numan_c1 - MT 3802 NUMERICAL ANALYSIS 2008/2009 Dr Clare E...

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Unformatted text preview: MT 3802 NUMERICAL ANALYSIS 2008/2009 Dr Clare E Parnell and Dr St ephane R egnier October 15, 2008 Contents 0 Handout 1 0.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.3 Books and Website . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Vector and Matrix Norms 4 1.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The Basis and Dimension of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Vector Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Inner Product Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.3 Commonly Used Norms and Normed Linear Spaces . . . . . . . . . . . . . . . 8 1.4 Sub-ordinate Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Commonly Used Sub-ordinate Matrix Norms . . . . . . . . . . . . . . . . . . 10 1.5 Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 The Condition Number and Ill-Conditioned Matrices . . . . . . . . . . . . . . . . . . 14 1.6.1 An Example of Using Norms to Solve a Linear System of Equations . . . . . 15 1.7 Some Useful Results for Determining the Condition Number . . . . . . . . . . . . . . 17 1.7.1 Finding Norms of Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7.2 Limits on the Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Examples of Ill-Conditioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Iterative Methods 21 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Single Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Sequences of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 i 2.2.1 The Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 Convergence of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.3 Spectral radius and rate of convergence . . . . . . . . . . . . . . . . . . . . . 25 2.2.4 Gerschgorins Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 The Jacobi Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Convergence of the Jacobi Iteration Method . . . . . . . . . . . . . . . . . . . 29 2.4 The Gauss-Seidel Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 The Successive Over Relaxation Iterative Method . . . . . . . . . . . . . . . . . . . .The Successive Over Relaxation Iterative Method ....
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This note was uploaded on 09/28/2009 for the course MATH MATH427 taught by Professor Dr.sharpey during the Spring '09 term at Monmouth IL.

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numan_c1 - MT 3802 NUMERICAL ANALYSIS 2008/2009 Dr Clare E...

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