lecture5 - CMPSC/MATH 451 Numerical Computations Lecture 5...

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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 5 Aug 31, 2011 Prof. Kamesh Madduri Systems of Linear Equations Covered on blackboard, corresponding slides from textbook follow. 2 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Systems of Linear Equations Given m n matrix A and m-vector b , find unknown n-vector x satisfying Ax = b System of equations asks Can b be expressed as linear combination of columns of A ? If so, coefficients of linear combination are given by components of solution vector x Solution may or may not exist, and may or may not be unique For now, we consider only square case, m = n Michael T. Heath Scientific Computing 3 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Singularity and Nonsingularity n n matrix A is nonsingular if it has any of following equivalent properties 1 Inverse of A , denoted by A- 1 , exists 2 det( A ) 6 = 0 3 rank( A ) = n 4 For any vector z 6 = , Az 6 = Michael T. Heath Scientific Computing 4 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Existence and Uniqueness...
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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 Pennsylvania State University, University Park.

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lecture5 - CMPSC/MATH 451 Numerical Computations Lecture 5...

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