NAG Library Function Document

nag_dptcon (f07jgc)

+− 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

1 Purpose

nag_dptcon (f07jgc) 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 nag_dpttrf (f07jdc).

2 Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dptcon (Integer n, const double d[], const double e[], double anorm, double rcond, NagError fail)

3 Description

nag_dptcon (f07jgc) should be preceded by a call to nag_dpttrf (f07jdc), 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. nag_dptcon (f07jgc) 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 Arguments

1: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 0$ .
2: d[ $\dim$ ] – const doubleInput: Note: the dimension, dim, 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[ $\dim$ ] – const doubleInput: Note: the dimension, dim, 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 – doubleInput: On entry: the $1$ -norm of the original matrix $A$ , which may be computed as shown in Section 10. anorm must be computed either before calling nag_dpttrf (f07jdc) or else from a copy of the original matrix $A$ .
Constraint: $anorm \geq 0.0$ .
5: rcond – double *Output: On exit: the reciprocal condition number, $1 / κ_{1} (A) = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $anorm = ⟨value⟩$ .
Constraint: $anorm \geq 0.0$ .

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 function is nag_zptcon (f07juc).

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 Function Documentnag_dptcon (f07jgc)