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In this assignment you will implement partial row pivoting on a coefficient matr
In this assignment you will implement partial row pivoting on a coefficient matrix during LU factorization. In notebook #21, we show an algorithm to transform a square coefficient matrix into upper triangular form U. Then we show a modified algorithm that transforms a square coefficient matrix into L+U+I form, where L is a canonical lower triangular matrix, U is the upper triangular form from the first algorithm, and I is the identity. Consider the coefficient matrix
�=[1030013−13−306024−6]
1. (20 points) Apply the initial upper triangular algorithm to this system. What issue do you observe? Demonstrate the problem in your notebook and explain the problem in a Markdown cell.
2. (20 points) Implement partial row pivoting in the upper triangular algorithm. The idea is to search all rows below the current row i in the row iteration to find the row j such that if rows i and j were swapped (exchanged), the swapped in row would have the largest diagonal element in absolute value.
3. (20 points) Implement partial row pivoting in the L+U+I form algorithm and save the indices of the rows that are pivoted in a data structure.
4. (20 points) Show A = LU. Note you need to first apply the same pivots to A.
5. (20 points) Use factors L and U to solve Ax = b for x where
�=[6043]
using the forward and backward substitution methods shown in notebook #21.
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