lecture12 - CMPSC/MATH 451 Numerical Computations Lecture...

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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 12 September 19, 2011 Prof. Kamesh Madduri Review of last class • Operation count for Gaussian Elimination • Studied Octave code: badlufact, lutx • Uniqueness of LU factorization • Viewing matrices as submatrices • Operation count for Inversion • Gaussian-Jordan Elimination 2 This class • Modified problems • Sherman-Morrison formula • Improving accuracy – Diagonal scaling – Iterative refinement • Cholesky factorization • Factoring banded systems • Covered on blackboard, corresponding slides from textbook follow. 3 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Solving Modified Problems If right-hand side of linear system changes but matrix does not, then LU factorization need not be repeated to solve new system Only forward- and back-substitution need be repeated for new right-hand side This is substantial savings in work, since additional triangular solutions cost only O ( n 2 ) work, in contrast to O ( n 3 ) cost of factorization Michael T. Heath Scientific Computing 67 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Sherman-Morrison Formula Sometimes refactorization can be avoided even when matrix does change Sherman-Morrison formula gives inverse of matrix resulting from rank-one change to matrix whose inverse is already known ( A- uv T )- 1 = A- 1 + A- 1 u (1- v T A- 1 u )- 1 v T A- 1 where u and v are n-vectors Evaluation of formula requires O ( n 2 ) work (for matrix-vector multiplications) rather than O ( n 3 ) work required for inversion Michael T. Heath Scientific Computing 68 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Rank-One Updating of Solution To solve linear system ( A- uv T ) x = b with new matrix, use Sherman-Morrison formula to obtain x = ( A- uv T )- 1 b = A- 1 b + A- 1 u (1- v T A- 1 u )- 1 v T A- 1 b which can be implemented by following steps Solve Az = u for z , so z = A- 1 u Solve Ay = b for y , so y = A- 1 b Compute x = y + (( v T y ) / (1- v T z )) z If A is already factored, procedure requires only triangular solutions and inner products, so only O ( n 2 ) work and no explicit inverses Michael T. Heath Scientific Computing 69 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Example: Rank-One Updating of Solution Consider rank-one modification 2 4- 2 4 9- 3- 2- 1 7 x 1 x 2 x 3 = 2 8 10...
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This note was uploaded on 01/19/2012 for the course CMPSC 451 taught by Professor Staff during the Spring '08 term at Penn State.

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lecture12 - CMPSC/MATH 451 Numerical Computations Lecture...

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