NAG Library Function Document
nag_herm_posdef_band_lin_solve (f04cfc)
1
Purpose
nag_herm_posdef_band_lin_solve (f04cfc) computes the solution to a complex system of linear equations , where is an by Hermitian positive definite band matrix of band width , 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_band_lin_solve (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer kd,
Integer nrhs,
Complex ab[],
Integer pdab,
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 band matrix with superdiagonals, and is a lower triangular band matrix with subdiagonals. 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 superdiagonals (and the number of subdiagonals) of the band matrix .
Constraint:
.
- 5:
– IntegerInput
-
On entry: the number of right-hand sides , i.e., the number of columns of the matrix .
Constraint:
.
- 6:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry:
- if then
- if , is stored in ;
- if , is stored in ,
for ; - if then
- if , is stored in ;
- if , is stored in ,
for ,
where
is the stride separating diagonal matrix elements in the array
ab.
See
Section 9 below for further details.
On exit: if
NE_NOERROR or
NE_RCOND, the factor
or
from the Cholesky factorization
or
, in the same storage format as
.
- 7:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 8:
– 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
.
- 9:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 10:
– double *Output
-
On exit: if
NE_NOERROR or
NE_RCOND, an estimate of the reciprocal of the condition number of the matrix
, computed as
.
- 11:
– 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.
- 12:
– 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: .
- 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_band_lin_solve (f04cfc) 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_band_lin_solve (f04cfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_herm_posdef_band_lin_solve (f04cfc) 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 band storage schemes for the array
ab are identical to the storage schemes for symmetric and Hermitian band matrices in
Chapter f07. See
Section 3.3.4 in the f07 Chapter Introduction for details of the storage schemes and an illustrated example.
If then the elements of the stored upper triangular part of are overwritten by the corresponding elements of the upper triangular matrix . Similarly, if then the elements of the stored lower triangular part of are overwritten by the corresponding elements of the lower triangular matrix .
Assuming that , the total number of floating-point operations required to solve the equations is approximately for the factorization and for the solution following the factorization. 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_band_lin_solve (f04cfc) is
nag_real_sym_posdef_band_lin_solve (f04bfc).
10
Example
This example solves the equations
where
is the Hermitian positive definite band 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 (f04cfce.c)
10.2
Program Data
Program Data (f04cfce.d)
10.3
Program Results
Program Results (f04cfce.r)