mase201sp09_Feb03_printable - Solving Systems of Linear...

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Solving Systems of Linear Equations using MATLAB (Review) MATLAB has built-in functions for solving systems of linear equations s These functions solve a system of linear equations without explicitly calculating a matrix inverse : • Gauss elimination method (with pivoting) • Gauss-Jordan method s We will: 2/03/2009 1 s Review how these methods and functions work s Apply them to engineering problems s Learn how they scale with problem size (i.e. the number of equations we want to solve) s Learn how to calculate a matrix inverse s Discuss limitations and caveats of the methods and functions MASE201 Spring 2009 Objective: “Look under the hood” and become intelligent users of MATLAB’s tools Why use matrix left divide rather than inverse of [A]? Given [A] [x] = [b], then [A] -1 [A] [x] = [A] -1 [b] Since [I]= [A] -1 [A] (def’n of a matrix inverse) then [I] [x] = [A] -1 [b] ince an entity matrix time an ector Since an R x R identity matrix time an R x 1 vector equals that R x 1 vector: then [x] = [A] -1 [b] MATLAB can compute a matrix inverse: a = [1 2; 3 4]; inv(a) % invert a inv(a)*a %result is a 2x2 identity matrix 2/03/2009 2 MASE201 Spring 2009 More computations are needed to invert a matrix • Computing an inverse takes more computational steps than using Gauss elimination – For small matrices, we won’t even notice the extra time – For a very large matrix or if the computation is repeated many times, the choice of method can make difference a difference 2/03/2009 MASE201 Spring 2009 3 500 1000 1500 2000 2500 0 10 20 30 square matrix size computation time (sec) a\b inv(a)*b Solving with Cramer’s rule • Cramer’s rule (2 x 2 matrix): = = = 12 11 2 12 1 11 2 12 11 22 2 12 1 1 2 1 2 1 22 21 12 11 et det t det : a a b a b a x a a a b a b x Find b b x x a a a a • Cramer’s rule (general form) where a j is the matrix formed by replacing the j th column of a with b 2/03/2009 4 22 21 22 21 det det a a a a ( ) ( ) a a det ' det j j x = MASE201 Spring 2009
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Calculating a determinant using MATLAB • If we want to use Cramer’s rule, then we need to calculate the determinant of a and every a j • How does MATLAB calculate a determinant? – It does NOT use cofactors and minors – It uses special properties of upper triangular and lower triangular matrices: • If [c] is an upper triangular matrix or a lower triangular matrix, det([c]) is equal to the product of the diagonal elements of [c] – The forward elimination step in the Gauss elimination
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mase201sp09_Feb03_printable - Solving Systems of Linear...

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