f08pkc computes selected left and/or right eigenvectors of a real upper Hessenberg matrix corresponding to specified eigenvalues, by inverse iteration.

2 Specification

#include <nag.h>

void

f08pkc (Nag_OrderType order, Nag_SideType side, Nag_EigValsSourceType eig_source, Nag_InitVeenumtype initv, Nag_Boolean select[], Integer n, const double h[], Integer pdh, double wr[], const double wi[], double vl[], Integer pdvl, double vr[], Integer pdvr, Integer mm, Integer *m, Integer ifaill[], Integer ifailr[], NagError *fail)

The function may be called by the names: f08pkc, nag_lapackeig_dhsein or nag_dhsein.

3 Description

f08pkc computes left and/or right eigenvectors of a real 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^{T} y = \bar{λ} y) .

Note that even though

H

is real,

λ

x

and

y

may be complex. If

x

is an eigenvector corresponding to a complex eigenvalue

λ

, then the complex conjugate vector

\bar{x}

is the eigenvector corresponding to the complex conjugate eigenvalue

\bar{λ}

The eigenvectors are computed by inverse iteration. They are scaled so that, for a real eigenvector

x

\max (| x_{i} |) = 1

, and for a complex eigenvector,

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

H

has been formed by reduction of a real general matrix

A

to upper Hessenberg form, then the eigenvectors of

H

may be transformed to eigenvectors of

A

by a call to f08ngc.

4 References

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: $side$ – Nag_SideType Input

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_EigValsSourceType Input

On entry: indicates whether the eigenvalues of

H

(stored in wr and wi) were found using f08pec.

$eig_source = Nag_HSEQRSource$: The eigenvalues of $H$ were found using f08pec; 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_InitVeenumtype Input

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: $select [\dim]$ – Nag_Boolean Input/Output

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 obtain the real eigenvector corresponding to the real eigenvalue

wr [j - 1]

select [j - 1]

must be set Nag_TRUE. To select the complex eigenvector corresponding to the complex eigenvalue

(wr [j - 1], wi [j - 1])

with complex conjugate (

wr [j], wi [j]

select [j - 1]

and/or

select [j]

must be set Nag_TRUE; the eigenvector corresponding to the first eigenvalue in the pair is computed.

On exit: if a complex eigenvector was selected as specified above,

select [j - 1]

is set to Nag_TRUE and

select [j]

to Nag_FALSE.

6: $n$ – Integer Input

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

7: $h [\dim]$ – const double Input

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 \times n

upper Hessenberg matrix

H

. If a NaN is detected in h, the function will return with

fail . code =

NE_BAD_PARAM.

Constraint: No element of h is equal to NaN.

8: $pdh$ – Integer Input

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: $wr [\dim]$ – double Input/Output

10: $wi [\dim]$ – const double Input

Note: the dimension, dim, of the arrays wr and wi must be at least

\max (1, n)

On entry: the real and imaginary parts, respectively, of the eigenvalues of the matrix

H

. Complex conjugate pairs of values must be stored in consecutive elements of the arrays. If

eig_source = Nag_HSEQRSource

, the arrays must be exactly as returned by f08pec.

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

11: $vl [\dim]$ – double Input/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

(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 rows or columns 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. Corresponding to each selected real eigenvalue is a real eigenvector, occupying one row or column. Corresponding to each selected complex eigenvalue is a complex eigenvector, occupying two rows or columns: the first row or column holds the real part and the second row or column holds the imaginary part.

side = Nag_RightSide

, vl is not referenced.

12: $pdvl$ – Integer Input

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 $side = Nag_RightSide$ , $pdvl \geq 1$ .

13: $vr [\dim]$ – double Input/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

(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 rows or columns 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 order argument), in the same order as their eigenvalues. Corresponding to each selected real eigenvalue is a real eigenvector, occupying one row or column. Corresponding to each selected complex eigenvalue is a complex eigenvector, occupying two rows or columns: the first row or column holds the real part and the second row or column holds the imaginary part.

side = Nag_LeftSide

, vr is not referenced.

14: $pdvr$ – Integer Input

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 $side = Nag_LeftSide$ , $pdvr \geq 1$ .

15: $mm$ – Integer Input

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,

m

, is obtained by counting

1

for each selected real eigenvector and

2

for each selected complex eigenvector (see select);

0 \leq m \leq n

Constraint:

mm \geq m

16: $m$ – Integer * Output

On exit:

m

, the number of rows or columns of vl and/or vr required to store the selected eigenvectors.

17: $ifaill [\dim]$ – Integer Output

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 as held in

(wr [j - 1], wi [j - 1])

failed to converge. If the

i

th and

(i + 1)

th rows or columns of vl contain a selected complex eigenvector, then

ifaill [i - 1]

and

ifaill [i]

are set to the same value.

side = Nag_RightSide

, ifaill is not referenced.

18: $ifailr [\dim]$ – Integer Output

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 row or column of vr (corresponding to the

j

th eigenvalue as held in

(wr [j - 1], wi [j - 1])

) failed to converge. If the

i

th and

(i + 1)

th rows or columns of vr contain a selected complex eigenvector, then

ifailr [i - 1]

and

ifailr [i]

are set to the same value.

side = Nag_LeftSide

, ifailr is not referenced.

19: $fail$ – 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: Constraint: No element of h is equal to NaN.

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, $side = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq \max (1, mm)$ ;
if $side = Nag_RightSide$ , $pdvl \geq 1$ .

On entry, $side = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq n$ ;
if $side = Nag_RightSide$ , $pdvl \geq 1$ .

On entry, $side = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq \max (1, mm)$ ;
if $side = Nag_LeftSide$ , $pdvr \geq 1$ .

On entry, $side = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq n$ ;
if $side = Nag_LeftSide$ , $pdvr \geq 1$ .
NE_INT: On entry, $mm = ⟨ value ⟩$ .
Constraint: $mm \geq m$ .

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.
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

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

f08pkc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08pkc 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.