void	f08yxc (Nag_OrderType order, Nag_SideType side, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const Complex a[], Integer pda, const Complex b[], Integer pdb, Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, Integer mm, Integer m, NagError fail)

The function may be called by the names: f08yxc, nag_lapackeig_ztgevc or nag_ztgevc.

3 Description

f08yxc computes some or all of the right and/or left generalized eigenvectors of the matrix pair

(A, B)

which is assumed to be in upper triangular form. If the matrix pair

(A, B)

is not upper triangular then the function f08xsc should be called before invoking f08yxc.

The right generalized eigenvector

x

and the left generalized eigenvector

y

(A, B)

corresponding to a generalized eigenvalue

λ

are defined by

(A - λ B) x = 0

and

y^{H} (A - λ B) = 0 .

If a generalized eigenvalue is determined as

0 / 0

, which is due to zero diagonal elements at the same locations in both

A

and

B

, a unit vector is returned as the corresponding eigenvector.

Note that the generalized eigenvalues are computed using f08xsc but f08yxc does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by f08yxc.

If all eigenvectors are requested, the function may either return the matrices

X

and/or

Y

of right or left eigenvectors of

(A, B)

, or the products

Z X

and/or

Q Y

, where

Z

and

Q

are two matrices supplied by you. Usually,

Q

and

Z

are chosen as the unitary matrices returned by f08xsc. Equivalently,

Q

and

Z

are the left and right Schur vectors of the matrix pair supplied to f08xsc. In that case,

Q Y

and

Z X

are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to f08xsc.

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

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

Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London

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: specifies the required sets of generalized eigenvectors.

$side = Nag_RightSide$: 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_BothSides

Nag_LeftSide

Nag_RightSide

3: $how_many$ – Nag_HowManyType Input

On entry: specifies further details of the required generalized eigenvectors.

$how_many = Nag_ComputeAll$: All right and/or left eigenvectors are computed.
$how_many = Nag_BackTransform$: All right and/or left eigenvectors are computed; they are backtransformed using the input matrices supplied in arrays vr and/or vl.
$how_many = Nag_ComputeSelected$: Selected right and/or left eigenvectors, defined by the array select, are computed.

Constraint:

how_many = Nag_ComputeAll

Nag_BackTransform

Nag_ComputeSelected

4: $select [\dim]$ – const Nag_Boolean Input

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

$n$ when $how_many = Nag_ComputeSelected$ ;
otherwise select may be NULL.

On entry: specifies the eigenvectors to be computed if

how_many = Nag_ComputeSelected

. To select the generalized eigenvector corresponding to the

j

th generalized eigenvalue, the

j

th element of select should be set to Nag_TRUE.

Constraint: if

how_many = Nag_ComputeSelected

select [j] = Nag_TRUE

Nag_FALSE

, for

j = 0, 1, \dots, n - 1

5: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

6: $a [\dim]$ – const Complex Input

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

pda \times n

The

(i, j)

th element of the matrix

A

is stored in

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

On entry: the matrix

A

must be in upper triangular form. Usually, this is the matrix

A

returned by f08xsc.

7: $pda$ – Integer Input

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

Constraint:

pda \geq \max (1, n)

8: $b [\dim]$ – const Complex Input

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

pdb \times n

The

(i, j)

th element of the matrix

B

is stored in

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

On entry: the matrix

B

must be in upper triangular form with non-negative real diagonal elements. Usually, this is the matrix

B

returned by f08xsc.

9: $pdb$ – Integer Input

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

Constraint:

pdb \geq \max (1, n)

10: $vl [\dim]$ – Complex Input/Output

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

$pdvl \times mm$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$n \times pdvl$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
otherwise vl may be NULL.

i

th element of the

j

th vector 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

how_many = Nag_BackTransform

and

side = Nag_LeftSide

Nag_BothSides

, vl must be initialized to an

n \times n

matrix

Q

. Usually, this is the unitary matrix

Q

of left Schur vectors returned by f08xsc.

On exit: if

side = Nag_LeftSide

Nag_BothSides

, vl contains:

if $how_many = Nag_ComputeAll$ , the matrix $Y$ of left eigenvectors of $(A, B)$ ;
if $how_many = Nag_BackTransform$ , the matrix $Q Y$ ;
if $how_many = Nag_ComputeSelected$ , the left eigenvectors of $(A, B)$ specified by select, stored consecutively in the rows or columns (depending on the value of order) of the array vl, in the same order as their corresponding eigenvalues.

11: $pdvl$ – Integer Input

On entry: the stride used in the array vl.

Constraints:

if $order = Nag_ColMajor$ ,
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq n$ ;
- if $side = Nag_RightSide$ , vl may be NULL;
if $order = Nag_RowMajor$ ,
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq mm$ ;
- if $side = Nag_RightSide$ , vl may be NULL.

12: $vr [\dim]$ – Complex Input/Output

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

$pdvr \times mm$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$n \times pdvr$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
otherwise vr may be NULL.

i

th element of the

j

th vector 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

how_many = Nag_BackTransform

and

side = Nag_RightSide

Nag_BothSides

, vr must be initialized to an

n \times n

matrix

Z

. Usually, this is the unitary matrix

Z

of right Schur vectors returned by f08xec.

On exit: if

side = Nag_RightSide

Nag_BothSides

, vr contains:

if $how_many = Nag_ComputeAll$ , the matrix $X$ of right eigenvectors of $(A, B)$ ;
if $how_many = Nag_BackTransform$ , the matrix $Z X$ ;
if $how_many = Nag_ComputeSelected$ , the right eigenvectors of $(A, B)$ specified by select, stored consecutively in the rows or columns (depending on the value of order) of the array vr, in the same order as their corresponding eigenvalues.

13: $pdvr$ – Integer Input

On entry: the stride used in the array vr.

Constraints:

if $order = Nag_ColMajor$ ,
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq n$ ;
- if $side = Nag_LeftSide$ , vr may be NULL;
if $order = Nag_RowMajor$ ,
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq mm$ ;
- if $side = Nag_LeftSide$ , vr may be NULL.

14: $mm$ – Integer Input

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

Constraints:

if $how_many = Nag_ComputeAll$ or $Nag_BackTransform$ , $mm \geq n$ ;
if $how_many = Nag_ComputeSelected$ , mm must not be less than the number of requested eigenvectors.

15: $m$ – Integer * Output

On exit: the number of columns in the arrays vl and/or vr actually used to store the eigenvectors. If

how_many = Nag_ComputeAll

Nag_BackTransform

, m is set to n. Each selected eigenvector occupies one column.

16: $fail$ – 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_CONSTRAINT: Constraint: if $how_many = Nag_ComputeSelected$ , $select [j] = Nag_TRUE$ or $Nag_FALSE$ , for $j = 0, 1, \dots, n - 1$ .
NE_ENUM_INT_2: On entry, $how_many = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $how_many = Nag_ComputeAll$ or $Nag_BackTransform$ , $mm \geq n$ ;
if $how_many = Nag_ComputeSelected$ , mm must not be less than the number of requested eigenvectors.

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

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

On entry, $side = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq mm$ .

On entry, $side = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq n$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .

On entry, $pdvl = ⟨ value ⟩$ .
Constraint: $pdvl > 0$ .

On entry, $pdvr = ⟨ value ⟩$ .
Constraint: $pdvr > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
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

It is beyond the scope of this manual to summarise the accuracy of the solution of the generalized eigenvalue problem. Interested readers should consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08yxc 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

f08yxc is the sixth step in the solution of the complex generalized eigenvalue problem and is usually called after f08xsc.

The real analogue of this function is f08ykc.

10 Example

This example computes the

α

and

β

arguments, which defines the generalized eigenvalues and the corresponding left and right eigenvectors, of the matrix pair

(A, B)

given by

A = (\begin{array}{r} 1.0 + 3.0 i & 1.0 + 4.0 i & 1.0 + 5.0 i & 1.0 + 6.0 i \\ 2.0 + 2.0 i & 4.0 + 3.0 i & 8.0 + 4.0 i & 16.0 + 5.0 i \\ 3.0 + 1.0 i & 9.0 + 2.0 i & 27.0 + 3.0 i & 81.0 + 4.0 i \\ 4.0 + 0.0 i & 16.0 + 1.0 i & 64.0 + 2.0 i & 256.0 + 3.0 i \end{array})

and

B = (\begin{array}{r} 1.0 + 0.0 i & 2.0 + 1.0 i & 3.0 + 2.0 i & 4.0 + 3.0 i \\ 1.0 + 1.0 i & 4.0 + 2.0 i & 9.0 + 3.0 i & 16.0 + 4.0 i \\ 1.0 + 2.0 i & 8.0 + 3.0 i & 27.0 + 4.0 i & 64.0 + 5.0 i \\ 1.0 + 3.0 i & 16.0 + 4.0 i & 81.0 + 5.0 i & 256.0 + 6.0 i \end{array}) .

To compute generalized eigenvalues, it is required to call five functions: f08wvc to balance the matrix, f08asc to perform the

Q R

factorization of

B

, f08auc to apply

Q

A

, f08wtc to reduce the matrix pair to the generalized Hessenberg form and f08xsc to compute the eigenvalues via the

Q Z

algorithm.

The computation of generalized eigenvectors is done by calling f08yxc to compute the eigenvectors of the balanced matrix pair. The function f08wwc is called to backward transform the eigenvectors to the user-supplied matrix pair. If both left and right eigenvectors are required then f08wwc must be called twice.

f08yx: FL CL CPP AD PY MB

NAG CL Interfacef08yxc (ztgevc)

▸▿ 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
f08yxc (ztgevc)