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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 1118 for the xTR and xTP paths, and types 1017 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 1213 and 1516 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:
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 

Next: Tests for the Orthogonal
Up: The Linear Equation Test
Previous: Tests for General and
Contents
Susan Blackford
20010813