NAG Library Routine Document
F07USF (ZTPTRS)
1 Purpose
F07USF (ZTPTRS) solves a complex triangular system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B, 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 AX=B, ATX=B or AHX=B, 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
A is upper or lower triangular.
- UPLO='U'
- A is upper triangular.
- UPLO='L'
- A is lower triangular.
Constraint:
UPLO='U' or 'L'.
- 2: TRANS – CHARACTER(1)Input
On entry: indicates the form of the equations.
- TRANS='N'
- The equations are of the form AX=B.
- TRANS='T'
- The equations are of the form ATX=B.
- TRANS='C'
- The equations are of the form AHX=B.
Constraint:
TRANS='N', 'T' or 'C'.
- 3: DIAG – CHARACTER(1)Input
On entry: indicates whether
A is a nonunit or unit triangular matrix.
- DIAG='N'
- A is a nonunit triangular matrix.
- DIAG='U'
- A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint:
DIAG='N' or 'U'.
- 4: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 5: NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint:
NRHS≥0.
- 6: AP(*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
AP
must be at least
max1,N×N+1/2.
On entry: the
n by
n triangular matrix
A, packed by columns.
More precisely,
- if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for i≤j;
- if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for i≥j.
If DIAG='U', the diagonal elements of A are assumed to be 1, and are not referenced; the same storage scheme is used whether DIAG='N' or ‘U’.
- 7: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
- 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:
LDB≥max1,N.
- 9: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
If INFO=i, ai,i is exactly zero; A 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
b, the computed solution
x is the exact solution of a perturbed system of equations
A+Ex=b, where
cn is a modest linear function of
n, and
ε is the
machine precision.
If
x^ is the true solution, then the computed solution
x satisfies a forward error bound of the form
where
condA,x=A-1Ax∞/x∞.
Note that condA,x≤condA=A-1A∞≤κ∞A; condA,x can be much smaller than condA and it is also possible for condAH, which is the same as condAT, to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling
F07UVF (ZTPRFS), and an estimate for
κ∞A can be obtained by calling
F07UUF (ZTPCON) with
NORM='I'.
8 Further Comments
The total number of real floating point operations is approximately 4n2r.
The real analogue of this routine is
F07UEF (DTPTRS).
9 Example
This example solves the system of equations
AX=B, where
and
using packed storage for
A.
9.1 Program Text
Program Text (f07usfe.f90)
9.2 Program Data
Program Data (f07usfe.d)
9.3 Program Results
Program Results (f07usfe.r)