NAG FL Interface
computes the solution to a real system of linear equations
symmetric matrix stored in packed format and
|Integer, Intent (In)
||n, nrhs, ldb
|Integer, Intent (Out)
|Real (Kind=nag_wp), Intent (Inout)
|Character (1), Intent (In)
|C Header Interface
f07paf_ (const char *uplo, const Integer *n, const Integer *nrhs, double ap, Integer ipiv, double b, const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07paf, nagf_lapacklin_dspsv or its LAPACK name dspsv.
f07paf 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 .
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 https://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
, the upper triangle of
If , the lower triangle of is stored.
On entry: , the number of linear equations, i.e., the order of the matrix .
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
– Real (Kind=nag_wp) array
the dimension of the array ap
must be at least
, packed by columns.
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
: the block diagonal matrix
and the multipliers used to obtain the factor
from the factorization
as computed by f07pdf
, stored as a packed triangular matrix in the same storage format as
– Integer array
: 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.
– Real (Kind=nag_wp) array
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 .
: the first dimension of the array b
as declared in the (sub)program from which f07paf
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix
is exactly singular, so the solution could not be computed.
The computed solution for a single right-hand side,
, satisfies an equation of the form
is the machine precision
. An approximate error bound for the computed solution is given by
, 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.
is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bjf
and returns a forward error bound and condition estimate. f04bjf
to solve the equations.
Parallelism and Performance
f07paf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction
for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note
for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately , where is the number of right-hand sides.
The complex analogues of f07paf
for Hermitian matrices, and f07qnf
for symmetric matrices.
This example solves the equations
is the symmetric matrix
Details of the factorization of are also output.