in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

2 Specification

#include <nag.h>

void	f08gpc (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Complex ap[], double vl, double vu, Integer il, Integer iu, double abstol, Integer m, double w[], Complex z[], Integer pdz, Integer jfail[], NagError fail)

The function may be called by the names: f08gpc, nag_lapackeig_zhpevx or nag_zhpevx.

3 Description

The Hermitian matrix

A

is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

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

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

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

5 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $job$ – Nag_JobType Input

On entry: indicates whether eigenvectors are computed.

$job = Nag_EigVals$: Only eigenvalues are computed.
$job = Nag_DoBoth$: Eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_EigVals

Nag_DoBoth

3: $range$ – Nag_RangeType Input

On entry: if

range = Nag_AllValues

, all eigenvalues will be found.

range = Nag_Interval

, all eigenvalues in the half-open interval

(vl, vu]

will be found.

range = Nag_Indices

, the ilth to iuth eigenvalues will be found.

Constraint:

range = Nag_AllValues

Nag_Interval

Nag_Indices

4: $uplo$ – Nag_UploType Input

On entry: if

uplo = Nag_Upper

, the upper triangular part of

A

is stored.

uplo = Nag_Lower

, the lower triangular part of

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $ap [\dim]$ – Complex Input/Output

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

\max (1, n \times (n + 1) / 2)

On entry: the upper or lower triangle of the

n \times n

Hermitian matrix

A

, packed by rows or columns.

The storage of elements

A_{i j}

depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ap [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ap [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ap [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ap [(i - 1) \times i / 2 + j - 1]$ , for $i \geq j$ .

On exit: ap is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of

A

7: $vl$ – double Input

8: $vu$ – double Input

On entry: if

range = Nag_Interval

, the lower and upper bounds of the interval to be searched for eigenvalues.

range = Nag_AllValues

Nag_Indices

, vl and vu are not referenced.

Constraint: if

range = Nag_Interval

vl < vu

9: $il$ – Integer Input

10: $iu$ – Integer Input

On entry: if

range = Nag_Indices

, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

range = Nag_AllValues

Nag_Interval

, il and iu are not referenced.

Constraints:

if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .

11: $abstol$ – double Input

On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval

[a, b]

of width less than or equal to

abstol + ε \max (| a |, | b |),

where

ε

is the machine precision. If abstol is less than or equal to zero, then

ε {‖ T ‖}_{1}

will be used in its place, where

T

is the tridiagonal matrix obtained by reducing

A

to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold

2 \times nag_real_safe_small_number

, not zero. If this function returns with

fail . code =

NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to

2 \times nag_real_safe_small_number

. See Demmel and Kahan (1990).

12: $m$ – Integer * Output

On exit: the total number of eigenvalues found.

0 \leq m \leq n

range = Nag_AllValues

m = n

range = Nag_Indices

m = iu - il + 1

13: $w [n]$ – double Output

On exit: the selected eigenvalues in ascending order.

14: $z [\dim]$ – Complex Output

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

$\max (1, pdz \times n)$ when $job = Nag_DoBoth$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

job = Nag_DoBoth

, then

if $fail . code =$ NE_NOERROR, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$ th column of $Z$ holding the eigenvector associated with $w [i - 1]$ ;
if an eigenvector fails to converge ( $fail . code =$ NE_CONVERGENCE), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

job = Nag_EigVals

, z is not referenced.

15: $pdz$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

16: $jfail [\dim]$ – Integer Output

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

\max (1, n)

On exit: if

job = Nag_DoBoth

, then

if $fail . code =$ NE_NOERROR, the first m elements of jfail are zero;
if $fail . code =$ NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge.

job = Nag_EigVals

, jfail is not referenced.

17: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: The algorithm failed to converge; $⟨ value ⟩$ eigenvectors did not converge. Their indices are stored in array jfail.
NE_ENUM_INT_2: On entry, $job = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_ENUM_INT_3: On entry, $range = ⟨ value ⟩$ , $il = ⟨ value ⟩$ , $iu = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .
NE_ENUM_REAL_2: On entry, $range = ⟨ value ⟩$ , $vl = ⟨ value ⟩$ and $vu = ⟨ value ⟩$ .
Constraint: if $range = Nag_Interval$ , $vl < vu$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

f08gpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08gpc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is proportional to

n^{3}

The real analogue of this function is f08gbc.

10 Example

This example finds the eigenvalues in the half-open interval

(- 2, 2]

, and the corresponding eigenvectors, of the Hermitian matrix

A = (\begin{array}{l} 1 & 2 - i & 3 - i & 4 - i \\ 2 + i & 2 & 3 - 2 i & 4 - 2 i \\ 3 + i & 3 + 2 i & 3 & 4 - 3 i \\ 4 + i & 4 + 2 i & 4 + 3 i & 4 \end{array}) .

f08gp: FL CL CPP AD

NAG CL Interfacef08gpc (zhpevx)

▸▿ 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

NAG CL Interface
f08gpc (zhpevx)