Tests Performed on the Singular Value Decomposition Routines

Finding the singular values and singular vectors of a dense, 3#3-by-4#4 matrix 16#16 is done in the following stages:

1. 16#16 is decomposed as 492#492, where 64#64 and 73#73 are unitary and 97#97 is real bidiagonal.

2. 97#97 is decomposed as 493#493, where 132#132 and 245#245 are real orthogonal and 494#494 is a positive real diagonal matrix of singular values. This is done three times to compute
   1. 495#495, where 496#496 is the diagonal matrix of singular values and the columns of the matrices 132#132 and 245#245 are the left and right singular vectors, respectively, of 97#97.
   2. Same as above, but the singular values are stored in 497#497 and the singular vectors are not computed.
   3. 498#498, the SVD of the original matrix 16#16.

For each pair of matrix dimensions 499#499 and each selected matrix type, an 3#3-by-4#4 matrix 16#16 and an 3#3-by-500#500 matrix 98#98 are generated. The problem dimensions are as follows

16#16	3#3-by-4#4
64#64	3#3-by-501#501 (but 3#3-by-3#3 if nrhs 502#502 0)
73#73	501#501-by-4#4
97#97	501#501-by-501#501
132#132, 245#245	501#501-by-501#501
503#503, 504#504	diagonal, order 501#501
98#98	3#3-by-500#500

where 505#505.

To check these calculations, the following test ratios are computed, where 506#506 and 507#507. Tests 1-3 test SGEBRD and SORGBR. Tests 4-10 test SBDSQR on a bidiagonal matrix B. Tests 11-14 test SBDSQR on a matrix A. Tests 15-19 test SBDSDC on a bidiagonal matrix B.

508#508

509#509

510#510

511#511

Tests 15-19 are the same as tests 4, 6, 7, 8, and 9, respectively, except that SBDSDC is tested. The subscript 133#133 indicates that 132#132 and 245#245 were computed at the same time as 494#494, and 53#53 that they were not. (All norms are 143#143.) The scalings in the test ratios assure that the ratios will be 124#124 (typically less than 10 or 100), independent of 144#144 and 9#9, and nearly independent of 3#3 or 4#4.