Chapt1 - ME 4905 Advanced Numerical Methods Solutions of Linear Systems Iterative Methods Lecturer T.Y Ng(PhD Email [email protected] 2 Equations

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Unformatted text preview: ME 4905 Advanced Numerical Methods Solutions of Linear Systems: Iterative Methods Lecturer: T.Y. Ng (PhD) Email: [email protected] 2 Equations – 2 Unknowns 3 possibilities: 1.No solution 2.one solution 3.Infinite many solutions Linear Systems               =                           n n nn n n n n b b b x x x a a a a a a a a a M M L M O M M L L 2 1 2 1 2 1 2 22 21 1 12 11 Many mathematical models in engineering and sciences are involved in solving linear system. Typical examples are linear circuits, system of spring-mass model, etc.             n nn n n n n b b b a a a a a a a a a M L M O M M L L 2 1 2 1 2 22 21 1 12 11 | | | | Upper-Triangular System                   =                                   n n nn n b b b x x x a a a a a M M M M L L M O O M M O O M M O O L L 2 1 2 1 22 1 12 11 ii n i j j ij n i i n n n n n n n n nn n n n a x a a x a x a a x a a x ∑ + = +--- +-- +- =- = = 1 1 , 1 , 1 , 1 1 , 1 1 1 , M How do we solve it?                 + + + 1 , 1 , 2 1 , 1 22 1 12 11 | | | | | n n n n nn n a a a a a a a a M M L L M O O M M O O M M O O L L 3 × 3 System Gaussian Elimination Elementary Row Operation               =                           n n nn n n n n b b b x x x a a a a a a a a a M M L M O M M L L 2 1 2 1 2 1 2 22 21 1 12 11                   =                                   n n nn b b b x x x a a a a a M M M M L L M O O M M O O M M O O L L 2 1 2 1 22 11 12 11 Geometric Aspect             = ⋅ = ⋅ = ⋅ ⇔               =                           n n n n nn n n n n b b b b b b x x x a a a a a a a a a x a x a x a M M M L M O M M L L 2 2...
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This note was uploaded on 01/17/2010 for the course ENG 91301 taught by Professor Lui during the Spring '08 term at Hong Kong Institute of Vocational Education.

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Chapt1 - ME 4905 Advanced Numerical Methods Solutions of Linear Systems Iterative Methods Lecturer T.Y Ng(PhD Email [email protected] 2 Equations

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