Question

I have implemented a Gauss-algorithm without pivoting.

import matplotlib.pyplot as plt
import numpy as np
import scipy as sp

def gauss_solve(A,b):
    """
    args: coefficient matrix A of dim(nxn) and vector b of dim(n) 
    of a system of linear equations with n unknowns.
    note: no zeroes on the main diagonal of A allowed!

    returns: vector x of dim(n) which solves the SLE
    """
    while np.ndim(A) != 2 or A.shape[0] != A.shape[1]:
        A = input(["The matrix you entered is not square, specify new input matrix A: "])
#    print "A ok."
    while np.ndim(b) != 1 or A.shape[1] != b.shape[0]:
        b = input(["The dimension of the constant vector b is incorrect, please specify new input vector b"])
#    print "b ok."
    if np.linalg.det(A) == 0:
        return "This linear system doesn't have a single unique solution."
#    print "System does have solution: "
    n = len(b)
    for i in xrange(n): # create triangular matrix
        if A[i,i] == 0:
            return "This implementation doesn't allow A to have zero entries on the main diagonal."
        A[i] = A[i]/float(A[i,i])
        b[i] = b[i]/float(A[i,i])
        for l in xrange(i+1,n):
            A[l] -= A[i]*A[l,i]
            b[l] -= b[i]*A[l,i]
    r = np.zeros(n) # result
    for i in xrange(n):
        r[-(i+1)] = b[-(i+1)] - np.dot(r,A[-(i+1)])
    return r

def test_gauss():
    m = 10
    e = 0.1
    A = sp.rand(m,m)
#    A,b = np.array([[e,1.],[1.,1.]]),np.array([1.,e])
    b = sp.rand(m)
    print gauss_solve(A,b)
    print "Build-in function says: \n", np.linalg.solve(A,b)

test_gauss()

The test-function can generate random entries for A and b. Everything works perfectly fine I think, but I have a matrix here which causes unexpected results:

A = [[e 1] [1 1]]
b = [1 e]

For e != 1 the analytical solution is

x = [-1 e+1]

But I tried some values for e and I just don’t get the analytical solutions. Even the build in function solve(A,b) fails. The first entry of x for instance is always 0 (though it should be -1, totally independent of e). Can anybody explain why this happens?

Answer 1

Your parallel updates to A and b are incorrect, since you’re updating b using the new values for A. You need to replace the lines:

A[i] = A[i]/float(A[i,i])
b[i] = b[i]/float(A[i,i])

with something like:

divisor = A[i,i]
A[i] = A[i]/float(divisor)
b[i] = b[i]/float(divisor)

and similarly, the lines:

A[l] -= A[i]*A[l,i]
b[l] -= b[i]*A[l,i]

with

multiplier = A[l,i]
A[l] -= A[i]*multiplier
b[l] -= b[i]*multiplier

In your original code, the lines for b do nothing (disregarding issues of floating-point precision): the first section of code divides b[i] by 1.0, while the second subtracts 0.0 times b[i] from b[l].