17-LU-Factorisation-4UP

17-LU-Factorisation-4UP - Easy systems Gaussian elimination...

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Easy systems Gaussian elimination LU factorisation LU Factorisation Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 1 / 27 Easy systems Gaussian elimination LU factorisation LU Factorisation 1 Easy-to-solve systems 2 Gaussian elimination 3 LU factorisation c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 2 / 27 Easy systems Gaussian elimination LU factorisation Linear systems of equations Rule for matrix multiplication permits representation of linear systems of equations using matrices and vectors e.g., linear system of equations 2 x 1 + x 2 + x 3 = 4 4 x 1 + 3 x 2 + 3 x 3 + x 4 = 11 8 x 1 + 7 x 2 + 9 x 3 + 5 x 4 = 29 6 x 1 + 7 x 2 + 9 x 3 + 8 x 4 = 30 can be written as A x = b with 2 1 1 0 4 3 3 1 8 7 9 5 6 7 9 8 | {z } A x 1 x 2 x 3 x 4 | {z } x 4 11 29 30 | {z } b c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 4 / 27 Easy systems Gaussian elimination LU factorisation Solving linear systems of equations in M ATLAB Want to solve n × n system of linear equations A x = b Backslash operator does this in M ATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b Read x = A \ b as x = A - 1 b c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 5 / 27
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Easy systems Gaussian elimination LU factorisation Solving linear systems in M ATLAB Present goal: to understand what \ does I Gaussian elimination I LU factorisation I Pivoting See doc mldivide left matrix division Solution of A x = b Never solve linear systems by computing A - 1 and x = A - 1 b ! Use backslash ( mldivide ) or some solver enclosing sensible algorithms c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 6 / 27 Easy systems Gaussian elimination LU factorisation Diagonal systems Given vector b = ( b 1 , . . . , b n ) T R n and diagonal matrix D = diag ( d ) with d = ( d 1 , . . . , d n ) T R n , wish to solve linear system of equations D x = b , i.e., d 1 d 2 . . . d n x 1 x 2 . . . x n = b 1 b 2 . . . b n Solution of D x = b directly computable: x k = b k d k ( d k 6 = 0, k = 1: n ) c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 7 / 27 Easy systems Gaussian elimination LU factorisation Diagonal system example Solve the linear system of equations 2 3 - 4 x 1 x 2 x 3 = 5 9 1 2 x 1 = 5 x 1 = 5 2 3 x 2 = 9 x 2 = 9 3 = 3 - 4 x 3 = 1 x 3 = - 1 4 Equations are completely decoupled c ± D. Aruliah (UOIT) LU Factorisation MATH 2070U 8 / 27 Easy systems Gaussian elimination LU factorisation
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This note was uploaded on 02/23/2010 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Spring '10 term at UOIT.

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17-LU-Factorisation-4UP - Easy systems Gaussian elimination...

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