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Consider the linear system Ax=b, where

A=[ 4 1 -2 [ -3
−3 7 −3 b= -5
5 1 7 ] and -1]

We want to approximate the solution using the Gauss- Seidel Method.
The matrix A can be split up as A=L+D+U, where
L=[ ]
D=[ ]
U=[ ]

1 Answer

1 vote

For the given linear system Ax=b with A=[4 1 -2; -3 7 -3; 5 1 7] and b=[-3; -5; -1], the Gauss-Seidel Method requires splitting A into L, D, and U matrices. The split is L=[0 0 0; 3 0 0; -5 -1 0], D=[4 0 0; 0 7 0; 0 0 7], and U=[0 1 -2; 0 0 -3; 0 0 0].

The linear system Ax = b using the Gauss-Seidel method, with each step explained in under 200 words:

Step 1: Splitting matrix A into L, D, and U

1. Identify the lower triangular matrix (L):This matrix consists of all elements below the diagonal of the original matrix A. In this case, L is:

[ 0 0 0 ]

[-3 0 0 ]

[ 5 1 0 ]

2. Identify the diagonal matrix (D):This matrix consists of only the diagonal elements of the original matrix A. In this case, D is:

[ 4 0 0 ]

[ 0 7 0 ]

[ 0 0 7 ]

3. Calculate the upper triangular matrix (U):This matrix is obtained by subtracting L and D from the original matrix A. In this case, U is:

[ 4 1 -2 ]

[ 0 7 -3 ]

[ 0 0 7 ]

Step 2: Initialize an initial guess for x

Set an initial guess for the solution vector x. A common practice is to start with all zeros, but you can also use any other approximation you might have. In this case, let's start with:

x = [0, 0, 0]

Step 3: Perform Gauss-Seidel iteration

1. Iterate through each row of the matrix A (i = 1 to n).

2. For each row, update the corresponding element of x using the formula:

x_i = (b_i - sum(A_ij * x_j for j < i)) / A_ii

where:

i is the current row index

b_i is the i-th element of the b vector

A_ij is the element at row i, column j of matrix A

x_j is the j-th element of the current approximation of x

3. Repeat steps 1 and 2 until the difference between subsequent iterations of x is less than a desired tolerance value. This indicates that the solution has converged.

User Hanky Panky
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