vcs: added .zip to gitignore

.gitignore

@@ -1 +1,2 @@
 /test*
+*.zip

make_latex.sh

@@ -15,8 +15,15 @@ cd workdir
 
 echo "Generating modified Markdown file..."
 
-# prefix every _ with an \
-sed -i -e 's/_/\\_/g' "$NEWNAME"
+#prefix every \ with a \
+sed -i -e 's/\\\\ *$/\\\\\\\\/g' "$NEWNAME"
+
+bat "$NEWNAME"
+
+# prefix every individual _ with an \
+sed -i -e 's/_([^_])/\\_\1/g' "$NEWNAME"
+
+bat "$NEWNAME"
 
 # remove tags section at the front
 sed -i -e 's/tags:.*//g' "$NEWNAME"
@@ -25,11 +32,32 @@ sed -i -e 's/---$//g' "$NEWNAME"
 #remove markdown links
 sed -i -e 's/\[.*\](.*)//g' "$NEWNAME"
 
-# remove html tags
+# remove code block lang declaration
+sed -i -e 's/```.*/```/g' "$NEWNAME"
+
+bat "$NEWNAME"
+
+# add newlines around code block declaration
+sed -i -e 's/``` *$/\n```\n/g' "$NEWNAME"
+
+bat "$NEWNAME"
+
+# replace special HTML tags
 sed -i -e 's/<div id="page-break-after"\/>/\\pagebreak/g' "$NEWNAME"
+sed -i -e 's/<br>/\\newline/g' "$NEWNAME"
+
+# remove html tags
 sed -i -e 's/<[a-zA-Z ]*>\(.*\)<\/[a-zA-Z =]*>/\1/g' "$NEWNAME"
 sed -i -e 's/<.*\/>//g' "$NEWNAME"
+sed -i -e 's/<.*>//g' "$NEWNAME"
 
+# replace arrows
+sed -i -e 's/->/$\\rightarrow$/g' "$NEWNAME"
+sed -i -e 's/<-/$\\leftarrow$/g' "$NEWNAME"
+sed -i -e 's/=>/$\\Rightarrow$/g' "$NEWNAME"
+sed -i -e 's/<=/$\\Leftarrow$/g' "$NEWNAME"
+
+bat "$NEWNAME"
 
 echo "Generating LaTeX file..."
 echo "\documentclass[a4paper, 11pt]{article}
@@ -46,5 +74,6 @@ tlmgr install markdown
 echo "Building PDF..."
 pdflatex -synctex=1 -interaction=nonstopmode --shell-escape file.tex
 
-echo "Done."
-okular file.pdf
+cp *.pdf ..
+
+echo "Done."

w1/matlab/convert2basis.m

@@ -0,0 +1,17 @@
+% EX2 a)
+
+function a = convert2basis(n, b)
+    if (b < 2)
+        disp('basis needs to be at least 2!')
+        return
+    end
+    
+    v = [];
+
+    while n > 0
+        v = [mod(n, b), v];
+        n = floor(n / b);
+    end
+
+    a = v
+end

w1/matlab/flp.m

@@ -0,0 +1,67 @@
+% EX3 c)
+function [d, v, t] = flp(b, m, n, x)
+    % GETS:
+    % b: basis
+    % m: mantissa length
+    % n: exponent length
+    % x: number to convert
+
+    % RETURNS: 
+    % d: mantissa coefficients
+    % v: coefficients of the exponent
+    % t: sign
+
+    # calculate exponent and mantissa
+    exponent = 0
+    mantissa = x
+    while mantissa > 1
+        mantissa /= b
+        exponent += 1
+    end
+    while (mantissa * b) < 1
+        mantissa *= b
+        exponent -= 1
+    end
+
+    inverted = convert2basis(mantissa^-1, b)
+    inverted(end)=[] 
+    val_man = flip(inverted)
+    val_exp = convert2basis(abs(exponent), b)
+    t = sign(exponent)
+
+    # pad result
+end
+
+function a = convert2basis(n, b)
+    if (b < 2)
+        disp('basis needs to be at least 2!')
+        return
+    end
+    
+    v = [];
+
+    while n > 0
+        v = [mod(n, b), v];
+        n = floor(n / b);
+    end
+
+    a = v
+end
+
+function b = convertDecimal(n, b)
+    if (b < 2)
+        disp('basis needs to be at least 2!')
+        return
+    end
+
+    while n < 1
+        
+
+end
+
+function res = padArr(arr, len)
+    for i = length(arr):len
+        arr(i+1) = 0
+    end
+    res = arr
+end

w1/matlab/interval.practise.m

@@ -0,0 +1,21 @@
+function interval(x, a, b)
+    if (nargin < 2)
+        a = 7;
+        b = 12;
+    end
+    if (a > b) 
+        bNew = a
+        a = b
+        b = bNew
+    end
+
+    if (a <= x && x <= b)
+        disp('x ist im Intervall')
+    elseif (x < a)
+        disp('x ist unterhalb des Intervals')
+    elseif (x > b)
+        disp('x ist überhalb des Intervals')
+    else
+        disp('x ist keine Nummer')
+    end
+end

w1/matlab/testConvert2basis.m

@@ -0,0 +1,91 @@
+% Angewandte Numerik 1, SoSe 2022
+% Uebungsblatt 01, Aufgabe 02: Darstellung natuerlicher Zahlen
+%
+% Testprogramm fuer die Funktion a = convert2basis(n, b)
+%
+% Letzte Aenderung: 22.04.2022
+
+
+
+%% Initialisierung
+
+clearvars;
+close all;
+clc;
+
+fprintf('\n');
+fprintf('Angewandte Numerik 1, Sommersemester 2022\n');
+fprintf('Uebungsblatt 1, Aufgabe 02: Darstellung natuerlicher Zahlen\n');
+fprintf('\n');
+
+
+
+%% Definition und Durchfuehrung der Testfaelle
+
+testfall = 0;
+while true                      % alle Testfaelle untersuchen
+    testfall = testfall + 1;    % naechster Testfall
+    
+    
+    %% alle Testfaelle definieren
+    
+    switch testfall
+        case 1                  % Testfall 1: b = 2, n = 30
+            b = 2;
+            n = 30;
+            a = [1 1 1 1 0];
+        case 2                  % Testfall 2: b = 2, n = 31
+            b = 2;
+            n = 31;
+            a = [1 1 1 1 1];
+        case 3                  % Testfall 3: b = 2, n = 32
+            b = 2;
+            n = 32;
+            a = [1 0 0 0 0 0];
+        case 4                  % Testfall 4: b = 2, n = 33
+            b = 2;
+            n = 33;
+            a = [1 0 0 0 0 1];
+        case 5                  % Testfall 5: b = 2, n = 42
+            b = 2;
+            n = 42;
+            a = [1 0 1 0 1 0];
+        case 6                  % Testfall 6: b = 2, n = 134110
+            b = 2;
+            n = 134110;
+            a = [1 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0];
+        case 7                  % Testfall 7: b = 8, n = 27
+            b = 8;
+            n = 27;
+            a = [3 3];
+        case 8                  % Testfall 8: b = 8, n = 3652
+            b = 8;
+            n = 3652;
+            a = [7 1 0 4];
+        case 9                  % Testfall 9: b = 8, n = 46807
+            b = 8;
+            n = 46807;
+            a = [1 3 3 3 2 7];
+        case 10                 % Testfall 10: b = 10, n = 3121
+            b = 10;
+            n = 3121;
+            a = [3 1 2 1];
+        case 11                 % Testfall 11: b = 10, n = 192310030133
+            b = 10;
+            n = 192310030133;
+            a = [1 9 2 3 1 0 0 3 0 1 3 3];
+        otherwise
+            break;              % keine Testfaelle mehr vorhanden
+    end
+    
+    
+    %% Testfall durchfuehren und Ergebnis ausgeben
+    
+    spezString = 'Testfall %2d (b = %2d, n = %12d): %s.\n';
+    if max(abs(a - convert2basis(n, b))) == 0
+        fprintf(spezString, testfall, b, n, 'Bestanden');
+    else
+        fprintf(2, spezString, testfall, b, n, 'Fehlgeschlagen')
+    end
+    
+end

w1/matlab/testFlp.m

@@ -0,0 +1,105 @@
+% Angewandte Numerik 1, SoSe 2022
+% Uebungsblatt 01, Aufgabe 03: Wert einer Gleitpunktdarstellung
+%
+% Testprogramm fuer die Funktion [d, v, t] = flp(b, m, n, x)
+%
+% Letzte Aenderung: 22.04.2022
+
+
+
+%% Initialisierung
+
+clearvars;
+close all;
+clc;
+
+fprintf('\n');
+fprintf('Angewandte Numerik 1, Sommersemester 2022\n');
+fprintf('Uebungsblatt 1, Aufgabe 3d: Gleitpunkt-Darstellung\n');
+fprintf('\n');
+
+
+
+%% Definition und Durchfuehrung der Testfaelle
+
+testfall = 0;
+while true                          % alle Testfaelle untersuchen
+    testfall = testfall + 1;        % naechster Testfall
+    
+    
+    %% alle Testfaelle definieren
+    
+    switch testfall
+        case 1                      % Testfall 1
+            b     = 2;                  % Basis
+            m     = 8;              	% Laenge der Mantisse
+            n     = 3;               	% Laenge des Exponenten
+            x     = 27.375;            	% zu konvertierende Zahl
+            dSoll = [1 1 0 1 1 0 1 1];  % Mantisse
+            vSoll = [1 0 1];            % Exponent
+            tSoll = 1;                  % Vorzeichen Exponent
+        case 2                      % Testfall 2
+            b     = 8;                  % Basis
+            m     = 8;              	% Laenge der Mantisse
+            n     = 3;               	% Laenge des Exponenten
+            x     = 27.375;            	% zu konvertierende Zahl
+            dSoll = [3 3 3 0 0 0 0 0];  % Mantisse
+            vSoll = [0 0 2];            % Exponent
+            tSoll = 1;                  % Vorzeichen Exponent
+        case 3                      % Testfall 3
+            b     = 2;                  % Basis
+            m     = 10;              	% Laenge der Mantisse
+            n     = 3;               	% Laenge des Exponenten
+            x     = 9.140625;          	% zu konvertierende Zahl
+            dSoll = [1 0 0 1 0 0 1 0 0 1];      % Mantisse
+            vSoll = [1 0 0];            % Exponent
+            tSoll = 1;                  % Vorzeichen Exponent
+        case 4                      % Testfall 4
+            b     = 8;                  % Basis
+            m     = 5;              	% Laenge der Mantisse
+            n     = 3;               	% Laenge des Exponenten
+            x     = 9.140625;          	% zu konvertierende Zahl
+            dSoll = [1 1 1 1 0];        % Mantisse
+            vSoll = [0 0 2];            % Exponent
+            tSoll = 1;                  % Vorzeichen Exponent
+        case 5                      % Testfall 5
+            b     = 2;                  % Basis
+            m     = 3;              	% Laenge der Mantisse
+            n     = 2;               	% Laenge des Exponenten
+            x     = 0.375;          	% zu konvertierende Zahl
+            dSoll = [1 1 0];            % Mantisse
+            vSoll = [0 1];              % Exponent
+            tSoll = -1;                 % Vorzeichen Exponent
+        case 6                      % Testfall 6
+            b     = 8;                  % Basis
+            m     = 5;              	% Laenge der Mantisse
+            n     = 2;               	% Laenge des Exponenten
+            x     = 0.0157470703125;   	% zu konvertierende Zahl
+            dSoll = [1 0 0 4 0];        % Mantisse
+            vSoll = [0 1];              % Exponent
+            tSoll = -1;                 % Vorzeichen Exponent
+        case 7                      % Testfall 7
+            b     = 2;                  % Basis
+            m     = 3;              	% Laenge der Mantisse
+            n     = 3;               	% Laenge des Exponenten
+            x     = 0.0625;          	% zu konvertierende Zahl
+            dSoll = [1 0 0];            % Mantisse
+            vSoll = [0 1 1];            % Exponent
+            tSoll = -1;                 % Vorzeichen Exponent
+        otherwise
+            break;                  % keine Testfaelle mehr vorhanden
+    end
+    
+    
+    %% Testfall durchfuehren und Ergebnis ausgeben
+
+    [d, v, t] = flp(b, m, n, x);
+    
+    spezString = 'Testfall %d: %s.\n';
+    if max(abs([d - dSoll, v - vSoll, t - tSoll])) == 0
+        fprintf(spezString, testfall, 'Bestanden');  
+    else
+        fprintf(2, spezString, testfall, 'Fehlgeschlagen');
+    end
+
+end

w1/matlab/testValue.m

@@ -0,0 +1,67 @@
+% Angewandte Numerik 1, SoSe 2022
+% Uebungsblatt 01, Aufgabe 03: Wert einer Gleitpunktdarstellung
+%
+% Testprogramm fuer die Funktion x = value(b, d, v, t)
+%
+% Letzte Aenderung: 22.04.2022
+
+
+
+%% Initialisierung
+
+clearvars;
+close all;
+clc;
+
+tol = 1e-14;                  % Geforderte Genauigkeit der Berechnungen
+
+fprintf('\n');
+fprintf('Angewandte Numerik 1, Sommersemester 2022\n');
+fprintf('Uebungsblatt 1, Aufgabe 3b: Gleitpunkt-Darstellung\n');
+fprintf('\n');
+
+
+
+%% Definition und Durchfuehrung der Testfaelle
+
+testfall = 0;
+while true                          % alle Testfaelle untersuchen
+    testfall = testfall + 1;        % naechster Testfall
+    
+    
+    %% alle Testfaelle definieren
+    
+    switch testfall
+        case 1                      % Testfall 1
+            b = 2;                      % Basis
+            d = [1 0 1 0 0 1 0 0];      % Mantisse
+            v = [0 0 1 0 0];            % Exponent
+            t = 1;                      % Vorzeichen Exponent
+            x = 10.25;                  % Wert der Zahl
+        case 2                      % Testfall 2
+            b = 2;                      % Basis
+            d = [1 0 1];                % Mantisse
+            v = [0 0 1];                % Exponent
+            t = -1;                     % Vorzeichen Exponent
+            x = 0.3125;                 % Wert der Zahl
+        case 3                      % Testfall 3
+            b = 8;                      % Basis
+            d = [7 5 0];                % Mantisse
+            v = [0 0 1];                % Exponent
+            t = -1;                     % Vorzeichen Exponent
+            x = 0.119140625;            % Wert der Zahl
+        otherwise
+            break;                  % keine Testfaelle mehr vorhanden
+    end
+    
+    
+    %% Testfall durchfuehren und Ergebnis ausgeben
+    
+    spezString = 'Testfall %d: %s.\n';
+    if abs(x - value(b,d,v,t)) < tol
+        fprintf(spezString, testfall, 'Bestanden');
+    else
+        fprintf(2, spezString, testfall, 'Fehlgeschlagen');
+    end
+    
+end

w1/matlab/value.m

@@ -0,0 +1,38 @@
+% EX3 a)
+
+function x = value(b, d, v, t)
+    % b: basis
+    % d: mantissa coefficients
+    % v: coefficients of the exponent
+    % t: sign
+    
+    if b < 2
+        disp('basis needs to be at least 2!')
+        return
+    elseif !(t == -1 || t == 1)
+        disp('t needs to be 1 or -1!')
+        return
+    end
+    exponent = (t * unConvert(v, b))
+    prefactor = b ^ exponent
+    mantissa = unConvertMantissa(d, b)
+    x = prefactor * mantissa
+end
+
+
+function val = unConvert(n, b)
+    nums = flip(n)
+    res = 0
+    for i = 1:length(n)
+        res += nums(i) * b^(i-1)
+    end
+    val = res
+end
+
+function val = unConvertMantissa(m, b)
+    res = 0
+    for i = 1:length(m)
+        res += m(i) * b^(-i)
+    end
+    val = res
+end

w3/gaussLR.m

@@ -0,0 +1,32 @@
+function A = gaussLR(A)
+    % Gaussian Elimination for reforming the matrix into left-lower and right-upper triangular matrix
+    % A: A R^(n \times n) matrix
+
+    % actually not used here (but it was planned to be used): bsxfun, which makes this possible: bsxfun(@times, [1,2;3,4], [1;2]) = [1,2;6,8]. It multiplies the first row by 1, and the second one by 2.
+    % newer (and actually (still not) used here): [1,2;3,4].*[1;2] = [1,2;6,8]
+
+    % for convenience
+    len = length(A)
+
+    for i = 1:len
+        prefactor = A(i,i)
+        if prefactor == 0
+            error("LR-Zerlegung failed. Please use pivotization, or check that the matrix is regular!")
+        end
+
+        % elimination (find R)
+        factors = zeros(len, 1);
+        for j = (i+1):len
+            factors(j) = A(j, i) / prefactor;
+        end
+
+        % A(:,i:len) describes the matrix A with only the last i cols (so, for example, a 3x3 matrix with (:,2,len) with len=3) is the last 2 cols of the matrix, and with that a 2x3 matrix 
+        
+        % extra step for traceability, could also be compressed to a single line
+        multmatrix = (-factors * A(i,i:len))
+        A(:,i:len) = A(:,i:len) + multmatrix;
+
+        % save factors for L
+        A(:,i) = A(:,i) + factors
+    end
+end

w3/lrPivot.m

@@ -0,0 +1,47 @@
+function [A, P] = lrPivot(A)
+    % Gaussian Elimination for reforming the matrix into left-lower and right-upper triangular matrix
+    % A: A R^(n \times n) matrix
+    % P: Permutations-Matrix
+
+    % actually not used here (but it was planned to be used): bsxfun, which makes this possible: bsxfun(@times, [1,2;3,4], [1;2]) = [1,2;6,8]. It multiplies the first row by 1, and the second one by 2.
+    % newer (and actually (still not) used here): [1,2;3,4].*[1;2] = [1,2;6,8]
+
+    % for convenience
+    len = length(A)
+    pvec = 1:len
+
+
+    % Generate LR-Zerlegung
+    for i = 1:len
+
+        % pivot
+        [maxVal, maxValPos] = max(abs(A(pvec(i:end), i)))
+
+        % swap
+        temp = pvec(i)
+        pvec(i) = pvec(maxValPos + i - 1)
+        pvec(maxValPos + i - 1) = temp
+
+        prefactor = A(pvec(i),i)
+        if prefactor == 0
+            error("LR-Zerlegung failed. Please check that the matrix is regular!")
+        end
+
+        % elimination (find R)
+        factors = zeros(len, 1)
+        for j = (i+1):len
+            factors(pvec(j)) = A(pvec(j), i) / prefactor
+        end
+
+        % A(:,i:len) describes the matrix A with only the last i cols (so, for example, a 3x3 matrix with (:,2,len) with len=3) is the last 2 cols of the matrix, and with that a 2x3 matrix 
+        
+        % extra step for traceability, could also be compressed to a single line
+        multmatrix = (-factors * A(pvec(i),i:len))
+        A(:,i:len) = A(:,i:len) + multmatrix
+
+        % save factors for L
+        A(:,i) = A(:,i) + factors
+    end
+
+    P = eye(len)(:,pvec)
+end

w3/solveLR.m

@@ -0,0 +1,30 @@
+function x_ret = solveLR(A, b)
+    len = length(b)
+    z = zeros(len,1)
+    
+    % solve Lz = b
+    for i = 1:len
+        z(i) = b(i);
+        
+        for j = 1:(i-1)
+            z(i) = z(i) - z(j) * A(i, j);
+        end
+
+        % no division necessary as L is unipotent
+    end
+
+    x = zeros(len,1)
+
+    % solve Rx = z
+    for i = len:-1:1
+        x(i) = z(i);
+
+        for j = len:-1:(i+1)
+            x(i) = x(i) - x(j) * A(i, j);
+        end
+        
+        x(i) = x(i) / A(i, i)
+    end
+
+    x_ret = x
+end

w3/solveLrPivot.m

@@ -0,0 +1,30 @@
+function x = solveLrPivot(A, P, b)
+    len = length(b)
+    z = zeros(len,1)
+    
+    % solve Lz = b
+    for i = 1:len
+        z(i) = b(i)
+        
+        for j = 1:(i-1)
+            z(i) = z(i) - z(j) * A(P(i), j)
+        end
+
+        % no division necessary as L is unipotent
+    end
+
+    x = zeros(len,1)
+
+    % solve Rx = z
+    for i = len:-1:1
+        x(i) = z(i)
+
+        for j = len:-1:(i+1)
+            x(i) = x(i) - x(j) * A(P(i), j)
+        end
+        
+        x(i) = x(i) / A(P(i), i)
+    end
+
+    x_ret = x
+end

w3/testGaussLR.m

@@ -0,0 +1,14 @@
+function testGaussLR()
+    success = testSingle( [ 1,1,1; 4,3,-1; 3,5,3 ], [1,1,1; 4,-1,-5; 3,-2,-10] )
+    success = success & testSingle( [1,0;0,1], [1,0;0,1] )
+    success = success & testSingle( [6,-4,7; -12,5,-12; 18,0,22], [6,-4,7; -2,-3,2; 3,-4,9] )
+    if success
+        disp("It works!")
+    else
+        disp("It doesn't work :/")
+    end
+end
+
+function success = testSingle(init, expected)
+    success = isequal(gaussLR(init), expected);
+end

w3/testSolve.m

@@ -0,0 +1,40 @@
+function testSolve
+     result = Test([0, 1; 1, 1], [1; 1])
+     result = result & Test([11,44,1; 0.1,0.4,3; 0,1,-1], [1; 1; 1])
+     result = result & Test([0.001,1,1; -1,0.004,0.004; -1000,0.004,0.000004], [1; 1; 1])
+
+    if result(1) % nopivot
+        disp("Pivot-less Solving worked!")
+    else
+        disp("Pivot-less Solving broke!")
+    end
+
+    if result(2) % pivot
+        disp("Pivot-full Solving worked!")
+    else
+        disp("Pivot-full Solving broke!")
+    end
+end
+
+function [nopivot, pivot] = Test(A, b)
+    try
+        nopivot = isequal(Solve(A, b), linsolve(A, b))
+    catch
+        nopivot = false
+    end
+
+    try
+        pivot = isequal(SolvePivot(A, b), linsolve(A, b))
+    catch
+        pivot = false
+    end
+end
+
+function x = Solve(A, b)
+    x = solveLR(gaussLR(A), b)
+end
+
+function x = SolvePivot(A, b)
+    [A, T] = lrPivot(A)
+    x = solveLrPivot(A, T, b)
+end

w3/testSolveLR.m

@@ -0,0 +1,14 @@
+function testSolveLR()
+    success = test( [1,3,2; 2,15,2; 1,3,4], [1;2;3] )
+    success = success & test ( [1,1,1; 4,3,-1; 3,5,3], [4;2;0] ) 
+    success = success & test ( [1, 0, 0; 0, 1, 0; 0, 0, 1], [4; 2; 0] )
+    if success
+        disp("It works!")
+    else 
+        disp("It broke!")
+    end
+end 
+
+function success = test(A, b)
+    success = isequal(linsolve(A, b), solveLR(gaussLR(A), b))
+end