void	nag_ztgevc (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)

3 Description

nag_ztgevc (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 nag_zhgeqz (f08xsc) should be called before invoking nag_ztgevc (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 nag_zhgeqz (f08xsc) but nag_ztgevc (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 nag_ztgevc (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 nag_zhgeqz (f08xsc). Equivalently,

Q

and

Z

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

Q Y

and

Z X

are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to nag_zhgeqz (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_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 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $side$ – Nag_SideTypeInput

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_HowManyTypeInput

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_BooleanInput

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: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

6: $a [\dim]$ – const ComplexInput

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 nag_zhgeqz (f08xsc).

7: $pda$ – IntegerInput

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 ComplexInput

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 nag_zhgeqz (f08xsc).

9: $pdb$ – IntegerInput

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: $vl [\dim]$ – ComplexInput/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.

The

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

n

matrix

Q

. Usually, this is the unitary matrix

Q

of left Schur vectors returned by nag_zhgeqz (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: $pdvl$ – IntegerInput

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: $vr [\dim]$ – ComplexInput/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.

The

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

n

matrix

Z

. Usually, this is the unitary matrix

Z

of right Schur vectors returned by nag_dhgeqz (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: $pdvr$ – IntegerInput

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: $mm$ – IntegerInput

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 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONSTRAINT: On entry, $how_many = 〈value〉$ and $select [j] = 〈value〉$ .
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, $side = 〈value〉$ , $pdvl = 〈value〉$ , $mm = 〈value〉$ .
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, $side = 〈value〉$ , $pdvr = 〈value〉$ , $mm = 〈value〉$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_ztgevc (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

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

The real analogue of this function is nag_dtgevc (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: nag_zggbal (f08wvc) to balance the matrix, nag_zgeqrf (f08asc) to perform the

Q R

factorization of

B

, nag_zunmqr (f08auc) to apply

Q

A

, nag_zgghrd (f08wsc) to reduce the matrix pair to the generalized Hessenberg form and nag_zhgeqz (f08xsc) to compute the eigenvalues via the

Q Z

algorithm.

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

NAG Library Function Documentnag_ztgevc (f08yxc)

▸▿ 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 Library Function Document

nag_ztgevc (f08yxc)