chap02_8up

chap02_8up - Existence, Uniqueness, and Conditioning...

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Unformatted text preview: Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Scientic 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 Scientic 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 Scientic 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 , nd unknown n-vector x satisfying Ax = b System of equations asks Can b be expressed as linear combination of columns of A ? If so, coefcients 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 Scientic 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 ) 6 = 0 3 rank( A ) = n 4 For any vector z 6 = , Az 6 = Michael T. Heath Scientic 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 ) innitely many singular b / span ( A ) none Michael T. Heath Scientic 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...
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chap02_8up - Existence, Uniqueness, and Conditioning...

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