feat: completed w3 contents

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

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+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

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+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

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+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