NAG Library Routine Document
F07GVF (ZPPRFS)
1 Purpose
F07GVF (ZPPRFS) returns error bounds for the solution of a complex Hermitian positive definite system of linear equations with multiple right-hand sides, , using packed storage. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
2 Specification
SUBROUTINE F07GVF ( |
UPLO, N, NRHS, AP, AFP, 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(*), AFP(*), B(LDB,*), X(LDX,*), WORK(2*N) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name zpprfs.
3 Description
F07GVF (ZPPRFS) returns the backward errors and estimated bounds on the forward errors for the solution of a complex Hermitian positive definite system of linear equations with multiple right-hand sides , using packed storage. The routine handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of F07GVF (ZPPRFS) in terms of a single right-hand side and solution .
Given a computed solution
, the routine computes the
component-wise backward error . This is the size of the smallest relative perturbation in each element of
and
such that
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
is the true solution.
For details of the method, see the
F07 Chapter Introduction.
4 References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: specifies whether the upper or lower triangular part of
is stored and how
is to be factorized.
- The upper triangular part of is stored and is factorized as , where is upper triangular.
- The lower triangular part of is stored and is factorized as , where is lower triangular.
Constraint:
or .
- 2: N – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
- 3: NRHS – INTEGERInput
On entry: , the number of right-hand sides.
Constraint:
.
- 4: AP() – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
AP
must be at least
.
On entry: the
by
original Hermitian positive definite matrix
as supplied to
F07GRF (ZPPTRF).
- 5: AFP() – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
AFP
must be at least
.
On entry: the Cholesky factor of
stored in packed form, as returned by
F07GRF (ZPPTRF).
- 6: B(LDB,) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
B
must be at least
.
On entry: the by right-hand side matrix .
- 7: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07GVF (ZPPRFS) is called.
Constraint:
.
- 8: X(LDX,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
X
must be at least
.
On entry: the
by
solution matrix
, as returned by
F07GSF (ZPPTRS).
On exit: the improved solution matrix .
- 9: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F07GVF (ZPPRFS) is called.
Constraint:
.
- 10: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: contains an estimated error bound for the th solution vector, that is, the th column of , for .
- 11: BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
- 12: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
- 13: RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
- 14: 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.
7 Accuracy
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.
For each right-hand side, computation of the backward error involves a minimum of real floating point operations. Each step of iterative refinement involves an additional real operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form ; the number is usually and never more than . Each solution involves approximately real operations.
The real analogue of this routine is
F07GHF (DPPRFS).
9 Example
This example solves the system of equations
using iterative refinement and to compute the forward and backward error bounds, where
and
Here
is Hermitian positive definite, stored in packed form, and must first be factorized by
F07GRF (ZPPTRF).
9.1 Program Text
Program Text (f07gvfe.f90)
9.2 Program Data
Program Data (f07gvfe.d)
9.3 Program Results
Program Results (f07gvfe.r)