NAG Library Routine Document
F04BJF
1 Purpose
F04BJF computes the solution to a real system of linear equations , where is an by symmetric matrix, stored in packed format and and are by matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.
2 Specification
INTEGER |
N, NRHS, IPIV(N), LDB, IFAIL |
REAL (KIND=nag_wp) |
AP(*), B(LDB,*), RCOND, ERRBND |
CHARACTER(1) |
UPLO |
|
3 Description
The diagonal pivoting method is used to factor as , if , or , if , where (or ) is a product of permutation and unit upper (lower) triangular matrices, and is symmetric and block diagonal with by and by diagonal blocks. The factored form of is then used to solve the system of equations .
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: if
, the upper triangle of the matrix
is stored.
If , the lower triangle of the matrix is stored.
Constraint:
or .
- 2: N – INTEGERInput
On entry: the number of linear equations , i.e., the order of the matrix .
Constraint:
.
- 3: NRHS – INTEGERInput
On entry: the number of right-hand sides , i.e., the number of columns of the matrix .
Constraint:
.
- 4: AP() – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
AP
must be at least
.
On entry: the
by
symmetric matrix
, packed column-wise in a linear array. The
th column of the matrix
is stored in the array
AP as follows:
- if , , for ;
- if , , for .
See
Section 8 below for further details.
On exit: if
, the block diagonal matrix
and the multipliers used to obtain the factor
or
from the factorization
or
as computed by
F07PDF (DSPTRF), stored as a packed triangular matrix in the same storage format as
.
- 5: IPIV(N) – INTEGER arrayOutput
On exit: if
, details of the interchanges and the block structure of
, as determined by
F07PDF (DSPTRF).
- If , then rows and columns and were interchanged, and is a by diagonal block;
- if and , then rows and columns and were interchanged and is a by diagonal block;
- if and , then rows and columns and were interchanged and is a by diagonal block.
- 6: B(LDB,) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
.
On entry: the by matrix of right-hand sides .
On exit: if or , the by solution matrix .
- 7: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F04BJF is called.
Constraint:
.
- 8: RCOND – REAL (KIND=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix , computed as .
- 9: ERRBND – REAL (KIND=nag_wp)Output
On exit: if
or
, an estimate of the forward error bound for a computed solution
, such that
, where
is a column of the computed solution returned in the array
B and
is the corresponding column of the exact solution
. If
RCOND is less than
machine precision, then
ERRBND is returned as unity.
- 10: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
,
. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
.
When the value is used it is essential to test the value of IFAIL on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
If , the th argument had an illegal value.
Allocation of memory failed. The integer allocatable memory required is
N, and the real allocatable memory required is
. Allocation failed before the solution could be computed.
If , is exactly zero. The factorization has been completed, but the block diagonal matrix is exactly singular, so the solution could not be computed.
RCOND is less than
machine precision, so that the matrix
is numerically singular. A solution to the equations
has nevertheless been computed.
7 Accuracy
The computed solution for a single right-hand side,
, satisfies an equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
with respect to the solution of the linear equations. F04BJF uses the approximation
to estimate
ERRBND. See Section 4.4 of
Anderson et al. (1999) for further details.
The packed storage scheme is illustrated by the following example when
and
. Two-dimensional storage of the symmetric matrix
:
Packed storage of the upper triangle of
:
The total number of floating point operations required to solve the equations is proportional to . The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The complex analogues of F04BJF are
F04CJF for complex Hermitian matrices, and
F04DJF for complex symmetric matrices.
9 Example
This example solves the equations
where
is the symmetric indefinite matrix
An estimate of the condition number of and an approximate error bound for the computed solutions are also printed.
9.1 Program Text
Program Text (f04bjfe.f90)
9.2 Program Data
Program Data (f04bjfe.d)
9.3 Program Results
Program Results (f04bjfe.r)