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Tests for the Orthogonal Factorization Routines

The orthogonal factorization routines are contained in the test paths xQR, xRQ, xLQ, xQL, xQP, and xTZ. The first four of these test the QR, RQ, LQ, and QL factorizations without pivoting. The subroutines to generate or multiply by the orthogonal matrix from the factorization are also tested in these paths. There is not a separate test path for the orthogonal transformation routines, since the important thing when generating an orthogonal matrix is not whether or not it is, in fact, orthogonal, but whether or not it is the orthogonal matrix we wanted. The xQP test path is used for QR with pivoting, and xTZ tests the reduction of a trapezoidal matrix by an RQ factorization.

The test paths xQR, xRQ, xLQ, and xQL all use the same set of test matrices and compute similar test ratios, so we will only describe the xQR path. Also, we will refer to the subroutines by their single precision real names, SGEQRF, SGEQRS, SORGQR, and SORMQR. In the complex case, the orthogonal matrices are unitary, so the names beginning with SOR- are changed to CUN-. Each of the orthogonal factorizations can operate on 3#3-by-4#4 matrices, where 60#60, 61#61, or 62#62.

Eight test matrices are used for SQR and the other orthogonal factorization test paths. All are generated with a predetermined condition number (by default, 63#63).


 		 1. 		 Diagonal 		 5. 		 Random, 24#24 

2. Upper triangular 6. Random, 25#25
3. Lower triangular 7. Scaled near underflow
4. Random, 63#63 8. Scaled near overflow

The tests for the SQR path are as follows:

In the SQP test path, we test the QR factorization with column pivoting (SGEQPF or SGEQP3), which decomposes a matrix 16#16 into a product of a permutation matrix 73#73, an orthogonal matrix 64#64, and an upper triangular matrix 74#74 such that 75#75. We generate three types of matrices 16#16 with singular values 52#52 as follows:

The following tests are performed:

In the STZ path, we test the trapezoidal reduction (STZRQF or STZRZF), which decomposes an 3#3-by-4#4 (m 82#82 n) upper trapezoidal matrix 74#74 (i.e. 83#83 if 84#84) into a product of a strictly upper triangular matrix 85#85 (i.e. 86#86 if 84#84 or 87#87) and an orthogonal matrix 88#88 such that 89#89. We generate matrices with the following three singular value distributions 52#52:

To obtain an upper trapezoidal matrix with the specified singular value distribution, we generate a dense matrix using SLATMS and reduce it to upper triangular form using SGEQR2. The following tests are performed:


next up previous contents
Next: Tests for the Least Up: The Linear Equation Test Previous: Tests for Triangular Matrices   Contents
Susan Blackford 2001-08-13