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# lec5 - Introduction to Simulation Lecture 5 QR...

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Introduction to Simulation - Lecture 5 QR Factorization Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

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SMA-HPC ©2003 MIT Singular Example QR Factorization LU Factorization Fails Strut Joint Load force The resulting nodal matrix is SINGULAR, but a solution exists!
SMA-HPC ©2003 MIT Singular Example QR Factorization LU Factorization Fails 1 v1 v2 v3 v4 1 The resulting nodal matrix is SINGULAR, but a solution exists!

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SMA-HPC ©2003 MIT Singular Example QR Factorization Recall weighted sum of columns view of systems of equations 1 1 2 2 1 2 N N N x b x b M M M x b = " G G G " # # " 1 1 2 2 N N x M x M x M b + + + = G G G " M is singular but b is in the span of the columns of M
SMA-HPC ©2003 MIT Orthogonalization QR Factorization If M has orthogonal columns 0 i j M M i j = G G Multiplying the weighted columns equation by ith column: Orthogonal columns implies: ( ) 1 1 2 2 i N N i M x M x M x M M b + + + = G G G G G " Simplifying using orthogonality: ( ) ( ) i i i i i i i i M b x M M M b x M M = = G G G G G G

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SMA-HPC ©2003 MIT Orthogonalization QR Factorization Orthonormal M - Picture 0 and 1 i j i i M M i j M M = = G G G G M is orthonormal if: Picture for the two-dimensional case 1 M G 1 M G b b Orthogonal Case 2 x 1 x 2 M G 2 M G Non-orthogonal Case
SMA-HPC ©2003 MIT Orthogonalization QR Factorization QR Algorithm Key Idea 1 1 2 2 1 2 N N N x b x b M M M x b Original Matrix = " G G G " # # " ±²²²²³²²²²´ 1 1 2 2 1 2 N N N y b y b Q Q Q y b Matrix with Orthonormal Columns = " G G G " # # " ±²²²³²²²´ T Qy b y Q b = = How to perform the conversion?

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SMA-HPC ©2003 MIT Orthogonalization QR Factorization Projection Formula 1 2 2 2 12 1 Given , find so that , = M M Q M r M G G G G G ( ) 1 2 1 2 12 1 0 M Q M M r M = = G G G G G 1 2 12 1 1 M M r M M = G G G G 2 M G 2 Q G 1 M G 12 r
SMA-HPC ©2003 MIT Orthogonalization QR Factorization Normalization 1 1 1 1 1 11 1 1 1 1 1 Q M M Q Q M M r = = = G G G G G G G Formulas simplify if we normalize 1 2 12 2 2 1 = Now find so that 0 r Q M Q Q Q = G G G G G µ µ 12 1 2 r Q M = G G 2 2 2 2 22 2 Fi 1 al 1 n ly r Q Q Q Q Q = = G G G µ µ G G µ µ

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SMA-HPC ©2003 MIT Orthogonalization QR Factorization How was a 2x2 matrix converted?
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