NAG Library Routine Document

F07JUF (ZPTCON)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

F07JUF (ZPTCON) computes the reciprocal condition number of a complex

n

n

Hermitian positive definite tridiagonal matrix

A

, using the

L D L^{H}

factorization returned by F07JRF (ZPTTRF).

2 Specification

SUBROUTINE F07JUF (

N, D, E, ANORM, RCOND, RWORK, INFO)

INTEGER	N, INFO
REAL (KIND=nag_wp)	D(*), ANORM, RCOND, RWORK(N)
COMPLEX (KIND=nag_wp)	E(*)

The routine may be called by its LAPACK name zptcon.

3 Description

F07JUF (ZPTCON) should be preceded by a call to F07JRF (ZPTTRF), which 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. F07JUF (ZPTCON) then utilizes the factorization to compute

{‖A^{- 1}‖}_{1}

by a direct method, from which the reciprocal of the condition number of

A

1 / κ (A)

is computed as

1 / κ_{1} (A) = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1}) .

1 / κ (A)

is returned, rather than

κ (A)

, since when

A

is singular

κ (A)

is infinite.

4 References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Parameters

1: $N$ – INTEGERInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $N \geq 0$ .
2: $D (*)$ – 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 diagonal matrix $D$ from the $L D L^{H}$ factorization of $A$ .
3: $E (*)$ – COMPLEX (KIND=nag_wp) arrayInput: 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 unit lower bidiagonal matrix $L$ . (E can also be regarded as the superdiagonal of the unit upper bidiagonal matrix $U$ from the $U^{H} D U$ factorization of $A$ .)
4: $ANORM$ – REAL (KIND=nag_wp)Input: On entry: the $1$ -norm of the original matrix $A$ , which may be computed by calling F06UPF with its parameter $NORM ='1'$ . ANORM must be computed either before calling F07JRF (ZPTTRF) or else from a copy of the original matrix $A$ .

Constraint: $ANORM \geq 0.0$ .
5: $RCOND$ – REAL (KIND=nag_wp)Output: On exit: the reciprocal condition number, $1 / κ_{1} (A) = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
6: $RWORK (N)$ – REAL (KIND=nag_wp) arrayWorkspace
7: $INFO$ – INTEGEROutput: 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 condition number will be the exact condition number for a closely neighbouring matrix.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The condition number estimation requires

O (n)

floating-point operations.

See Section 15.6 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.

The real analogue of this routine is F07JGF (DPTCON).

10 Example

This example computes the condition number of the Hermitian positive definite tridiagonal matrix

A

given by

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}) .

NAG Library Routine DocumentF07JUF (ZPTCON)