Test Matrices for the Singular Value Decomposition Routines

Sixteen different types of test matrices may be generated for the singular value decomposition routines. Table 11 shows the types available, along with the numbers used to refer to the matrix types. Except as noted, all matrix types other than the random bidiagonal matrices have 124#124 entries.

Table 11: Test matrices for the singular value decomposition

	Singular Value Distribution
Type	Arithmetic	Geometric	Clustered	Other
Zero				1
Identity				2
Diagonal	3, 626#26, 7126#126	4	5
489#489	8, 1126#26, 12126#126	9	10
Random entries				13, 1426#26, 15126#126
Random bidiagonal				16
26#26- matrix entries are 129#129
126#126- matrix entries are 130#130

Matrix types identified as ``Zero'', ``Diagonal'', and ``Random entries'' should be self-explanatory. The other matrix types have the following meanings:

Identity:

A 490#490-by- 490#490 identity matrix with zero rows or columns added to the bottom or right to make it 487#487-by-118#118

489#489:

Real 487#487-by-118#118 diagonal matrix 282#282 with 124#124 entries multiplied by unitary (or real orthogonal) matrices on the left and right

Random bidiagonal:

Upper bidiagonal matrix whose entries are randomly chosen from a logarithmic distribution on 491#491

The QR algorithm used in xBDSQR (or xBDSDC) should compute all singular values, even small ones, to good relative accuracy, even of matrices with entries varying over many orders of magnitude, and the random bidiagonal matrix is intended to test this. Thus, unlike the other matrix types, the random bidiagonal matrix is neither 124#124, nor an 124#124 matrix scaled to some other magnitude.

The singular value distributions are analogous to the eigenvalue distributions in the nonsymmetric eigenvalue problem (see Section 6.2.1).