(global) Must be

'N'

or

'V'

.

Specifies if it is necessary to compute the eigenvectors:

If

jobz

=

'N'

, then only eigenvalues are computed.

If

jobz

=

'V'

, then eigenvalues and eigenvectors are computed.

(global) Must be

'A'

,

'V'

, or

'I'

.

If

range

=

'A'

, all eigenvalues will be found.

If

range

=

'V'

, all eigenvalues in the half-open interval

[

vl

,

vu

]

will be found.

If

range

=

'I'

, the eigenvalues with indices

il

through

iu

will be found.

(global) Must be

'U'

or

'L'

.

Specifies whether the upper or lower triangular part of the Hermitian matrix

A

is stored:

If

uplo

=

'U'

,

a

stores the upper triangular part of

A

.

If

uplo

=

'L'

,

a

stores the lower triangular part of

A

.

(global) The number of rows and columns of the matrix

A

(

n

≥

0)

.

(local).

Block cyclic array of global size

n

*

n

and local size

lld_a

*

LOC

c

(

ja

+

n

-1)

. On entry, the Hermitian matrix

A

.

If

uplo

=

'U'

, only the upper triangular part of

A

is used to define the elements of the Hermitian matrix.

If

uplo

=

'L'

, only the lower triangular part of

A

is used to define the elements of the Hermitian matrix.

(global) The row and column indices in the global matrix

A

indicating the first row and the first column of the submatrix

A

, respectively.

(global and local) array of size

dlen_

. The array descriptor for the distributed matrix

A

. If

desca

[

ctxt_

- 1]

is incorrect,

p?heevx

cannot guarantee correct error reporting.

(global)

If

range

=

'V'

, the lower and upper bounds of the interval to be searched for eigenvalues; not referenced if

range

=

'A'

or

'I'

.

(global)

If

range

=

'I'

, the indices of the smallest and largest eigenvalues to be returned.

Constraints:

il

≥ 1;

min(

il

,

n

) ≤

iu

≤

n

.

Not referenced if

range

=

'A'

or

'V'

.

(global).

If

jobz

=

'V'

, setting

abstol

to

p?lamch

(

context

,

'U'

) yields the most orthogonal eigenvectors.

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

+

eps

*max(|

a

|,|

b

|)

, where

eps

is the machine precision. If

abstol

is less than or equal to zero, then

eps

*norm(

T

)

will be used in its place, where

norm(

T

)

is the 1-norm of the tridiagonal matrix obtained by reducing

A

to tridiagonal form.

Eigenvalues are computed most accurately when

abstol

is set to twice the underflow threshold

2*

p?lamch

('S')

, not zero. If this

function

returns with

((

mod

(

info

,2)

≠

0).

or

.(

mod

(

info

/8,2)

≠

0))

, indicating that some eigenvalues or eigenvectors did not converge, try setting

abstol

to

2*

p?lamch

('S')

.

(global).

Specifies which eigenvectors should be reorthogonalized. Eigenvectors that correspond to eigenvalues which are within

tol

=

orfac

*norm(

A

) of each other are to be reorthogonalized. However, if the workspace is insufficient (see

lwork

),

tol

may be decreased until all eigenvectors to be reorthogonalized can be stored in one process. No reorthogonalization will be done if

orfac

equals zero. A default value of 1.0e-3 is used if

orfac

is negative.

orfac

should be identical on all processes.

(global) The row and column indices in the global matrix

Z

indicating the first row and the first column of the submatrix

Z

, respectively.

(global and local) array of size

dlen_

. The array descriptor for the distributed matrix

Z

.

descz

[

ctxt_

- 1]

must equal

desca

[

ctxt_

- 1]

.

(local).

Array of size

lwork

.

(local) The size of the array

work

.

If only eigenvalues are requested:

If eigenvectors are requested:

with

nq

0 =

numroc

(

nn

,

nb

, 0, 0,

NPCOL

)

.

For optimal performance, greater workspace is needed, that is

where

lwork

is as defined above, and

nhetrd_lwork

=

n

+ 2*(

anb

+1)*(4*

nps

+2) + (

nps

+1)*

nps

If

lwork

= -1

, then

lwork

is global input and a workspace query is assumed; the

function

only calculates the size required for optimal performance for all work arrays. Each of these values is returned in the first entry of the corresponding work arrays, and no error message is issued by

pxerbla

.

(local)

Workspace array of size

lrwork

.

(local) The size of the array

work

.

See below for definitions of variables used to define

lwork

.

If no eigenvectors are requested (

jobz

=

'N'

), then

lrwork

≥

5*

nn

+4*

n.

If eigenvectors are requested (

jobz

=

'V'

), then the amount of workspace required to guarantee that all eigenvectors are computed is:

The computed eigenvectors may not be orthogonal if the minimal workspace is supplied and

orfac

is too small. If you want to guarantee orthogonality (at the cost of potentially poor performance) you should add the following values to

lrwork

:

(

clustersize

-1)*

n

,

where

clustersize

is the number of eigenvalues in the largest cluster, where a cluster is defined as a set of close eigenvalues:

Variable definitions:

neig

= number of eigenvectors requested;

nb

=

desca

[

mb_

- 1] =

desca

[

nb_

- 1] =

descz

[

mb_

- 1] =

descz

[

nb_

- 1]

;

nn

=

max

(

n

,

NB

, 2)

;

desca

[

rsrc_

- 1] =

desca

[

nb_

- 1] =

descz

[

rsrc_

- 1] =

descz

[

csrc_

- 1] = 0

;

iceil

(

x

,

y

)

is a ScaLAPACK function returning ceiling(

x

/

y

)

When

lrwork

is too small:

If

lwork

is too small to guarantee orthogonality,

p?heevx

attempts to maintain orthogonality in the clusters with the smallest spacing between the eigenvalues. If

lwork

is too small to compute all the eigenvectors requested, no computation is performed and

info

= -23 is returned. Note that when

range

=

'V'

,

p?heevx

does not know how many eigenvectors are requested until the eigenvalues are computed. Therefore, when

range

=

'V'

and as long as

lwork

is large enough to allow

p?heevx

to compute the eigenvalues,

p?heevx

will compute the eigenvalues and as many eigenvectors as it can.

Relationship between workspace, orthogonality and performance:

If

clustersize

≥

n

/

sqrt

(

NPROW

*

NPCOL

)

, then providing enough space to compute all the eigenvectors orthogonally will cause serious degradation in performance. In the limit (that is,

clustersize

=

n

-1)

p?stein

will perform no better than

?stein

on 1 processor.

For

clustersize

=

n

/

sqrt

(

NPROW

*

NPCOL

)

reorthogonalizing all eigenvectors will increase the total execution time by a factor of 2 or more.

For

clustersize

>

n

/

sqrt

(

NPROW

*

NPCOL

)

execution time will grow as the square of the cluster size, all other factors remaining equal and assuming enough workspace. Less workspace means less reorthogonalization but faster execution.

If

lwork

= -1

, then

lwork

is global input and a workspace query is assumed; the

function

only calculates the size required for optimal performance for all work arrays. Each of these values is returned in the first entry of the corresponding work arrays, and no error message is issued by

pxerbla

.

(local) Workspace array.

(local), size of

iwork

.

Where:

nnp

=

max

(

n

,

NPROW

*

NPCOL

+1, 4)

If

liwork

= -1

, then

liwork

is global input and a workspace query is assumed; the

function

only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by

pxerbla

.