matrices, using the modified Cholesky factorization returned by f07jrf and an initial solution returned by f07jsf. Iterative refinement is used to reduce the backward error as much as possible.

2 Specification

Fortran Interface

Subroutine f07jvf (

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

Integer, Intent (In)	::	n, nrhs, ldb, ldx
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	d(), df()
Real (Kind=nag_wp), Intent (Out)	::	ferr(nrhs), berr(nrhs), rwork(n)
Complex (Kind=nag_wp), Intent (In)	::	e(), ef(), b(ldb,*)
Complex (Kind=nag_wp), Intent (Inout)	::	x(ldx,*)
Complex (Kind=nag_wp), Intent (Out)	::	work(n)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void

f07jvf_ (const char *uplo, const Integer *n, const Integer *nrhs, const double d[], const Complex e[], const double df[], const Complex ef[], const Complex b[], const Integer *ldb, Complex x[], const Integer *ldx, double ferr[], double berr[], Complex work[], double rwork[], Integer *info, const Charlen length_uplo)

The routine may be called by the names f07jvf, nagf_lapacklin_zptrfs or its LAPACK name zptrfs.

3 Description

f07jvf should normally be preceded by calls to f07jrf and f07jsf. f07jrf computes a modified Cholesky factorization of the matrix

A

A = L D L^{H},

where

L

is a unit lower bidiagonal matrix and

D

is a diagonal matrix, with positive diagonal elements. f07jsf then utilizes the factorization to compute a solution,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, f07jvf 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^{H} 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: $uplo$ – Character(1) Input

On entry: specifies the form of the factorization as follows:

$uplo ='U'$: $A = U^{H} D U$ .
$uplo ='L'$: $A = L D L^{H}$ .

Constraint:

uplo ='U'

'L'

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: $d (*)$ – 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

5: $e (*)$ – Complex (Kind=nag_wp) array Input

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

\max (1, n - 1)

On entry: if

uplo ='U'

, e must contain the

(n - 1)

superdiagonal elements of the matrix

A

uplo ='L'

, e must contain the

(n - 1)

subdiagonal elements of the matrix

A

6: $df (*)$ – 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

7: $ef (*)$ – Complex (Kind=nag_wp) array Input

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

\max (1, n - 1)

On entry: if

uplo ='U'

, ef must contain the

(n - 1)

superdiagonal elements of the unit upper bidiagonal matrix

U

from the

U^{H} D U

factorization of

A

uplo ='L'

, ef must contain the

(n - 1)

subdiagonal elements of the unit lower bidiagonal matrix

L

from the

L D L^{H}

factorization of

A

8: $b (ldb, *)$ – Complex (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

9: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07jvf is called.

Constraint:

ldb \geq \max (1, n)

10: $x (ldx, *)$ – Complex (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

11: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which f07jvf is called.

Constraint:

ldx \geq \max (1, n)

12: $ferr (nrhs)$ – 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.

13: $berr (nrhs)$ – 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

B

that makes

{\hat{x}}_{j}

an exact solution).

14: $work (n)$ – Complex (Kind=nag_wp) array Workspace

15: $rwork (n)$ – Real (Kind=nag_wp) array Workspace

16: $info$ – 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 f07juf 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.

f07jvf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07jvf 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 real analogue of this routine is f07jhf.

10 Example

This example solves the equations

A X = B,

where

A

is the Hermitian positive definite tridiagonal matrix

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array})

and

B = (\begin{array}{r} 64.0 + 16.0 i & - 16.0 - 32.0 i \\ 93.0 + 62.0 i & 61.0 - 66.0 i \\ 78.0 - 80.0 i & 71.0 - 74.0 i \\ 14.0 - 27.0 i & 35.0 + 15.0 i \end{array}) .

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

f07jv: FL CL CPP AD PY MB

NAG FL Interfacef07jvf (zptrfs)

▸▿ 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
f07jvf (zptrfs)