Tests for Triangular Matrices

The triangular test paths, xTR, xTP, and xTB, include a number of pathological test matrices for testing the auxiliary routines xLATRS, xLATPS, and xLATBS, which are robust triangular solves used in condition estimation. The triangular test matrices are summarized in Table 3. To generate unit triangular matrices of predetermined condition number, we choose a special unit triangular matrix and use plane rotations to fill in the zeros without destroying the ones on the diagonal. For the xTB path, all combinations of the values 0, 1, 17#17, 18#18, and 19#19 are used for the number of offdiagonals 22#22, so the diagonal type is not necessary.

Types 11-18 for the xTR and xTP paths, and types 10-17 for xTB, are used only to test the scaling options in xLATRS, xLATPS, and xLATBS. These subroutines solve a scaled triangular system 50#50 or 51#51, where 52#52 is allowed to underflow to 53#53 in order to prevent overflow in 54#54. A growth factor is computed using the norms of the columns of 16#16, and if the solution can not overflow, the Level 2 BLAS routine is called. Types 11 and 18 test the scaling of 55#55 when 55#55 is initially large, types 12-13 and 15-16 test scaling when the diagonal of 16#16 is small or zero, and type 17 tests the scaling if overflow occurs when adding multiples of the columns to the right hand side. In type 14, no scaling is done, but the growth factor is too large to call the equivalent BLAS routine.

Table 3: Test matrices for triangular linear systems

Test matrix type	TR, TP	TB
Diagonal	1
Random, 23#23	2	1
Random, 24#24	3	2
Random, 25#25	4	3
Scaled near underflow	5	4
Scaled near overflow	6	5
Identity	7	6
Unit triangular, 23#23	8	7
Unit triangular, 24#24	9	8
Unit triangular, 25#25	10	9
Matrix elements are O(1), large right hand side	11	10
First diagonal causes overflow, offdiagonal column norms 56#56	12	11
First diagonal causes overflow, offdiagonal column norms 57#57	13	12
Growth factor underflows, solution does not overflow	14	13
Small diagonal causes gradual overflow	15	14
One zero diagonal element	16	15
Large offdiagonals cause overflow when adding a column	17	16
Unit triangular with large right hand side	18	17

The tests performed for the triangular routines are similar to those for the general and symmetric routines, including tests of the inverse, solve, iterative refinement, and condition estimation routines. One additional test ratio is computed for the robust triangular solves:

• 58#58

Table 4 shows the test ratios computed for the triangular test paths.

Table 4: Tests performed for triangular linear systems

Test ratio	TR, TP	TB
33#33	1
35#35	2	1
36#36	3	2
36#36, refined	4	3
(backward error)38#38	5	4
48#48	6	5
49#49	7	6
59#59	8	7