F07UVF (ZTPRFS) returns error bounds for the solution of a complex triangular system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B, using packed storage.
SUBROUTINE F07UVF ( |
UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) |
INTEGER |
N, NRHS, LDB, LDX, INFO |
REAL (KIND=nag_wp) |
FERR(NRHS), BERR(NRHS), RWORK(N) |
COMPLEX (KIND=nag_wp) |
AP(*), B(LDB,*), X(LDX,*), WORK(2*N) |
CHARACTER(1) |
UPLO, TRANS, DIAG |
|
F07UVF (ZTPRFS) returns the backward errors and estimated bounds on the forward errors for the solution of a complex triangular system of linear equations with multiple right-hand sides AX=B, ATX=B or AHX=B, using packed storage. The routine handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of F07UVF (ZTPRFS) in terms of a single right-hand side b and solution x.
Given a computed solution
x, the routine computes the
component-wise backward error
β. This is the size of the smallest relative perturbation in each element of
A and
b such that
x is the exact solution of a perturbed system
Then the routine estimates a bound for the
component-wise forward error in the computed solution, defined by:
where
x^ is the true solution.
For details of the method, see the
F07 Chapter Introduction.
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 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
-
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.
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07UVF (ZTPRFS) is called.
Constraint:
LDB≥max1,N.
- 9: X(LDX,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
X
must be at least
max1,NRHS.
On entry: the
n by
r solution matrix
X, as returned by
F07USF (ZTPTRS).
- 10: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F07UVF (ZTPRFS) is called.
Constraint:
LDX≥max1,N.
- 11: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: FERRj contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,…,r.
- 12: BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: BERRj contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,…,r.
- 13: WORK(2×N) – COMPLEX (KIND=nag_wp) arrayWorkspace
- 14: RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
- 15: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
The bounds returned in
FERR are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.
A call to F07UVF (ZTPRFS), for each right-hand side, involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 4n2 real floating point operations.
The real analogue of this routine is
F07UHF (DTPRFS).
This example solves the system of equations
AX=B and to compute forward and backward error bounds, where
and
using packed storage for
A.