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

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CMPSC/MATH 451 Numerical Computations Lecture 5 Aug 31, 2011 Prof. Kamesh Madduri
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Systems of Linear Equations Covered on blackboard, corresponding slides from textbook follow. 2
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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
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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 = 0 , Az 6 = 0 Michael T. Heath Scientific Computing 4 / 88
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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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