NAG Library Function Document
nag_herm_posdef_lin_solve (f04cdc)
1
Purpose
nag_herm_posdef_lin_solve (f04cdc) computes the solution to a complex system of linear equations , where is an by Hermitian positive definite matrix and and are by matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.
2
Specification
#include <nag.h> |
#include <nagf04.h> |
void |
nag_herm_posdef_lin_solve (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
Complex a[],
Integer pda,
Complex b[],
Integer pdb,
double *rcond,
double *errbnd,
NagError *fail) |
|
3
Description
The Cholesky factorization is used to factor as , if , or , if , where is an upper triangular matrix and is a lower triangular matrix. 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
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_UploTypeInput
-
On entry: if
, the upper triangle of the matrix
is stored.
If , the lower triangle of the matrix is stored.
Constraint:
or .
- 3:
– IntegerInput
-
On entry: the number of linear equations , i.e., the order of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: the number of right-hand sides , i.e., the number of columns of the matrix .
Constraint:
.
- 5:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the
by
Hermitian matrix
.
If
, the leading
n by
n upper triangular part of
a contains the upper triangular part of the matrix
, and the strictly lower triangular part of
a is not referenced.
If
, the leading
n by
n lower triangular part of
a contains the lower triangular part of the matrix
, and the strictly upper triangular part of
a is not referenced.
On exit: if
NE_NOERROR or
NE_RCOND, the factor
or
from the Cholesky factorization
or
.
- 6:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 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 matrix of right-hand sides .
On exit: if
NE_NOERROR or
NE_RCOND, 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:
– double *Output
-
On exit: if
NE_NOERROR or
NE_RCOND, an estimate of the reciprocal of the condition number of the matrix
, computed as
.
- 10:
– double *Output
-
On exit: if
NE_NOERROR or
NE_RCOND, an estimate of the forward error bound for a computed solution
, such that
, where
is a column of the computed solution returned in the array
b and
is the corresponding column of the exact solution
. If
rcond is less than
machine precision,
errbnd is returned as unity.
- 11:
– 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.
The double allocatable memory required is
n, and the Complex allocatable memory required is
. Allocation failed before the solution could be computed.
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: .
- NE_INT_2
-
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_POS_DEF
-
The principal minor of order of the matrix is not positive definite. The factorization has not been completed and the solution could not be computed.
- NE_RCOND
-
A solution has been computed, but
rcond is less than
machine precision so that the matrix
is numerically singular.
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.
nag_herm_posdef_lin_solve (f04cdc) uses the approximation
to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
nag_herm_posdef_lin_solve (f04cdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_herm_posdef_lin_solve (f04cdc) 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 floating-point operations required to solve the equations is proportional to . The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The real analogue of
nag_herm_posdef_lin_solve (f04cdc) is
nag_real_sym_posdef_lin_solve (f04bdc).
10
Example
This example solves the equations
where
is the Hermitian positive definite matrix
and
An estimate of the condition number of and an approximate error bound for the computed solutions are also printed.
10.1
Program Text
Program Text (f04cdce.c)
10.2
Program Data
Program Data (f04cdce.d)
10.3
Program Results
Program Results (f04cdce.r)