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Task/Gaussian-elimination/Python/gaussian-elimination.py

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+# The 'gauss' function takes two matrices, 'a' and 'b', with 'a' square, and it return the determinant of 'a' and a matrix 'x' such that a*x = b.
+# If 'b' is the identity, then 'x' is the inverse of 'a'.
+
+import copy
+from fractions import Fraction
+
+def gauss(a, b):
+    a = copy.deepcopy(a)
+    b = copy.deepcopy(b)
+    n = len(a)
+    p = len(b[0])
+    det = 1
+    for i in range(n - 1):
+        k = i
+        for j in range(i + 1, n):
+            if abs(a[j][i]) > abs(a[k][i]):
+                k = j
+        if k != i:
+            a[i], a[k] = a[k], a[i]
+            b[i], b[k] = b[k], b[i]
+            det = -det
+
+        for j in range(i + 1, n):
+            t = a[j][i]/a[i][i]
+            for k in range(i + 1, n):
+                a[j][k] -= t*a[i][k]
+            for k in range(p):
+                b[j][k] -= t*b[i][k]
+
+    for i in range(n - 1, -1, -1):
+        for j in range(i + 1, n):
+            t = a[i][j]
+            for k in range(p):
+                b[i][k] -= t*b[j][k]
+        t = 1/a[i][i]
+        det *= a[i][i]
+        for j in range(p):
+            b[i][j] *= t
+    return det, b
+
+def zeromat(p, q):
+    return [[0]*q for i in range(p)]
+
+def matmul(a, b):
+    n, p = len(a), len(a[0])
+    p1, q = len(b), len(b[0])
+    if p != p1:
+        raise ValueError("Incompatible dimensions")
+    c = zeromat(n, q)
+    for i in range(n):
+        for j in range(q):
+                c[i][j] = sum(a[i][k]*b[k][j] for k in range(p))
+    return c
+
+
+def mapmat(f, a):
+    return [list(map(f, v)) for v in a]
+
+def ratmat(a):
+    return mapmat(Fraction, a)
+
+# As an example, compute the determinant and inverse of 3x3 magic square
+
+a = [[2, 9, 4], [7, 5, 3], [6, 1, 8]]
+b = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
+det, c = gauss(a, b)
+
+det
+-360.0
+
+c
+[[-0.10277777777777776, 0.18888888888888888, -0.019444444444444438],
+[0.10555555555555554, 0.02222222222222223, -0.061111111111111116],
+[0.0638888888888889, -0.14444444444444446, 0.14722222222222223]]
+
+# Check product
+matmul(a, c)
+[[1.0, 0.0, 0.0], [5.551115123125783e-17, 1.0, 0.0],
+[1.1102230246251565e-16, -2.220446049250313e-16, 1.0]]
+
+# Same with fractions, so the result is exact
+
+det, c = gauss(ratmat(a), ratmat(b))
+
+det
+Fraction(-360, 1)
+
+c
+[[Fraction(-37, 360), Fraction(17, 90), Fraction(-7, 360)],
+[Fraction(19, 180), Fraction(1, 45), Fraction(-11, 180)],
+[Fraction(23, 360), Fraction(-13, 90), Fraction(53, 360)]]
+
+matmul(a, c)
+[[Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)],
+[Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)],
+[Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)]]