The routine may be called by the names f07pef, nagf_lapacklin_dsptrs or its LAPACK name dsptrs.
3Description
f07pef is used to solve a real symmetric indefinite system of linear equations , the routine must be preceded by a call to f07pdf which computes the Bunch–Kaufman factorization of , using packed storage.
If , , where is a permutation matrix, is an upper triangular matrix and is a symmetric block diagonal matrix with and blocks; the solution is computed by solving and then .
If , , where is a lower triangular matrix; the solution is computed by solving and then .
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Character(1)Input
On entry: specifies how has been factorized.
, where is upper triangular.
, where is lower triangular.
Constraint:
or .
2: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array ap
must be at least
.
On entry: the factorization of stored in packed form, as returned by f07pdf.
5: – Integer arrayInput
Note: the dimension of the array ipiv
must be at least
.
On entry: details of the interchanges and the block structure of , as returned by f07pdf.
6: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
On exit: the solution matrix .
7: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07pef is called.
Constraint:
.
8: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
For each right-hand side vector , the computed solution is the exact solution of a perturbed system of equations , where
if , ;
if , ,
is a modest linear function of , and is the machine precision.
If is the true solution, then the computed solution satisfies a forward error bound of the form
where .
Note that can be much smaller than .
Forward and backward error bounds can be computed by calling f07phf, and an estimate for () can be obtained by calling f07pgf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07pef 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.
9Further Comments
The total number of floating-point operations is approximately .
This routine may be followed by a call to f07phf to refine the solution and return an error estimate.
The complex analogues of this routine are f07psf for Hermitian matrices and f07qsf for symmetric matrices.
10Example
This example solves the system of equations , where
Here is symmetric indefinite, stored in packed form, and must first be factorized by f07pdf.