dsyevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the

Q L

Q R

algorithm.

2 Specification

#include "f08/nagcpp_f08fc.hpp"

template <typename A, typename W>

void function dsyevd(const string job, const string uplo, A &&a, W &&w, OptionalF08FC opt)

template <typename A, typename W>

void function dsyevd(const string job, const string uplo, A &&a, W &&w)

3 Description

dsyevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix

A

. In other words, it can compute the spectral factorization of

A

A = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues

λ_{i}

, and

Z

is the orthogonal matrix whose columns are the eigenvectors

z_{i}

. Thus

A z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, n .

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

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

5 Arguments

1: $job$ – string Input

On entry: indicates whether eigenvectors are computed.

$job ='N'$: Only eigenvalues are computed.
$job ='V'$: Eigenvalues and eigenvectors are computed.

Constraint:

job ='N'

'V'

2: $uplo$ – string Input

On entry: indicates whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored.
$uplo ='L'$: The lower triangular part of $A$ is stored.

Constraint:

uplo ='U'

'L'

3: $a (n, n)$ – double array Input/Output

On entry: the

n

n

symmetric matrix

A

If $uplo ='U'$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

job ='V'

, a is overwritten by the orthogonal matrix

Z

which contains the eigenvectors of

A

4: $w (n)$ – double array Output

On exit: the eigenvalues of the matrix

A

in ascending order.

5: $opt$ – OptionalF08FC Input/Output

Optional parameter container, derived from Optional.

5.1Additional Quantities

1: $n$: $n$ , the order of the matrix $A$
2: $info$

6 Exceptions and Warnings

All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).

Raises: ErrorException

$errorid = - 1$: On entry, error in parameter job.
Constraint: $job = "N" or "V"$ .

$errorid = - 2$: On entry, error in parameter uplo.
Constraint: $uplo = "U" or "L"$ .

$errorid = - 3$: On entry, error in parameter $n$ .
Constraint: $n \geq 0$ .

$errorid > 0$: If $info = ⟨ value ⟩$ and $job = "N"$ , the algorithm failed to
converge; $⟨ value ⟩$ elements of an intermediate tridiagonal
form did not converge to zero; if $info = ⟨ value ⟩$ and
$job = "V"$ , then the algorithm failed to compute an eigenvalue while working
on the submatrix lying in rows and column $⟨ value ⟩ / (n + 1)$
through $\mod (⟨ value ⟩, n + 1)$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument has $⟨ value ⟩$ dimensions.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument was a vector of size $⟨ value ⟩$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
The size for the supplied array could not be ascertained.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a $⟨ value ⟩$ x $⟨ value ⟩$ array.
Supplied argument has $⟨ value ⟩$ dimensions.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a $⟨ value ⟩$ x $⟨ value ⟩$ array.
Supplied argument was a $⟨ value ⟩$ x $⟨ value ⟩$ array.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a $⟨ value ⟩$ x $⟨ value ⟩$ array.
Not all of the sizes for the supplied array could be ascertained.

$errorid = 10602$: On entry, the raw data component of $⟨ value ⟩$ is null.

$errorid = 10603$: On entry, unable to ascertain a value for $⟨ value ⟩$ .

$errorid = 10604$: On entry, the data in $⟨ value ⟩$ is stored in $⟨ value ⟩$ Major Order.
The data was expected to be in $⟨ value ⟩$ Major Order.

$errorid = - 99$: An unexpected error has been triggered by this routine.

$errorid = - 999$: Dynamic memory allocation failed.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(A + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ A ‖}_{2},

and

ε

is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Please see the description for the underlying computational routine in this section of the FL Interface documentation.

9 Further Comments

The complex analogue of this function is f08fqf (no CPP interface).

10 Example

Examples for the NAG CPP Interface are not currently available.

NAG Library Manual, Mark 27

Interfaces: FL CL CPP AD

NAG CPP Interface Introduction

F08 (Lapackeig) Chapter Contents

f08fc: FL CL CPP AD

Namespace: nagcpp

Namespace: lapackeig

Function: dsyevd

NAG CPP Interfacenagcpp::lapackeig::dsyevd (f08fc)

▸▿ Contents

1 Purpose