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.