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NUMERICAL METHODS & OPTIMISATION Part II: Linear Algebraic Equations Raihana Edros Faculty of Engineering Technology [email protected] Linear Algebraic Equation By Raihana Edros For updated version, please click on
Chapter Description Aims Apply numerical methods in solving engineering problem and optimisation Expected Outcomes Solve simultaneous equations by using LU Decomposition and Matrix Inversion methods Apply linear algebraic equations to solve engineering problems References Steven C. Chapra and Raymond P. Canale (2009), Numerical Methods for Engineers, McGraw-Hill, 6 th Edition Linear Algebraic Equation By Raihana Edros
27/08/2017 RZE/2015/BTP2412 3 LU Decomposition Given the system of linear algebraic equation in matrix form: In LU decomposition, [A] is decomposed into upper triangle form known as [U] and lower triangle form known as [L] [ ] ú ú ú û ù ê ê ê ë é = 1 0 0 1 0 1 32 31 21 f f f L [ ] ú ú ú û ù ê ê ê ë é ¢ ¢ ¢ ¢ = 33 23 13 22 12 11 0 0 0 a a a a a a U [ ] { } { } B X A = Linear Algebraic Equation By Raihana Edros
Given [A] in the form of: This matrix can be decomposed into [L] & [U] by using the principles of Naïve-Gauss Elimination. [U] is a direct product of forward elimination: The factors used to eliminate a 21 , a 31 and a 32 are stored in [L] in the form of: [ ] ú ú ú û ù ê ê ê ë é = 33 23 13 32 22 12 31 21 11 a a a a a a a a a A Decomposition of [A] into [U] and [L] [ ] ú ú ú û ù ê ê ê ë é = 33 23 13 22 12 11 ' ' ' 0 ' 0 0 a a a a a a U [ ] ú ú ú û ù ê ê ê ë é = 1 0 1 0 0 1 32 31 21 f f f L Linear Algebraic Equation By Raihana Edros
85 . 7 2 . 0 1 . 0 3 3 2 1 = - - x x x 3 . 19 3 . 0 7 1 . 0 3 2 1 - = - + x x x 4 . 71 10 2 . 0 3 . 0 3 2 1 = + - x x x ú ú ú û ù ê ê ê ë é - - - - 10 3 . 0 2 . 0 2 . 0 7 1 . 0 3 . 0 1 . 0 3 1. Use Naïve-Gauss Elimination to obtain [U]: LU Decomposition: Example 10.1 & 10.2 ] [ 0120 . 10 293333 . 0 2 . 0 0 00333 . 7 1 . 0 0 0 3 U = ú ú ú û ù ê ê ê ë é - - - [ ] ú ú ú û ù ê ê ê ë é = 1 0 0 1 0 1 32 31 21 f f f L 11 21 21 a a f = 11 31 31 a a f = 22 32 32 a a f ¢ ¢ = Linear Algebraic Equation By Raihana Edros =608&notifyeditingon=1
] [ 1 0 0 0271300 . 0
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