LDMt - A = LDM t If A is nonsingular and A = LU then we can...

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A = LDM t If A is nonsingular and A = LU , then we can set D = diag( U ) and (since D is nonsinglar) M t D - 1 U is a unit upper triangular matrix and A = LDM t . There is no inherent beneﬁt to this factorization over LU , but it can give us a perspective from which to develop other algorithms. The idea is not to compute LDM t from LU , but to derive a method to compute L , D and M directly. To that end, consider the k th column of A = LDM t : a Ae k = LDM t e k Ly. (1) Suppose we have already know the ﬁrst k - 1 columns of L and consider the blocked treatment of the unit lower triangular system a = Ly : ± L 11 0 L 21 L 22 ² ± y 1 y 2 ² = ± a 1 a 2 ² , (2) where L 11 is k × k and known , but the last column of L 21 is something we would like to compute. Forward substitution gives y 1 as the solution to L 11 y 1 = a 1 . Now we know the ﬁrst k elements of y , and (from (1)) y = DM t e k , (3) giving e t k y = e t k DM t e k = d kk e t k M t e k = d kk (right?). D - 1 y = M t e k with M t unit
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