F07AHF (DGERFS) returns error bounds for the solution of a real system of linear equations with multiple right-hand sides, AX=B or ATX=B. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
| SUBROUTINE F07AHF ( | TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) |
| INTEGER | N, NRHS, LDA, LDAF, IPIV(*), LDB, LDX, IWORK(N), INFO |
| double precision | A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), FERR(NRHS), BERR(NRHS), WORK(3*N) |
| CHARACTER*1 | TRANS |
F07AHF (DGERFS) returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides AX=B or ATX=B. The routine handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of F07AHF (DGERFS) 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: TRANS – CHARACTER*1Input
-
On entry: indicates the form of the linear equations for which
X is the computed solution.
- TRANS='N'
- The linear equations are of the form AX=B.
- TRANS='T' or 'C'
- The linear equations are of the form ATX=B.
Constraint:
TRANS='N', 'T' or 'C'.
- 2: N – INTEGERInput
-
On entry:
n, the order of the matrix A.
Constraint:
N≥0.
- 3: NRHS – INTEGERInput
-
On entry:
r, the number of right-hand sides.
Constraint:
NRHS≥0.
- 4: A(LDA,*) – double precision arrayInput
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
n by
n original
matrix
A as supplied to
F07ADF (DGETRF).
- 5: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint:
LDA≥max1,N.
- 6: AF(LDAF,*) – double precision arrayInput
Note: the second dimension of the array
AF
must be at least
max1,N.
On entry: the
LU factorization of
A, as returned by
F07ADF (DGETRF).
- 7: LDAF – INTEGERInput
-
On entry: the first dimension of the array
AF as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint:
LDAF≥max1,N.
- 8: IPIV(*) – INTEGER arrayInput
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: the pivot indices, as returned by
F07ADF (DGETRF).
- 9: B(LDB,*) – double precision 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.
- 10: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint:
LDB≥max1,N.
- 11: X(LDX,*) – double precision arrayInput/Output
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
F07AEF (DGETRS).
On exit: the improved solution matrix X.
- 12: LDX – INTEGERInput
-
On entry: the first dimension of the array
X as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint:
LDX≥max1,N.
- 13: FERR(NRHS) – double precision 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.
- 14: BERR(NRHS) – double precision 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.
- 15: WORK(3×N) – double precision arrayWorkspace
- 16: IWORK(N) – INTEGER arrayWorkspace
- 17: 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.
For each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point operations. Each step of iterative refinement involves an additional 6n2 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 Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2 operations.
The complex analogue of this routine is
F07AVF (ZGERFS).
This example solves the system of equations
AX=B using iterative refinement and to compute the forward and backward error bounds, where
Here
A is nonsymmetric and must first be factorized by
F07ADF (DGETRF).