NAG Library Routine Document

F07JVF (ZPTRFS)

matrices, using the modified Cholesky factorization returned by F07JRF (ZPTTRF) and an initial solution returned by F07JSF (ZPTTRS). Iterative refinement is used to reduce the backward error as much as possible.

2 Specification

SUBROUTINE F07JVF (

UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)

INTEGER	N, NRHS, LDB, LDX, INFO
REAL (KIND=nag_wp)	D(), DF(), FERR(NRHS), BERR(NRHS), RWORK(N)
COMPLEX (KIND=nag_wp)	E(), EF(), B(LDB,), X(LDX,), WORK(N)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zptrfs.

3 Description

F07JVF (ZPTRFS) should normally be preceded by calls to F07JRF (ZPTTRF) and F07JSF (ZPTTRS). F07JRF (ZPTTRF) 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 (ZPTTRS) then utilizes the factorization to compute a solution,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, F07JVF (ZPTRFS) 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 http://www.netlib.org/lapack/lug

5 Parameters

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 – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: NRHS – INTEGERInput

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) arrayInput

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) arrayInput

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) arrayInput

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) arrayInput

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) arrayInput

Note: the second dimension of the array B must be at least

\max (1, NRHS)

On entry: the

n

r

matrix of right-hand sides

B

9: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07JVF (ZPTRFS) is called.

Constraint:

LDB \geq \max (1, N)

10: X(LDX, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array X must be at least

\max (1, NRHS)

On entry: the

n

r

initial solution matrix

X

On exit: the

n

r

refined solution matrix

X

11: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07JVF (ZPTRFS) is called.

Constraint:

LDX \geq \max (1, N)

12: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

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) arrayOutput

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) arrayWorkspace

15: RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace

16: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th argument 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 (ZPTCON) can be used to compute the condition number of

A

8 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 (DPTRFS).

9 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.

NAG Library Routine DocumentF07JVF (ZPTRFS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07JVF (ZPTRFS)