NAG Library Function Document
nag_dtbtrs (f07vec)
1
Purpose
nag_dtbtrs (f07vec) solves a real triangular band system of linear equations with multiple right-hand sides, or .
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_dtbtrs (Nag_OrderType order,
Nag_UploType uplo,
Nag_TransType trans,
Nag_DiagType diag,
Integer n,
Integer kd,
Integer nrhs,
const double ab[],
Integer pdab,
double b[],
Integer pdb,
NagError *fail) |
|
3
Description
nag_dtbtrs (f07vec) solves a real triangular band system of linear equations or .
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265
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: specifies whether
is upper or lower triangular.
- is upper triangular.
- is lower triangular.
Constraint:
or .
- 3:
– Nag_TransTypeInput
-
On entry: indicates the form of the equations.
- The equations are of the form .
- or
- The equations are of the form .
Constraint:
, or .
- 4:
– Nag_DiagTypeInput
-
On entry: indicates whether
is a nonunit or unit triangular matrix.
- is a nonunit triangular matrix.
- is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .
Constraint:
or .
- 5:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 6:
– IntegerInput
-
On entry: , the number of superdiagonals of the matrix if , or the number of subdiagonals if .
Constraint:
.
- 7:
– IntegerInput
-
On entry: , the number of right-hand sides.
Constraint:
.
- 8:
– const doubleInput
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the
by
triangular band matrix
.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of
, depends on the
order and
uplo arguments as follows:
- if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and .
If , the diagonal elements of are assumed to be , and are not referenced.
- 9:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 10:
– 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: the by solution matrix .
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 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_SINGULAR
-
Element of the diagonal is exactly zero.
is singular and the solution has not been computed.
7
Accuracy
The solutions of triangular systems of equations are usually computed to high accuracy. See
Higham (1989).
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.
If
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
.
Note that ; can be much smaller than and it is also possible for to be much larger (or smaller) than .
Forward and backward error bounds can be computed by calling
nag_dtbrfs (f07vhc), and an estimate for
can be obtained by calling
nag_dtbcon (f07vgc) with
.
8
Parallelism and Performance
nag_dtbtrs (f07vec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dtbtrs (f07vec) 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 total number of floating-point operations is approximately if .
The complex analogue of this function is
nag_ztbtrs (f07vsc).
10
Example
This example solves the system of equations
, where
Here
is treated as a lower triangular band matrix with one subdiagonal.
10.1
Program Text
Program Text (f07vece.c)
10.2
Program Data
Program Data (f07vece.d)
10.3
Program Results
Program Results (f07vece.r)