nag_zhsein (f08pxc) : NAG Library, Mark 24

2 Specification

#include <nag.h>

#include <nagf08.h>

void

nag_zhsein (Nag_OrderType order, Nag_SideType side, Nag_EigValsSourceType eig_source, Nag_InitVeenumtype initv, const Nag_Boolean select[], Integer n, const Complex h[], Integer pdh, Complex w[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, Integer mm, Integer *m, Integer ifaill[], Integer ifailr[], NagError *fail)

3 Description

nag_zhsein (f08pxc) computes left and/or right eigenvectors of a complex upper Hessenberg matrix

H

, corresponding to selected eigenvalues.

The right eigenvector

x

, and the left eigenvector

y

, corresponding to an eigenvalue

λ

, are defined by:

H x = λ x and y^{H} H = λ y^{H} (or H^{H} y = \bar{λ} y) .

The eigenvectors are computed by inverse iteration. They are scaled so that

\max (|Re (x_{i})| + |Im x_{i}|) = 1

H

has been formed by reduction of a complex general matrix

A

to upper Hessenberg form, then the eigenvectors of

H

may be transformed to eigenvectors of

A

by a call to nag_zunmhr (f08nuc).

5 Arguments

1: order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: side – Nag_SideTypeInput

On entry: indicates whether left and/or right eigenvectors are to be computed.

$side = Nag_RightSide$: Only right eigenvectors are computed.
$side = Nag_LeftSide$: Only left eigenvectors are computed.
$side = Nag_BothSides$: Both left and right eigenvectors are computed.

Constraint:

side = Nag_RightSide

Nag_LeftSide

Nag_BothSides

3: eig_source – Nag_EigValsSourceTypeInput

On entry: indicates whether the eigenvalues of

H

(stored in w) were found using nag_zhseqr (f08psc).

$eig_source = Nag_HSEQRSource$: The eigenvalues of $H$ were found using nag_zhseqr (f08psc); thus if $H$ has any zero subdiagonal elements (and so is block triangular), then the $j$ th eigenvalue can be assumed to be an eigenvalue of the block containing the $j$ th row/column. This property allows the function to perform inverse iteration on just one diagonal block.
$eig_source = Nag_NotKnown$: No such assumption is made and the function performs inverse iteration using the whole matrix.

Constraint:

eig_source = Nag_HSEQRSource

Nag_NotKnown

4: initv – Nag_InitVeenumtypeInput

On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.

$initv = Nag_NoVec$: No initial estimates are supplied.
$initv = Nag_UserVec$: Initial estimates are supplied in vl and/or vr.

Constraint:

initv = Nag_NoVec

Nag_UserVec

5: select[

\dim

] – const Nag_BooleanInput

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

\max (1, n)

On entry: specifies which eigenvectors are to be computed. To select the eigenvector corresponding to the eigenvalue

w [j - 1]

select [j - 1]

must be set to Nag_TRUE.

6: n – IntegerInput

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

7: h[

\dim

] – const ComplexInput

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

\max (1, pdh \times n)

The

(i, j)

th element of the matrix

H

is stored in

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

On entry: the

n

n

upper Hessenberg matrix

H

8: pdh – IntegerInput

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

Constraint:

pdh \geq \max (1, n)

9: w[

\dim

] – ComplexInput/Output

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

\max (1, n)

On entry: the eigenvalues of the matrix

H

. If

eig_source = Nag_HSEQRSource

, the array must be exactly as returned by nag_zhseqr (f08psc).

On exit: the real parts of some elements of w may be modified, as close eigenvalues are perturbed slightly in searching for independent eigenvectors.

10: vl[

\dim

] – ComplexInput/Output

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

$\max (1, pdvl \times mm)$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$\max (1, n \times pdvl)$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
$1$ when $side = Nag_RightSide$ .

The dimension, dim, of the array vl must be at least

$\max (1, pdvl \times m)$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$\max (1, n \times pdvl)$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
$1$ when $side = Nag_RightSide$ .

The

(i, j)

th element of the matrix is stored in

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

On entry: if

initv = Nag_UserVec

and

side = Nag_LeftSide

Nag_BothSides

, vl must contain starting vectors for inverse iteration for the left eigenvectors. Each starting vector must be stored in the same row or column as will be used to store the corresponding eigenvector (see below).

initv = Nag_NoVec

, vl need not be set.

On exit: if

side = Nag_LeftSide

Nag_BothSides

, vl contains the computed left eigenvectors (as specified by select). The eigenvectors are stored consecutively in the rows or columns of the array (depending on the value of order), in the same order as their eigenvalues.

side = Nag_RightSide

, vl is not referenced.

11: pdvl – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq n$ ;
- if $side = Nag_RightSide$ , $pdvl \geq 1$ ;
if order=Nag_RowMajor,
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq \max (1, mm)$ ;
- if $side = Nag_RightSide$ , $pdvl \geq 1$ .
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq \max (1, m)$ ;
- if $side = Nag_RightSide$ , $pdvl \geq 1$ .

12: vr[

\dim

] – ComplexInput/Output

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

$\max (1, pdvr \times mm)$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$\max (1, n \times pdvr)$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
$1$ when $side = Nag_LeftSide$ .

The dimension, dim, of the array vr must be at least

$\max (1, pdvr \times m)$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$\max (1, n \times pdvr)$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
$1$ when $side = Nag_LeftSide$ .

The

(i, j)

th element of the matrix is stored in

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

On entry: if

initv = Nag_UserVec

and

side = Nag_RightSide

Nag_BothSides

, vr must contain starting vectors for inverse iteration for the right eigenvectors. Each starting vector must be stored in the same row or column as will be used to store the corresponding eigenvector (see below).

initv = Nag_NoVec

, vr need not be set.

On exit: if

side = Nag_RightSide

Nag_BothSides

, vr contains the computed right eigenvectors (as specified by select). The eigenvectors are stored consecutively in the rows or columns of the array (depending on the value of order), in the same order as their eigenvalues.

side = Nag_LeftSide

, vr is not referenced.

13: pdvr – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq n$ ;
- if $side = Nag_LeftSide$ , $pdvr \geq 1$ ;
if order=Nag_RowMajor,
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq \max (1, mm)$ ;
- if $side = Nag_LeftSide$ , $pdvr \geq 1$ .
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq \max (1, m)$ ;
- if $side = Nag_LeftSide$ , $pdvr \geq 1$ .

14: mm – IntegerInput

On entry: the number of columns in the arrays vl and/or vr if

order = Nag_ColMajor

or the number of rows in the arrays if

order = Nag_RowMajor

. The actual number of rows or columns required,

required_rowcol

, is obtained by counting

1

for each selected real eigenvector and

2

for each selected complex eigenvector (see select);

0 \leq required_rowcol \leq n

Constraint:

mm \geq required_rowcol

15: m – Integer *Output

On exit:

required_rowcol

, the number of selected eigenvectors.

16: ifaill[

\dim

] – IntegerOutput

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

$\max (1, mm)$ when $side = Nag_LeftSide$ or $Nag_BothSides$ ;
$1$ when $side = Nag_RightSide$ .

On exit: if

side = Nag_LeftSide

Nag_BothSides

, then

ifaill [i - 1] = 0

if the selected left eigenvector converged and

ifaill [i - 1] = j \geq 0

if the eigenvector stored in the

i

th row or column of vl (corresponding to the

j

th eigenvalue) failed to converge.

side = Nag_RightSide

, ifaill is not referenced.

17: ifailr[

\dim

] – IntegerOutput

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

$\max (1, mm)$ when $side = Nag_RightSide$ or $Nag_BothSides$ ;
$1$ when $side = Nag_LeftSide$ .

On exit: if

side = Nag_RightSide

Nag_BothSides

, then

ifailr [i - 1] = 0

if the selected right eigenvector converged and

ifailr [i - 1] = j \geq 0

if the eigenvector stored in the

i

th column of vr (corresponding to the

j

th eigenvalue) failed to converge.

side = Nag_LeftSide

, ifailr is not referenced.

18: 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_CONVERGENCE

⟨value⟩

eigenvectors (as indicated by arguments ifaill and/or ifailr) failed to converge. The corresponding columns of vl and/or vr contain no useful information.

NE_ENUM_INT_2

On entry,

side = ⟨value⟩

pdvl = ⟨value⟩

m = ⟨value⟩

.
Constraint: if

side = Nag_LeftSide

Nag_BothSides

pdvl \geq \max (1, m)

;
if

side = Nag_RightSide

pdvl \geq 1

On entry,

side = ⟨value⟩

pdvl = ⟨value⟩

mm = ⟨value⟩

.
Constraint: if

side = Nag_LeftSide

Nag_BothSides

pdvl \geq \max (1, mm)

;
if

side = Nag_RightSide

pdvl \geq 1

On entry,

side = ⟨value⟩

pdvl = ⟨value⟩

and

n = ⟨value⟩

.
Constraint: if

side = Nag_LeftSide

Nag_BothSides

pdvl \geq n

;
if

side = Nag_RightSide

pdvl \geq 1

On entry,

side = ⟨value⟩

pdvr = ⟨value⟩

m = ⟨value⟩

.
Constraint: if

side = Nag_RightSide

Nag_BothSides

pdvr \geq \max (1, m)

;
if

side = Nag_LeftSide

pdvr \geq 1

On entry,

side = ⟨value⟩

pdvr = ⟨value⟩

mm = ⟨value⟩

.
Constraint: if

side = Nag_RightSide

Nag_BothSides

pdvr \geq \max (1, mm)

;
if

side = Nag_LeftSide

pdvr \geq 1

On entry,

side = ⟨value⟩

pdvr = ⟨value⟩

and

n = ⟨value⟩

.
Constraint: if

side = Nag_RightSide

Nag_BothSides

pdvr \geq n

;
if

side = Nag_LeftSide

pdvr \geq 1

NE_INT

On entry,

mm = ⟨value⟩

.
Constraint:

mm \geq required_rowcol

, where

required_rowcol

is the number of selected eigenvectors.

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 0

On entry,

pdh = ⟨value⟩

.
Constraint:

pdh > 0

On entry,

pdvl = ⟨value⟩

.
Constraint:

pdvl > 0

On entry,

pdvr = ⟨value⟩

.
Constraint:

pdvr > 0

NE_INT_2

On entry,

pdh = ⟨value⟩

and

n = ⟨value⟩

.
Constraint:

pdh \geq \max (1, n)

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.

7 Accuracy

Each computed right eigenvector

x_{i}

is the exact eigenvector of a nearby matrix

A + E_{i}

, such that

‖E_{i}‖ = O (ε) ‖A‖

. Hence the residual is small:

‖A x_{i} - λ_{i} x_{i}‖ = O (ε) ‖A‖ .

However, eigenvectors corresponding to close or coincident eigenvalues may not accurately span the relevant subspaces.

Similar remarks apply to computed left eigenvectors.

8 Parallelism and Performance

nag_zhsein (f08pxc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zhsein (f08pxc) 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 Users' Note for your implementation for any additional implementation-specific information.

NAG Library Function Document

nag_zhsein (f08pxc)

+− 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

NAG Library Function Documentnag_zhsein (f08pxc)

+− 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

NAG Library Function Document

nag_zhsein (f08pxc)