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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 • GaussianJordan Elimination 2 This class • Modified problems • ShermanMorrison 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 righthand side of linear system changes but matrix does not, then LU factorization need not be repeated to solve new system Only forward and backsubstitution need be repeated for new righthand 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 ShermanMorrison Formula Sometimes refactorization can be avoided even when matrix does change ShermanMorrison formula gives inverse of matrix resulting from rankone 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 nvectors Evaluation of formula requires O ( n 2 ) work (for matrixvector 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 RankOne Updating of Solution To solve linear system ( A uv T ) x = b with new matrix, use ShermanMorrison 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: RankOne Updating of Solution Consider rankone 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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