TITLE: Most efficient method for computing Singular Value Decomposition of a triangular matrix?
QUESTION [14 upvotes]: There are several methods available for computing SVD of a general matrix. I am interested in knowing about the best approach which could be used for computing SVD of an upper triangular matrix. Please suggest me an algorithm which could be optimized for this special case of matrices.

REPLY [3 votes]: Stardard algorithms for computing SVD factorization of an $m \times n$ matrix A process in two steps:

bidiagonalization of A
SVD factorization of bidiagonal matrix using QR method or divide-and-conquer.

The bidiagonalization procedure cannot utilize the triangular structure of A, and therefore standard SVD solvers cannot be optimized for triangular matrices.
However the Jacobi method can be optimized for triangular matrices, see
MLA Drmac, Zlatko, and Krešimir Veselic. "New fast and accurate Jacobi SVD algorithm. II." SIAM Journal on Matrix Analysis and Applications 29.4 (2008): 1343-1362; also Lapack Working Note 170.
To my knowledge, this optimization was not implemented and it is hard to say if this method would be faster, than methods for general matrices. Jacobi method is slower than divide-and-conquer for general matrices, so it may be still slower for triangular matrices.