Tests for General and Symmetric Matrices

For each LAPACK test path specified in the input file, the test program generates test matrices, calls the LAPACK routines in that path, and computes a number of test ratios to verify that each operation has performed correctly. The test matrices used in the test paths for general and symmetric matrices are shown in Table 1. Both the computational routines and the driver routines are tested with the same set of matrix types. In this context, 9#9 is the machine epsilon and 15#15 is the condition number of the matrix 16#16. Matrix types with one or more columns set to zero (or rows and columns, if the matrix is symmetric) are used to test the error return codes. For band matrices, all combinations of the values 0, 1, 17#17, 18#18, and 19#19 are used for 20#20 and 21#21 in the GB path, and for 22#22 in the PB path. For the tridiagonal test paths xGT and xPT, types 1-6 use matrices of predetermined condition number, while types 7-12 use random tridiagonal matrices.

Table 1: Test matrices for general and symmetric linear systems

Test matrix type	GE	GB	GT	PO, PP	PB	PT	SY, SP, HE, HP
Diagonal	1		1	1		1	1
Upper triangular	2
Lower triangular	3
Random, 23#23	4	1	2	2	1	2	2
Random, 24#24	8	5	3	6	5	3	7
Random, 25#25	9	6	4	7	6	4	8
First column zero	5	2	8	3	2	8	3
Last column zero	6	3	9	4	3	9	4
Middle column zero				5	4	10	5
Last 4#4/2 columns zero	7	4	10				6
Scaled near underflow	10	7	5, 11	8	7	5, 11	9
Scaled near overflow	11	8	6, 12	9	8	6, 12	10
Random, unspecified 15#15			7			7
Block diagonal							1126#26
26#26- complex symmetric test paths only

For the LAPACK test paths shown in Table 1, each test matrix is subjected to the following tests:

• Factor the matrix using xxxTRF, and compute the ratio
  • 27#27
  This form is for the paths xGE, xGB, and xGT. For the paths xPO, xPP, or xPB, replace 2#2 by 28#28 or 29#29; for xPT, replace 2#2 by 30#30 or 31#31, where D is diagonal; and for the paths xSY, xSP, xHE, or xHP, replace 2#2 by 30#30 or 32#32, where D is diagonal with 1-by-1 and 2-by-2 diagonal blocks.

• Invert the matrix 16#16 using xxxTRI, and compute the ratio
  • 33#33
  For tridiagonal and banded matrices, inversion routines are not available because the inverse would be dense.

• Solve the system 34#34 using xxxTRS, and compute the ratios
  • 35#35
  • 36#36
  where 37#37 is the exact solution and 15#15 is the condition number of 16#16.

• Use iterative refinement (xxxRFS) to improve the solution, and compute the ratios
  • 36#36
  • (backward error) 38#38
  • 39#39 (error bound) 40#40

• Compute the reciprocal condition number RCOND using xxxCON, and compare to the value RCONDC which was computed as 1/(ANORM * AINVNM) where AINVNM is the explicitly computed norm of 41#41. The larger of the ratios
  • 42#42 and 43#43
  is returned. Since the same value of ANORM is used in both cases, this test measures the accuracy of the estimate computed for 41#41.

The solve and iterative refinement steps are also tested with 16#16 replaced by 44#44 or 45#45 where applicable. The test ratios computed for the general and symmetric test paths are listed in Table 2. Here we use 46#46 to describe the difference in the recomputed matrix, even though it is actually 47#47 or some other form for other paths.

Table 2: Tests performed for general and symmetric linear systems

	GE, PO, PP, SY, SP		GB, GT, PB, PT
Test ratio	routines	drivers	routines	drivers
27#27	1	1	1	1
33#33	2
35#35	3	2	2	2
36#36	4		3
36#36, refined	5	3	4	3
(backward error)38#38	6	4	5	4
48#48	7	5	6	5
49#49	8	6	7	6