NAG Library Routine Document
F07PAF (DSPSV)
1 Purpose
F07PAF (DSPSV) 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.
2 Specification
INTEGER |
N, NRHS, IPIV(N), LDB, INFO |
REAL (KIND=nag_wp) |
AP(*), B(LDB,*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dspsv.
3 Description
F07PAF (DSPSV) uses the diagonal pivoting method to factor as if or if , where (or ) is a product of permutation and unit upper (lower) triangular matrices, 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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
is stored.
If , the lower triangle of 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 by columns.
More precisely,
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
On exit: 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: details of the interchanges and the block structure of
. More precisely,
- if , is a by pivot block and the th row and column of were interchanged with the th row and column;
- if and , is a by pivot block and the th row and column of were interchanged with the th row and column;
- if and , is a by pivot block and the th row and column of were interchanged with the th row and column.
- 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 right-hand side matrix .
On exit: if , the by solution matrix .
- 7: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07PAF (DSPSV) is called.
Constraint:
.
- 8: 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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , is exactly zero. The factorization has been completed, but the block diagonal matrix is exactly singular, so the solution could not be 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. See Section 4.4 of
Anderson et al. (1999) and Chapter 11 of
Higham (2002) for further details.
F07PBF (DSPSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively,
F04BJF solves
and returns a forward error bound and condition estimate.
F04BJF calls F07PAF (DSPSV) to solve the equations.
The total number of floating point operations is approximately , where is the number of right-hand sides.
The complex analogues of F07PAF (DSPSV) are
F07PNF (ZHPSV) for Hermitian matrices, and
F07QNF (ZSPSV) for symmetric matrices.
9 Example
This example solves the equations
where
is the symmetric matrix
Details of the factorization of are also output.
9.1 Program Text
Program Text (f07pafe.f90)
9.2 Program Data
Program Data (f07pafe.d)
9.3 Program Results
Program Results (f07pafe.r)