Solving Systems of Equations by the Gaussian Elimination ...
Solving Systems of Equations by the Gaussian Elimination Method
Introduction:
The Gaussian Elimination Method is used to solve a system of N equations, expressed as A•x = b. b and x are matrices of length N and A is a NxN matrix. We desire to find x.
The A matrix can be decomposed into lower (L) and upper (U) matrices,
(A) • x = b (1)
(L•U) • x = b (2)
Multiplying (2) by L-1, the inverse of L,
U • x = L-1 • b ≡ f (3)
Where f is a new matrix of length N.
To get (3) it is not necessary to find L-1. Instead, we start by combining A and b into an augmented matrix A+, which is the A matrix with b in the N+1 column. A+ is then manipulated through row exchanges and row addition/subtraction to create U (an upper diagonal matrix) in the A part of the augmented matrix. f is the newly created N+1 column.
U • x = f can then be solved by,
1. Solving the Nth row and then substituting xN back into the (N-1)th row to get xN-1.
2. Repeat this process of back substitution until all elements of x are known.
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