NAG CL Interface
f07cbc (dgtsvx)
1
Purpose
f07cbc uses the
factorization to compute the solution to a real system of linear equations
where
is a tridiagonal matrix of order
and
and
are
by
matrices. Error bounds on the solution and a condition estimate are also provided.
2
Specification
void |
f07cbc (Nag_OrderType order,
Nag_FactoredFormType fact,
Nag_TransType trans,
Integer n,
Integer nrhs,
const double dl[],
const double d[],
const double du[],
double dlf[],
double df[],
double duf[],
double du2[],
Integer ipiv[],
const double b[],
Integer pdb,
double x[],
Integer pdx,
double *rcond,
double ferr[],
double berr[],
NagError *fail) |
|
The function may be called by the names: f07cbc, nag_lapacklin_dgtsvx or nag_dgtsvx.
3
Description
f07cbc performs the following steps:
-
1.If , the decomposition is used to factor the matrix as , where is a product of permutation and unit lower bidiagonal matrices and is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
-
2.If some , so that is exactly singular, then the function returns with . 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
https://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_OrderType
Input
-
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_FactoredFormType
Input
-
On entry: specifies whether or not the factorized form of the matrix
has been supplied.
- dlf, df, duf, du2 and ipiv contain the factorized form of the matrix . dlf, df, duf, du2 and ipiv will not be modified.
- The matrix will be copied to dlf, df and duf and factorized.
Constraint:
or .
-
3:
– Nag_TransType
Input
-
On entry: specifies the form of the system of equations.
- (No transpose).
- or
- (Transpose).
Constraint:
, or .
-
4:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
5:
– Integer
Input
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
-
6:
– const double
Input
-
Note: the dimension,
dim, of the array
dl
must be at least
.
On entry: the subdiagonal elements of .
-
7:
– const double
Input
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: the diagonal elements of .
-
8:
– const double
Input
-
Note: the dimension,
dim, of the array
du
must be at least
.
On entry: the superdiagonal elements of .
-
9:
– double
Input/Output
-
Note: the dimension,
dim, of the array
dlf
must be at least
.
On entry: if
,
dlf contains the
multipliers that define the matrix
from the
factorization of
.
On exit: if
,
dlf contains the
multipliers that define the matrix
from the
factorization of
.
-
10:
– double
Input/Output
-
Note: the dimension,
dim, of the array
df
must be at least
.
On entry: if
,
df contains the
diagonal elements of the upper triangular matrix
from the
factorization of
.
On exit: if
,
df contains the
diagonal elements of the upper triangular matrix
from the
factorization of
.
-
11:
– double
Input/Output
-
Note: the dimension,
dim, of the array
duf
must be at least
.
On entry: if
,
duf contains the
elements of the first superdiagonal of
.
On exit: if
,
duf contains the
elements of the first superdiagonal of
.
-
12:
– double
Input/Output
-
Note: the dimension,
dim, of the array
du2
must be at least
.
On entry: if
,
du2 contains the (
) elements of the second superdiagonal of
.
On exit: if
,
du2 contains the (
) elements of the second superdiagonal of
.
-
13:
– Integer
Input/Output
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: if
,
ipiv contains the pivot indices from the
factorization of
.
On exit: if
,
ipiv contains the pivot indices from the
factorization of
; row
of the matrix was interchanged with row
.
will always be either
or
;
indicates a row interchange was not required.
-
14:
– const double
Input
-
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 .
-
15:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
-
16:
– double
Output
-
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
.
-
17:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
-
18:
– double *
Output
-
On exit: the estimate of the reciprocal condition number of the matrix
. If
, the matrix may be exactly singular. This condition is indicated by
NE_SINGULAR. Otherwise, if
rcond is less than the
machine precision, the matrix is singular to working precision. This condition is indicated by
NE_SINGULAR_WP.
-
19:
– double
Output
-
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, an estimate of the forward error bound for each computed solution vector, such that
where
is the
th column of the computed solution returned in the array
x and
is the corresponding column of the exact solution
. The estimate is as reliable as the estimate for
rcond, and is almost always a slight overestimate of the true error.
-
20:
– double
Output
-
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, an estimate of the component-wise relative backward error of each computed solution vector
(i.e., the smallest relative change in any element of
or
that makes
an exact solution).
-
21:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_SINGULAR
-
Element of the diagonal is exactly zero.
The factorization has been completed, but the factor
is exactly singular, so the solution and error bounds could not be computed.
is returned.
Element of the diagonal is exactly zero.
The factorization has not been completed, but the factor is exactly
singular, so the solution and error bounds could not be computed.
is returned.
- 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 9.3 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
f07cbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07cbc 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 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. The solution is then refined, and the errors estimated, using iterative refinement.
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 analogue of this function is
f07cpc.
10
Example
This example solves the equations
where
is the tridiagonal matrix
and
Estimates for the backward errors, forward errors and condition number are also output.
10.1
Program Text
10.2
Program Data
10.3
Program Results