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!
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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 11 22 12 N NN xb MM M ⎤⎡⎤ ⎡⎤ ↑↑ ⎥⎢⎥ ⎢⎥ = ↓↓ ⎣⎦ ⎦⎣⎦ " GG G " ## " 2 2 xM x M b + ++ = G " M is singular but b is in the span of the columns of M
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SMA-HPC ©2003 MIT Orthogonalization QR Factorization If M has orthogonal columns 0 ij M Mi j =≠ G G Multiplying the weighted columns equation by ith column: Orthogonal columns implies: ( ) 11 2 2 iN N i M xM x M M b •++ + = GGG GG " Simplifying using orthogonality: () i ii i i i ii M b xM M M b x MM •=•⇒= G G G G
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SMA-HPC ©2003 MIT Orthogonalization QR Factorization Orthonormal M - Picture 0a n d 1 ij ii MM •= GG 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
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SMA-HPC ©2003 MIT Orthogonalization QR Factorization QR Algorithm Key Idea 11 22 12 N NN xb MM M Original Matrix ⎡⎤⎡⎤ ⎡⎤ ↑↑ ⎢⎥⎢⎥ ⎢⎥ = ↓↓ ⎣⎦ ⎣⎦⎣⎦ " GG G " ## " ±²²²²³²²²²´ N yb QQ Q Matrix with Orthonormal Columns ⎤⎡⎤ ⎥⎢⎥ = ⎦⎣⎦ " G " " ±²²²³ ² ² ²´ T Qy b y Q b =⇒= How to perform the conversion?
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SMA-HPC ©2003 MIT Orthogonalization QR Factorization Projection Formula 12 22 1 2 1 Given , find so that ,= MM QM rM G GG G G ( ) 12 1 21 2 1 0 MQM M rM •= • = G G G 12 11 M M r M M = G G G G 2 M G 2 Q G 1 M G 12 r
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SMA-HPC ©2003 MIT Orthogonalization QR Factorization Normalization 11 1 1 1 11 1 1 1 1 1 QM M Q Q MM r == = GG G G G Formulas simplify if we normalize 12 12 22 1 = Now find so that 0 r Q QQ •= G G G ±± 12 1 2 rQM =• G G 2 2 22 2 Fi 1 al 1 nl y r Q G
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