NAG Library Function Document
nag_real_sym_packed_lin_solve (f04bjc)
1
Purpose
nag_real_sym_packed_lin_solve (f04bjc) computes the solution to a real system of linear equations , where is an by symmetric 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_real_sym_packed_lin_solve (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
double ap[],
Integer ipiv[],
double 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 symmetric 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:
– 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:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the
by
symmetric matrix
, packed column-wise in a linear array. The
th column of the matrix
is stored in the array
ap as follows:
The storage of elements
depends on the
order and
uplo arguments as follows:
- if and ,
is stored in , for ; - if and ,
is stored in , for ; - if and ,
is stored in , for ; - if and ,
is stored in , for .
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_dsptrf (f07pdc), stored as a packed triangular matrix in the same storage format as
.
- 6:
– IntegerOutput
-
On exit: if
NE_NOERROR, details of the interchanges and the block structure of
, as determined by
nag_dsptrf (f07pdc).
- 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:
– doubleInput/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 no constraints are violated, 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 Integer allocatable memory required is
n, and the double 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: .
- 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.
- 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_real_sym_packed_lin_solve (f04bjc) uses the approximation
to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
nag_real_sym_packed_lin_solve (f04bjc) 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 packed storage scheme is illustrated by the following example when
and
. Two-dimensional storage of the symmetric 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.
The complex analogues of
nag_real_sym_packed_lin_solve (f04bjc) are
nag_herm_packed_lin_solve (f04cjc) for complex Hermitian matrices, and
nag_complex_sym_packed_lin_solve (f04djc) for complex symmetric matrices.
10
Example
This example solves the equations
where
is the symmetric indefinite matrix
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 (f04bjce.c)
10.2
Program Data
Program Data (f04bjce.d)
10.3
Program Results
Program Results (f04bjce.r)