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Unformatted text preview: Gaussian Elimination LU Factorisation Pivoting LU Factorisation Dhavide Aruliah Faculty of Science MATH 2070/2072 Gaussian Elimination LU Factorisation Pivoting Outline Gaussian Elimination EasytoSolve Systems Elimination LU Factorisation Pivoting Gaussian Elimination LU Factorisation Pivoting Outline Gaussian Elimination EasytoSolve Systems Elimination LU Factorisation Pivoting Gaussian Elimination LU Factorisation Pivoting I Want to solve n n system of linear equations Ax = b Gaussian Elimination LU Factorisation Pivoting I Want to solve n n system of linear equations Ax = b I Backslash operator does this in MATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b Gaussian Elimination LU Factorisation Pivoting I Want to solve n n system of linear equations Ax = b I Backslash operator does this in MATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b I Read x = A \ B as x = A 1 b Gaussian Elimination LU Factorisation Pivoting I Want to solve n n system of linear equations Ax = b I Backslash operator does this in MATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b I Read x = A \ B as x = A 1 b I Present goal: to understand what \ does I Gaussian elimination I LU factorisation I Pivoting Gaussian Elimination LU Factorisation Pivoting I Want to solve n n system of linear equations Ax = b I Backslash operator does this in MATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b I Read x = A \ B as x = A 1 b I Present goal: to understand what \ does I Gaussian elimination I LU factorisation I Pivoting I See doc mldivide left matrix division Gaussian Elimination LU Factorisation Pivoting I Want to solve n n system of linear equations Ax = b I Backslash operator does this in MATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b I Read x = A \ B as x = A 1 b I Present goal: to understand what \ does I Gaussian elimination I LU factorisation I Pivoting I See doc mldivide left matrix division Solution of Ax = b Never solve linear systems by computing A 1 and x = A 1 b ! Gaussian Elimination LU Factorisation Pivoting Easy Systems Diagonal Systems I 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 Dx = b , i.e., d 1 d 2 . . . d n x 1 x 2 . . . x n = b 1 b 2 . . . b n Gaussian Elimination LU Factorisation Pivoting Easy Systems Diagonal Systems I 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 Dx = b , i.e., d 1 d 2 ....
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 Spring '10
 aruliahdhavidhe
 Gaussian Elimination

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