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chap02_8up

chap02_8up - Existence Uniqueness and Conditioning Solving...

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Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Scientific Computing: An Introductory Survey Chapter 2 – Systems of Linear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 87 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Outline 1 Existence, Uniqueness, and Conditioning 2 Solving Linear Systems 3 Special Types of Linear Systems 4 Software for Linear Systems Michael T. Heath Scientific Computing 2 / 87 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 / 87 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 ) = 0 3 rank( A ) = n 4 For any vector z = 0 , Az = 0 Michael T. Heath Scientific Computing 4 / 87 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 Existence and uniqueness of solution to Ax = b depend on whether A is singular or nonsingular Can also depend on b , but only in singular case If b span ( A ) , system is consistent A b # solutions nonsingular arbitrary one (unique) singular b span ( A ) infinitely many singular b / span ( A ) none Michael T. Heath Scientific Computing 5 / 87 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Geometric Interpretation In two dimensions, each equation determines straight line in plane Solution is intersection point of two lines If two straight lines are not parallel (nonsingular), then intersection point is unique If two straight lines are parallel (singular), then lines either do not intersect (no solution) or else coincide (any point along line is solution) In higher dimensions, each equation determines hyperplane; if matrix is nonsingular, intersection of hyperplanes is unique solution Michael T. Heath

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chap02_8up - Existence Uniqueness and Conditioning Solving...

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