NAG Library Function Document
nag_dptsv (f07jac)
1
Purpose
nag_dptsv (f07jac) computes the solution to a real system of linear equations
where
is an
by
symmetric positive definite tridiagonal matrix, and
and
are
by
matrices.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_dptsv (Nag_OrderType order,
Integer n,
Integer nrhs,
double d[],
double e[],
double b[],
Integer pdb,
NagError *fail) |
|
3
Description
nag_dptsv (f07jac) factors as . 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
- 4:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
On exit: the diagonal elements of the diagonal matrix from the factorization .
- 5:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: the subdiagonal elements of the tridiagonal matrix .
On exit: the
subdiagonal elements of the unit bidiagonal factor
from the
factorization of
. (
e can also be regarded as the superdiagonal of the unit bidiagonal factor
from the
factorization of
.)
- 6:
– 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 right-hand side matrix .
On exit: if NE_NOERROR, the by solution matrix .
- 7:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 8:
– 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: .
- 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_MAT_NOT_POS_DEF
-
The leading minor of order is not positive definite,
and the solution has not been computed.
The factorization has not been completed unless .
- 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.
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. See Section 4.4 of
Anderson et al. (1999) for further details.
nag_dptsvx (f07jbc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively,
nag_real_sym_posdef_tridiag_lin_solve (f04bgc) solves
and returns a forward error bound and condition estimate.
nag_real_sym_posdef_tridiag_lin_solve (f04bgc) calls
nag_dptsv (f07jac) to solve the equations.
8
Parallelism and Performance
nag_dptsv (f07jac) 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 of is proportional to , and the number of floating-point operations required for the solution of the equations is proportional to , where is the number of right-hand sides.
The complex analogue of this function is
nag_zptsv (f07jnc).
10
Example
This example solves the equations
where
is the symmetric positive definite tridiagonal matrix
Details of the factorization of are also output.
10.1
Program Text
Program Text (f07jace.c)
10.2
Program Data
Program Data (f07jace.d)
10.3
Program Results
Program Results (f07jace.r)