NAG Library Routine Document

F07JGF (DPTCON)

▸▿ 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

F07JGF (DPTCON) computes the reciprocal condition number of a real

n

n

symmetric positive definite tridiagonal matrix

A

, using the

L D L^{T}

factorization returned by F07JDF (DPTTRF).

2 Specification

SUBROUTINE F07JGF (

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

INTEGER	N, INFO
REAL (KIND=nag_wp)	D(), E(), ANORM, RCOND, WORK(N)

The routine may be called by its LAPACK name dptcon.

3 Description

F07JGF (DPTCON) should be preceded by a call to F07JDF (DPTTRF), which 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. F07JGF (DPTCON) 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 $L D L^{T}$ factorization of $A$ .
3: $E (*)$ – REAL (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 $U^{T} D U$ 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 F06RPF with its parameter $NORM ='1'$ . ANORM must be computed either before calling F07JDF (DPTTRF) 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: $WORK (N)$ – 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 complex analogue of this routine is F07JUF (ZPTCON).

10 Example

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

A

given by

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}) .

NAG Library Routine DocumentF07JGF (DPTCON)