NAG Library Function Document
nag_herm_packed_lin_solve (f04cjc)
1 Purpose
nag_herm_packed_lin_solve (f04cjc) computes the solution to a complex system of linear equations , where is an by complex Hermitian matrix, stored in packed format 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_packed_lin_solve (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
Complex ap[],
Integer ipiv[],
Complex b[],
Integer pdb,
double *rcond,
double *errbnd,
NagError *fail) |
|
3 Description
The diagonal pivoting method is used to factor as , 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. 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:
order – 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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
uplo – 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:
n – IntegerInput
On entry: the number of linear equations , i.e., the order of the matrix .
Constraint:
.
- 4:
nrhs – IntegerInput
On entry: the number of right-hand sides , i.e., the number of columns of the matrix .
Constraint:
.
- 5:
ap[] – ComplexInput/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the
by
Hermitian matrix
, packed column-wise in a linear array. The
th column of the matrix
is stored in the array
ap as follows:
- if , for ;
- if , for .
See
Section 9 below for further details.
On exit: if
NE_NOERROR, the block diagonal matrix
and the multipliers used to obtain the factor
or
from the factorization
or
as computed by
nag_zhptrf (f07prc), stored as a packed triangular matrix in the same storage format as
.
- 6:
ipiv[n] – IntegerOutput
On exit: if
NE_NOERROR, details of the interchanges and the block structure of
, as determined by
nag_zhptrf (f07prc).
- If , then rows and columns and were interchanged, and is a by diagonal block;
- if and , then rows and columns and were interchanged and is a by diagonal block;
- if and , then rows and columns and were interchanged and is a by diagonal block.
- 7:
b[] – 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:
pdb – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 9:
rcond – double *Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix , computed as .
- 10:
errbnd – 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, then
errbnd is returned as unity.
- 11:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- 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: .
- 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.
- NE_RCOND
-
A solution has been computed, but
rcond is less than
machine precision so that the matrix
is numerically singular.
- NE_SINGULAR
-
Diagonal block of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.
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_packed_lin_solve (f04cjc) uses the approximation
to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8 Parallelism and Performance
nag_herm_packed_lin_solve (f04cjc) is not threaded by NAG in any implementation.
nag_herm_packed_lin_solve (f04cjc) 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
Users' Note for your implementation for any additional implementation-specific information.
The packed storage scheme is illustrated by the following example when
and
. Two-dimensional storage of the Hermitian matrix
:
Packed storage of the upper triangle of
:
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.
Function
nag_complex_sym_packed_lin_solve (f04djc) is for complex symmetric matrices, and the real analogue of nag_herm_packed_lin_solve (f04cjc) is
nag_real_sym_packed_lin_solve (f04bjc).
10 Example
This example solves the equations
where
is the Hermitian indefinite 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 (f04cjce.c)
10.2 Program Data
Program Data (f04cjce.d)
10.3 Program Results
Program Results (f04cjce.r)