NAG Library Function Document
nag_dptrfs (f07jhc)
1
Purpose
nag_dptrfs (f07jhc) computes error bounds and refines the solution to a real system of linear equations
, where
is an
by
symmetric positive definite tridiagonal matrix and
and
are
by
matrices, using the modified Cholesky factorization returned by
nag_dpttrf (f07jdc) and an initial solution returned by
nag_dpttrs (f07jec). Iterative refinement is used to reduce the backward error as much as possible.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_dptrfs (Nag_OrderType order,
Integer n,
Integer nrhs,
const double d[],
const double e[],
const double df[],
const double ef[],
const double b[],
Integer pdb,
double x[],
Integer pdx,
double ferr[],
double berr[],
NagError *fail) |
|
3
Description
nag_dptrfs (f07jhc) should normally be preceded by calls to
nag_dpttrf (f07jdc) and
nag_dpttrs (f07jec).
nag_dpttrf (f07jdc) computes a modified Cholesky factorization of the matrix
as
where
is a unit lower bidiagonal matrix and
is a diagonal matrix, with positive diagonal elements.
nag_dpttrs (f07jec) then utilizes the factorization to compute a solution,
, to the required equations. Letting
denote a column of
,
nag_dptrfs (f07jhc) computes a
component-wise backward error,
, the smallest relative perturbation in each element of
and
such that
is the exact solution of a perturbed system
The function also estimates a bound for the component-wise forward error in the computed solution defined by , where is the corresponding column of the exact solution, .
Note that the modified Cholesky factorization of
can also be expressed as
where
is unit upper bidiagonal.
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
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:
– const doubleInput
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: must contain the diagonal elements of the matrix of .
- 5:
– const doubleInput
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: must contain the subdiagonal elements of the matrix .
- 6:
– const doubleInput
-
Note: the dimension,
dim, of the array
df
must be at least
.
On entry: must contain the diagonal elements of the diagonal matrix from the factorization of .
- 7:
– const doubleInput
-
Note: the dimension,
dim, of the array
ef
must be at least
.
On entry: must contain the subdiagonal elements of the unit bidiagonal matrix from the factorization of .
- 8:
– 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 matrix of right-hand sides .
- 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:
– doubleInput/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 entry: the by initial solution matrix .
On exit: the by refined solution matrix .
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 12:
– doubleOutput
-
On exit: 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 almost always a slight overestimate of the true error.
- 13:
– doubleOutput
-
On exit: 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).
- 14:
– 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_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.
Function
nag_dptcon (f07jgc) can be used to compute the condition number of
.
8
Parallelism and Performance
nag_dptrfs (f07jhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dptrfs (f07jhc) 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 . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this function is
nag_zptrfs (f07jvc).
10
Example
This example solves the equations
where
is the symmetric positive definite tridiagonal matrix
Estimates for the backward errors and forward errors are also output.
10.1
Program Text
Program Text (f07jhce.c)
10.2
Program Data
Program Data (f07jhce.d)
10.3
Program Results
Program Results (f07jhce.r)