NAG Library Function Document

nag_ddisna (f08flc)

+− 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

1 Purpose

nag_ddisna (f08flc) computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian

m

m

matrix

A

, or for the left or right singular vectors of a general

m

n

matrix

A

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_ddisna (Nag_JobType job, Integer m, Integer n, const double d[], double sep[], NagError *fail)

3 Description

The bound on the error, measured by the angle in radians, for the

i

th computed vector is given by

ε {‖A‖}_{2} / {sep}_{i}

, where

ε

is the machine precision and

{sep}_{i}

is the reciprocal condition number for the vectors, returned in the array element

sep [i - 1]

sep [i - 1]

is restricted to be at least

ε {‖A‖}_{2}

in order to limit the size of the error bound.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: job – Nag_JobTypeInput

On entry: specifies for which problem the reciprocal condition number should be computed.

$job = Nag_EigVecs$: The eigenvectors of a symmetric or Hermitian matrix.
$job = Nag_LeftSingVecs$: The left singular vectors of a general matrix.
$job = Nag_RightSingVecs$: The right singular vectors of a general matrix.

Constraint:

job = Nag_EigVecs

Nag_LeftSingVecs

Nag_RightSingVecs

2: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: n – IntegerInput

On entry:

n

, the number of columns of the matrix when

job = Nag_LeftSingVecs

Nag_RightSingVecs

job = Nag_EigVecs

, n is not referenced.

Constraint: if

job = Nag_LeftSingVecs

Nag_RightSingVecs

n \geq 0

4: d[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array d must be at least

$\max (1, m)$ when $job = Nag_EigVecs$ ;
$\max (1, \min (m, n))$ when $job = Nag_LeftSingVecs$ or $Nag_RightSingVecs$ .

On entry: the eigenvalues if

job = Nag_EigVecs

, or singular values if

job = Nag_LeftSingVecs

Nag_RightSingVecs

of the matrix

A

Constraints:

the elements of the array d must be in either increasing or decreasing order;
if $job = Nag_LeftSingVecs$ or $Nag_RightSingVecs$ the elements of d must be non-negative.

5: sep[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array sep must be at least

$\max (1, m)$ when $job = Nag_EigVecs$ ;
$\max (1, \min (m, n))$ when $job = Nag_LeftSingVecs$ or $Nag_RightSingVecs$ .

On exit: the reciprocal condition numbers of the vectors.

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_ENUM_INT: On entry, $job = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $job = Nag_LeftSingVecs$ or $Nag_RightSingVecs$ , $n \geq 0$ .
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
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_NOT_MONOTONIC: Constraint: the elements of the array d must be in either increasing or decreasing order.
if $job = Nag_LeftSingVecs$ or $Nag_RightSingVecs$ the elements of d must be non-negative.

7 Accuracy

The reciprocal condition numbers are computed to machine precision relative to the size of the eigenvalues, or singular values.

8 Parallelism and Performance

Not applicable.

9 Further Comments

nag_ddisna (f08flc) may also be used towards computing error bounds for the eigenvectors of the generalized symmetric or Hermitian definite eigenproblem. See Golub and Van Loan (1996) for further details on the error bounds.

10 Example

The use of nag_ddisna (f08flc) in computing error bounds for eigenvectors of the symmetric eigenvalue problem is illustrated in Section 10 in nag_dsyev (f08fac); its use in computing error bounds for singular vectors is illustrated in Section 10 in nag_dgesvd (f08kbc); and its use in computing error bounds for eigenvectors of the generalized symmetric definite eigenvalue problem is illustrated in Section 10 in nag_dsygv (f08sac).

nag_ddisna (f08flc) (PDF version)

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NAG Library Manual

NAG Library Function Documentnag_ddisna (f08flc)