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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 9 September 12, 2011 Prof. Kamesh Madduri Review of last class • Gaussian Elimination/LU decomposition 2 Improved Gaussian Elimination • 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 Row Interchanges Gaussian elimination breaks down if leading diagonal entry of remaining unreduced matrix is zero at any stage Easy fix: if diagonal entry in column k is zero, then interchange row k with some subsequent row having nonzero entry in column k and then proceed as usual If there is no nonzero on or below diagonal in column k , then there is nothing to do at this stage, so skip to next column Zero on diagonal causes resulting upper triangular matrix U to be singular, but LU factorization can still be completed Subsequent backsubstitution will fail, however, as it should for singular matrix Michael T. Heath Scientific Computing 48 / 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 Partial Pivoting In principle, any nonzero value will do as pivot, but in practice pivot should be chosen to minimize error propagation To avoid amplifying previous rounding errors when multiplying remaining portion of matrix by elementary elimination matrix, multipliers should not exceed 1 in magnitude This can be accomplished by choosing entry of largest magnitude on or below diagonal as pivot at each stage Such partial pivoting is essential in practice for numerically stable implementation of Gaussian elimination for general linear systems < interactive example > Michael T. Heath Scientific Computing 49 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination...
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 Spring '08
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 Determinant, Linear Systems, Types of Linear Systems

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