NAG Library Routine Document
F07USF (ZTPTRS)
1 Purpose
F07USF (ZTPTRS) solves a complex triangular system of linear equations with multiple right-hand sides, , or , using packed storage.
2 Specification
INTEGER |
N, NRHS, LDB, INFO |
COMPLEX (KIND=nag_wp) |
AP(*), B(LDB,*) |
CHARACTER(1) |
UPLO, TRANS, DIAG |
|
The routine may be called by its
LAPACK
name ztptrs.
3 Description
F07USF (ZTPTRS) solves a complex triangular system of linear equations , or , using packed storage.
4 References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: specifies whether
is upper or lower triangular.
- is upper triangular.
- is lower triangular.
Constraint:
or .
- 2: TRANS – CHARACTER(1)Input
On entry: indicates the form of the equations.
- The equations are of the form .
- The equations are of the form .
- The equations are of the form .
Constraint:
, or .
- 3: DIAG – CHARACTER(1)Input
On entry: indicates whether
is a nonunit or unit triangular matrix.
- is a nonunit triangular matrix.
- is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .
Constraint:
or .
- 4: N – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
- 5: NRHS – INTEGERInput
On entry: , the number of right-hand sides.
Constraint:
.
- 6: AP() – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
AP
must be at least
.
On entry: the
by
triangular matrix
, packed by columns.
More precisely,
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
If , the diagonal elements of are assumed to be , and are not referenced; the same storage scheme is used whether or ‘U’.
- 7: B(LDB,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
.
On entry: the by right-hand side matrix .
On exit: the by solution matrix .
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07USF (ZTPTRS) is called.
Constraint:
.
- 9: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , is exactly zero; is singular and the solution has not been computed.
7 Accuracy
The solutions of triangular systems of equations are usually computed to high accuracy. See
Higham (1989).
For each right-hand side vector
, the computed solution
is the exact solution of a perturbed system of equations
, where
is a modest linear function of
, and
is the
machine precision.
If
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
.
Note that ; can be much smaller than and it is also possible for , which is the same as , to be much larger (or smaller) than .
Forward and backward error bounds can be computed by calling
F07UVF (ZTPRFS), and an estimate for
can be obtained by calling
F07UUF (ZTPCON) with
.
The total number of real floating point operations is approximately .
The real analogue of this routine is
F07UEF (DTPTRS).
9 Example
This example solves the system of equations
, where
and
using packed storage for
.
9.1 Program Text
Program Text (f07usfe.f90)
9.2 Program Data
Program Data (f07usfe.d)
9.3 Program Results
Program Results (f07usfe.r)