feat: completed w3 contents

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