├── Skript ├── cs.bib ├── analysis.pdf ├── fourier.tex ├── ableitung.tex ├── anwendungen.tex ├── einleitung.tex ├── integration.tex ├── irrational.tex ├── analysis-2015.pdf ├── einheitskreis.png ├── inconsolatag.tfm ├── inconsolatag.ttf ├── reelle-zahlen.tex ├── Figures │ ├── strange.png │ ├── circle-sine.gif │ ├── mean-value-theorem.png │ ├── circle-eps-converted-to.pdf │ ├── concav-eps-converted-to.pdf │ ├── convex-eps-converted-to.pdf │ ├── umkehr-eps-converted-to.pdf │ ├── strange-eps-converted-to.pdf │ ├── saegezahn-eps-converted-to.pdf │ ├── trapezoidal-eps-converted-to.pdf │ ├── xhoch2int-eps-converted-to.pdf │ ├── Winkelfunktionen_einheitskreis.PNG │ ├── regula-falsi-eps-converted-to.pdf │ ├── mean-value-theorem-eps-converted-to.pdf │ ├── xhoch2RiemannRight-eps-converted-to.pdf │ ├── umkehr.fig │ ├── convex.py │ ├── concav.py │ ├── saegezahn.eps │ ├── trapezoidal.eps │ └── xhoch2int.eps ├── rundungsfehler.tex ├── Aufgaben │ ├── aufgaben.pdf │ ├── loesungen.pdf │ ├── aufgaben.tex │ └── test.eps ├── folgen-und-reihen.tex ├── Maple │ ├── fourierAnalyse.maple │ ├── zeta3.maple │ ├── keplerFass.maple │ ├── simpson.maple │ ├── Fourier.mws │ └── Maple-6.mws ├── dhbw-logo-eps-converted-to.pdf ├── t1inconsolatag.fd ├── Makefile ├── T1-WGL4.enc └── analysis.tex ├── Programs ├── harmonic ├── solution.c ├── solution-fast.c ├── riemannInt.maple ├── sqrt.mpi ├── solution-smart.c ├── harmonic.c ├── saege2.maple ├── saegezahn.maple ├── ramanujan.stl ├── interpolate.maple ├── SecantMethod.java ├── Jacobi.java ├── Seidel.java ├── RegulaFalsi.java ├── Bisection.java └── Solver.java ├── SymPy-SciPy ├── sympy-scipy.pdf ├── sine-and-cosine.py ├── induction.py ├── naked-sine.eps └── naked-sine-and-cosine.eps ├── SetlX ├── cosXisX-naive.stlx ├── cosXisX-newton.stlx ├── cosXisX.stlx ├── maschinen-konstante.stlx ├── xExpX2.stlx ├── xExpX1.stlx ├── geometric-mean.stlx ├── harmonic-mean.stlx ├── arithmetic-mean.stlx ├── solve.stlx ├── harmonic.stlx ├── sinXisX.stlx ├── sqrt.stlx ├── machin.stlx ├── plot-f.stlx ├── kepler-rule.stlx ├── secant.stlx ├── plot-aufgabe-76.stlx ├── plot-saegezahn.stlx ├── midpoint-rule.stlx ├── gauss-seidel.stlx ├── jacobi-method.stlx ├── trapezoidal-rule.stlx ├── simpson-rule.stlx ├── regulaFalsi.stlx ├── bisection.stlx ├── illinois.stlx └── newton-zero.stlx ├── README.md ├── .gitignore └── Java ├── Harmonic.java └── MaschinenKonstante.java /Skript/cs.bib: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/karlstroetmann/Analysis/HEAD/Skript/cs.bib -------------------------------------------------------------------------------- /Programs/harmonic: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/karlstroetmann/Analysis/HEAD/Programs/harmonic -------------------------------------------------------------------------------- /Skript/analysis.pdf: 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-------------------------------------------------------------------------------- 1 | x := 0; 2 | alpha := read("input alpha: "); 3 | for (i in [1 .. 14]) { 4 | x := 1 / (1 + alpha) * (cos(x) + alpha * x); 5 | print("$i$: $x$"); 6 | } 7 | 8 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | Analysis 2 | ======== 3 | 4 | The lecture notes and programs for my class on real analysis. This is very much work in progress! 5 | If you find an error in these notes, you are welcome to send me a pull request. 6 | -------------------------------------------------------------------------------- /Skript/t1inconsolatag.fd: -------------------------------------------------------------------------------- 1 | \ProvidesFile{t1inconsolatag.fd} 2 | \DeclareFontFamily{T1}{inconsolatag}{} 3 | \DeclareFontShape{T1}{inconsolatag}{m}{n}{ <-> inconsolatag}{} 4 | \pdfmapline{+inconsolatag\space 2 | #include 3 | #include 4 | 5 | int main() { 6 | float x; 7 | x = 0.7; 8 | for (int i = 0; i < 50; ++i) { 9 | x = cos(x); 10 | printf("x%-2d = %f\n", i+1, x); 11 | } 12 | } 13 | -------------------------------------------------------------------------------- /SetlX/maschinen-konstante.stlx: -------------------------------------------------------------------------------- 1 | maschinenKonstante := procedure() { 2 | eps := 1.0; 3 | old := eps; 4 | while (1.0 + eps > 1.0) { 5 | old := eps; 6 | eps /= 2; 7 | } 8 | return old; 9 | }; 10 | 11 | print("eps = $maschinenKonstante()$"); 12 | -------------------------------------------------------------------------------- /Skript/Maple/keplerFass.maple: -------------------------------------------------------------------------------- 1 | # Die Kepler'sche Fassregel 2 | restart; 3 | intKepler := 1/6 * (f(a) + 4 * f((a+b)/2) + f(b)) * (b - a); 4 | 5 | a := 0; 6 | b := 1; 7 | f := (t -> exp(-t^2)); 8 | evalf(intKepler); 9 | err := evalf( intKepler - int( exp(-t^2), t = 0 .. 1 ) ); 10 | -------------------------------------------------------------------------------- /SetlX/xExpX2.stlx: -------------------------------------------------------------------------------- 1 | // Solve the equation 2 | // x * exp(x) = y 3 | // for small values of y by using the fixpoint equation 4 | // x := y * exp(-x) 5 | // for y = 10**(-6). 6 | x := 1; 7 | y := 1 / 10 ** 6; 8 | for (i in [1 .. 5]) { 9 | x := y * exp(-x); 10 | print("$i$: $x$"); 11 | } 12 | 13 | -------------------------------------------------------------------------------- /Skript/Makefile: -------------------------------------------------------------------------------- 1 | open: analysis.pdf 2 | open analysis.pdf 3 | 4 | %.pdf: %.ps 5 | ps2pdf $< 6 | 7 | %.ps: %.dvi 8 | dvips -o $@ $< 9 | 10 | analysis.dvi: analysis.tex folgen-und-reihen.tex stetige-funktionen.tex fourier.tex \ 11 | anwendungen.tex integration.tex 12 | latex analysis.tex -------------------------------------------------------------------------------- /Programs/solution-fast.c: -------------------------------------------------------------------------------- 1 | #include 2 | #include 3 | #include 4 | 5 | int main() { 6 | double x = 0.0; 7 | double alpha = sin(0.739); 8 | for (int i = 1; i <= 50; ++i) { 9 | x = 1 / (1 + alpha) * (cos(x) + alpha * x); 10 | printf("x%-2d = %17.15f\n", i, x); 11 | } 12 | } 13 | -------------------------------------------------------------------------------- /SetlX/xExpX1.stlx: -------------------------------------------------------------------------------- 1 | // Solve the equation 2 | // x * exp(x) = y 3 | // for big values of y by using the fixpoint equation 4 | // x := ln(y) - ln(x). 5 | // for y = 10**6. 6 | x := 6; 7 | y := 10 ** 6; 8 | ln := log; 9 | for (i in [1 .. 16]) { 10 | x := ln(y) - ln(x); 11 | print("$i$: $x$"); 12 | } 13 | 14 | -------------------------------------------------------------------------------- /Programs/riemannInt.maple: -------------------------------------------------------------------------------- 1 | riemannInt := 2 | proc(f, x, a, b, n) 3 | local h, i; 4 | description "compute an approximation to int(f, x=a..b) using Riemann sums"; 5 | h := (b - a) / n; 6 | return sum( subs(x = a+i*h, f) * h, i = 1..n ); 7 | end proc; 8 | 9 | r := riemannInt(exp(t), t, 0, x, n); 10 | limit(r, n = infinity); 11 | 12 | -------------------------------------------------------------------------------- /Programs/sqrt.mpi: -------------------------------------------------------------------------------- 1 | mySqrt := proc(a) 2 | local b, i; 3 | 4 | if a >= 1 then 5 | b := a; 6 | else 7 | b := 1; 8 | end if; 9 | for i from 1 to 10 do 10 | b := evalf(0.5 * (b + a/b), 60); 11 | printf("i = %2g: b = %-63.60g\n", i, b); 12 | end do; 13 | return b; 14 | end proc; 15 | 16 | 17 | -------------------------------------------------------------------------------- /SetlX/geometric-mean.stlx: -------------------------------------------------------------------------------- 1 | mean := procedure(a, b) { 2 | while (abs(a - b) > 1/10**17) { 3 | [b, a] := [sqrt(a * b), b]; 4 | print(b); 5 | } 6 | return b; 7 | }; 8 | 9 | test := procedure(a, b) { 10 | print("a = $a$, $b$ = b"); 11 | mean(a, b); 12 | }; 13 | 14 | 15 | test(1, 2); 16 | test(2, 3); 17 | test(1, 4); 18 | -------------------------------------------------------------------------------- /Programs/solution-smart.c: -------------------------------------------------------------------------------- 1 | #include 2 | #include 3 | #include 4 | 5 | int main() { 6 | double x = 0.0; 7 | double alpha = sin(0.0); 8 | for (int i = 1; i <= 50; ++i) { 9 | alpha = sin(x); 10 | x = 1 / (1 + alpha) * (cos(x) + alpha * x); 11 | printf("x%-2d = %17.15f\n", i, x); 12 | } 13 | } 14 | -------------------------------------------------------------------------------- /SetlX/harmonic-mean.stlx: -------------------------------------------------------------------------------- 1 | mean := procedure(a, b) { 2 | while (abs(a - b) > 1/10**17) { 3 | [b, a] := [1/(0.5 * (1/a + 1/b)), b]; 4 | print(b); 5 | } 6 | return b; 7 | }; 8 | 9 | test := procedure(a, b) { 10 | print("a = $a$, $b$ = b"); 11 | mean(a, b); 12 | }; 13 | 14 | 15 | test(1, 2); 16 | test(2, 2); 17 | test(2, 4); 18 | 19 | -------------------------------------------------------------------------------- /Programs/harmonic.c: -------------------------------------------------------------------------------- 1 | #include 2 | 3 | int main() { 4 | double s = 0; 5 | double i = 1; 6 | while (1) { 7 | float oldS = s; 8 | s += 1/i; 9 | ++i; 10 | if (oldS == s) { 11 | break; 12 | } 13 | } 14 | printf("s = %f\n", s); 15 | printf("i = %f\n", i); 16 | } 17 | 18 | 19 | 20 | -------------------------------------------------------------------------------- /SetlX/arithmetic-mean.stlx: -------------------------------------------------------------------------------- 1 | mean := procedure(a, b) { 2 | while (abs(a - b) > 1/10**17) { 3 | [b, a] := [0.5 * (a + b), b]; 4 | print(b); 5 | } 6 | return b; 7 | }; 8 | 9 | test := procedure(a, b) { 10 | print("a = $a$, $b$ = b"); 11 | mean(a, b); 12 | }; 13 | 14 | test(0, 1); 15 | test(1, 0); 16 | test(0, 2); 17 | test(1, 2); 18 | test(-1, 1); 19 | -------------------------------------------------------------------------------- /SetlX/solve.stlx: -------------------------------------------------------------------------------- 1 | // compute the sequence (x_n)-N defined inductively as follows: 2 | // 1. x_1 := 1.0, 3 | // 2. x_{n+1} := cos(x_n) 4 | solve := procedure(k) { 5 | x := []; // x[n] stores x_{n} 6 | x[1] := 1.0; 7 | for (n in [1 .. k]) { 8 | x[n+1] := cos(x[n]); 9 | print("x_{$n$} = $x[n+1]$"); 10 | } 11 | }; 12 | 13 | solve(99); 14 | 15 | -------------------------------------------------------------------------------- /SymPy-SciPy/sine-and-cosine.py: -------------------------------------------------------------------------------- 1 | import matplotlib.pyplot as plt 2 | import numpy as np 3 | xs = np.linspace(-2*np.pi, 2*np.pi, 200) 4 | ys = [np.sin(x) for x in xs] 5 | plt.ion() 6 | plt.plot(xs, ys) 7 | zs = np.cos(xs) 8 | plt.plot(xs, zs) 9 | plt.title("The functions sin(x) and cos(x)") 10 | plt.xlabel("x") 11 | plt.ylabel("sin(x) vs. cos(x)") 12 | plt.savefig("sine-and-cosine.eps") 13 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | *.o 2 | *~ 3 | *.aux 4 | *.bbl 5 | *.blg 6 | *.dvi 7 | *.log 8 | *.out 9 | *.toc 10 | *.class 11 | *.el 12 | *.synctex.gz 13 | _region_.tex 14 | auto/ 15 | .DS_Store 16 | Script/_region_.prv 17 | Script/analysis.synctex.gz 18 | Script/analysis.prv 19 | Script/dhbw-logo-eps-converted-to.pdf 20 | Sympy-Scipy/_region_.prv 21 | Sympy-Scipy/sympy-scipy.synctex.gz 22 | Sympy-Scipy/sympy-scipy.prv 23 | .dropbox -------------------------------------------------------------------------------- /Programs/saege2.maple: -------------------------------------------------------------------------------- 1 | saege := 2 | proc(x) 3 | local y; 4 | description "computes chain tooth function"; 5 | if evalb(x <= 0) then 6 | y := x + 2 * Pi; 7 | return saegeAux(y); 8 | end if; 9 | return saegeAux(x); 10 | end proc; 11 | 12 | saegeAux := proc(x); 13 | y := x / (2*Pi); 14 | return evalf(4*Pi*min( frac(y)^2, frac(1 - y)^2)); 15 | end proc; 16 | 17 | plot( saege, -2*Pi .. 2*Pi); 18 | 19 | -------------------------------------------------------------------------------- /Programs/saegezahn.maple: -------------------------------------------------------------------------------- 1 | saege := 2 | proc(x) 3 | local y; 4 | description "computes chain tooth function"; 5 | if evalb(x <= 0) then 6 | y := x + 2 * Pi; 7 | return saegeAux(y); 8 | end if; 9 | return saegeAux(x); 10 | end proc; 11 | 12 | saegeAux := proc(x); 13 | y := x / (2*Pi); 14 | return evalf(2*Pi*min( frac(y), frac(1 - y))); 15 | end proc; 16 | 17 | plot( saege, -2*Pi .. 2*Pi); 18 | 19 | -------------------------------------------------------------------------------- /SetlX/harmonic.stlx: -------------------------------------------------------------------------------- 1 | // call this program as setlx --real32 harmonic.stlx 2 | harmonic := procedure() { 3 | oldSum := 0.0; 4 | sum := 1.0; 5 | n := 1; 6 | while (oldSum < sum) { 7 | oldSum := sum; 8 | n += 1; 9 | sum += 1/n; 10 | if (n % 100000 == 0) { 11 | print("n = $n$, sum = $sum$"); 12 | } 13 | } 14 | print("sum = $sum$, n = $n$"); 15 | }; 16 | 17 | harmonic(); 18 | -------------------------------------------------------------------------------- /SetlX/sinXisX.stlx: -------------------------------------------------------------------------------- 1 | // compute the sequence (x_n)_n defined inductively as follows: 2 | // 1. x_1 := 1.0, 3 | // 2. x_{n+1} := sin(x_n) 4 | solve := procedure(k) { 5 | n := 1; 6 | x := 1; 7 | powers := { 10 ** i : i in [0 .. k] }; 8 | while (n <= 10 ** k) { 9 | n += 1; 10 | x := sin(x); 11 | if (n in powers) { 12 | print("x_{$n$} = $x$"); 13 | } 14 | } 15 | }; 16 | 17 | solve(9); 18 | 19 | -------------------------------------------------------------------------------- /SetlX/sqrt.stlx: -------------------------------------------------------------------------------- 1 | pad := procedure(k, n) { 2 | s := "" + k; 3 | if (#s < n) { 4 | return " " * (n - #s) + s; 5 | } 6 | return s; 7 | }; 8 | 9 | mySqrt := procedure(a) { 10 | if (a <= 1) { 11 | b := 1; 12 | } else { 13 | b := a; 14 | } 15 | for (n in [1 .. 12]) { 16 | b := 1/2 * (b + a/b); 17 | print("$pad(n, 2)$: $nDecimalPlaces(b, 85)$"); 18 | } 19 | return b; 20 | }; 21 | 22 | mySqrt(2); 23 | 24 | -------------------------------------------------------------------------------- /Java/Harmonic.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | 3 | public class Harmonic { 4 | public static void main(String[] args) { 5 | harmonic(); 6 | } 7 | 8 | public static float harmonic() { 9 | float oldSum = 0.0F; 10 | float sum = 1.0F; 11 | int n = 1; 12 | while (oldSum < sum) { 13 | oldSum = sum; 14 | sum += 1.0F/n; 15 | n += 1; 16 | if (n % 100000 == 0) { 17 | System.out.println("n = " + n + ", sum = " + sum); 18 | } 19 | } 20 | System.out.println("n = " + n + ", sum = " + sum); 21 | return sum; 22 | }; 23 | } 24 | -------------------------------------------------------------------------------- /SetlX/machin.stlx: -------------------------------------------------------------------------------- 1 | // compute pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348 2 | machin := procedure(n) { 3 | return 4 * (4 * series(1/5, n) - series(1/239, n)); 4 | }; 5 | 6 | series := procedure(x, n) { 7 | result := 0; 8 | sign := 1; 9 | power := x; 10 | for (i in [0 .. n]) { 11 | result += sign * power / (2 * i + 1); 12 | power *= x * x; 13 | sign := -sign; 14 | } 15 | return result; 16 | }; 17 | 18 | print("pi = $nDecimalPlaces(machin(144), 101)$"); 19 | 20 | -------------------------------------------------------------------------------- /Skript/Maple/simpson.maple: -------------------------------------------------------------------------------- 1 | # Derivation of simpsons rule for integration 2 | # read "/home/stroetma/Kurse/Analysis/Script/Maple/simpson.maple"; 3 | restart; 4 | 5 | x[i-1] := x[i] - h; 6 | x[i+1] := x[i] + h; 7 | p := f[i-1] * (t - x[i]) * (t - x[i+1]) / ( (x[i-1] - x[i] ) * (x[i-1] - x[i+1]) ) + 8 | f[i] * (t - x[i-1]) * (t - x[i+1]) / ( (x[i] - x[i-1]) * (x[i] - x[i+1]) ) + 9 | f[i+1] * (t - x[i-1]) * (t - x[i]) / ( (x[i+1] - x[i-1]) * (x[i+1] - x[i] ) ); 10 | IntSimpson := int( p, t = x[i-1] .. x[i+1] ); 11 | IntSimpson := normal(IntSimpson); 12 | 13 | -------------------------------------------------------------------------------- /Programs/ramanujan.stl: -------------------------------------------------------------------------------- 1 | program main; 2 | 3 | n := 1000; 4 | M1 := { [q,p] : q in {2 .. n}, p in {2 .. n} | (p*p-1)*(q+1) = 2 * p*(q-1) }; 5 | print(M1); 6 | 7 | for p in {2 .. n} loop 8 | alpha := 4.0 * p * (p*p - 1) / (p**4 - 6*p**2 + 1); 9 | qApprox := (alpha + 1) / (alpha -1); 10 | for q in { floor(qApprox - 0.5) .. ceil(qApprox + 0.5) } loop 11 | if (q+1)*(p**4 - 6*p**2 + 1) = 4*p*(p**2 - 1)*(q-1) then 12 | print(p, " ", q); 13 | end if; 14 | end loop; 15 | end loop; 16 | 17 | end main; -------------------------------------------------------------------------------- /SetlX/plot-f.stlx: -------------------------------------------------------------------------------- 1 | f := procedure(x) { 2 | if (x == 0) { 3 | return 0; 4 | } 5 | return exp(-1/(x*x)); 6 | }; 7 | 8 | c := plot_createCanvas("A strange function."); 9 | 10 | width := 5; 11 | height := 1; 12 | 13 | plot_addListGraph(c, [[-width, 0], [width, 0]], "x-axis", [0, 0, 0]); 14 | plot_addListGraph(c, [[ 0, 0], [0, height]], "y-axis"); 15 | plot_addGraph(c, "f(x)", "y = f(x)", [0, 0, 255]); 16 | plot_modScale(c, [-width, width],[0, height]); 17 | 18 | // The next line is needed if this program is run standalone. 19 | get("Press enter to continue."); 20 | -------------------------------------------------------------------------------- /SetlX/kepler-rule.stlx: -------------------------------------------------------------------------------- 1 | // Compute the integral \int_a^b f(x) dx using Kepler's rule. 2 | // The intervall [a,b] is divided into n subintervals. 3 | kepler := procedure(f, a, b) { 4 | x := a; 5 | sum := evalTerm(f); 6 | x := (a + b) / 2; 7 | sum += 4 * evalTerm(f); 8 | x := b; 9 | sum += evalTerm(f); 10 | return sum * (b - a) / 6; 11 | }; 12 | 13 | interactive := procedure() { 14 | s := read("enter function"); 15 | f := parse(s); 16 | a := read("enter left boundary"); 17 | b := read("enter right boundary"); 18 | integral := kepler(f, a, b); 19 | print("The approximate value of the integral is $integral$."); 20 | }; 21 | 22 | interactive(); 23 | -------------------------------------------------------------------------------- /SetlX/secant.stlx: -------------------------------------------------------------------------------- 1 | secant := procedure(f, a, b, n) { 2 | fa := f(a); 3 | fb := f(b); 4 | for (k in [1 .. n]) { 5 | c := (fb * a - fa * b) / (fb - fa); 6 | if (isRational(a) && isRational(b)) { 7 | print("c = $nDecimalPlaces(c, 80)$"); 8 | } else { 9 | print("c = $c$"); 10 | } 11 | a := b; b := c; fa := fb; fb := f(c); 12 | } 13 | return b; 14 | }; 15 | 16 | f := procedure(x) { 17 | return x ** 2 - 2; 18 | }; 19 | 20 | // x := secant(f, 1, 2, 14); 21 | 22 | print("x = $nDecimalPlaces(x, 80)$"); 23 | print("\n\n\n"); 24 | 25 | 26 | strange := procedure(x) { 27 | return 2 / (x * x + 1) - 1; 28 | }; 29 | 30 | x := secant(strange, 0, 5, 10); 31 | 32 | 33 | 34 | -------------------------------------------------------------------------------- /SetlX/plot-aufgabe-76.stlx: -------------------------------------------------------------------------------- 1 | parabel := procedure(x) { 2 | Pi := mathConst("Pi"); 3 | switch { 4 | case -Pi <= x && x <= Pi: return x * x; 5 | case x < Pi : return parabel(x + 2 * Pi); 6 | case x > Pi : return parabel(x - 2 * Pi); 7 | } 8 | }; 9 | 10 | c := plot_createCanvas("Die Saegezahn-Funktion."); 11 | 12 | width := 3.0 * mathConst("Pi"); 13 | height := 1.1 * mathConst("Pi") * mathConst("Pi"); 14 | 15 | plot_addListGraph(c, [[-width, 0], [width, 0]], "x-axis", [0, 0, 0]); 16 | plot_addListGraph(c, [[ 0, 0], [0, height]], "y-axis"); 17 | plot_addGraph(c, "parabel(x)", "y = p(x)", [0, 0, 255]); 18 | plot_modScale(c, [-width, width], [0, height]); 19 | 20 | // The next line is needed if this program is run standalone. 21 | get("Press enter to continue."); 22 | -------------------------------------------------------------------------------- /Java/MaschinenKonstante.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | 3 | public class MaschinenKonstante { 4 | public static void main(String[] args) { 5 | float eps1 = singlePrecisionEps(); 6 | double eps2 = doublePrecisionEps(); 7 | System.out.println("eps single precission = " + eps1); 8 | System.out.println("eps double precission = " + eps2); 9 | } 10 | 11 | public static float singlePrecisionEps() { 12 | float eps = 1.0F; 13 | float old = eps; 14 | while (1.0F + eps > 1.0F) { 15 | old = eps; 16 | eps *= 0.5F; 17 | } 18 | return old; 19 | }; 20 | public static double doublePrecisionEps() { 21 | double eps = 1.0; 22 | double old = eps; 23 | while (1.0 + eps > 1.0) { 24 | old = eps; 25 | eps *= 0.5; 26 | } 27 | return old; 28 | }; 29 | } 30 | -------------------------------------------------------------------------------- /Programs/interpolate.maple: -------------------------------------------------------------------------------- 1 | interpolate := 2 | proc(f, a, b, n) 3 | local h, p, q, xi, xq, i, j; 4 | description "create a polynom interpolation the function f at n+1 equidistant points between a and b"; 5 | h := (b - a) / n; 6 | p := 0; 7 | xq := a; 8 | for i from 0 to n do 9 | q := 1; 10 | xi := a; 11 | for j from 0 to n do 12 | if i <> j then 13 | q := q * (x - xi) / (xq - xi); 14 | end if; 15 | xi := xi + h; 16 | end do; 17 | p := p + q * subs(x = xq, f): 18 | xq := xq + h; 19 | end do; 20 | return simplify(normal(p)); 21 | end proc; 22 | 23 | a := -5; 24 | b := 5; 25 | f := 1 / (1 + x^2); 26 | for n from 3 to 25 do 27 | print(n); 28 | p := interpolate( f, a, b, n): 29 | plot( [ f, p ], x = a..b ); 30 | plot( f - p, x = a..b ); 31 | end do; 32 | -------------------------------------------------------------------------------- /SetlX/plot-saegezahn.stlx: -------------------------------------------------------------------------------- 1 | saegezahn := procedure(x) { 2 | Pi := mathConst("Pi"); 3 | switch { 4 | case 0 <= x && x <= Pi: return x; 5 | case Pi <= x && x <= 2 * Pi: return 2 * Pi - x; 6 | case 2 * Pi <= x : return saegezahn(x - 2 * Pi); 7 | case x < 0 : return saegezahn(x + 2 * Pi); 8 | } 9 | }; 10 | 11 | c := plot_createCanvas("Die Saegezahn-Funktion."); 12 | 13 | width := 2.0 * mathConst("Pi"); 14 | height := 1.1 * mathConst("Pi"); 15 | 16 | plot_addListGraph(c, [[-width, 0], [width, 0]], "x-axis", [0, 0, 0]); 17 | plot_addListGraph(c, [[ 0, 0], [0, height]], "y-axis"); 18 | plot_addGraph(c, "saegezahn(x)", "y = s(x)", [0, 0, 255]); 19 | plot_modScale(c, [-width, width], [0, height]); 20 | 21 | // The next line is needed if this program is run standalone. 22 | get("Press enter to continue."); 23 | -------------------------------------------------------------------------------- /SetlX/midpoint-rule.stlx: -------------------------------------------------------------------------------- 1 | // Compute the integral \int_a^b f(x) dx using the trapeziodal rule. 2 | // The intervall [a,b] is divided into n subintervals. 3 | midPoint := procedure(f, a, b, n) { 4 | sum := 0; 5 | h := (b - a) / n; 6 | x := a + h/2; 7 | for (i in [1 .. n]) { 8 | sum += evalTerm(f); 9 | x += h; 10 | } 11 | return sum * h; 12 | }; 13 | 14 | interactive := procedure() { 15 | s := read("enter function"); 16 | f := parse(s); 17 | a := read("enter left boundary"); 18 | b := read("enter right boundary"); 19 | n := read("enter number of intervals"); 20 | integral := midPoint(f, a, b, n); 21 | print("The approximate value of the integral is $integral$."); 22 | }; 23 | 24 | demo := procedure() { 25 | f := parse("exp(-x*x)"); 26 | a := 0; 27 | b := 1; 28 | n := 1; 29 | while (n < 1000000) { 30 | print("n = $n$: $midPoint(f, a, b, n)$"); 31 | n *= 2; 32 | } 33 | }; 34 | 35 | interactive(); 36 | -------------------------------------------------------------------------------- /SymPy-SciPy/induction.py: -------------------------------------------------------------------------------- 1 | from sympy import * 2 | 3 | n = symbols("n") 4 | i = symbols("i") 5 | 6 | def verifySum(s, e, i, n): 7 | """ 8 | check by induction whether the folowing equation holds: 9 | sum(e(i), i=1..n) == s 10 | """ 11 | lhs = e.subs(i, 1) 12 | rhs = s.subs(n, 1) 13 | base_case = simplify(lhs - rhs) 14 | lhs = s + e.subs(i, n+1) 15 | rhs = s.subs(n, n + 1) 16 | induction_step = simplify(lhs - rhs) 17 | return base_case == 0 and induction_step == 0 18 | 19 | def test(s, e, i, n): 20 | if verifySum(s, e, i, n): 21 | print "sum(" + str(e) + ", " + str(i) + "= 1.." + str(n) + ") = " + str(s) 22 | else: 23 | print "unable to prove:" 24 | print "sum(" + str(e) + ", " + str(i) + "= 1.." + str(n) + ") == " + str(s) 25 | 26 | s = n * (n + 1) / 2 27 | test(s, i, i, n) 28 | s = n * (2 * n + 1) * (n + 1) / 6 29 | test(s, i*i, i, n) 30 | 31 | s = n / (n + 1) 32 | test(s, 1/(i*(i+1)), i, n) 33 | 34 | s = n / (n + 1) 35 | test(s, 1/(i*i), i, n) 36 | -------------------------------------------------------------------------------- /Programs/SecantMethod.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | import static java.lang.Math.*; 3 | 4 | public class SecantMethod { 5 | // This function locates the zero of the function f in the intervall [a, b] 6 | // using the secant method. 7 | static double findZero(double a, double b, double eps) { 8 | int count = 0; 9 | double fa = f(a); ++count; 10 | double fb = f(b); ++count; 11 | for (int i = 1; abs(b - a) > eps; ++i) { 12 | double c = (fb * a - fa * b) / (fb - fa); 13 | System.out.printf(java.util.Locale.ENGLISH, 14 | "%3d: b = %-13.11g, f(b) = %+10.8e\n", 15 | i, b, f(b)); 16 | a = b; b = c; fa = fb; fb = f(c); ++count; 17 | } 18 | System.out.println("number of function evaluations: " + count); 19 | return b; 20 | } 21 | 22 | static double f(double x) { 23 | // return x - cos(x); 24 | // return x * x * x - 1; 25 | return cos(x); 26 | } 27 | 28 | public static void main(String args[]) { 29 | System.out.println("Zero is: " + findZero(2.0, 0.713661915, 1e-12) ); 30 | } 31 | 32 | } 33 | 34 | -------------------------------------------------------------------------------- /SetlX/gauss-seidel.stlx: -------------------------------------------------------------------------------- 1 | // given a matrix a and a vector b this function performs k steps 2 | // of the gauss-seidel method to solve the linear system of equations 3 | // a x = b 4 | gaussSeidel := procedure(a, b, k) { 5 | n := #b; 6 | assert(#a == n, "wrong number of equations"); 7 | assert(#a[1] == n, "wrong number of variables"); 8 | x := [ 0 : i in [ 1 .. n ] ]; 9 | for (l in [1 .. k]) { 10 | for (i in [1 .. n]) { 11 | x[i] := b[i]; 12 | for (j in [ 1 .. n ]) { 13 | if (i != j) { 14 | x[i] -= a[i][j] * x[j]; 15 | } 16 | } 17 | x[i] /= a[i][i]; 18 | } 19 | print("$l$: $x$"); 20 | } 21 | return x; 22 | }; 23 | 24 | demo := procedure() { 25 | a := [ [ 4.0, 1.0, 0.0 ], 26 | [ 1.0, 4.0, 1.0 ], 27 | [ 0.0, 1.0, 4.0 ] ]; 28 | b := [ 5.0, 6.0, 5.0 ]; 29 | k := 18; 30 | x := gaussSeidel(a, b, k); 31 | print("x = $x$"); 32 | }; 33 | 34 | demo(); 35 | 36 | -------------------------------------------------------------------------------- /SetlX/jacobi-method.stlx: -------------------------------------------------------------------------------- 1 | // given a matrix a and a vector b this function performs k steps 2 | // of the jacobi method to solve the linear system of equations 3 | // a x = b 4 | jacobi := procedure(a, b, k) { 5 | n := #b; 6 | assert(#a == n, "wrong number of equations"); 7 | assert(#a[1] == n, "wrong number of variables"); 8 | x := xNew := [ 0 : i in [ 1 .. n ] ]; 9 | for (l in [1 .. k]) { 10 | for (i in [1 .. n]) { 11 | xNew[i] := b[i]; 12 | for (j in [ 1 .. n ]) { 13 | if (i != j) { 14 | xNew[i] -= a[i][j] * x[j]; 15 | } 16 | } 17 | xNew[i] /= a[i][i]; 18 | } 19 | x := xNew; 20 | print("$l$: $x$"); 21 | } 22 | return x; 23 | }; 24 | 25 | demo := procedure() { 26 | a := [ [ 4.0, 1.0, 0.0 ], 27 | [ 1.0, 4.0, 1.0 ], 28 | [ 0.0, 1.0, 4.0 ] ]; 29 | b := [ 5.0, 6.0, 5.0 ]; 30 | k := 35; 31 | x := jacobi(a, b, k); 32 | print("x = $x$"); 33 | }; 34 | 35 | demo(); 36 | 37 | -------------------------------------------------------------------------------- /SetlX/trapezoidal-rule.stlx: -------------------------------------------------------------------------------- 1 | // Compute the integral \int_a^b f(x) dx using the trapeziodal rule. 2 | // The intervall [a,b] is divided into n subintervals. 3 | trapezoidal := procedure(f, a, b, n) { 4 | sum := 0; 5 | x := a; 6 | h := (b - a) / n; 7 | sum += evalTerm(f) / 2; 8 | for (i in [1 .. n - 1]) { 9 | x += h; 10 | sum += evalTerm(f); 11 | } 12 | x := b; 13 | sum += evalTerm(f) / 2; 14 | return sum * h; 15 | }; 16 | 17 | interactive := procedure() { 18 | s := read("enter function"); 19 | f := parse(s); 20 | a := read("enter left boundary"); 21 | b := read("enter right boundary"); 22 | n := read("enter number of intervals"); 23 | integral := trapezoidal(f, a, b, n); 24 | print("The approximate value of the integral is $integral$."); 25 | }; 26 | 27 | demo := procedure() { 28 | f := parse("exp(-x*x)"); 29 | a := 0; 30 | b := 1; 31 | n := 1; 32 | while (n < 1000000) { 33 | print("n = $n$: $trapezoidal(f, a, b, n)$"); 34 | n *= 2; 35 | } 36 | }; 37 | 38 | interactive(); 39 | -------------------------------------------------------------------------------- /Skript/Figures/umkehr.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 5 1 0 1 0 7 50 -1 -1 0.000 0 1 0 0 4950.687 3362.634 5040 7110 7335 6255 8460 4680 11 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 12 | 2970 8325 10035 8325 13 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 7 0 0 2 14 | 2970 2070 2970 8325 15 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 16 | 2970 2070 3150 2250 17 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 18 | 2970 2070 2790 2250 19 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 3 20 | 9900 8190 10080 8325 9900 8460 21 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 22 | 5040 7110 5040 8325 23 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 24 | 8460 4680 8460 8325 25 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 26 | 2970 7110 5040 7110 27 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 28 | 8460 4680 2970 4680 29 | 4 0 0 50 -1 0 16 0.0000 4 105 105 4995 8595 a\001 30 | 4 0 0 50 -1 0 16 0.0000 4 150 120 8460 8595 b\001 31 | 4 0 0 50 -1 0 16 0.0000 4 105 120 10260 8460 x\001 32 | 4 0 0 50 -1 0 16 0.0000 4 150 120 2880 1935 y\001 33 | 4 0 0 50 -1 0 16 0.0000 4 210 330 2475 7245 f(a)\001 34 | 4 0 0 50 -1 0 16 0.0000 4 210 345 2430 4770 f(b)\001 35 | -------------------------------------------------------------------------------- /SetlX/simpson-rule.stlx: -------------------------------------------------------------------------------- 1 | // Compute the integral \int_a^b f(x) dx using Simpson's rule. 2 | // The intervall [a,b] is divided into n subintervals. 3 | simpson := procedure(f, a, b, n) { 4 | x := a; 5 | sum := evalTerm(f); 6 | h := (b - a) / n; 7 | for (i in [1 .. n - 1]) { 8 | x += h; 9 | v := evalTerm(f); 10 | if (i % 2 == 1) { 11 | sum += 4 * v; 12 | } else { 13 | sum += 2 * v; 14 | } 15 | } 16 | x := b; 17 | sum += evalTerm(f); 18 | return sum * h / 3; 19 | }; 20 | 21 | interactive := procedure() { 22 | s := read("enter function"); 23 | f := parse(s); 24 | a := read("enter left boundary"); 25 | b := read("enter right boundary"); 26 | n := read("enter number of intervals (must be even)"); 27 | n := (n + 1) \ 2 * 2; // ensure n is even 28 | print(n); 29 | integral := simpson(f, a, b, n); 30 | print("The approximate value of the integral is $integral$."); 31 | }; 32 | 33 | demo := procedure() { 34 | f := parse("exp(-x*x)"); 35 | a := 0; 36 | b := 1; 37 | n := 2; 38 | while (n < 100000) { 39 | print("n = $n$: $simpson(f, a, b, n) - 0.7468241328124270$"); 40 | n *= 2; 41 | } 42 | }; 43 | 44 | interactive(); 45 | -------------------------------------------------------------------------------- /Programs/Jacobi.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | 3 | public class Jacobi 4 | { 5 | Double[][] mA; 6 | Double[] mB; 7 | Double[] mX; 8 | 9 | public Jacobi(Double[][] A, Double[] b) { 10 | mA = A; 11 | mB = b; 12 | mX = new Double[b.length]; 13 | for (int i = 0; i < b.length; ++i) { 14 | mX[i] = 0.0; 15 | } 16 | } 17 | 18 | public void iterate() { 19 | int n = mB.length; 20 | Double[] xNew = new Double[n]; 21 | for (int i = 0; i < n; ++i) { 22 | xNew[i] = mB[i]; 23 | for (int j = 0; j < n; ++j) { 24 | if (i != j) { 25 | xNew[i] -= mA[i][j] * mX[j]; 26 | } 27 | } 28 | xNew[i] /= mA[i][i]; 29 | } 30 | mX = xNew; 31 | } 32 | 33 | public String toString() { 34 | String result = "("; 35 | for (int i = 0; i < mX.length - 1; ++i) { 36 | result += mX[i] + ", "; 37 | } 38 | result += mX[mX.length-1] + ")"; 39 | return result; 40 | } 41 | 42 | public static void main(String args[]) { 43 | Double[][] A = { { 4.0, 1.0, 0.0}, 44 | { 1.0, 4.0, 1.0}, 45 | { 0.0, 1.0, 4.0} }; 46 | Double[] b = { 5.0, 6.0, 5.0 }; 47 | Jacobi j = new Jacobi(A, b); 48 | for (int i = 0; i < 50; ++i) { 49 | j.iterate(); 50 | System.out.println(j.toString()); 51 | } 52 | } 53 | } 54 | 55 | -------------------------------------------------------------------------------- /SetlX/regulaFalsi.stlx: -------------------------------------------------------------------------------- 1 | // This function finds a root of the equation f(x) = 0. 2 | // The function works by using the regula falsi in the interval [a,b]. 3 | // The number n is the number of iterations. 4 | regulaFalsi := procedure(f, a, b, n) { 5 | assert(a < b, "Error: !(a < b)"); 6 | assert(f(a) < 0 && f(b) > 0, "Error: !(f(a) < 0 && f(b) > 0)"); 7 | fa := f(a); fb := f(b); 8 | for (i in [1 .. n]) { 9 | c := (fb * a - fa * b) / (fb - fa); 10 | fc := f(c); 11 | if (fc <= 0) { 12 | a := c; fa := fc; 13 | } else { 14 | b := c; fb := fc; 15 | } 16 | if (isRational(a) && isRational(b)) { 17 | print("a = $nDecimalPlaces(a, 40)$, b = $nDecimalPlaces(b, 40)$"); 18 | } else { 19 | print("a = $a$, b = $b$"); 20 | } 21 | } 22 | if (-fa < fb) { // -fa = |fa| 23 | return a; 24 | } else { 25 | return b; 26 | } 27 | }; 28 | 29 | sqrt := procedure(x) { 30 | return x * x - 2; 31 | }; 32 | 33 | // r2 := regulaFalsi(sqrt, 0, 2, 60); 34 | 35 | print("2 ** (1/2) = $nDecimalPlaces(r2, 42)$"); 36 | print("\n\n\n"); 37 | 38 | cosXEqualsX := procedure(x) { 39 | return x - cos(x); 40 | }; 41 | 42 | // x := regulaFalsi(cosXEqualsX, 0.0, 1.0, 14); 43 | 44 | // print("x = $x$\n\n\n"); 45 | 46 | f4 := x |-> x ** 4 - 1; 47 | 48 | x := regulaFalsi(f4, 0.0, 10.0, 1000); 49 | 50 | -------------------------------------------------------------------------------- /Programs/Seidel.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | 3 | public class Seidel 4 | { 5 | Double[][] mA; Double[] mB; Double[] mX; 6 | 7 | public Seidel(Double[][] A, Double[] b) { 8 | mA = A; 9 | mB = b; 10 | mX = new Double[b.length]; 11 | for (int i = 0; i < b.length; ++i) { 12 | mX[i] = 0.0; 13 | } 14 | } 15 | public void iterate() { 16 | int n = mB.length; 17 | for (int i = 0; i < n; ++i) { 18 | mX[i] = mB[i]; 19 | for (int j = 0; j < n; ++j) { 20 | if (i != j) { 21 | mX[i] -= mA[i][j] * mX[j]; 22 | } 23 | } 24 | mX[i] /= mA[i][i]; 25 | } 26 | } 27 | public String toString() { 28 | String result = "("; 29 | for (int i = 0; i < mX.length - 1; ++i) { 30 | result += mX[i] + ", "; 31 | } 32 | result += mX[mX.length-1] + ")"; 33 | return result; 34 | } 35 | public static void main(String args[]) { 36 | Double[][] A = { { 4.0, 1.0, 0.0}, 37 | { 1.0, 4.0, 1.0}, 38 | { 0.0, 1.0, 4.0} }; 39 | Double[] b = { 5.0, 6.0, 5.0 }; 40 | Seidel j = new Seidel(A, b); 41 | for (int i = 0; i < 30; ++i) { 42 | j.iterate(); 43 | System.out.println(j.toString()); 44 | } 45 | } 46 | } 47 | 48 | -------------------------------------------------------------------------------- /Programs/RegulaFalsi.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | import static java.lang.Math.*; 3 | 4 | public class RegulaFalsi { 5 | // This function locates the zero of the function f in the intervall [a, b]. 6 | // It is assumed that a < b and f(a) < f(b) holds initially. 7 | // The argument n specifies the number of iterations. 8 | static double findZero(double a, double b, int n) { 9 | assert a < b : "Error: a < b"; 10 | assert f(a) < 0 && f(b) > 0 : "Error: f(a) < 0 && f(b) > 0"; 11 | int count = 0; 12 | double fa = f(a); ++count; 13 | double fb = f(b); ++count; 14 | for (int i = 1; fa != 0.0 && fb != 0.0 && i <= n; ++i) { 15 | double c = (fb * a - fa * b) / (fb - fa); 16 | double fc = f(c); ++count; 17 | System.out.printf(java.util.Locale.ENGLISH, 18 | "%3d: a = %-13.11f, b = %-13.11f, c = %-13.11f, " + 19 | "f(a) = %-10.8e, f(b) = %-10.8e, f(c) = %-10.8e\n", 20 | i, a, b, c, f(a), f(b), f(c)); 21 | if (fc <= 0.0) { 22 | a = c; fa = fc; 23 | } else { 24 | b = c; fb = fc; 25 | } 26 | } 27 | System.out.println("number of function evaluations: " + count); 28 | return -fa < fb ? a : b; 29 | } 30 | 31 | static double f(double x) { 32 | return -cos(x); 33 | // return x * x * x * x - 1; 34 | } 35 | 36 | public static void main(String args[]) { 37 | System.out.println("Zero is: " + findZero(1.0, 2.0, 4) ); 38 | } 39 | 40 | } 41 | 42 | -------------------------------------------------------------------------------- /SetlX/bisection.stlx: -------------------------------------------------------------------------------- 1 | // Find the zero of the function f in the intervall [a, b]. The number 2 | // digits is the number of digits required. 3 | // This function does only work if either f(a) <= 0 <= f(b) or 4 | // f(a) >= 0 >= f(b). Also, the function f is required to be smooth. 5 | findZero := procedure(f, a, b, n) { 6 | assert(a < b, "a has to be less than b"); 7 | assert(f(a) < 0 && 0 < f(b), "we need f($a$) < 0 and f($b$) > 0"); 8 | for (k in [1 .. n]) { 9 | c := 1/2 * (a + b); 10 | if (f(c) < 0) { 11 | a := c; 12 | } else if (f(c) > 0) { 13 | b := c; 14 | } else { 15 | return c; 16 | } 17 | if (isRational(a) && isRational(b)) { 18 | print("a = $nDecimalPlaces(a, 40)$"); 19 | print("b = $nDecimalPlaces(b, 40)$"); 20 | } else { 21 | print("a = $a$, b = $b$, $f(a)$, $f(b)$"); 22 | } 23 | } 24 | return 1/2 * (a + b); 25 | }; 26 | root3 := procedure(x) { 27 | return x * x * x - 2; 28 | }; 29 | r3 := findZero(root3, 1, 2, 136); 30 | 31 | print("2 ** (1/3) = $nDecimalPlaces(r3, 42)$"); 32 | print("\n\n\n"); 33 | 34 | // root2 := procedure(x) { 35 | // return x * x - 2; 36 | // }; 37 | // 38 | // r2 := findZero(root2, 1, 2, 136); 39 | // 40 | // print("sqrt(2) = $nDecimalPlaces(r2, 42)$"); 41 | // print("\n\n\n"); 42 | // 43 | // cosXEqualsX := procedure(x) { 44 | // return x - cos(x); 45 | // }; 46 | 47 | // x := findZero(cosXEqualsX, 0.0, 1.0, 40); 48 | // print("solution of x = cos(x), $x$"); 49 | 50 | // x := findZero(x |-> -cos(x), 1.0, 2.0, 40); 51 | // pi := mathConst("pi"); 52 | 53 | // print("solution of 0 = cos(x), $x$"); 54 | // print("difference: $x - pi/2$"); 55 | -------------------------------------------------------------------------------- /Programs/Bisection.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | import static java.lang.Math.*; 3 | 4 | public class Bisection { 5 | // This function locates the zero of the function f in the intervall [a, b]. 6 | // It is assumed that a < b and that the function changes the sign in the 7 | // interval [a, b], i.e. either 8 | // f(a) < 0 and f(b) > 0 or f(a) > 0 and f(b) < 0 9 | // holds initially. The argument eps specifies the required accuracy. 10 | static double findZero(double a, double b, double eps) { 11 | assert a < b : "a has to be less than b"; 12 | assert f(a) * f(b) < 0 : "no sign change in interval [a, b]"; 13 | int count = 0; 14 | double fa = f(a); ++count; 15 | double fb = f(b); ++count; 16 | boolean sign = (fa < 0); 17 | for (int i = 1; fa != 0.0 && fb != 0.0 && b - a > eps; ++i) { 18 | double c = 0.5 * (a + b); 19 | double fc = f(c); ++count; 20 | System.out.printf(java.util.Locale.ENGLISH, 21 | "%3d: a = %-12.9f, b = %-12.9f, c = %-12.9f, " + 22 | "f(a) = %-10.8e, f(b) = %-10.8e, f(c) = %-10.8e\n", 23 | i, a, b, c, f(a), f(b), f(c)); 24 | if ((sign && fc < 0.0) || (!sign && fc > 0)) { 25 | a = c; fa = fc; 26 | } else { 27 | b = c; fb = fc; 28 | } 29 | } 30 | System.out.println("number of function evaluations: " + count); 31 | if (fa == 0.0) { 32 | return a; 33 | } 34 | if (fb == 0.0) { 35 | return b; 36 | } 37 | return 0.5 * (a + b); 38 | } 39 | 40 | static double f(double x) { 41 | return x - cos(x); 42 | //return cos(x) - x; 43 | //return x * x * x * x - 1; 44 | } 45 | 46 | public static void main(String args[]) { 47 | System.out.println("Zero is: " + findZero(0.0, 1.0, 0.5e-9) ); 48 | } 49 | 50 | } 51 | 52 | -------------------------------------------------------------------------------- /SetlX/illinois.stlx: -------------------------------------------------------------------------------- 1 | // This function implements the Illinois-method to compute the zero of 2 | // the function f in the interval [a,b]. 3 | illinois := procedure(f, a, b, n) { 4 | assert(a < b, "a has to be less than b"); 5 | assert(f(a) < 0 && 0 < f(b), "We need f(a) < 0 and 0 < f(b)!"); 6 | [ fa, fb ] := [ f(a), f(b) ]; 7 | oldA1 := om; oldB1 := om; 8 | oldA2 := om; oldB2 := om; 9 | alpha := 1; beta := 1; 10 | for (k in [1 .. n]) { 11 | c := (beta * fb * a - alpha * fa * b) / (beta * fb - alpha * fa); 12 | fc := f(c); 13 | if (fc < 0) { 14 | a := c; fa := fc; alpha := 1; 15 | if (oldB2 == b) { 16 | beta /= 2; 17 | } 18 | } else if (fc > 0) { 19 | b := c; fb := fc; beta := 1; 20 | if (oldA2 == a) { 21 | alpha /= 2; 22 | } 23 | } else { return c; } 24 | oldA2 := oldA1; oldB2 := oldB1; 25 | oldA1 := a; oldB1 := b; 26 | if (isRational(a) && isRational(b)) { 27 | print("$k$: a = $nDecimalPlaces(a, 36)$"); 28 | print(" b = $nDecimalPlaces(b, 36)$"); 29 | print(" alpha = $alpha$, beta = $beta$"); 30 | } else { 31 | print("$k$: a = $a$, b = $b$, alpha = $alpha$, beta = $beta$"); 32 | } 33 | } 34 | return (a + b) / 2; 35 | }; 36 | 37 | f := procedure(x) { 38 | return x ** 4 - 1; 39 | }; 40 | 41 | x := illinois(f, 0.0, 10.0, 35); 42 | print("solution to x ** 4 - 1 = 0 is $x$\n"); 43 | 44 | g := procedure(x) { 45 | return x ** 2 - 2; 46 | }; 47 | 48 | x := illinois(g, 1, 2, 15); 49 | print("solution to x ** 2 = 2 is $nDecimalPlaces(x, 41)$\n"); 50 | 51 | strange := procedure(x) { 52 | return 1 - 2 / (x * x + 1); 53 | }; 54 | 55 | x := illinois(strange, 0.0, 5.0, 20); 56 | print("solution to 1 - 2 / (x * x + 1) is $x$\n"); 57 | 58 | // x := illinois(x |-> -cos(x), 1, 2, 23); 59 | pi := mathConst("pi"); 60 | 61 | // print("solution of 0 = cos(x): x = $x$"); 62 | // print("difference: $x - pi/2$"); 63 | 64 | -------------------------------------------------------------------------------- /Skript/Figures/convex.py: -------------------------------------------------------------------------------- 1 | import matplotlib.pyplot as plt 2 | import numpy as np 3 | 4 | def f(x): 5 | return (x-1)**2 + 1 6 | 7 | def g(x): 8 | a = -1.0 9 | b = 4.0 10 | return (x-a)/(b-a)*f(b)+(x-b)/(a-b)*f(a) 11 | 12 | # plt.ion() # plotting interactively is slow 13 | 14 | x = np.linspace(-2, 5, 256, endpoint = True) 15 | y = f(x) 16 | plt.plot(x, y, color="blue", label="f") 17 | 18 | lx = np.linspace(-1,4, 200, endpoint = True) 19 | ly = g(lx) 20 | plt.plot(lx, ly, color="red", label="g") 21 | 22 | plt.xlim(-3, 6) 23 | plt.ylim(-5,18) 24 | 25 | plt.legend(loc="upper left") 26 | 27 | ax = plt.gca() 28 | ax.spines['right'].set_color('none') 29 | ax.spines['top'].set_color('none') 30 | ax.xaxis.set_ticks_position('bottom') 31 | ax.spines['bottom'].set_position(('data',0)) 32 | ax.yaxis.set_ticks_position('left') 33 | ax.spines['left'].set_position(('data',0)) 34 | 35 | t = -1 36 | plt.scatter([t,],[f(t),], 50, color ='blue') 37 | plt.annotate(r'$\langle x_1, f(x_1) \rangle$', xy=(t, f(t)), xycoords='data', xytext=(-80, -50), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.1,shrink=0.2)) 38 | 39 | t = 4 40 | plt.scatter([t,],[f(t),], 50, color ='blue') 41 | plt.annotate(r'$\langle x_2, f(x_2) \rangle$', xy=(t, f(t)), xycoords='data', xytext=(+10, -50), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.11,shrink=0.2)) 42 | 43 | t = 1.5 44 | plt.scatter([t,],[f(t),], 50, color ='blue') 45 | plt.annotate(r'$\langle t\cdot x_1 + (1-t)\cdot x_2, f(t\cdot x_2 + (1-t)\cdot x_2) \rangle$', xy=(t, f(t)), xycoords='data', xytext=(+20, -10), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.1,shrink=0.3,headwidth=5)) 46 | plt.scatter([t,],[g(t),], 50, color ='red') 47 | plt.annotate(r'$\langle t\cdot x_1 + (1-t)\cdot x_2, g(t\cdot x_1 + (1-t)\cdot x_2) \rangle$', xy=(t, g(t)), xycoords='data', xytext=(+25, +40), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.1,shrink=0.2),horizontalalignment='center') 48 | 49 | plt.plot([t,t],[0,f(t)], color ='blue', linestyle="--") 50 | plt.plot([t,t],[f(t),g(t)], color ='red', linestyle="--") 51 | 52 | plt.savefig("convex.eps",dpi=300) 53 | -------------------------------------------------------------------------------- /Skript/Figures/concav.py: -------------------------------------------------------------------------------- 1 | import matplotlib.pyplot as plt 2 | import numpy as np 3 | 4 | def f(x): 5 | return 12 - (x-1)**2 6 | 7 | def g(x): 8 | a = -1.0 9 | b = 4.0 10 | return (x-a)/(b-a)*f(b)+(x-b)/(a-b)*f(a) 11 | 12 | # plt.ion() # plotting interactively is slow 13 | 14 | x = np.linspace(-2, 5, 256, endpoint = True) 15 | y = f(x) 16 | plt.plot(x, y, color="blue", label="f") 17 | 18 | lx = np.linspace(-1,4, 200, endpoint = True) 19 | ly = g(lx) 20 | plt.plot(lx, ly, color="red", label="g") 21 | 22 | plt.xlim(-3, 6) 23 | plt.ylim(-5,18) 24 | 25 | plt.legend(loc="upper left") 26 | 27 | ax = plt.gca() 28 | ax.spines['right'].set_color('none') 29 | ax.spines['top'].set_color('none') 30 | ax.xaxis.set_ticks_position('bottom') 31 | ax.spines['bottom'].set_position(('data',0)) 32 | ax.yaxis.set_ticks_position('left') 33 | ax.spines['left'].set_position(('data',0)) 34 | 35 | t = -1 36 | plt.scatter([t,],[f(t),], 50, color ='blue') 37 | plt.annotate(r'$\langle x_1, f(x_1) \rangle$', xy=(t, f(t)), xycoords='data', xytext=(-80, 20), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.1,shrink=0.2)) 38 | 39 | t = 4 40 | plt.scatter([t,],[f(t),], 50, color ='blue') 41 | plt.annotate(r'$\langle x_2, f(x_2) \rangle$', xy=(t, f(t)), xycoords='data', xytext=(+20, 20), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.11,shrink=0.2)) 42 | 43 | t = 1.5 44 | plt.scatter([t,],[f(t),], 50, color ='blue') 45 | plt.annotate(r'$\langle t\cdot x_1 + (1-t)\cdot x_2, f(t\cdot x_2 + (1-t)\cdot x_2) \rangle$', xy=(t, f(t)), xycoords='data', xytext=(-60, 20), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.1,shrink=0.3,headwidth=5)) 46 | plt.scatter([t,],[g(t),], 50, color ='red') 47 | plt.annotate(r'$\langle t\cdot x_1 + (1-t)\cdot x_2, g(t\cdot x_2 + (1-t)\cdot x_2) \rangle$', xy=(t, g(t)), xycoords='data', xytext=(+25, -55), textcoords='offset points', fontsize=12, arrowprops=dict(width=0.1,shrink=0.2),horizontalalignment='center') 48 | 49 | plt.plot([t,t],[0,g(t)], color ='red', linestyle="--") 50 | plt.plot([t,t],[g(t),f(t)], color ='blue', linestyle="--") 51 | 52 | 53 | plt.savefig("concav.eps",dpi=300) 54 | -------------------------------------------------------------------------------- /Programs/Solver.java: -------------------------------------------------------------------------------- 1 | import java.util.*; 2 | import static java.lang.Math.*; 3 | 4 | public class Solver { 5 | double mXn, mXnMinus1; 6 | double mFXn, mFXnMinus1; 7 | double mA, mB, mC; 8 | double mFA, mFB, mFC; 9 | double mEps; 10 | int mCount; // count the number of function invocations 11 | 12 | public Solver(double a, double b, double eps) { 13 | assert a < b : "a has to be less than b"; 14 | mA = a; 15 | mB = b; 16 | mEps = eps; 17 | mCount = 0; 18 | mFA = f(a); ++mCount; 19 | mFB = f(b); ++mCount; 20 | assert mFA <= 0 : "f(a) is positive"; 21 | assert mFB >= 0 : "f(b) is negative"; 22 | mXnMinus1 = mA; 23 | mXn = mB; 24 | mFXnMinus1 = mFA; 25 | mFXn = mFB; 26 | } 27 | 28 | 29 | // This function locates the zero of the function f in the intervall [mA, mB]. 30 | // The algorithm used is a combination of the secant method and the 31 | // bisection algorithm which guarantess that the algorithm does not diverge. 32 | double findZero() { 33 | for (int i = 1; mFXn != 0.0 && mB - mA > mEps; ++i) { 34 | mC = (mFXn * mXnMinus1 - mFXnMinus1 * mXn) / (mFXn - mFXnMinus1); 35 | // Use the secant method if the new value is within the bounds of 36 | // the intervall [ma, mB]. 37 | if (mA < mC && mC < mB) { 38 | System.out.println("Secant method: x = " + mC); 39 | mFC = f(mC); ++mCount; 40 | updateIntervall(); 41 | continue; 42 | } 43 | // Otherwise, use bisection. 44 | mC = 0.5 * (mA + mB); 45 | mFC = f(mC); ++mCount; 46 | System.out.println("Bisection: c = " + mC); 47 | updateIntervall(); 48 | } 49 | System.out.println("number of function evaluations: " + mCount); 50 | return -mFA < mFB ? mA : mB; 51 | } 52 | 53 | void updateIntervall() { 54 | mXnMinus1 = mXn; 55 | mXn = mC; 56 | mFXnMinus1 = mFXn; 57 | mFXn = mFC; 58 | if (mFC <= 0.0) { 59 | mA = mC; mFA = mFC; 60 | } else { 61 | mB = mC; mFB = mFC; 62 | } 63 | } 64 | 65 | static double f(double x) { 66 | return x - cos(x); 67 | // return x * x * x * x - 1; 68 | // return x * x - 2; 69 | // return 1 - 2 / (x * x + 1); 70 | } 71 | 72 | public static void main(String args[]) { 73 | Solver solver = new Solver(0.0, 10.0, 1e-10); 74 | System.out.println("Zero is: " + solver.findZero() ); 75 | } 76 | 77 | } 78 | 79 | -------------------------------------------------------------------------------- /SetlX/newton-zero.stlx: -------------------------------------------------------------------------------- 1 | newton := procedure(t, x, n) { 2 | ts := diff(t, "x"); 3 | for (i in [1 .. n]) { 4 | x -= evalTerm(t) / evalTerm(ts); 5 | print("i = $i$: $myString(x)$"); 6 | } 7 | return x; 8 | }; 9 | 10 | // differentiate the term t with respect to the variable x 11 | diff := procedure(t, x) { 12 | match (t) { 13 | case t1 + t2 : 14 | return diff(t1, x) + diff(t2, x); 15 | case - a : 16 | return - diff(a, x); 17 | case t1 - t2 : 18 | return diff(t1, x) - diff(t2, x); 19 | case t1 * t2 : 20 | return diff(t1, x) * t2 + t1 * diff(t2, x); 21 | case t1 / t2 : 22 | return ( diff(t1, x) * t2 - t1 * diff(t2, x) ) / t2 ** 2; 23 | case f ** g : 24 | if (isNumber(g)) { 25 | g1 := g - 1; 26 | return g * diff(f, x) * f ** g1; 27 | } else { 28 | return diff( @exp(g * @ln(f)), x); 29 | } 30 | case sqrt(a) : 31 | return 1 / 2 * diff(a, x) / @sqrt(a); 32 | case sin(a) : 33 | return diff(a, x) * @cos(a); 34 | case cos(a) : 35 | return -diff(a, x) * @sin(a); 36 | case tan(a) : 37 | return diff(@sin(a) / @cos(a), x); 38 | case asin(a) : 39 | return diff(a, x) / @sqrt(1 - a ** 2); 40 | case acos(a) : 41 | return - diff(a, x) / @sqrt(1 - a ** 2); 42 | case atan(a) : 43 | return diff(a, x) / (1 + a ** 2); 44 | case ln(a) : 45 | return diff(a, x) / a; 46 | case exp(a) : 47 | return diff(a, x) * @exp(a); 48 | case ^variable(y) : 49 | if (y == x) { 50 | return 1; 51 | } else { 52 | return 0; 53 | } 54 | default: 55 | if (isNumber(t)) { 56 | return 0; 57 | } else { 58 | abort("error: diff($t$, $x$)"); 59 | } 60 | } 61 | }; 62 | 63 | myString := procedure(x) { 64 | if (isRational(x)) { 65 | return nDecimalPlaces(x,52); 66 | } else { 67 | return x; 68 | } 69 | }; 70 | 71 | interactive := procedure() { 72 | s := read("enter function: "); 73 | t := parse(s); 74 | x := read("enter start value: "); 75 | n := read("enter number of iterations: "); 76 | x := newton(t, x, n); 77 | }; 78 | 79 | interactive(); 80 | 81 | -------------------------------------------------------------------------------- /Skript/T1-WGL4.enc: -------------------------------------------------------------------------------- 1 | % T1-WGL4.enc 2 | % 3 | % 4 | % This is LaTeX T1 encoding for WGL4 encoded TrueType fonts 5 | % (e.g. from Windows 95) 6 | % 7 | % 8 | % Note that /hyphen appears twice (for the T1 code points `hyphen' 0x2d 9 | % and `hyphenchar' 0x7f). 10 | % 11 | % 12 | % LIGKERN space l =: lslash ; 13 | % LIGKERN space L =: Lslash ; 14 | % LIGKERN question quoteleft =: questiondown ; 15 | % LIGKERN exclam quoteleft =: exclamdown ; 16 | % LIGKERN hyphen hyphen =: endash ; 17 | % LIGKERN endash hyphen =: emdash ; 18 | % LIGKERN quoteleft quoteleft =: quotedblleft ; 19 | % LIGKERN quoteright quoteright =: quotedblright ; 20 | % LIGKERN comma comma =: quotedblbase ; 21 | % LIGKERN less less =: guillemotleft ; 22 | % LIGKERN greater greater =: guillemotright ; 23 | % 24 | % LIGKERN f i =: fi ; 25 | % LIGKERN f l =: fl ; 26 | % LIGKERN f f =: ff ; 27 | % LIGKERN ff i =: ffi ; 28 | % LIGKERN ff l =: ffl ; 29 | % 30 | % We blow away kerns to and from spaces (TeX doesn't have a 31 | % space) and also remove any kerns from the numbers. 32 | % 33 | % LIGKERN space {} * ; * {} space ; 34 | % LIGKERN zero {} * ; * {} zero ; 35 | % LIGKERN one {} * ; * {} one ; 36 | % LIGKERN two {} * ; * {} two ; 37 | % LIGKERN three {} * ; * {} three ; 38 | % LIGKERN four {} * ; * {} four ; 39 | % LIGKERN five {} * ; * {} five ; 40 | % LIGKERN six {} * ; * {} six ; 41 | % LIGKERN seven {} * ; * {} seven ; 42 | % LIGKERN eight {} * ; * {} eight ; 43 | % LIGKERN nine {} * ; * {} nine ; 44 | 45 | /T1Encoding [ % now 256 chars follow 46 | % 0x00 47 | /grave /acute /circumflex /tilde 48 | /dieresis /hungarumlaut /ring /caron 49 | /breve /macron /dotaccent /cedilla 50 | /ogonek /quotesinglbase /guilsinglleft /guilsinglright 51 | % 0x10 52 | /quotedblleft /quotedblright /quotedblbase /guillemotleft 53 | /guillemotright /endash /emdash /compwordmark 54 | /perthousandzero /dotlessi /dotlessj /ff 55 | /fi /fl /ffi /ffl 56 | % 0x20 57 | /visualspace /exclam /quotedbl /numbersign 58 | /dollar /percent /ampersand /quoteright 59 | /parenleft /parenright /asterisk /plus 60 | /comma /hyphen /period /slash 61 | % 0x30 62 | /zero /one /two /three 63 | /four /five /six /seven 64 | /eight /nine /colon /semicolon 65 | /less /equal /greater /question 66 | % 0x40 67 | /at /A /B /C 68 | /D /E /F /G 69 | /H /I /J /K 70 | /L /M /N /O 71 | % 0x50 72 | /P /Q /R /S 73 | /T /U /V /W 74 | /X /Y /Z /bracketleft 75 | /backslash /bracketright /asciicircum /underscore 76 | % 0x60 77 | /quoteleft /a /b /c 78 | /d /e /f /g 79 | /h /i /j /k 80 | /l /m /n /o 81 | % 0x70 82 | /p /q /r /s 83 | /t /u /v /w 84 | /x /y /z /braceleft 85 | /bar /braceright /asciitilde /hyphen 86 | % 0x80 87 | /Abreve /Aogonek /Cacute /Ccaron 88 | /Dcaron /Ecaron /Eogonek /Gbreve 89 | /Lacute /Lcaron /Lslash /Nacute 90 | /Ncaron /Eng /Odblacute /Racute 91 | % 0x90 92 | /Rcaron /Sacute /Scaron /Scedilla 93 | /Tcaron /Tcedilla /Udblacute /Uring 94 | /Ydieresis /Zacute /Zcaron /Zdot 95 | /IJ /Idot /dmacron /section 96 | % 0xA0 97 | /abreve /aogonek /cacute /ccaron 98 | /dcaron /ecaron /eogonek /gbreve 99 | /lacute /lcaron /lslash /nacute 100 | /ncaron /eng /odblacute /racute 101 | % 0xB0 102 | /rcaron /sacute /scaron /scedilla 103 | /tcaron /tcedilla /udblacute /uring 104 | /ydieresis /zacute /zcaron /zdot 105 | /ij /exclamdown /questiondown /sterling 106 | % 0xC0 107 | /Agrave /Aacute /Acircumflex /Atilde 108 | /Adieresis /Aring /AE /Ccedilla 109 | /Egrave /Eacute /Ecircumflex /Edieresis 110 | /Igrave /Iacute /Icircumflex /Idieresis 111 | % 0xD0 112 | /Eth /Ntilde /Ograve /Oacute 113 | /Ocircumflex /Otilde /Odieresis /OE 114 | /Oslash /Ugrave /Uacute /Ucircumflex 115 | /Udieresis /Yacute /Thorn /Germandbls 116 | % 0xE0 117 | /agrave /aacute /acircumflex /atilde 118 | /adieresis /aring /ae /ccedilla 119 | /egrave /eacute /ecircumflex /edieresis 120 | /igrave /iacute /icircumflex /idieresis 121 | % 0xF0 122 | /eth /ntilde /ograve /oacute 123 | /ocircumflex /otilde /odieresis /oe 124 | /oslash /ugrave /uacute /ucircumflex 125 | /udieresis /yacute /thorn /germandbls 126 | ] def 127 | 128 | % eof 129 | -------------------------------------------------------------------------------- /Skript/Maple/Fourier.mws: -------------------------------------------------------------------------------- 1 | {VERSION 6 0 "IBM INTEL LINUX" "6.0" } 2 | {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 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Defaults, Optio 39 | ns): \n# print(Options); # (nur zur Fehlersuche)\ngraph := Options[plo 40 | t]:\nonlylast := Options[last]:\n \nT := right-left; # Periodenl\344 41 | nge\nomega := 2*Pi/T; # Kreisfrequenz\n\n# Berechnung der Koeffizient 42 | en\na[0] := 2/T*int(func(x), x=left..right);\nfor i from 1 to nmax by \+ 43 | 1 do\n a[i] := 2/T*int(func(x)*cos(i*omega*x), x=left..right);\n b[i 44 | ] := 2/T*int(func(x)*sin(i*omega*x), x=left..right);\nend do;\n\n# Aus 45 | gabe der Koeffizienten\nprintf(\"a[0] = %12a\\n\", a[0]);\nfor i from \+ 46 | 1 to nmax by 1 do\n printf(\"a[%d] = %12a \\t\", i, a[i]);\n print 47 | f(\"b[%d] = %12a\\n\", i, b[i]);\nend do;\n\n# Grafik\nif (graph) then 48 | \n P[-1] := plot(func(x), x=left..right, color=green, thickness=2): 49 | \n if (onlylast) then\n P[0] := plot(a[0]/2+sum(a[k]*cos(k*omega* 50 | x)+\n b[k]*sin(k*omega*x),k=1..nmax), x=left..right): \n \+ 51 | display(P[-1], P[0]);\n else \n P[0] := plot(a[0]/2, x=left..rig 52 | ht):\n for i from 1 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}}}{EXCHG {PARA 0 "> 81 | " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 82 | {PARA 4 "" 0 "" {TEXT -1 11 "Anwendung 4" }}{EXCHG {PARA 0 "" 0 "" 83 | {TEXT -1 0 "" }{TEXT 266 43 "Hier k\366nnte Ihr eigenes Beispiel stehe 84 | n ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 85 | 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 86 | 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 87 | 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 88 | 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 89 | 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "1 2" 0 }{VIEWOPTS 1 1 0 1 90 | 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } 91 | -------------------------------------------------------------------------------- /Skript/analysis.tex: -------------------------------------------------------------------------------- 1 | \documentclass{report} 2 | \usepackage[latin1]{inputenc} 3 | \usepackage[ngerman]{babel} 4 | \usepackage{a4wide} 5 | \usepackage{epsfig} 6 | \usepackage{amssymb} 7 | \usepackage{amsmath} 8 | \usepackage{enumerate} 9 | \usepackage{fancyvrb} 10 | \usepackage{alltt} 11 | \usepackage{fleqn} 12 | \usepackage{epic} 13 | \usepackage{color} 14 | \usepackage{theorem} 15 | \usepackage{hyperref} 16 | \usepackage[all]{hypcap} 17 | \hypersetup{ 18 | colorlinks = true, % comment this to make xdvi work 19 | linkcolor = blue, 20 | citecolor = red, 21 | filecolor = [rgb]{0.1, 0.1, 1.0}, 22 | urlcolor = [rgb]{0.7, 0.0, 0.7}, 23 | pdfborder = {0 0 0} 24 | } 25 | 26 | \usepackage{fancyhdr} 27 | \usepackage{lastpage} 28 | 29 | \definecolor{amber}{rgb}{1.0, 0.75, 0.0} 30 | \definecolor{gold}{rgb}{1.0, 0.843, 0.0} 31 | 32 | \renewcommand*{\familydefault}{\sfdefault} 33 | 34 | \pagestyle{fancy} 35 | 36 | \fancyfoot[C]{--- \thepage/\pageref{LastPage}\ ---} 37 | 38 | \fancypagestyle{plain}{% 39 | \fancyhf{} 40 | \fancyfoot[C]{--- \thepage/\pageref{LastPage}\ ---} 41 | \renewcommand{\headrulewidth}{0pt} 42 | } 43 | 44 | \renewcommand{\chaptermark}[1]{\markboth{\chaptername \ \thechapter.\ #1}{}} 45 | \renewcommand{\sectionmark}[1]{\markright{\thesection. \ #1}{}} 46 | \fancyhead[R]{\leftmark} 47 | \fancyhead[L]{\rightmark} 48 | 49 | \definecolor{amethyst}{rgb}{0.2, 0.4, 0.6} 50 | \definecolor{orange}{rgb}{1, 0.9, 0.0} 51 | 52 | \usepackage[fixlanguage]{babelbib} 53 | \bibliographystyle{babunsrt} 54 | 55 | {\theorembodyfont{\sf} 56 | \newtheorem{Definition}{Definition} 57 | \newtheorem{Axiom}[Definition]{Axiom} 58 | \newtheorem{Notation}[Definition]{Notation} 59 | \newtheorem{Korollar}[Definition]{Korollar} 60 | \newtheorem{Lemma}[Definition]{Lemma} 61 | \newtheorem{Satz}[Definition]{Satz} 62 | \newtheorem{Theorem}[Definition]{Theorem} 63 | } 64 | 65 | \newcommand{\blue}[1]{{\color{blue}#1}} 66 | 67 | \newcommand{\proof}{\vspace*{0.2cm} 68 | 69 | \noindent 70 | \textbf{Beweis}: } 71 | 72 | \newcommand{\proofStar}{\vspace*{0.2cm} 73 | 74 | \noindent 75 | \textbf{Beweis}$^*$: } 76 | 77 | \newcommand{\hint}{\vspace*{0.2cm} 78 | 79 | \noindent 80 | \textbf{Hinweis}: } 81 | 82 | \newcommand{\qed}{\hspace*{\fill} $\Box$ 83 | \vspace*{0.2cm} 84 | 85 | } 86 | 87 | \newcommand{\eod}{\hspace*{\fill} $\diamond$} 88 | 89 | \newcommand{\eox}{\hspace*{\fill} $\diamond$} 90 | 91 | \newcommand{\edx}{\hspace*{\fill} $\diamond$} 92 | 93 | % set the monospace-font to Inconsalata-g 94 | % font-source: http://leonardo-m.livejournal.com/77079.html 95 | \renewcommand{\encodingdefault}{T1} 96 | \renewcommand{\ttdefault}{inconsolatag} 97 | 98 | \title{\epsfig{file=dhbw-logo.eps, scale=1.5}\\[0.3cm] 99 | Analysis\\[0.3cm] 100 | --- Sommersemester 2017 ---} 101 | \author{Prof.~Dr.~Karl Stroetmann} 102 | 103 | 104 | \date{\today \\[2.5cm] 105 | \begin{minipage}[t]{1.0\linewidth} 106 | \noindent 107 | Dieses Skript ist einschlie{\ss}lich der \LaTeX-Quellen sowie der in diesem Skript diskutierten 108 | Programme unter 109 | \\[0.2cm] 110 | \hspace*{\fill} 111 | \href{https://github.com/karlstroetmann/Analysis}{\texttt{https://github.com/karlstroetmann/Analysis}} 112 | \hspace*{\fill} 113 | \\[0.2cm] 114 | im Netz verf\"ugbar. Das Skript wird laufend \"uberarbeitet. Wenn Sie auf Ihrem Rechner \href{http://git-scm.com/download}{\texttt{git}} 115 | installieren und mein Repository mit Hilfe des Befehls 116 | \\[0.2cm] 117 | \hspace*{1.3cm} 118 | \texttt{git clone https://github.com/karlstroetmann/Analysis.git} 119 | \\[0.2cm] 120 | klonen, dann k\"onnen Sie durch Absetzen des Befehls 121 | \\[0.2cm] 122 | \hspace*{1.3cm} 123 | \texttt{git pull} 124 | \\[0.2cm] 125 | die aktuelle Version meines Skripts aus dem Netz laden. 126 | \end{minipage} 127 | } 128 | 129 | 130 | \newcommand{\solution}{\vspace*{0.2cm} 131 | 132 | \noindent 133 | \textbf{L\"osung}: } 134 | 135 | \newcounter{aufgabe} 136 | \newcommand{\exercise}{\vspace*{0.2cm} 137 | \stepcounter{aufgabe} 138 | 139 | \noindent 140 | \textbf{Aufgabe \arabic{aufgabe}}: } 141 | 142 | \newcommand{\exerciseStar}{\vspace*{0.2cm} 143 | \stepcounter{aufgabe} 144 | 145 | \noindent 146 | \textbf{Aufgabe \arabic{aufgabe}$^*$}: } 147 | 148 | \newcommand{\exercises}{\vspace*{0.2cm} 149 | \stepcounter{aufgabe} 150 | 151 | \noindent 152 | \textbf{Aufgabe \arabic{aufgabe}$^*$}: } 153 | 154 | \newcommand{\example}{\vspace*{0.2cm} 155 | 156 | \noindent 157 | \textbf{Beispiel}: \ } 158 | 159 | \newcommand{\examples}{\vspace*{0.2cm} 160 | 161 | \noindent 162 | \textbf{Beispiele}: \ } 163 | 164 | \newcommand{\remark}{\vspace*{0.2cm} 165 | \noindent 166 | \textbf{Bemerkung}: } 167 | 168 | \newcommand{\lb}{\hspace*{\fill} \linebreak} 169 | 170 | \newcommand{\ds}{\displaystyle} 171 | \newcommand{\bruch}[2]{\displaystyle\frac{\;\displaystyle#1\;}{\;\displaystyle#2\;}} 172 | \newcommand{\bruchs}[2]{\textstyle\frac{\;\textstyle#1\;}{\;\textstyle#2\;}} 173 | \newcommand{\folge}[1]{\bigl(#1\bigr)_{n\in\mathbb{N}^*}} 174 | \newcommand{\folgez}[1]{\bigl(#1\bigr)_{n\in\mathbb{N}}} 175 | \newcommand{\folgea}[1]{\bigl(#1\bigr)_{n\in\mathbb{N}^*}} 176 | \newcommand{\Folge}[1]{\left(#1\right)_{n\in\mathbb{N}^*}} 177 | \newcommand{\Folgez}[1]{\left(#1\right)_{n\in\mathbb{N}}} 178 | \newcommand{\Reihe}[1]{\left(\sum\limits_{i=0}^n #1\right)_{n\in\mathbb{N}}} 179 | \newcommand{\ReiheEins}[1]{\left(\sum\limits_{i=1}^n #1\right)_{n\in\mathbb{N}}} 180 | \newcommand{\bint}{\displaystyle\int} 181 | \newcommand{\dint}[2]{\displaystyle\int_{#1}^{#2}\hspace{-0.2cm}} 182 | \newcommand{\Oh}{\mathcal{O}} 183 | \newcommand{\df}[1]{\displaystyle\frac{\textrm{d}#1}{\textrm{d}x}} 184 | \newcommand{\dfo}{\displaystyle\frac{\textrm{d}\;}{\textrm{d}x}} 185 | \newcommand{\dr}{\textrm{d}} 186 | \newcommand{\err}[1]{\textsl{error}_n(#1)} 187 | \newcommand{\erri}[2]{\textsl{error}^{(#2)}_n(#1)} 188 | \newcommand{\norm}[1]{\big\|#1\bigr\|_{\infty}} 189 | 190 | \def\pair(#1,#2){\langle #1, #2 \rangle} 191 | 192 | \newlength{\mylength} 193 | \setlength{\mathindent}{1.3cm} 194 | 195 | %\includeonly{folgen-und-reihen} 196 | %\includeonly{rundungsfehler} 197 | 198 | \begin{document} 199 | 200 | \maketitle 201 | \tableofcontents 202 | \include{einleitung} 203 | \include{reelle-zahlen} 204 | \include{folgen-und-reihen} 205 | \include{stetige-funktionen} 206 | \include{ableitung} 207 | \include{anwendungen} 208 | \include{integration} 209 | \include{irrational} 210 | \include{fourier} 211 | \include{rundungsfehler} 212 | 213 | %\bibliographystyle{alpha} 214 | \bibliography{cs} 215 | %\bibliography{/Users/stroetma/Dropbox/Kurse/cs} 216 | 217 | \end{document} 218 | 219 | 220 | 221 | -------------------------------------------------------------------------------- /Skript/Aufgaben/aufgaben.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage{german} 3 | \usepackage[latin1]{inputenc} 4 | \usepackage{a4wide} 5 | \usepackage{epsfig} 6 | \usepackage{amssymb} 7 | \usepackage{fancyvrb} 8 | \usepackage{alltt} 9 | \usepackage{fleqn} 10 | \usepackage{epic} 11 | 12 | \newtheorem{Definition}{Definition} 13 | \newtheorem{Axiom}[Definition]{Axiom} 14 | \newtheorem{Notation}[Definition]{Notation} 15 | \newtheorem{Korollar}[Definition]{Korollar} 16 | \newtheorem{Lemma}[Definition]{Lemma} 17 | \newtheorem{Satz}[Definition]{Satz} 18 | \newtheorem{Theorem}[Definition]{Theorem} 19 | 20 | \renewcommand{\labelenumi}{(\alph{enumi})} 21 | 22 | \title{Analysis} 23 | \author{Karl Stroetmann} 24 | 25 | \newcommand{\bruch}[2]{\displaystyle\frac{\;\displaystyle#1\;}{\;\displaystyle#2\;}} 26 | \newcommand{\folge}[1]{\bigl(#1\bigr)_{n\in\mathbb{N}}} 27 | \newcommand{\Folge}[1]{\left(#1\right)_{n\in\mathbb{N}}} 28 | \newcommand{\Reihe}[1]{\left(\sum\limits_{i=0}^n #1\right)_{n\in\mathbb{N}}} 29 | \newcommand{\Oh}{\mathcal{O}} 30 | \newcommand{\df}{\displaystyle\frac{d\;}{dx}} 31 | \newcommand{\err}[1]{\textsl{error}_n(#1)} 32 | \newcommand{\erri}[2]{\textsl{error}^{(#2)}_n(#1)} 33 | \newcommand{\norm}[1]{\big\|#1\bigr\|_{\infty}} 34 | 35 | \def\pair(#1,#2){\langle #1, #2 \rangle} 36 | 37 | \newlength{\mylength} 38 | \setlength{\mathindent}{1.3cm} 39 | 40 | 41 | \begin{document} 42 | \noindent 43 | \section{Taylor-Reihe} 44 | \textbf{Aufgabe}: Leiten Sie die folgende Formel aus dem Additions-Theorem des 45 | Arcus-Tangens her und berechnen Sie damit $\pi$ auf eine Genauigkeit von $10^{-9}$: 46 | \begin{equation} 47 | \label{eq:Pi} 48 | \bruch{\pi}{4} = 2 * \arctan\Bigl(\frac{1}{2}\Bigr) -\arctan\Bigl(\frac{1}{7}\Bigr). 49 | \end{equation} 50 | 51 | 52 | \section{Interpolation} 53 | \textbf{Aufgabe}: F\"ur die Funktion $x \mapsto \sin(x)$ soll im Intervall 54 | $[0,\frac{\pi}{2}]$ ein Tabelle erstellt werden, so dass der bei linearer Interpolation entstehende 55 | Interpolations-Fehler kleiner als $10^{-5}$ ist. Das Intervall $[0,\frac{\pi}{2}]$ soll zu diesem 56 | Zweck in gleich gro\3e Intervalle aufgeteilt werden. Berechnen Sie die Anzahl der 57 | Eintr\"age, die f\"ur die Erstellung der Tabelle notwendig ist. 58 | \vspace*{0.3cm} 59 | 60 | 61 | \noindent 62 | \textbf{Aufgabe}: L\"osen Sie f\"ur $y=10^6$ und $y=10^{-6}$ die Gleichung $x*\exp(x) = y$ 63 | durch eine einfache Fixpunkt-Iteration. Berechnen Sie die L\"osung $x$ jeweils auf eine 64 | Genauigkeit von $10^{-3}$. 65 | \vspace*{0.3cm} 66 | 67 | 68 | \noindent 69 | \textbf{Aufgabe}: Untersuchen Sie mit Hilfe des Integral-Vergleichskriteriums, ob die 70 | Reihe 71 | \\[0.1cm] 72 | \hspace*{1.3cm} 73 | $\displaystyle \sum\limits_{n=1}^\infty \bruch{1}{n*(n+1)}$ 74 | \\[0.3cm] 75 | konvergiert. 76 | \vspace*{0.3cm} 77 | \pagebreak 78 | 79 | \noindent 80 | \textbf{Aufgabe}: Berechnen Sie, in wieviele Teil-Intervalle das Intervall $[0,1]$ 81 | aufgeteilt werden muss, wenn das Integral 82 | \\[0.1cm] 83 | \hspace*{1.3cm} 84 | $\displaystyle \int_0^1 e^{-x^2}\, dx$ 85 | \\[0.1cm] 86 | mit Hilfe der Trapez-Regel mit einer Genauigkeit von $10^{-6}$ berechnet werden soll. 87 | \vspace*{0.3cm} 88 | 89 | \noindent 90 | \textbf{Aufgabe}: 91 | \begin{enumerate} 92 | \item Berechnen Sie mit Hilfe der Kepler'schen Fass-Regel eine Approximation 93 | f\"ur das Integral 94 | \\[0.1cm] 95 | \hspace*{1.3cm}$\displaystyle \int_0^{\frac{1}2} \sin(x)\, dx$. 96 | \item Geben Sie eine m\"oglichst genaue Absch\"atzung f\"ur den Approximations-Fehler. 97 | \item Vergleichen Sie ihr Ergebnis mit dem exakten Wert. 98 | \end{enumerate} 99 | \vspace*{0.3cm} 100 | 101 | 102 | \noindent 103 | \textbf{Aufgabe}: Gegenstand dieser Aufgabe ist die numerische Berechnung der Summe 104 | \\[0.1cm] 105 | \hspace*{1.3cm} $\displaystyle \sum\limits_{k=1}^\infty \bruch{1}{k^3}$. 106 | \\[0.1cm] 107 | Gehen Sie zur Berechnung dieser Summe in folgenden Schritten vor. 108 | \begin{enumerate} 109 | \item Approximieren Sie die Rest-Summe $\sum\limits_{k=n}^\infty \bruch{1}{k^3}$ 110 | durch ein geeignetes Integral. 111 | 112 | \textbf{Hinweis}: Es gilt 113 | \\[0.1cm] 114 | \hspace*{1.3cm} $\displaystyle f(k) = \int_{k-\frac{1}{2}}^{k+\frac{1}{2}} f(k) \, dt \approx \int_{k-\frac{1}{2}}^{k+\frac{1}{2}} f(t) \, dt$. 115 | \item Berechnen Sie eine Absch\"atzung f\"ur den Approximations-Fehler, 116 | den Sie bei der Integration in Teil (a) erhalten. 117 | 118 | \textbf{Hinweis}: Approximieren Sie die auftretenden Summen durch Integrale. 119 | \item Berechnen Sie nun, wir gro\3 Sie $n$ w\"ahlen m\"ussen, damit der Approximations-Fehler 120 | kleiner als $10^{-6}$ bleibt. 121 | \item Geben Sie nun einen N\"aherungs-Wert f\"ur die Summe $\sum\limits_{k=1}^\infty \bruch{1}{k^3}$, 122 | der sich von dem exakten Ergebnis um weniger als $10^{-6}$ unterscheidet. 123 | \end{enumerate} 124 | \vspace*{0.3cm} 125 | 126 | \noindent 127 | \textbf{Aufgabe}: 128 | Die Funktion $p$ sei auf dem Intervall $[-\pi,\pi]$ definiert durch 129 | \\[0.1cm] 130 | \hspace*{1.3cm} 131 | $p(x) = x^2$. 132 | \\[0.1cm] 133 | Die Funktion werde so auf $\mathbb{R}$ fortgesetzt, dass die resultierende Funktion die Periode 134 | $2\!\cdot\!\pi$ hat. 135 | \begin{enumerate} 136 | \item Berechnen Sie die Fourier-Reihe von $p$. 137 | \item Berechnen Sie mit Hilfe der Fourier-Reihe von $p$ einen Wert f\"ur die Summe 138 | \\[0.1cm] 139 | \hspace*{1.3cm} 140 | $\displaystyle \sum\limits_{n=1}^\infty \bruch{1}{n^2}$. 141 | \end{enumerate} 142 | 143 | \pagebreak 144 | \noindent 145 | \textbf{Aufgabe}: Die Lambert'sche W-Funktion (Johann Heinrich Lambert; 1728 - 1777) 146 | $x \mapsto W(x)$ ist f\"ur $x\geq 0$ definiert als die Umkehr-Funktion der Funktion $x \mapsto x\cdot e^x$, 147 | es gilt also 148 | \\[0.1cm] 149 | \hspace*{1.3cm} $\displaystyle W(x) \cdot e^{W(x)} = x$ \quad f\"ur alle $x \in \mathbb{R}_+$. 150 | \begin{enumerate} 151 | \item Berechnen Sie die Ableitung der Lambert'schen W-Funktion. 152 | \item Formen Sie den Ausdruck f\"ur $W'(x)$ so um, dass der Term $e^{W(x)}$ nicht mehr 153 | auftritt. 154 | \item Berechnen Sie eine Stamm-Funktion der Lambert'schen W-Funktion. 155 | \item Nehmen Sie an, dass Sie die Lambert'sche W-Funktion berechnen k\"onnen und 156 | bestimmen Sie unter dieser Annahme f\"ur ein gegebenes $\varepsilon$ die L\"osung der Gleichung 157 | \\[0.1cm] 158 | \hspace*{1.3cm} 159 | $\bruch{1}{n} * \Bigl(\frac{1}{2}\Bigr)^n = \varepsilon$ 160 | \\[0.1cm] 161 | durch algebraische Umformungen. 162 | 163 | \noindent 164 | \textbf{Hinweis}: Invertieren Sie die Gleichung und bringen Sie die Gleichung dann 165 | auf die Form $\alpha * e^\alpha = \beta$ f\"ur geeignete $\alpha$ und $\beta$, denn 166 | dann gilt $\alpha = W(\beta)$. 167 | \end{enumerate} 168 | \vspace*{0.3cm} 169 | 170 | 171 | \noindent 172 | \textbf{Aufgabe}: Zeigen Sie anhand der Definition des Grenzwerts, dass 173 | $\lim\limits_{n \rightarrow \infty} \bruch{1}{\sqrt{n}} = 0$ ist. 174 | %\item Zeigen Sie, dass die Folge $\folge{\sin(n)}$ nicht konvergent ist. 175 | % leider zu schwer 176 | \vspace*{0.3cm} 177 | 178 | \noindent 179 | \textbf{Aufgabe}: 180 | Es seien $a,b\in\mathbb{R}$ gegeben. 181 | Die Folge $\folge{c_n}$ werde induktiv definiert durch $c_0 = a$, $c_1 = b$ und 182 | \\[0.1cm] 183 | \hspace*{1.3cm} $c_{n+2} = \frac{1}{2}*(c_n + c_{n+1})$. 184 | \begin{enumerate} 185 | \item Zeigen Sie durch Induktion nach $n$, dass gilt: 186 | \\[0.1cm] 187 | \hspace*{1.3cm} 188 | $c_n = \bruch{1}{3}\cdot a \biggl(1-\Bigl(\frac{-1}{2}\Bigr)^{n-1}\biggr) + \bruch{2}{3}\cdot b \biggl(1-\Bigl(\frac{-1}{2}\Bigr)^{n}\biggr)$ 189 | \item Berechnen Sie den Grenzwert $\lim\limits_{n \rightarrow \infty} c_n$. 190 | \end{enumerate} 191 | 192 | \end{document} 193 | 194 | 195 | 196 | 197 | 198 | %%% Local Variables: 199 | %%% mode: latex 200 | %%% TeX-master: t 201 | %%% End: 202 | -------------------------------------------------------------------------------- /Skript/Maple/Maple-6.mws: -------------------------------------------------------------------------------- 1 | {VERSION 6 0 "IBM INTEL LINUX" "6.0" } 2 | {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 3 | 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } 4 | {CSTYLE "" -1 257 "" 1 10 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 5 | 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 6 | 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 7 | 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } 8 | {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 9 | 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 10 | 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 11 | 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } 12 | {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 13 | 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 14 | 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 15 | 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } 16 | {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 17 | 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 18 | 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 19 | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } 20 | {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 21 | 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " 22 | " -1 -1 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 23 | 0 -1 0 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 24 | 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} 25 | {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 23 "Einf\374 26 | hrung in Maple (6)" }{TEXT 257 0 "" }{TEXT -1 0 "" }}}{SECT 1 {PARA 3 27 | "" 0 "" {TEXT -1 21 "Folgen und Grenzwerte" }}{EXCHG {PARA 0 "> " 0 " 28 | " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 13 "Za 29 | hlenfolgen:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Wir definieren ein 30 | e Folge (in der Schreibweise wie eine Funktion):" }}{PARA 0 "> " 0 "" 31 | {MPLTEXT 1 0 21 "a := n -> sin(n)/n^2;" }}}{EXCHG {PARA 0 "" 0 "" 32 | {TEXT -1 157 "Diese wollen wir zun\344chst grafisch darstellen (vgl. a 33 | uch das Arbeitsblatt zur grafischen Darstellung von Funktionen). Dazu \+ 34 | erzeugen wir zun\344chst mittels des " }{TEXT 267 3 "seq" }{TEXT -1 35 | 203 "-Befehls eine Liste, bestehend aus Paaren (Index des Folgengliede 36 | s, Wert), im Beispiel f\374r n=1 bis 20. Wenn Sie den Doppelpunkt im \+ 37 | n\344chsten Befehl durch ein Semikolon ersetzten, sehen Sie die Liste \+ 38 | ... " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Folge := [seq([n,a(n)], n=1 39 | ..20)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Dann kommt die Grafik: 40 | " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(Folge, style=point);" }}} 41 | {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Maple berechnet mittels des " } 42 | {TEXT 268 5 "limit" }{TEXT -1 23 "-Befehls den Grenzwert:" }}{PARA 0 " 43 | > " 0 "" {MPLTEXT 1 0 23 "limit(a(n),n=infinity);" }}}{EXCHG {PARA 0 " 44 | " 0 "" {TEXT -1 24 "Es gibt auch den Befehl " }{TEXT 269 5 "Limit" } 45 | {TEXT -1 68 ". 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JcEC_r;b"?J,~> 310 | JcEairrCRK!!)BcJ,~> 311 | JcEairrDoqr;baT!!)BcJ,~> 312 | JcEF`r;at>J,~> 313 | JcEairrCRK!!)BcJ,~> 314 | JcEairrDoqr;baT!!)BcJ,~> 315 | JcEF`r;at>J,~> 316 | JcEairrCRK!!)BcJ,~> 317 | JcEairrDusr;b[R!!)BcJ,~> 318 | JcELbr;an 319 | JcEairrCRK!!)BcJ,~> 320 | JcEairrE#tr;bXQ!!)BcJ,~> 321 | JcEOcr;ak;J,~> 322 | JcEairrCRK!!)BcJ,~> 323 | JcEairrE#tr;bXQ!!)BcJ,~> 324 | JcEOcr;ak;J,~> 325 | JcEairrCRK!!)BcJ,~> 326 | JcEairr<'!qZ,CN!!)BcJ,~> 327 | JcEXfqZ+V8J,~> 328 | JcEjlqZ,1H!!)BcJ,~> 329 | JcEjlp]/tH!!)BcJ,~> 330 | JcEjlp]/22J,~> 331 | JcEmmqZ$Zs!7q/M!:9`9~> 332 | JcEmmq#K"G!!)BcJ,~> 333 | JcEmmr;cltrrBk7J,~> 334 | JcEmmqZ$Zs!7q/M!:9`9~> 335 | JcEmmq#K"G!!)BcJ,~> 336 | JcEmmr;cltrrBk7J,~> 337 | JcGECdf9LM!!*&u!8.;O!:9`9~> 338 | JcGECbQ-;q!!)BcJ,~> 339 | JcEmmr;cltr;a_7J,~> 340 | JcGHD!!(IIqZ,:K!!)BcJ,~> 341 | JcGHD!!(UMpAikG!!)BcJ,~> 342 | JcEmmr;cltr;a_7J,~> 343 | JcGHD!!(IIqZ,:K!!)BcJ,~> 344 | JcGHD!!(UMpAikG!!)BcJ,~> 345 | JcEmmr;cltr;a_7J,~> 346 | JcGHD!!(LJquG=J!!)BcJ,~> 347 | JcGHD!!([OpAieE!!)BcJ,~> 348 | JcEsoqZ-ZrrrBk7J,~> 349 | JcGHD!!(RLqZ,1H!!)BcJ,~> 350 | JcGHD!!(^PpAibD!!)BcJ,~> 351 | JcF!pq>gQq!!'e6J,~> 352 | JcGHD!!(IIrrCRK!!)BcJ,~> 353 | JcGHD!!(aQr;cfrrrCRK!!)BcJ,~> 354 | JcF$qr;aA-J,~> 355 | JcGHD!!(IIrrCRK!!)BcJ,~> 356 | JcGHD!!(aQr;cfrrrCRK!!)BcJ,~> 357 | JcF$qr;aA-J,~> 358 | JcGHD!!(IIrrCRK!!)BcJ,~> 359 | JcGHD!!(gSr;c`prrCRK!!)BcJ,~> 360 | JcF*sr;a;+J,~> 361 | JcGHD!!(IIrrCRK!!)BcJ,~> 362 | JcGHD!!(jTr;c]orrCRK!!)BcJ,~> 363 | JcF-tr;a8*J,~> 364 | JcGHD!!(IIrrCRK!!)BcJ,~> 365 | JcGHD!!(jTr;c]orrCRK!!)BcJ,~> 366 | JcF-tr;a8*J,~> 367 | JcGHD!!(IIrrCRK!!)BcJ,~> 368 | JcGHD!!(pVquHQmrrCRK!!)BcJ,~> 369 | JcF4!quF,(J,~> 370 | JcGHD!!(IIrrCRK!!)BcJ,~> 371 | JcGHD!!(sWr;cTlrrCRK!!)BcJ,~> 372 | JcF7"r;a/'J,~> 373 | JcGHD!!(IIrrCRK!!)BcJ,~> 374 | JcGHD!!(sWr;cTlrrCRK!!)BcJ,~> 375 | JcF7"r;a/'J,~> 376 | JcGHD!!(IIrrCRK!!)BcJ,~> 377 | JcGHD!!)!Xr;cQkrrCRK!!)BcJ,~> 378 | JcF:#r;a,&J,~> 379 | JcGHD!!(IIrrCRK!!)BcJ,~> 380 | 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http://matplotlib.org/ 4 | %%CreationDate: Sat Jun 7 13:35:54 2014 5 | %%Orientation: portrait 6 | %%BoundingBox: 18 180 594 612 7 | %%EndComments 8 | %%BeginProlog 9 | /mpldict 8 dict def 10 | mpldict begin 11 | /m { moveto } bind def 12 | /l { lineto } bind def 13 | /r { rlineto } bind def 14 | /c { curveto } bind def 15 | /cl { closepath } bind def 16 | /box { 17 | m 18 | 1 index 0 r 19 | 0 exch r 20 | neg 0 r 21 | cl 22 | } bind def 23 | /clipbox { 24 | box 25 | clip 26 | newpath 27 | } bind def 28 | %!PS-Adobe-3.0 Resource-Font 29 | %%Title: Bitstream Vera Sans 30 | %%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. 31 | %%Creator: Converted from TrueType to type 3 by PPR 32 | 25 dict begin 33 | /_d{bind def}bind def 34 | /_m{moveto}_d 35 | /_l{lineto}_d 36 | /_cl{closepath eofill}_d 37 | /_c{curveto}_d 38 | /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d 39 | /_e{exec}_d 40 | /FontName /BitstreamVeraSans-Roman def 41 | /PaintType 0 def 42 | /FontMatrix[.001 0 0 .001 0 0]def 43 | /FontBBox[-183 -236 1287 928]def 44 | /FontType 3 def 45 | /Encoding [ /period /zero /one /two /four /five /six /eight /minus ] def 46 | /FontInfo 10 dict dup begin 47 | /FamilyName (Bitstream Vera Sans) def 48 | /FullName (Bitstream Vera Sans) def 49 | /Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. 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| showpage 1219 | --------------------------------------------------------------------------------