matrices, using the modified Cholesky factorization returned by f07jdc and an initial solution returned by f07jec. Iterative refinement is used to reduce the backward error as much as possible.

2 Specification

#include <nag.h>

void	f07jhc (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)

The function may be called by the names: f07jhc, nag_lapacklin_dptrfs or nag_dptrfs.

3 Description

f07jhc should normally be preceded by calls to f07jdc and f07jec. f07jdc computes a modified Cholesky factorization of the matrix

A

A = L D L^{T},

where

L

is a unit lower bidiagonal matrix and

D

is a diagonal matrix, with positive diagonal elements. f07jec then utilizes the factorization to compute a solution,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, f07jhc computes a component-wise backward error,

β

, the smallest relative perturbation in each element of

A

and

b

such that

\hat{x}

is the exact solution of a perturbed system

(A + E) \hat{x} = b + f, with | e_{i j} | \leq β | a_{i j} |, and | f_{j} | \leq β | b_{j} | .

The function also estimates a bound for the component-wise forward error in the computed solution defined by

\max | x_{i} - \hat{x_{i}} | / \max | \hat{x_{i}} |

, where

x

is the corresponding column of the exact solution,

X

Note that the modified Cholesky factorization of

A

can also be expressed as

A = U^{T} D U,

where

U

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 https://www.netlib.org/lapack/lug

5 Arguments

1: $order$ – 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

order = Nag_RowMajor

. 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:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

4: $d [\dim]$ – const double Input

Note: the dimension, dim, of the array d must be at least

\max (1, n)

On entry: must contain the

n

diagonal elements of the matrix of

A

5: $e [\dim]$ – const double Input

Note: the dimension, dim, of the array e must be at least

\max (1, n - 1)

On entry: must contain the

(n - 1)

subdiagonal elements of the matrix

A

6: $df [\dim]$ – const double Input

Note: the dimension, dim, of the array df must be at least

\max (1, n)

On entry: must contain the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{T}

factorization of

A

7: $ef [\dim]$ – const double Input

Note: the dimension, dim, of the array ef must be at least

\max (1, n)

On entry: must contain the

(n - 1)

subdiagonal elements of the unit bidiagonal matrix

L

from the

L D L^{T}

factorization of

A

8: $b [\dim]$ – const double Input

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times r

matrix of right-hand sides

B

9: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

10: $x [\dim]$ – double Input/Output

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

X

is stored in

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times r

initial solution matrix

X

On exit: the

n \times r

refined solution matrix

X

11: $pdx$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdx \geq \max (1, nrhs)$ .

12: $ferr [nrhs]$ – double Output

On exit: estimate of the forward error bound for each computed solution vector, such that

{‖ {\hat{x}}_{j} - x_{j} ‖}_{\infty} / {‖ {\hat{x}}_{j} ‖}_{\infty} \leq ferr [j - 1]

, where

{\hat{x}}_{j}

is the

j

th column of the computed solution returned in the array x and

x_{j}

is the corresponding column of the exact solution

X

. The estimate is almost always a slight overestimate of the true error.

13: $berr [nrhs]$ – double Output

On exit: estimate of the component-wise relative backward error of each computed solution vector

{\hat{x}}_{j}

(i.e., the smallest relative change in any element of

A

B

that makes

{\hat{x}}_{j}

an exact solution).

14: $fail$ – 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 $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = ⟨ value ⟩$ .
Constraint: $nrhs \geq 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .

On entry, $pdx = ⟨ value ⟩$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, nrhs)$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq \max (1, n)$ .

On entry, $pdx = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdx \geq \max (1, nrhs)$ .
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.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖ E ‖}_{\infty} = O (ε) {‖ A ‖}_{\infty}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖ \hat{x} - x ‖}_{\infty}}{{‖ x ‖}_{\infty}} \leq κ (A) \frac{{‖ E ‖}_{\infty}}{{‖ A ‖}_{\infty}},

where

κ (A) = {‖ A^{- 1} ‖}_{\infty} {‖ A ‖}_{\infty}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

Function f07jgc can be used to compute the condition number of

A

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f07jhc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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.

9 Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

. 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 f07jvc.

10 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{matrix} 4.0 & - 2.0 & 0 & 0 & 0 \\ - 2.0 & 10.0 & - 6.0 & 0 & 0 \\ 0 & - 6.0 & 29.0 & 15.0 & 0 \\ 0 & 0 & 15.0 & 25.0 & 8.0 \\ 0 & 0 & 0 & 8.0 & 5.0 \end{matrix}) and B = (\begin{matrix} 6.0 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{matrix}) .

Estimates for the backward errors and forward errors are also output.

f07jh: FL CL CPP AD PY MB

NAG CL Interfacef07jhc (dptrfs)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f07jhc (dptrfs)