LU-Factorisations

LU-Factorisations - Gaussian Elimination LU Factorisation...

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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 Easy-to-Solve Systems Elimination LU Factorisation Pivoting Gaussian Elimination LU Factorisation Pivoting Outline Gaussian Elimination Easy-to-Solve 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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LU-Factorisations - Gaussian Elimination LU Factorisation...

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