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LU-iter

# LU-iter - Week 10 Tri-diagonal Matrices(brief LU...

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Week 10 - Tri-diagonal Matrices (brief) - LU factorization - Iterative computations to solve a linear system

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Tridiagonal Matrices Naive/Scaled pivot Runtime: O(n 3 ) Tridiagonal/banded Runtime: O(n)
Matrix review 1 0 0 -1 1 0 -1 0 1 [ ][ ] [ ] 1 1 1 1 2 3 = 1 3 3 1 1 1 0 1 2 0 2 2 R2= R2 - R1 R3= R3 - R1

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Matrix review 1 0 0 0 1 0 0 -2 1 [ ] [ ] 1 1 1 0 1 2 0 2 2 R3= R3 - 2*R2 =
Matrix review 1 0 0 0 1 0 0 -2 1 [ ] [ ] 1 1 1 0 1 2 0 0 -2 R3= R3 - 2*R2 = [ ] 1 1 1 0 1 2 0 2 2

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LU factorization 1 0 0 0 1 0 0 -2 1 [ ] [ ] 1 1 1 0 1 2 0 0 -2 = [ ] 1 0 0 -1 1 0 -1 0 1 [ ] 1 1 1 1 2 3 1 3 3 A U L' L'*A = U
LU factorization Have: L'*A = U Want: A = L*U Multiply by L' -1 : L' -1 *L'*A = L' -1 *U I*A = L' -1 *U A = L' -1 *U

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LU factorization 1 0 0 0 1 0 0 -2 1 ([ ] [ ]) 1 0 0 -1 1 0 -1 0 1 -1 [ ] 1 0 0 0 1 0 0 -2 1 1 0 0 -1 1 0 -1 0 1 [ ] -1 -1 =
LU factorization [ ] 1 0 0 0 1 0 0 -2 1 1 0 0 -1 1 0 -1 0 1 [ ] -1 -1 = [ ] 1 0 0 0 1 0 0 2 1 1 0 0 1 1 0 1 0 1 [ ] = [ ] 1 0 0 1 1 0 1 2 1

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LU factorization 1 0 0 1 1 0 1 2 1 [ ][ ] 1 1 1 0 1 2 0 0 -2 [ ] 1 1 1 1 2 3 1 3 3 A U L = “L” is the inverse of the product of row-reduction matrices “U” is the row reduced matrix
LU factorization If A is symmetric, then U*(L T ) -1 = D is diagonal.

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LU-iter - Week 10 Tri-diagonal Matrices(brief LU...

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