NAG Library Function Document
nag_zsptrs (f07qsc)
1
Purpose
nag_zsptrs (f07qsc) solves a complex symmetric system of linear equations with multiple right-hand sides,
where
has been factorized by
nag_zsptrf (f07qrc), using packed storage.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_zsptrs (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
const Complex ap[],
const Integer ipiv[],
Complex b[],
Integer pdb,
NagError *fail) |
|
3
Description
nag_zsptrs (f07qsc) is used to solve a complex symmetric system of linear equations
, the function must be preceded by a call to
nag_zsptrf (f07qrc) 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 by and by blocks; the solution is computed by solving and then .
If , , where is a lower triangular matrix; the solution is computed by solving and then .
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_UploTypeInput
-
On entry: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
- 3:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of right-hand sides.
Constraint:
.
- 5:
– const ComplexInput
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the factorization of
stored in packed form, as returned by
nag_zsptrf (f07qrc).
- 6:
– const IntegerInput
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: details of the interchanges and the block structure of
, as returned by
nag_zsptrf (f07qrc).
- 7:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by right-hand side matrix .
On exit: the by solution matrix .
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 9:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
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
nag_zsprfs (f07qvc), and an estimate for
(
) can be obtained by calling
nag_zspcon (f07quc).
8
Parallelism and Performance
nag_zsptrs (f07qsc) 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of real floating-point operations is approximately .
This function may be followed by a call to
nag_zsprfs (f07qvc) to refine the solution and return an error estimate.
The real analogue of this function is
nag_dsptrs (f07pec).
10
Example
This example solves the system of equations
, where
and
Here
is symmetric, stored in packed form, and must first be factorized by
nag_zsptrf (f07qrc).
10.1
Program Text
Program Text (f07qsce.c)
10.2
Program Data
Program Data (f07qsce.d)
10.3
Program Results
Program Results (f07qsce.r)