Question

Solve the following system of equations using LU decomposition with partial pivoting. Show all steps of the computation. Show all steps of the computation. 2x1−6x2−x3=−38−3x1−x2+7x3=−34−8x1+x2−2x3=−20

Answer 1

To solve the system of equations using LU decomposition with partial pivoting, form the augmented matrix, reorder rows for numerical stability, decompose to L and U matrices using forward elimination, and solve for the solution vector using forward and back substitution.

Step-by-step explanation:

The system of equations given is:

• 2x1 - 6x2 - x3 = -38
• -3x1 - x2 + 7x3 = -34
• -8x1 + x2 - 2x3 = -20

The steps to solve this system using LU decomposition with partial pivoting are:

1. Form the augmented matrix of the system.
2. Begin the LU decomposition process with partial pivoting, which involves reordering the rows based on the magnitude of the elements in the first column to ensure numerical stability.
3. Use forward elimination to decompose the matrix into a lower triangular matrix L and an upper triangular matrix U.
4. Apply the forward substitution method to solve Ly = b where b is the constant vector.
5. Use back substitution to solve Ux = y to obtain the solution vector x.

After finding the matrices L and U, the intermediate variable vector y is determined, followed by the solution vector x.