NAG Library Function Document
nag_dptsvx (f07jbc)
1
Purpose
nag_dptsvx (f07jbc) uses the factorization
to compute the solution to a real system of linear equations
where
is an
by
symmetric positive definite tridiagonal 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_dptsvx (Nag_OrderType order,
Nag_FactoredFormType fact,
Integer n,
Integer nrhs,
const double d[],
const double e[],
double df[],
double ef[],
const double b[],
Integer pdb,
double x[],
Integer pdx,
double *rcond,
double ferr[],
double berr[],
NagError *fail) |
|
3
Description
nag_dptsvx (f07jbc) performs the following steps:
1. |
If , the matrix is factorized as , where is a unit lower bidiagonal matrix and is diagonal. The factorization can also be regarded as having the form . |
2. |
If the leading by principal minor is not positive definite, then the function returns with and NE_MAT_NOT_POS_DEF. 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.
- df and ef contain the factorized form of the matrix . df and ef will not be modified.
- The matrix will be copied to df and ef and factorized.
Constraint:
or .
- 3:
– IntegerInput
-
On entry: , 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:
– const doubleInput
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
- 6:
– const doubleInput
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: the subdiagonal elements of the tridiagonal matrix .
- 7:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
df
must be at least
.
On entry: if
,
df must contain the
diagonal elements of the diagonal matrix
from the
factorization of
.
On exit: if
,
df contains the
diagonal elements of the diagonal matrix
from the
factorization of
.
- 8:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
ef
must be at least
.
On entry: if
,
ef must contain the
subdiagonal elements of the unit bidiagonal factor
from the
factorization of
.
On exit: if
,
ef contains the
subdiagonal elements of the unit bidiagonal factor
from the
factorization of
.
- 9:
– const doubleInput
-
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 .
- 10:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 11:
– doubleOutput
-
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
.
- 12:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 13:
– double *Output
-
On exit: the reciprocal condition number of the matrix
. If
rcond is less than the
machine precision (in particular, if
), the matrix is singular to working precision. This condition is indicated by a return code of
NE_SINGULAR_WP.
- 14:
– doubleOutput
-
On exit: the forward error bound for each solution vector (the th column of the solution matrix ). If is the true solution corresponding to , is an estimated upper bound for the magnitude of the largest element in () divided by the magnitude of the largest element in .
- 15:
– doubleOutput
-
On exit: the component-wise relative backward error of each solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
- 16:
– 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: .
- NE_INT_2
-
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_MAT_NOT_POS_DEF
-
The leading minor of order of is not positive
definite, so the factorization could not be completed, and the solution has
not been computed. is returned.
- 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_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
is a modest linear function of
, and
is the
machine precision. See Section 10.1 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_dptsvx (f07jbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dptsvx (f07jbc) 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 number of floating-point operations required for the factorization, and for the estimation of the condition number of is proportional to . The number of floating-point operations required for the solution of the equations, and for the estimation of the forward and backward error is proportional to , where is the number of right-hand sides.
The condition estimation is based upon Equation (15.11) of
Higham (2002). For further details of the error estimation, see Section 4.4 of
Anderson et al. (1999).
The complex analogue of this function is
nag_zptsvx (f07jpc).
10
Example
This example solves the equations
where
is the symmetric positive definite tridiagonal matrix
and
Error estimates for the solutions and an estimate of the reciprocal of the condition number of are also output.
10.1
Program Text
Program Text (f07jbce.c)
10.2
Program Data
Program Data (f07jbce.d)
10.3
Program Results
Program Results (f07jbce.r)