matrices, using the modified Cholesky factorization returned by f07jdf and an initial solution returned by f07jef. Iterative refinement is used to reduce the backward error as much as possible.

2 Specification

Fortran Interface

Subroutine f07jhf (

n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)

Integer, Intent (In)	::	n, nrhs, ldb, ldx
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	d(), e(), df(), ef(), b(ldb,*)
Real (Kind=nag_wp), Intent (Inout)	::	x(ldx,*)
Real (Kind=nag_wp), Intent (Out)	::	ferr(nrhs), berr(nrhs), work(2*n)

C Header Interface

#include <nag.h>

void	f07jhf_ (const Integer n, const Integer nrhs, const double d[], const double e[], const double df[], const double ef[], const double b[], const Integer ldb, double x[], const Integer ldx, double ferr[], double berr[], double work[], Integer *info)

The routine may be called by the names f07jhf, nagf_lapacklin_dptrfs or its LAPACK name dptrfs.

3 Description

f07jhf should normally be preceded by calls to f07jdf and f07jef. f07jdf 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. f07jef then utilizes the factorization to compute a solution,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, f07jhf 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 routine 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: $n$ – Integer Input: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs$ – Integer Input: On entry: $r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs \geq 0$ .
3: $d (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension of the array d must be at least $\max (1, n)$ .

On entry: must contain the $n$ diagonal elements of the matrix of $A$ .
4: $e (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension of the array e must be at least $\max (1, n - 1)$ .

On entry: must contain the $(n - 1)$ subdiagonal elements of the matrix $A$ .
5: $df (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension 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 $A$ .
6: $ef (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension 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 $A$ .
7: $b (ldb, *)$ – Real (Kind=nag_wp) array Input: Note: the second dimension of the array b must be at least $\max (1, nrhs)$ .

On entry: the $n \times r$ matrix of right-hand sides $B$ .
8: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f07jhf is called.

Constraint: $ldb \geq \max (1, n)$ .
9: $x (ldx, *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array x must be at least $\max (1, nrhs)$ .

On entry: the $n \times r$ initial solution matrix $X$ .

On exit: the $n \times r$ refined solution matrix $X$ .
10: $ldx$ – Integer Input: On entry: the first dimension of the array x as declared in the (sub)program from which f07jhf is called.

Constraint: $ldx \geq \max (1, n)$ .
11: $ferr (nrhs)$ – Real (Kind=nag_wp) array Output: On exit: estimate of the forward error bound for each computed solution vector, such that ${‖ {\hat{x}}_{j} - x_{j} ‖}_{\infty} / {‖ {\hat{x}}_{j} ‖}_{\infty} \leq ferr (j)$ , 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.
12: $berr (nrhs)$ – Real (Kind=nag_wp) array 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 $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
13: $work (2 \times n)$ – Real (Kind=nag_wp) array Workspace
14: $info$ – Integer Output: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

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.

Routine f07jgf can be used to compute the condition number of

A

8 Parallelism and Performance

f07jhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07jhf 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 routine. 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 routine is f07jvf.

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

NAG FL Interfacef07jhf (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 FL Interface
f07jhf (dptrfs)