Unformatted text preview: LDL t factorization for spd matrices. The classical algorithm appears easily by looking at the columns of A = GG t with lower triangular G . Assume we know the ﬁrst k-1 columns of G , and look at the k th column of A = GG t : a k ≡ Ae k = GG t e k = [ g 1 ,g 2 ,...,g n ] z. Here z t is the k th row of (lower triangular) G , so we can write a k = k X i =1 g ki g i , or g kk g k = a k-k-1 X i =1 g ki g i . In particular g 2 kk = a kk-∑ k-1 j =1 g 2 kj , so we take the positive root and solve for g k : g k = ( a k-k-1 X i =1 g ki g i ) /g kk . This method runs to completion (no zero or complex roots) iﬀ A is spd. It requires 1 3 n 3 + O( n 2 ) ﬂops, and no extra memory if the lower triangular of A is overwritten by that of G . With respect to rounding errors, the computed ˜ G satisﬁes ˜ G ˜ G t = A + δA, where k δA k ≤ 12 n 2 μ k A k ....
View Full Document
- Fall '10
- Cholesky Decomposition, Diagonal matrix, symmetric matrices, Cholesky factorization, gki gi