NAG Library Function Document
nag_zhesvx (f07mpc)
1
Purpose
nag_zhesvx (f07mpc) uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations
where
is an
by
Hermitian matrix and
and
are
by
matrices. Error bounds on the solution and a condition estimate are also provided.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_zhesvx (Nag_OrderType order,
Nag_FactoredFormType fact,
Nag_UploType uplo,
Integer n,
Integer nrhs,
const Complex a[],
Integer pda,
Complex af[],
Integer pdaf,
Integer ipiv[],
const Complex b[],
Integer pdb,
Complex x[],
Integer pdx,
double *rcond,
double ferr[],
double berr[],
NagError *fail) |
|
3
Description
nag_zhesvx (f07mpc) performs the following steps:
1. |
If , the diagonal pivoting method is used to factor . The form of the factorization is if or if , where (or ) is a product of permutation and unit upper (lower) triangular matrices, and is Hermitian and block diagonal with by and by diagonal blocks. |
2. |
If some , so that is exactly singular, then the function returns with and NE_SINGULAR. Otherwise, the factored form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision, NE_SINGULAR_WP is returned as a warning, but the function still goes on to solve for and compute error bounds as described below. |
3. |
The system of equations is solved for using the factored form of . |
4. |
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it. |
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
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_FactoredFormTypeInput
-
On entry: specifies whether or not the factorized form of the matrix
has been supplied.
- af and ipiv contain the factorized form of the matrix . af and ipiv will not be modified.
- The matrix will be copied to af and factorized.
Constraint:
or .
- 3:
– Nag_UploTypeInput
-
On entry: if
, the upper triangle of
is stored.
If , the lower triangle of is stored.
Constraint:
or .
- 4:
– IntegerInput
-
On entry: , the number of linear equations, i.e., the order of the matrix .
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
- 6:
– const ComplexInput
-
Note: the dimension,
dim, of the array
a
must be at least
.
On entry: the
by
Hermitian matrix
.
If , is stored in .
If , is stored in .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
- 7:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
a.
Constraint:
.
- 8:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
af
must be at least
.
The
th element of the matrix is stored in
- when ;
- when .
On entry: if
,
af contains the block diagonal matrix
and the multipliers used to obtain the factor
or
from the factorization
or
as computed by
nag_zhetrf (f07mrc).
On exit: if
,
af returns the block diagonal matrix
and the multipliers used to obtain the factor
or
from the factorization
or
.
- 9:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
af.
Constraint:
.
- 10:
– IntegerInput/Output
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: if
,
ipiv contains details of the interchanges and the block structure of
, as determined by
nag_zhetrf (f07mrc).
- 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.
On exit: if
,
ipiv contains details of the interchanges and the block structure of
, as determined by
nag_zhetrf (f07mrc), as described above.
- 11:
– const ComplexInput
-
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 .
- 12:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 13:
– ComplexOutput
-
Note: the dimension,
dim, of the array
x
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, the
by
solution matrix
.
- 14:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 15:
– double *Output
-
On exit: the estimate of the reciprocal condition number of the matrix
. If
, the matrix may be exactly singular. This condition is indicated by
NE_SINGULAR. Otherwise, if
rcond is less than the
machine precision, the matrix is singular to working precision. This condition is indicated by
NE_SINGULAR_WP.
- 16:
– doubleOutput
-
Note: the dimension,
dim, of the array
ferr
must be at least
.
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, an estimate of the forward error bound for each computed solution vector, such that
where
is the
th column of the computed solution returned in the array
x and
is the corresponding column of the exact solution
. The estimate is as reliable as the estimate for
rcond, and is almost always a slight overestimate of the true error.
- 17:
– doubleOutput
-
Note: the dimension,
dim, of the array
berr
must be at least
.
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, an estimate of the component-wise relative backward error of each computed solution vector
(i.e., the smallest relative change in any element of
or
that makes
an exact solution).
- 18:
– 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: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
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.
- NE_SINGULAR
-
Element of the diagonal is exactly zero.
The factorization has been completed, but the factor is exactly singular,
so the solution and error bounds could not be computed.
is returned.
- NE_SINGULAR_WP
-
is nonsingular, but
rcond is less than
machine precision, meaning that the matrix is singular to working precision.
Nevertheless, the solution and error bounds are computed because there
are a number of situations where the computed solution can be more accurate
than the value of
rcond would suggest.
7
Accuracy
For each right-hand side vector
, the computed solution
is the exact solution of a perturbed system of equations
, where
where
is the
machine precision. See Chapter 11 of
Higham (2002) for further details.
If
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
.
If
is the
th column of
, then
is returned in
and a bound on
is returned in
. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
nag_zhesvx (f07mpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhesvx (f07mpc) 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 factorization of requires approximately floating-point operations.
For each right-hand side, computation of the backward error involves a minimum of floating-point operations. Each step of iterative refinement involves an additional operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form ; the number is usually or and never more than . Each solution involves approximately operations.
The real analogue of this function is
nag_dsysvx (f07mbc). The complex symmetric analogue of this function is
nag_zsysvx (f07npc).
10
Example
This example solves the equations
where
is the Hermitian matrix
and
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix are also output.
10.1
Program Text
Program Text (f07mpce.c)
10.2
Program Data
Program Data (f07mpce.d)
10.3
Program Results
Program Results (f07mpce.r)