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4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). (a) Verify that A = 4 -4 8 3 -5 7 -8 Is a product of a L = 00 -1 1 2 0 1 and U = 4 3 -5 0 -2 2 0 0 2 (b) Use the LU factorization to solve Az = -4 for . To do so, first solve Ly=b, and after Uz = y. 6 (c) Find L¹ and U-1, then use these to find A showing that A is invertible.

4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). (a) Verify that A = 4 -4 8 3 -5 7 -8 Is a product of a L = 00 -1 1 2 0 1 and U = 4 3 -5 0 -2 2 0 0 2 (b) Use the LU factorization to solve Az = -4 for . To do so, first solve Ly=b, and after Uz = y. 6 (c) Find L¹ and U-1, then use these to find A showing that A is invertible.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). (a) Verify that A = 4 -4 8 3 -5 -5 7 6 -8 Is a product of a L = 0 -1 1 0 and U = 2 0 (b) Use the LU factorization to solve Az = H (c) Find L-¹ and U-¹, then use these to find A showing that A is invertible. 4 3 -5 0 -2 2 0 0 2 for . To do so, first solve Lij= 6, and after Uz = ý. (d) (Optional) To find the LU factorization, reduce A to U by using only row replacements operations. Second, place entries in L so that the same sequence of row operations reduces L to I. Follow this algorithm to find the LU decomposition of A. (Consult Section 2.5 for more details.)
Transcribed Image Text:4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). (a) Verify that A = 4 -4 8 3 -5 -5 7 6 -8 Is a product of a L = 0 -1 1 0 and U = 2 0 (b) Use the LU factorization to solve Az = H (c) Find L-¹ and U-¹, then use these to find A showing that A is invertible. 4 3 -5 0 -2 2 0 0 2 for . To do so, first solve Lij= 6, and after Uz = ý. (d) (Optional) To find the LU factorization, reduce A to U by using only row replacements operations. Second, place entries in L so that the same sequence of row operations reduces L to I. Follow this algorithm to find the LU decomposition of A. (Consult Section 2.5 for more details.)
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