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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

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 condition number will be the exact condition number for a closely neighbouring matrix.

8 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).

9 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)