n", 410 | "diam(a) => BigFloat[1.13654201492032948403e-02,7.57694676613553032719e-03]\n", 411 | "k += 1 => 7\n", 412 | "a => [Interval(6.13793095773444506483e-02 with 64 bits of precision,6.3892608842076667713e-02 with 64 bits of precision),Interval(7.57404927438615525144e-01 with 64 bits of precision,7.59080460281770336714e-01 with 64 bits of precision)]\n", 413 | "diam(a) => BigFloat[2.51329926473221706469e-03,1.67553284315481157071e-03]\n", 414 | "k += 1 => 8\n", 415 | "a => [Interval(6.25749570475021556183e-02 with 64 bits of precision,6.26969613719402846155e-02 with 64 bits of precision),Interval(7.58202025752039780542e-01 with 64 bits of precision,7.58283361968331866775e-01 with 64 bits of precision)]\n", 416 | "diam(a) => BigFloat[1.22004324438128997207e-04,8.13362162920862330484e-05]\n", 417 | "k += 1 => 9\n", 418 | "a => " 419 | ] 420 | }, 421 | { 422 | "output_type": "stream", 423 | "stream": "stdout", 424 | "text": [ 425 | "[Interval(6.26358157579486172535e-02 with 64 bits of precision,6.26361026614938229531e-02 with 64 bits of precision),Interval(7.58242598225670755047e-01 with 64 bits of precision,7.58242789494700892487e-01 with 64 bits of precision)]\n", 426 | "diam(a) => BigFloat[2.8690354520569962514e-07,1.91269030137440274042e-07]\n", 427 | "Unique zero in [Interval(6.26359592089280223205e-02 with 64 bits of precision,6.26359592105144178929e-02 with 64 bits of precision),Interval(7.58242693859657025137e-01 with 64 bits of precision,7.58242693860714622343e-01 with 64 bits of precision)]\n" 428 | ] 429 | }, 430 | { 431 | "metadata": {}, 432 | "output_type": "pyout", 433 | "prompt_number": 13, 434 | "text": [ 435 | "1-element Array{Array{Interval,1},1}:\n", 436 | " [Interval(6.26359592089280223205e-02 with 64 bits of precision,6.26359592105144178929e-02 with 64 bits of precision),Interval(7.58242693859657025137e-01 with 64 bits of precision,7.58242693860714622343e-01 with 64 bits of precision)]" 437 | ] 438 | } 439 | ], 440 | "prompt_number": 13 441 | }, 442 | { 443 | "cell_type": "markdown", 444 | "metadata": {}, 445 | "source": [ 446 | "Increasing the first interval destroys the root finding capability, and the second one in this example doesn't seem to affect it." 447 | ] 448 | } 449 | ], 450 | "metadata": {} 451 | } 452 | ] 453 | } -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | 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Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | {one line to give the program's name and a brief idea of what it does.} 635 | Copyright (C) {year} {name of author} 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | {project} Copyright (C) {year} {fullname} 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /newton2d-examples.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "metadata": { 3 | "language": "Julia", 4 | "name": "", 5 | "signature": "sha256:f5659f10e8db0fe182f61bd8e6d5925f7ae80cbdbf66d269a920e00eb03518df" 6 | }, 7 | "nbformat": 3, 8 | "nbformat_minor": 0, 9 | "worksheets": [ 10 | { 11 | "cells": [ 12 | { 13 | "cell_type": "code", 14 | "collapsed": false, 15 | "input": [ 16 | "include(\"newton2d.jl\")" 17 | ], 18 | "language": "python", 19 | "metadata": {}, 20 | "outputs": [ 21 | { 22 | "metadata": {}, 23 | "output_type": "pyout", 24 | "prompt_number": 1, 25 | "text": [ 26 | "newton2d (generic function with 2 methods)" 27 | ] 28 | } 29 | ], 30 | "prompt_number": 1 31 | }, 32 | { 33 | "cell_type": "code", 34 | "collapsed": false, 35 | "input": [ 36 | "f(x) = [x[1]^2 + x[2]^2 - 1, x[1] - x[2]^2]" 37 | ], 38 | "language": "python", 39 | "metadata": {}, 40 | "outputs": [ 41 | { 42 | "metadata": {}, 43 | "output_type": "pyout", 44 | "prompt_number": 2, 45 | "text": [ 46 | "f (generic function with 1 method)" 47 | ] 48 | } 49 | ], 50 | "prompt_number": 2 51 | }, 52 | { 53 | "cell_type": "code", 54 | "collapsed": false, 55 | "input": [ 56 | "a = [Interval(0.5, 0.8), Interval(0.6, 0.9)]" 57 | ], 58 | "language": "python", 59 | "metadata": {}, 60 | "outputs": [ 61 | { 62 | "metadata": {}, 63 | "output_type": "pyout", 64 | "prompt_number": 3, 65 | "text": [ 66 | "2-element Array{Interval,1}:\n", 67 | " Interval(5e-01 with 256 bits of precision,8.000000000000000000000000000000000000000000000000000000000000000000000000000017e-01 with 256 bits of precision) \n", 68 | " Interval(5.999999999999999999999999999999999999999999999999999999999999999999999999999948e-01 with 256 bits of precision,9.000000000000000000000000000000000000000000000000000000000000000000000000000052e-01 with 256 bits of precision)" 69 | ] 70 | } 71 | ], 72 | "prompt_number": 3 73 | }, 74 | { 75 | "cell_type": "code", 76 | "collapsed": false, 77 | "input": [ 78 | "roots = newton2d(f, a, 64)" 79 | ], 80 | "language": "python", 81 | "metadata": {}, 82 | "outputs": [ 83 | { 84 | "output_type": "stream", 85 | "stream": "stdout", 86 | "text": [ 87 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(5.69951923076923076632e-01 with 64 bits of precision,6.43645833333333333595e-01 with 64 bits of precision),Interval(7.76909722222222222138e-01 with 64 bits of precision,8.02564102564102564328e-01 with 64 bits of precision)]]" 88 | ] 89 | }, 90 | { 91 | "output_type": "stream", 92 | "stream": "stdout", 93 | "text": [ 94 | "\n", 95 | "roots_array = roots_array_new => [[Interval(5.69951923076923076632e-01 with 64 bits of precision,6.43645833333333333595e-01 with 64 bits of precision),Interval(7.76909722222222222138e-01 with 64 bits of precision,8.02564102564102564328e-01 with 64 bits of precision)]]\n", 96 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(6.16649962609692997909e-01 with 64 bits of precision,6.19676057175074426167e-01 with 64 bits of precision),Interval(7.85711856788002042517e-01 with 64 bits of precision,7.86710356930029978083e-01 with 64 bits of precision)]]\n", 97 | "roots_array = roots_array_new => [[Interval(6.16649962609692997909e-01 with 64 bits of precision,6.19676057175074426167e-01 with 64 bits of precision),Interval(7.85711856788002042517e-01 with 64 bits of precision,7.86710356930029978083e-01 with 64 bits of precision)]]\n", 98 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(6.18033201420526032209e-01 with 64 bits of precision,6.18034789347726672202e-01 with 64 bits of precision),Interval(7.86151061673094964761e-01 with 64 bits of precision,7.86151706419289133176e-01 with 64 bits of precision)]]\n", 99 | "roots_array = roots_array_new => [[Interval(6.18033201420526032209e-01 with 64 bits of precision,6.18034789347726672202e-01 with 64 bits of precision),Interval(7.86151061673094964761e-01 with 64 bits of precision,7.86151706419289133176e-01 with 64 bits of precision)]]\n", 100 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(6.18033988749863042703e-01 with 64 bits of precision,6.18033988749926693068e-01 with 64 bits of precision),Interval(7.86151377757410522269e-01 with 64 bits of precision,7.86151377757436125161e-01 with 64 bits of precision)]]\n", 101 | "roots_array = roots_array_new => [[Interval(6.18033988749863042703e-01 with 64 bits of precision,6.18033988749926693068e-01 with 64 bits of precision),Interval(7.86151377757410522269e-01 with 64 bits of precision,7.86151377757436125161e-01 with 64 bits of precision)]]\n", 102 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(6.18033988749894848045e-01 with 64 bits of precision,6.18033988749894848424e-01 with 64 bits of precision),Interval(7.86151377757423285931e-01 with 64 bits of precision,7.8615137775742328631e-01 with 64 bits of precision)]]\n", 103 | "roots_array = roots_array_new => [[Interval(6.18033988749894848045e-01 with 64 bits of precision,6.18033988749894848424e-01 with 64 bits of precision),Interval(7.86151377757423285931e-01 with 64 bits of precision,7.8615137775742328631e-01 with 64 bits of precision)]]\n", 104 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(6.18033988749894848045e-01 with 64 bits of precision,6.18033988749894848424e-01 with 64 bits of precision),Interval(7.86151377757423285931e-01 with 64 bits of precision,7.8615137775742328631e-01 with 64 bits of precision)]]\n", 105 | "Function calls: 5\n" 106 | ] 107 | }, 108 | { 109 | "metadata": {}, 110 | "output_type": "pyout", 111 | "prompt_number": 5, 112 | "text": [ 113 | "1-element Array{Array{Interval,1},1}:\n", 114 | " [Interval(6.18033988749894848045e-01 with 64 bits of precision,6.18033988749894848424e-01 with 64 bits of precision),Interval(7.86151377757423285931e-01 with 64 bits of precision,7.8615137775742328631e-01 with 64 bits of precision)]" 115 | ] 116 | } 117 | ], 118 | "prompt_number": 5 119 | }, 120 | { 121 | "cell_type": "code", 122 | "collapsed": false, 123 | "input": [ 124 | "f(roots[1])" 125 | ], 126 | "language": "python", 127 | "metadata": {}, 128 | "outputs": [ 129 | { 130 | "metadata": {}, 131 | "output_type": "pyout", 132 | "prompt_number": 7, 133 | "text": [ 134 | "2-element Array{Interval,1}:\n", 135 | " Interval(-5.42101086242752217051e-19 with 64 bits of precision,8.673617379884035473e-19 with 64 bits of precision) \n", 136 | " Interval(-5.96311194867027438704e-19 with 64 bits of precision,4.87890977618476995303e-19 with 64 bits of precision)" 137 | ] 138 | } 139 | ], 140 | "prompt_number": 7 141 | }, 142 | { 143 | "cell_type": "code", 144 | "collapsed": false, 145 | "input": [ 146 | "# Works." 147 | ], 148 | "language": "python", 149 | "metadata": {}, 150 | "outputs": [], 151 | "prompt_number": 5 152 | }, 153 | { 154 | "cell_type": "code", 155 | "collapsed": false, 156 | "input": [ 157 | "# 2nd example from Moore book" 158 | ], 159 | "language": "python", 160 | "metadata": {}, 161 | "outputs": [], 162 | "prompt_number": 6 163 | }, 164 | { 165 | "cell_type": "code", 166 | "collapsed": false, 167 | "input": [ 168 | "f(x) = [20 - 20x[1] - x[2], x[1] - x[2]/20 - 1e-9*exp(x[2]/0.052)]" 169 | ], 170 | "language": "python", 171 | "metadata": {}, 172 | "outputs": [ 173 | { 174 | "metadata": {}, 175 | "output_type": "pyout", 176 | "prompt_number": 8, 177 | "text": [ 178 | "f (generic function with 1 method)" 179 | ] 180 | } 181 | ], 182 | "prompt_number": 8 183 | }, 184 | { 185 | "cell_type": "code", 186 | "collapsed": false, 187 | "input": [ 188 | "a = [Interval(0.5, 1.2), Interval(0.6, 1.2)]" 189 | ], 190 | "language": "python", 191 | "metadata": {}, 192 | "outputs": [ 193 | { 194 | "metadata": {}, 195 | "output_type": "pyout", 196 | "prompt_number": 9, 197 | "text": [ 198 | "2-element Array{Interval,1}:\n", 199 | " Interval(5e-01 with 64 bits of precision,1.20000000000000000004e+00 with 64 bits of precision) \n", 200 | " Interval(5.99999999999999999967e-01 with 64 bits of precision,1.20000000000000000004e+00 with 64 bits of precision)" 201 | ] 202 | } 203 | ], 204 | "prompt_number": 9 205 | }, 206 | { 207 | "cell_type": "code", 208 | "collapsed": false, 209 | "input": [ 210 | "roots = newton2d(f, a, 64)" 211 | ], 212 | "language": "python", 213 | "metadata": {}, 214 | "outputs": [ 215 | { 216 | "output_type": "stream", 217 | "stream": "stdout", 218 | "text": [ 219 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(5.24913596541994085176e-01 with 64 bits of precision,9.54783404920287672335e-01 with 64 bits of precision),Interval(9.04331901594246556598e-01 with 64 bits of precision,1.20000000000000000004e+00 with 64 bits of precision)]]" 220 | ] 221 | }, 222 | { 223 | "output_type": "stream", 224 | "stream": "stdout", 225 | "text": [ 226 | "\n", 227 | "roots_array = roots_array_new => [[Interval(5.24913596541994085176e-01 with 64 bits of precision,9.54783404920287672335e-01 with 64 bits of precision),Interval(9.04331901594246556598e-01 with 64 bits of precision,1.20000000000000000004e+00 with 64 bits of precision)]]\n", 228 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.29487402579674734042e-01 with 64 bits of precision,9.47322133828186144428e-01 with 64 bits of precision),Interval(1.05355732343627711643e+00 with 64 bits of precision,1.20000000000000000004e+00 with 64 bits of precision)]]\n", 229 | "roots_array = roots_array_new => [[Interval(9.29487402579674734042e-01 with 64 bits of precision,9.47322133828186144428e-01 with 64 bits of precision),Interval(1.05355732343627711643e+00 with 64 bits of precision,1.20000000000000000004e+00 with 64 bits of precision)]]\n", 230 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.4405739404211960064e-01 with 64 bits of precision,9.47322133828186144428e-01 with 64 bits of precision),Interval(1.05355732343627711643e+00 with 64 bits of precision,1.11885211915760798037e+00 with 64 bits of precision)]]\n", 231 | "roots_array = roots_array_new => [[Interval(9.4405739404211960064e-01 with 64 bits of precision,9.47322133828186144428e-01 with 64 bits of precision),Interval(1.05355732343627711643e+00 with 64 bits of precision,1.11885211915760798037e+00 with 64 bits of precision)]]\n", 232 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46028161295262554455e-01 with 64 bits of precision,9.46870644360673580542e-01 with 64 bits of precision),Interval(1.06258711278652839491e+00 with 64 bits of precision,1.07943677409474890645e+00 with 64 bits of precision)]]\n", 233 | "roots_array = roots_array_new => [[Interval(9.46028161295262554455e-01 with 64 bits of precision,9.46870644360673580542e-01 with 64 bits of precision),Interval(1.06258711278652839491e+00 with 64 bits of precision,1.07943677409474890645e+00 with 64 bits of precision)]]\n", 234 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46407827444742915182e-01 with 64 bits of precision,9.46419276900890626275e-01 with 64 bits of precision),Interval(1.07161446198218747721e+00 with 64 bits of precision,1.07184345110514169289e+00 with 64 bits of precision)]]\n", 235 | "roots_array = roots_array_new => [[Interval(9.46407827444742915182e-01 with 64 bits of precision,9.46419276900890626275e-01 with 64 bits of precision),Interval(1.07161446198218747721e+00 with 64 bits of precision,1.07184345110514169289e+00 with 64 bits of precision)]]\n", 236 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46414245222457044325e-01 with 64 bits of precision,9.46414248263380555859e-01 with 64 bits of precision),Interval(1.07171503473238888629e+00 with 64 bits of precision,1.07171509555085910917e+00 with 64 bits of precision)]]\n", 237 | "roots_array = roots_array_new => [[Interval(9.46414245222457044325e-01 with 64 bits of precision,9.46414248263380555859e-01 with 64 bits of precision),Interval(1.07171503473238888629e+00 with 64 bits of precision,1.07171509555085910917e+00 with 64 bits of precision)]]\n", 238 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46414246833517562878e-01 with 64 bits of precision,9.46414246833517668533e-01 with 64 bits of precision),Interval(1.07171506332964663205e+00 with 64 bits of precision,1.07171506332964873995e+00 with 64 bits of precision)]]\n", 239 | "roots_array = roots_array_new => [[Interval(9.46414246833517562878e-01 with 64 bits of precision,9.46414246833517668533e-01 with 64 bits of precision),Interval(1.07171506332964663205e+00 with 64 bits of precision,1.07171506332964873995e+00 with 64 bits of precision)]]\n", 240 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46414246833517617088e-01 with 64 bits of precision,9.46414246833517617467e-01 with 64 bits of precision),Interval(1.07171506332964765413e+00 with 64 bits of precision,1.07171506332964765488e+00 with 64 bits of precision)]]\n", 241 | "roots_array = roots_array_new => [[Interval(9.46414246833517617088e-01 with 64 bits of precision,9.46414246833517617467e-01 with 64 bits of precision),Interval(1.07171506332964765413e+00 with 64 bits of precision,1.07171506332964765488e+00 with 64 bits of precision)]]\n", 242 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46414246833517617142e-01 with 64 bits of precision,9.46414246833517617467e-01 with 64 bits of precision),Interval(1.07171506332964765413e+00 with 64 bits of precision,1.07171506332964765488e+00 with 64 bits of precision)]]\n", 243 | "roots_array = roots_array_new => [[Interval(9.46414246833517617142e-01 with 64 bits of precision,9.46414246833517617467e-01 with 64 bits of precision),Interval(1.07171506332964765413e+00 with 64 bits of precision,1.07171506332964765488e+00 with 64 bits of precision)]]\n", 244 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.46414246833517617142e-01 with 64 bits of precision,9.46414246833517617467e-01 with 64 bits of precision),Interval(1.07171506332964765413e+00 with 64 bits of precision,1.07171506332964765488e+00 with 64 bits of precision)]]\n", 245 | "Function calls: 9\n" 246 | ] 247 | }, 248 | { 249 | "metadata": {}, 250 | "output_type": "pyout", 251 | "prompt_number": 11, 252 | "text": [ 253 | "1-element Array{Array{Interval,1},1}:\n", 254 | " [Interval(9.46414246833517617142e-01 with 64 bits of precision,9.46414246833517617467e-01 with 64 bits of precision),Interval(1.07171506332964765413e+00 with 64 bits of precision,1.07171506332964765488e+00 with 64 bits of precision)]" 255 | ] 256 | } 257 | ], 258 | "prompt_number": 11 259 | }, 260 | { 261 | "cell_type": "code", 262 | "collapsed": false, 263 | "input": [ 264 | "f(ans[1])" 265 | ], 266 | "language": "python", 267 | "metadata": {}, 268 | "outputs": [ 269 | { 270 | "metadata": {}, 271 | "output_type": "pyout", 272 | "prompt_number": 12, 273 | "text": [ 274 | "2-element Array{Interval,1}:\n", 275 | " Interval(-6.39679281766447616064e-18 with 64 bits of precision,4.87890977618476995303e-18 with 64 bits of precision)\n", 276 | " Interval(-8.83624770575686113716e-18 with 64 bits of precision,1.20888542232133744399e-17 with 64 bits of precision)" 277 | ] 278 | } 279 | ], 280 | "prompt_number": 12 281 | }, 282 | { 283 | "cell_type": "code", 284 | "collapsed": false, 285 | "input": [ 286 | "# Works. Krawczyk2d is not yet capable of doing this on such an interval (only on a tiny one)" 287 | ], 288 | "language": "python", 289 | "metadata": {}, 290 | "outputs": [], 291 | "prompt_number": 10 292 | }, 293 | { 294 | "cell_type": "code", 295 | "collapsed": false, 296 | "input": [ 297 | "# 3rd example from Moore" 298 | ], 299 | "language": "python", 300 | "metadata": {}, 301 | "outputs": [], 302 | "prompt_number": 11 303 | }, 304 | { 305 | "cell_type": "heading", 306 | "level": 2, 307 | "metadata": {}, 308 | "source": [ 309 | "Complicated case with 3 zeros:" 310 | ] 311 | }, 312 | { 313 | "cell_type": "code", 314 | "collapsed": false, 315 | "input": [ 316 | "f(x) = [0.5*(-(17.76x[1] - 103.79x[1]^2 + 229.62x[1]^3 - 226.31x[1]^4 + 83.72x[1]^5) + x[2]), 0.2*(-x[1] - 1.5x[2] + 1.2)]" 317 | ], 318 | "language": "python", 319 | "metadata": {}, 320 | "outputs": [ 321 | { 322 | "metadata": {}, 323 | "output_type": "pyout", 324 | "prompt_number": 61, 325 | "text": [ 326 | "f (generic function with 1 method)" 327 | ] 328 | } 329 | ], 330 | "prompt_number": 61 331 | }, 332 | { 333 | "cell_type": "heading", 334 | "level": 3, 335 | "metadata": {}, 336 | "source": [ 337 | "True zeros:" 338 | ] 339 | }, 340 | { 341 | "cell_type": "markdown", 342 | "metadata": {}, 343 | "source": [ 344 | "There are three real roots: {x1 -> 0.062636, x2 -> 0.758243}, {x1 -> 0.88443, x2 -> 0.21038}, {x1 -> 0.285369, x2 -> 0.609754}" 345 | ] 346 | }, 347 | { 348 | "cell_type": "code", 349 | "collapsed": false, 350 | "input": [ 351 | "a = [Interval(0.01, 5), Interval(0.1, 5)]" 352 | ], 353 | "language": "python", 354 | "metadata": {}, 355 | "outputs": [ 356 | { 357 | "metadata": {}, 358 | "output_type": "pyout", 359 | "prompt_number": 62, 360 | "text": [ 361 | "2-element Array{Interval,1}:\n", 362 | " Interval(9.9999999999999999998e-03 with 64 bits of precision,5e+00 with 64 bits of precision) \n", 363 | " Interval(9.99999999999999999946e-02 with 64 bits of precision,5e+00 with 64 bits of precision)" 364 | ] 365 | } 366 | ], 367 | "prompt_number": 62 368 | }, 369 | { 370 | "cell_type": "code", 371 | "collapsed": false, 372 | "input": [ 373 | "newton2d(f, a, 64)" 374 | ], 375 | "language": "python", 376 | "metadata": {}, 377 | "outputs": [ 378 | { 379 | "output_type": "stream", 380 | "stream": "stdout", 381 | "text": [ 382 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.52558389946483697317e+00 with 64 bits of precision),Interval(9.99999999999999999946e-02 with 64 bits of precision,5e+00 with 64 bits of precision)]]" 383 | ] 384 | }, 385 | { 386 | "output_type": "stream", 387 | "stream": "stdout", 388 | "text": [ 389 | "\n", 390 | "roots_array = roots_array_new => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.52558389946483697317e+00 with 64 bits of precision),Interval(9.99999999999999999946e-02 with 64 bits of precision,5e+00 with 64 bits of precision)]]\n", 391 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,1.2686662376273644769e+00 with 64 bits of precision),Interval(9.99999999999999999946e-02 with 64 bits of precision,5e+00 with 64 bits of precision)]]\n", 392 | "roots_array = roots_array_new => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,1.2686662376273644769e+00 with 64 bits of precision),Interval(9.99999999999999999946e-02 with 64 bits of precision,5e+00 with 64 bits of precision)]]\n", 393 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,6.41495905915175664469e-01 with 64 bits of precision),Interval(3.72336062723216222939e-01 with 64 bits of precision,4.84710334659612871984e+00 with 64 bits of precision)]]\n", 394 | "roots_array = roots_array_new => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,6.41495905915175664469e-01 with 64 bits of precision),Interval(3.72336062723216222939e-01 with 64 bits of precision,4.84710334659612871984e+00 with 64 bits of precision)]]\n", 395 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,3.37257266081398510295e-01 with 64 bits of precision),Interval(5.75161822612400992316e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 396 | "roots_array = roots_array_new => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,3.37257266081398510295e-01 with 64 bits of precision),Interval(5.75161822612400992316e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 397 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.13368376853307920932e-01 with 64 bits of precision),Interval(6.57754415431128052253e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 398 | "roots_array = roots_array_new => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.13368376853307920932e-01 with 64 bits of precision),Interval(6.57754415431128052253e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 399 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.0409028710522643644e-01 with 64 bits of precision),Interval(6.63939808596515708319e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 400 | "roots_array = roots_array_new => [[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.0409028710522643644e-01 with 64 bits of precision),Interval(6.63939808596515708319e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 401 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => " 402 | ] 403 | }, 404 | { 405 | "output_type": "stream", 406 | "stream": "stdout", 407 | "text": [ 408 | "[[Interval(9.9999999999999999998e-03 with 64 bits of precision,2.0409028710522643644e-01 with 64 bits of precision),Interval(6.63939808596515708319e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]]\n", 409 | "Function calls: 6\n" 410 | ] 411 | }, 412 | { 413 | "metadata": {}, 414 | "output_type": "pyout", 415 | "prompt_number": 63, 416 | "text": [ 417 | "1-element Array{Array{Interval,1},1}:\n", 418 | " [Interval(9.9999999999999999998e-03 with 64 bits of precision,2.0409028710522643644e-01 with 64 bits of precision),Interval(6.63939808596515708319e-01 with 64 bits of precision,4.73470830452635709981e+00 with 64 bits of precision)]" 419 | ] 420 | } 421 | ], 422 | "prompt_number": 63 423 | }, 424 | { 425 | "cell_type": "code", 426 | "collapsed": false, 427 | "input": [ 428 | "# Works. Another interval:" 429 | ], 430 | "language": "python", 431 | "metadata": {}, 432 | "outputs": [], 433 | "prompt_number": 15 434 | }, 435 | { 436 | "cell_type": "code", 437 | "collapsed": false, 438 | "input": [ 439 | "a = [Interval(0.7, 0.9), Interval(0.1, 0.3)]" 440 | ], 441 | "language": "python", 442 | "metadata": {}, 443 | "outputs": [ 444 | { 445 | "metadata": {}, 446 | "output_type": "pyout", 447 | "prompt_number": 20, 448 | "text": [ 449 | "2-element Array{Interval,1}:\n", 450 | " Interval(6.99999999999999999989e-01 with 64 bits of precision,9.00000000000000000033e-01 with 64 bits of precision)\n", 451 | " Interval(9.99999999999999999946e-02 with 64 bits of precision,3.00000000000000000011e-01 with 64 bits of precision)" 452 | ] 453 | } 454 | ], 455 | "prompt_number": 20 456 | }, 457 | { 458 | "cell_type": "code", 459 | "collapsed": false, 460 | "input": [ 461 | "newton2d(f, a, 64)" 462 | ], 463 | "language": "python", 464 | "metadata": {}, 465 | "outputs": [ 466 | { 467 | "output_type": "stream", 468 | "stream": "stdout", 469 | "text": [ 470 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(7.99533569108615181306e-01 with 64 bits of precision,9.00000000000000000087e-01 with 64 bits of precision),Interval(1.32402716810851698009e-01 with 64 bits of precision,2.66977620594256545897e-01 with 64 bits of precision)]]" 471 | ] 472 | }, 473 | { 474 | "output_type": "stream", 475 | "stream": "stdout", 476 | "text": [ 477 | "\n", 478 | "roots_array = roots_array_new => [[Interval(7.99533569108615181306e-01 with 64 bits of precision,9.00000000000000000087e-01 with 64 bits of precision),Interval(1.32402716810851698009e-01 with 64 bits of precision,2.66977620594256545897e-01 with 64 bits of precision)]]\n", 479 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.49310448278202196964e-01 with 64 bits of precision,9.00000000000000000141e-01 with 64 bits of precision),Interval(1.64941883459212836004e-01 with 64 bits of precision,2.33793034481198535481e-01 with 64 bits of precision)]]\n", 480 | "roots_array = roots_array_new => [[Interval(8.49310448278202196964e-01 with 64 bits of precision,9.00000000000000000141e-01 with 64 bits of precision),Interval(1.64941883459212836004e-01 with 64 bits of precision,2.33793034481198535481e-01 with 64 bits of precision)]]\n", 481 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.74368868193437198753e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.80847351067254425328e-01 with 64 bits of precision,2.17087421204375200919e-01 with 64 bits of precision)]]\n", 482 | "roots_array = roots_array_new => [[Interval(8.74368868193437198753e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.80847351067254425328e-01 with 64 bits of precision,2.17087421204375200919e-01 with 64 bits of precision)]]\n", 483 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.86713661126687165365e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.88028751271188605487e-01 with 64 bits of precision,2.08857559248875223218e-01 with 64 bits of precision)]]\n", 484 | "roots_array = roots_array_new => [[Interval(8.86713661126687165365e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.88028751271188605487e-01 with 64 bits of precision,2.08857559248875223218e-01 with 64 bits of precision)]]\n", 485 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.92042290862508345105e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.05305139424994436743e-01 with 64 bits of precision)]]\n", 486 | "roots_array = roots_array_new => [[Interval(8.92042290862508345105e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.05305139424994436743e-01 with 64 bits of precision)]]\n", 487 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.93576641696621274086e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.04282238868919150697e-01 with 64 bits of precision)]]\n", 488 | "roots_array = roots_array_new => [[Interval(8.93576641696621274086e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.04282238868919150697e-01 with 64 bits of precision)]]\n", 489 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.93683850490391404895e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.04210766339739063622e-01 with 64 bits of precision)]]\n", 490 | "roots_array = roots_array_new => [[Interval(8.93683850490391404895e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.04210766339739063622e-01 with 64 bits of precision)]]\n", 491 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(8.93683850490391404895e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.04210766339739063622e-01 with 64 bits of precision)]]\n", 492 | "Function calls: 7\n" 493 | ] 494 | }, 495 | { 496 | "metadata": {}, 497 | "output_type": "pyout", 498 | "prompt_number": 21, 499 | "text": [ 500 | "1-element Array{Array{Interval,1},1}:\n", 501 | " [Interval(8.93683850490391404895e-01 with 64 bits of precision,9.00000000000000000195e-01 with 64 bits of precision),Interval(1.89953354713197845736e-01 with 64 bits of precision,2.04210766339739063622e-01 with 64 bits of precision)]" 502 | ] 503 | } 504 | ], 505 | "prompt_number": 21 506 | }, 507 | { 508 | "cell_type": "code", 509 | "collapsed": false, 510 | "input": [ 511 | "# For some reason, the intervals are still relatively large" 512 | ], 513 | "language": "python", 514 | "metadata": {}, 515 | "outputs": [], 516 | "prompt_number": 18 517 | }, 518 | { 519 | "cell_type": "code", 520 | "collapsed": false, 521 | "input": [ 522 | "a = [Interval(0.11, 0.3), Interval(0.5, 0.69)]" 523 | ], 524 | "language": "python", 525 | "metadata": {}, 526 | "outputs": [ 527 | { 528 | "metadata": {}, 529 | "output_type": "pyout", 530 | "prompt_number": 13, 531 | "text": [ 532 | "2-element Array{Interval,1}:\n", 533 | " Interval(1.09999999999999999999e-01 with 64 bits of precision,3.00000000000000000011e-01 with 64 bits of precision)\n", 534 | " Interval(5e-01 with 64 bits of precision,6.90000000000000000052e-01 with 64 bits of precision) " 535 | ] 536 | } 537 | ], 538 | "prompt_number": 13 539 | }, 540 | { 541 | "cell_type": "code", 542 | "collapsed": false, 543 | "input": [ 544 | "newton2d(f, a, 64)" 545 | ], 546 | "language": "python", 547 | "metadata": {}, 548 | "outputs": [ 549 | { 550 | "output_type": "stream", 551 | "stream": "stdout", 552 | "text": [ 553 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(1.98959862448933060819e-01 with 64 bits of precision,3.00000000000000000038e-01 with 64 bits of precision),Interval(5.22511857040455090476e-01 with 64 bits of precision,6.67360091700711293024e-01 with 64 bits of precision)]]" 554 | ] 555 | }, 556 | { 557 | "output_type": "stream", 558 | "stream": "stdout", 559 | "text": [ 560 | "\n", 561 | "roots_array = roots_array_new => [[Interval(1.98959862448933060819e-01 with 64 bits of precision,3.00000000000000000038e-01 with 64 bits of precision),Interval(5.22511857040455090476e-01 with 64 bits of precision,6.67360091700711293024e-01 with 64 bits of precision)]]\n", 562 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.44405284093183400559e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.47605281726672503634e-01 with 64 bits of precision,6.37063143937877733179e-01 with 64 bits of precision)]]\n", 563 | "roots_array = roots_array_new => [[Interval(2.44405284093183400559e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.47605281726672503634e-01 with 64 bits of precision,6.37063143937877733179e-01 with 64 bits of precision)]]\n", 564 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.67881780072239415627e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.21412146618507056422e-01 with 64 bits of precision)]]\n", 565 | "roots_array = roots_array_new => [[Interval(2.67881780072239415627e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.21412146618507056422e-01 with 64 bits of precision)]]\n", 566 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.8018905070704416428e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.13207299528637223896e-01 with 64 bits of precision)]]\n", 567 | "roots_array = roots_array_new => [[Interval(2.8018905070704416428e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.13207299528637223896e-01 with 64 bits of precision)]]\n", 568 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.86802017717913006298e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.08798654854724662804e-01 with 64 bits of precision)]]\n", 569 | "roots_array = roots_array_new => [[Interval(2.86802017717913006298e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.08798654854724662804e-01 with 64 bits of precision)]]\n", 570 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => " 571 | ] 572 | }, 573 | { 574 | "output_type": "stream", 575 | "stream": "stdout", 576 | "text": [ 577 | "[[Interval(2.90699552997113188816e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.06200298001924541035e-01 with 64 bits of precision)]]\n", 578 | "roots_array = roots_array_new => [[Interval(2.90699552997113188816e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.06200298001924541035e-01 with 64 bits of precision)]]\n", 579 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.9119045375552936604e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05873030829647089444e-01 with 64 bits of precision)]]\n", 580 | "roots_array = roots_array_new => [[Interval(2.9119045375552936604e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05873030829647089444e-01 with 64 bits of precision)]]\n", 581 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.91199878305938713083e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866747796040858154e-01 with 64 bits of precision)]]\n", 582 | "roots_array = roots_array_new => [[Interval(2.91199878305938713083e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866747796040858154e-01 with 64 bits of precision)]]\n", 583 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.91199911011646166213e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725992235889437e-01 with 64 bits of precision)]]\n", 584 | "roots_array = roots_array_new => [[Interval(2.91199911011646166213e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725992235889437e-01 with 64 bits of precision)]]\n", 585 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.91199911115097032901e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923268644924e-01 with 64 bits of precision)]]\n", 586 | "roots_array = roots_array_new => [[Interval(2.91199911115097032901e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923268644924e-01 with 64 bits of precision)]]\n", 587 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => " 588 | ] 589 | }, 590 | { 591 | "output_type": "stream", 592 | "stream": "stdout", 593 | "text": [ 594 | "[[Interval(2.91199911115424146346e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923050569258e-01 with 64 bits of precision)]]\n", 595 | "roots_array = roots_array_new => [[Interval(2.91199911115424146346e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923050569258e-01 with 64 bits of precision)]]\n", 596 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.91199911115425180485e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923049879868e-01 with 64 bits of precision)]]\n", 597 | "roots_array = roots_array_new => [[Interval(2.91199911115425180485e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923049879868e-01 with 64 bits of precision)]]\n", 598 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.91199911115425183792e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923049877591e-01 with 64 bits of precision)]]\n", 599 | "roots_array = roots_array_new => [[Interval(2.91199911115425183792e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923049877591e-01 with 64 bits of precision)]]\n", 600 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(2.91199911115425183792e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923049877591e-01 with 64 bits of precision)]]\n", 601 | "Function calls: 13\n" 602 | ] 603 | }, 604 | { 605 | "metadata": {}, 606 | "output_type": "pyout", 607 | "prompt_number": 14, 608 | "text": [ 609 | "1-element Array{Array{Interval,1},1}:\n", 610 | " [Interval(2.91199911115425183792e-01 with 64 bits of precision,3.00000000000000000065e-01 with 64 bits of precision),Interval(5.55929366475467126932e-01 with 64 bits of precision,6.05866725923049877591e-01 with 64 bits of precision)]" 611 | ] 612 | } 613 | ], 614 | "prompt_number": 14 615 | }, 616 | { 617 | "cell_type": "markdown", 618 | "metadata": {}, 619 | "source": [] 620 | }, 621 | { 622 | "cell_type": "code", 623 | "collapsed": false, 624 | "input": [ 625 | "a = [Interval(0.01, 1), Interval(0.01, 1)]" 626 | ], 627 | "language": "python", 628 | "metadata": {}, 629 | "outputs": [ 630 | { 631 | "metadata": {}, 632 | "output_type": "pyout", 633 | "prompt_number": 70, 634 | "text": [ 635 | "2-element Array{Interval,1}:\n", 636 | " Interval(9.9999999999999999998e-03 with 64 bits of precision,1e+00 with 64 bits of precision)\n", 637 | " Interval(9.9999999999999999998e-03 with 64 bits of precision,1e+00 with 64 bits of precision)" 638 | ] 639 | } 640 | ], 641 | "prompt_number": 70 642 | }, 643 | { 644 | "cell_type": "code", 645 | "collapsed": false, 646 | "input": [ 647 | "newton2d(f, a, 64)" 648 | ], 649 | "language": "python", 650 | "metadata": {}, 651 | "outputs": [ 652 | { 653 | "output_type": "stream", 654 | "stream": "stdout", 655 | "text": [ 656 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(3.77986706010765157344e-01 with 64 bits of precision,5.05395803514839542323e-01 with 64 bits of precision),Interval(4.63069464323440305111e-01 with 64 bits of precision,5.48008862659489895115e-01 with 64 bits of precision)]]" 657 | ] 658 | }, 659 | { 660 | "output_type": "stream", 661 | "stream": "stdout", 662 | "text": [ 663 | "\n", 664 | "roots_array = roots_array_new => [[Interval(3.77986706010765157344e-01 with 64 bits of precision,5.05395803514839542323e-01 with 64 bits of precision),Interval(4.63069464323440305111e-01 with 64 bits of precision,5.48008862659489895115e-01 with 64 bits of precision)]]\n", 665 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(4.38274314608033558981e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 666 | "roots_array = roots_array_new => [[Interval(4.38274314608033558981e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 667 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 668 | "roots_array = roots_array_new => [[Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 669 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 670 | "Function calls: 3\n" 671 | ] 672 | }, 673 | { 674 | "metadata": {}, 675 | "output_type": "pyout", 676 | "prompt_number": 71, 677 | "text": [ 678 | "1-element Array{Array{Interval,1},1}:\n", 679 | " [Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]" 680 | ] 681 | } 682 | ], 683 | "prompt_number": 71 684 | }, 685 | { 686 | "cell_type": "code", 687 | "collapsed": false, 688 | "input": [ 689 | "# This one looks like a false positive. Why?" 690 | ], 691 | "language": "python", 692 | "metadata": {}, 693 | "outputs": [], 694 | "prompt_number": 17 695 | }, 696 | { 697 | "cell_type": "code", 698 | "collapsed": false, 699 | "input": [ 700 | "sol = newton2d(f, a, 64)" 701 | ], 702 | "language": "python", 703 | "metadata": {}, 704 | "outputs": [ 705 | { 706 | "output_type": "stream", 707 | "stream": "stdout", 708 | "text": [ 709 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(3.77986706010765157344e-01 with 64 bits of precision,5.05395803514839542323e-01 with 64 bits of precision),Interval(4.63069464323440305111e-01 with 64 bits of precision,5.48008862659489895115e-01 with 64 bits of precision)]]" 710 | ] 711 | }, 712 | { 713 | "output_type": "stream", 714 | "stream": "stdout", 715 | "text": [ 716 | "\n", 717 | "roots_array = roots_array_new => [[Interval(3.77986706010765157344e-01 with 64 bits of precision,5.05395803514839542323e-01 with 64 bits of precision),Interval(4.63069464323440305111e-01 with 64 bits of precision,5.48008862659489895115e-01 with 64 bits of precision)]]\n", 718 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(4.38274314608033558981e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 719 | "roots_array = roots_array_new => [[Interval(4.38274314608033558981e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 720 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 721 | "roots_array = roots_array_new => [[Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 722 | "roots_array_new = push!(roots_array_new,isectext(roots_array[i],N(roots_array[i]))) => [[Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]]\n", 723 | "Function calls: 3\n" 724 | ] 725 | }, 726 | { 727 | "metadata": {}, 728 | "output_type": "pyout", 729 | "prompt_number": 72, 730 | "text": [ 731 | "1-element Array{Array{Interval,1},1}:\n", 732 | " [Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]" 733 | ] 734 | } 735 | ], 736 | "prompt_number": 72 737 | }, 738 | { 739 | "cell_type": "code", 740 | "collapsed": false, 741 | "input": [ 742 | "center(x) = make_intervals(mid(x))\n", 743 | "N(f, x) = center(x) - inv(jacobian(f, x))*f(center(x))" 744 | ], 745 | "language": "python", 746 | "metadata": {}, 747 | "outputs": [ 748 | { 749 | "metadata": {}, 750 | "output_type": "pyout", 751 | "prompt_number": 73, 752 | "text": [ 753 | "N (generic function with 1 method)" 754 | ] 755 | } 756 | ], 757 | "prompt_number": 73 758 | }, 759 | { 760 | "cell_type": "code", 761 | "collapsed": false, 762 | "input": [ 763 | "NNN = N(f, sol[1])" 764 | ], 765 | "language": "python", 766 | "metadata": {}, 767 | "outputs": [ 768 | { 769 | "metadata": {}, 770 | "output_type": "pyout", 771 | "prompt_number": 74, 772 | "text": [ 773 | "2-element Array{Interval,1}:\n", 774 | " Interval(3.87637464474657538128e-01 with 64 bits of precision,5.20025515047235507198e-01 with 64 bits of precision)\n", 775 | " Interval(4.5331632330184299542e-01 with 64 bits of precision,5.41575023683561641169e-01 with 64 bits of precision) " 776 | ] 777 | } 778 | ], 779 | "prompt_number": 74 780 | }, 781 | { 782 | "cell_type": "code", 783 | "collapsed": false, 784 | "input": [ 785 | "inside(sol[1][1], NNN[1])" 786 | ], 787 | "language": "python", 788 | "metadata": {}, 789 | "outputs": [ 790 | { 791 | "metadata": {}, 792 | "output_type": "pyout", 793 | "prompt_number": 34, 794 | "text": [ 795 | "true" 796 | ] 797 | } 798 | ], 799 | "prompt_number": 34 800 | }, 801 | { 802 | "cell_type": "code", 803 | "collapsed": false, 804 | "input": [ 805 | "inside(sol[1][2], NNN[2])" 806 | ], 807 | "language": "python", 808 | "metadata": {}, 809 | "outputs": [ 810 | { 811 | "metadata": {}, 812 | "output_type": "pyout", 813 | "prompt_number": 35, 814 | "text": [ 815 | "true" 816 | ] 817 | } 818 | ], 819 | "prompt_number": 35 820 | }, 821 | { 822 | "cell_type": "code", 823 | "collapsed": false, 824 | "input": [ 825 | "J = jacobian(f, sol[1])" 826 | ], 827 | "language": "python", 828 | "metadata": {}, 829 | "outputs": [ 830 | { 831 | "metadata": {}, 832 | "output_type": "pyout", 833 | "prompt_number": 38, 834 | "text": [ 835 | "2x2 Array{Interval,2}:\n", 836 | " Interval(-1.76984841281877910762e+00 with 64 bits of precision,3.37773217687788244361e+00 with 64 bits of precision) \u2026 Interval(5e-01 with 64 bits of precision,5e-01 with 64 bits of precision) \n", 837 | " Interval(-2.00000000000000000016e-01 with 64 bits of precision,-1.99999999999999999976e-01 with 64 bits of precision) Interval(-3.00000000000000000065e-01 with 64 bits of precision,-2.99999999999999999984e-01 with 64 bits of precision)" 838 | ] 839 | } 840 | ], 841 | "prompt_number": 38 842 | }, 843 | { 844 | "cell_type": "code", 845 | "collapsed": false, 846 | "input": [ 847 | "inv(J)" 848 | ], 849 | "language": "python", 850 | "metadata": {}, 851 | "outputs": [ 852 | { 853 | "metadata": {}, 854 | "output_type": "pyout", 855 | "prompt_number": 39, 856 | "text": [ 857 | "2x2 Array{Interval,2}:\n", 858 | " Interval(-4.75470083281939574619e-01 with 64 bits of precision,3.2847207327004322624e-01 with 64 bits of precision) \u2026 Interval(-1.30300529972210691028e+01 with 64 bits of precision,5.4745345545007204264e-01 with 64 bits of precision) \n", 859 | " Interval(-2.18981382180028817362e-01 with 64 bits of precision,3.16980055521293049565e-01 with 64 bits of precision) Interval(-3.69830230363338136068e+00 with 64 bits of precision,5.35336866481404606349e+00 with 64 bits of precision)" 860 | ] 861 | } 862 | ], 863 | "prompt_number": 39 864 | }, 865 | { 866 | "cell_type": "code", 867 | "collapsed": false, 868 | "input": [ 869 | "left(x::Interval) = Interval(x.lo, mid(x))\n", 870 | "right(x::Interval) = Interval(mid(x), x.hi)\n", 871 | "\n", 872 | "\n", 873 | "function bisect(xx::Vector{Interval})\n", 874 | " \n", 875 | " if length(xx) != 2\n", 876 | " error(\"Only works for 2 at the moment\")\n", 877 | " end\n", 878 | " \n", 879 | " x, y = xx\n", 880 | " \n", 881 | " @show x\n", 882 | " \n", 883 | " intervals = Vector{Interval}[]\n", 884 | " \n", 885 | " push!(intervals, [left(x), left(y)])\n", 886 | " push!(intervals, [left(x), right(y)])\n", 887 | " push!(intervals, [right(x), left(y)])\n", 888 | " push!(intervals, [right(x), right(y)])\n", 889 | " \n", 890 | " intervals\n", 891 | "end" 892 | ], 893 | "language": "python", 894 | "metadata": {}, 895 | "outputs": [ 896 | { 897 | "metadata": {}, 898 | "output_type": "pyout", 899 | "prompt_number": 75, 900 | "text": [ 901 | "bisect (generic function with 1 method)" 902 | ] 903 | } 904 | ], 905 | "prompt_number": 75 906 | }, 907 | { 908 | "cell_type": "code", 909 | "collapsed": false, 910 | "input": [ 911 | "x = sol[1]" 912 | ], 913 | "language": "python", 914 | "metadata": {}, 915 | "outputs": [ 916 | { 917 | "metadata": {}, 918 | "output_type": "pyout", 919 | "prompt_number": 76, 920 | "text": [ 921 | "2-element Array{Interval,1}:\n", 922 | " Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision)\n", 923 | " Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)" 924 | ] 925 | } 926 | ], 927 | "prompt_number": 76 928 | }, 929 | { 930 | "cell_type": "code", 931 | "collapsed": false, 932 | "input": [ 933 | "y = bisect(x)" 934 | ], 935 | "language": "python", 936 | "metadata": {}, 937 | "outputs": [ 938 | { 939 | "output_type": "stream", 940 | "stream": "stdout", 941 | "text": [ 942 | "x => Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision)" 943 | ] 944 | }, 945 | { 946 | "output_type": "stream", 947 | "stream": "stdout", 948 | "text": [ 949 | "\n" 950 | ] 951 | }, 952 | { 953 | "metadata": {}, 954 | "output_type": "pyout", 955 | "prompt_number": 78, 956 | "text": [ 957 | "4-element Array{Array{Interval,1},1}:\n", 958 | " [Interval(4.38274314608033558954e-01 with 64 bits of precision,4.41728143876593134902e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.05514570748937910167e-01 with 64 bits of precision)]\n", 959 | " [Interval(4.38274314608033558954e-01 with 64 bits of precision,4.41728143876593134902e-01 with 64 bits of precision),Interval(5.05514570748937910113e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]\n", 960 | " [Interval(4.41728143876593134875e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.03212017903231526256e-01 with 64 bits of precision,5.05514570748937910167e-01 with 64 bits of precision)]\n", 961 | " [Interval(4.41728143876593134875e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision),Interval(5.05514570748937910113e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)]" 962 | ] 963 | } 964 | ], 965 | "prompt_number": 78 966 | }, 967 | { 968 | "cell_type": "code", 969 | "collapsed": false, 970 | "input": [ 971 | "import Base.show\n", 972 | "show(io::IO, x::Interval) = print(io, \"[$(round(float(x.lo), 5)), $(x.hi)]\")" 973 | ], 974 | "language": "python", 975 | "metadata": {}, 976 | "outputs": [ 977 | { 978 | "metadata": {}, 979 | "output_type": "pyout", 980 | "prompt_number": 53, 981 | "text": [ 982 | "show (generic function with 90 methods)" 983 | ] 984 | } 985 | ], 986 | "prompt_number": 53 987 | }, 988 | { 989 | "cell_type": "code", 990 | "collapsed": false, 991 | "input": [ 992 | "y = bisect(x)\n", 993 | "after_newton = [N(f,x) for x in y]" 994 | ], 995 | "language": "python", 996 | "metadata": {}, 997 | "outputs": [ 998 | { 999 | "output_type": "stream", 1000 | "stream": "stdout", 1001 | "text": [ 1002 | "x => Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision)" 1003 | ] 1004 | }, 1005 | { 1006 | "output_type": "stream", 1007 | "stream": "stdout", 1008 | "text": [ 1009 | "\n" 1010 | ] 1011 | }, 1012 | { 1013 | "metadata": {}, 1014 | "output_type": "pyout", 1015 | "prompt_number": 79, 1016 | "text": [ 1017 | "4-element Array{Any,1}:\n", 1018 | " [Interval(3.47097933805788840721e-01 with 64 bits of precision,6.56963251841997641835e-01 with 64 bits of precision),Interval(3.62024498772001572446e-01 with 64 bits of precision,5.68601377462807439783e-01 with 64 bits of precision)]\n", 1019 | " [Interval(3.47097933805788839555e-01 with 64 bits of precision,6.45856424941334515417e-01 with 64 bits of precision),Interval(3.69429050039110323301e-01 with 64 bits of precision,5.68601377462807440108e-01 with 64 bits of precision)]\n", 1020 | " [Interval(3.48964360037304871157e-01 with 64 bits of precision,6.41213649073018553885e-01 with 64 bits of precision),Interval(3.72524233951320964331e-01 with 64 bits of precision,5.67357093308463419131e-01 with 64 bits of precision)]\n", 1021 | " [Interval(3.36248746659157818085e-01 with 64 bits of precision,6.43246950176062365161e-01 with 64 bits of precision),Interval(3.71168699882625089759e-01 with 64 bits of precision,5.75834168893894787701e-01 with 64 bits of precision)]" 1022 | ] 1023 | } 1024 | ], 1025 | "prompt_number": 79 1026 | }, 1027 | { 1028 | "cell_type": "code", 1029 | "collapsed": false, 1030 | "input": [ 1031 | "display(y[1])\n", 1032 | "display(after_newton[1])" 1033 | ], 1034 | "language": "python", 1035 | "metadata": {}, 1036 | "outputs": [ 1037 | { 1038 | "metadata": {}, 1039 | "output_type": "display_data", 1040 | "text": [ 1041 | "2-element Array{Interval,1}:\n", 1042 | " Interval(4.38274314608033558954e-01 with 64 bits of precision,4.41728143876593134902e-01 with 64 bits of precision)\n", 1043 | " Interval(5.03212017903231526256e-01 with 64 bits of precision,5.05514570748937910167e-01 with 64 bits of precision)" 1044 | ] 1045 | }, 1046 | { 1047 | "metadata": {}, 1048 | "output_type": "display_data", 1049 | "text": [ 1050 | "2-element Array{Interval,1}:\n", 1051 | " Interval(3.47097933805788840721e-01 with 64 bits of precision,6.56963251841997641835e-01 with 64 bits of precision)\n", 1052 | " Interval(3.62024498772001572446e-01 with 64 bits of precision,5.68601377462807439783e-01 with 64 bits of precision)" 1053 | ] 1054 | } 1055 | ], 1056 | "prompt_number": 82 1057 | }, 1058 | { 1059 | "cell_type": "code", 1060 | "collapsed": false, 1061 | "input": [ 1062 | "x" 1063 | ], 1064 | "language": "python", 1065 | "metadata": {}, 1066 | "outputs": [ 1067 | { 1068 | "metadata": {}, 1069 | "output_type": "pyout", 1070 | "prompt_number": 55, 1071 | "text": [ 1072 | "2-element Array{Interval,1}:\n", 1073 | " Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision)\n", 1074 | " Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)" 1075 | ] 1076 | } 1077 | ], 1078 | "prompt_number": 55 1079 | }, 1080 | { 1081 | "cell_type": "code", 1082 | "collapsed": false, 1083 | "input": [ 1084 | "f(x)" 1085 | ], 1086 | "language": "python", 1087 | "metadata": {}, 1088 | "outputs": [ 1089 | { 1090 | "metadata": {}, 1091 | "output_type": "pyout", 1092 | "prompt_number": 83, 1093 | "text": [ 1094 | "2-element Array{Interval,1}:\n", 1095 | " Interval(-4.19940929233143423632e-01 with 64 bits of precision,7.49196193356633135163e-01 with 64 bits of precision)\n", 1096 | " Interval(-1.38153170742383035755e-03 with 64 bits of precision,1.38153170742383035755e-03 with 64 bits of precision)" 1097 | ] 1098 | } 1099 | ], 1100 | "prompt_number": 83 1101 | }, 1102 | { 1103 | "cell_type": "code", 1104 | "collapsed": false, 1105 | "input": [ 1106 | "using NewtonMethod" 1107 | ], 1108 | "language": "python", 1109 | "metadata": {}, 1110 | "outputs": [ 1111 | { 1112 | "output_type": "stream", 1113 | "stream": "stdout", 1114 | "text": [ 1115 | "Syntax: newton(function, Interval(lo, hi), precision [default is 64])\n" 1116 | ] 1117 | } 1118 | ], 1119 | "prompt_number": 84 1120 | }, 1121 | { 1122 | "cell_type": "code", 1123 | "collapsed": false, 1124 | "input": [ 1125 | "using KrawczykMethod" 1126 | ], 1127 | "language": "python", 1128 | "metadata": {}, 1129 | "outputs": [ 1130 | { 1131 | "output_type": "stream", 1132 | "stream": "stdout", 1133 | "text": [ 1134 | "Syntax: krawczyk(function, Interval(lo, hi), precision [default is 64])\n" 1135 | ] 1136 | } 1137 | ], 1138 | "prompt_number": 85 1139 | }, 1140 | { 1141 | "cell_type": "code", 1142 | "collapsed": false, 1143 | "input": [ 1144 | " x" 1145 | ], 1146 | "language": "python", 1147 | "metadata": {}, 1148 | "outputs": [ 1149 | { 1150 | "metadata": {}, 1151 | "output_type": "pyout", 1152 | "prompt_number": 86, 1153 | "text": [ 1154 | "2-element Array{Interval,1}:\n", 1155 | " Interval(4.38274314608033558954e-01 with 64 bits of precision,4.45181973145152710768e-01 with 64 bits of precision)\n", 1156 | " Interval(5.03212017903231526256e-01 with 64 bits of precision,5.07817123594644293915e-01 with 64 bits of precision)" 1157 | ] 1158 | } 1159 | ], 1160 | "prompt_number": 86 1161 | }, 1162 | { 1163 | "cell_type": "code", 1164 | "collapsed": false, 1165 | "input": [], 1166 | "language": "python", 1167 | "metadata": {}, 1168 | "outputs": [] 1169 | } 1170 | ], 1171 | "metadata": {} 1172 | } 1173 | ] 1174 | } --------------------------------------------------------------------------------