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chapt1 - ME 4905 Advanced Numerical Methods Solutions of...

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ME 4905 Advanced Numerical Methods Solutions of Linear Systems: Iterative Methods Lecturer: T.Y. Ng (PhD) Email: [email protected]
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2 Equations – 2 Unknowns 3 possibilities: 1.No solution 2.one solution 3.Infinite many solutions
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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 | | | |
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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 0 0 0 0 0 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 | | | | | 0 0 0 0 0 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
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3 × 3 System
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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 0 0 0 0 0
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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 1 1 2 1 2 1 2 1 2 22 21 1 12 11 n i b x a i j ij , , 2 , 1 K = = Or in a more compact form The vector x is the solution . Alternatively, we can think about the projection of all a i onto x .
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