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lecture13 - Lecture 13 Forward and Back Substitution Daniel...

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Lecture 13: Forward and Back Substitution Daniel Frances c 2012 Contents 1 Forward/Backward Substitution for triangular matrices 1 1.1 Forward Substitution for lower triangular matrices . . . . . . . . . . . . . . . 1 1.2 Backward Substitution for upper triangular matrices . . . . . . . . . . . . . 3 1.3 O n analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Forward/Backward Substitution for triangular ma- trices From the previous treatment we quickly learn that inverting a triangular matrix is much simpler than inverting a non-triangular matrix, i.e. of order n 3 versus 2 n 3 . Thus before inverting a matrix, it’s highly advisable to check! It turns out that the procedure for inverting a triangular matrix will be at the heart of the LU factorization approach, which in turn is at the heart of state-of-the-art LP solver codes. These recursive procedures are Forward Substitution (FS) for inverting a lower triangular matrix, and Backward Substitution (BS) for inverting an upper triangular matrix. 1.1 Forward Substitution for lower triangular matrices Consider a lower triangular matrix A = a 11 0 0 . . . 0
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