void	nag_zgghd3 (Nag_OrderType order, Nag_ComputeQType compq, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, Complex a[], Integer pda, Complex b[], Integer pdb, Complex q[], Integer pdq, Complex z[], Integer pdz, NagError *fail)

3

Description

nag_zgghd3 (f08wtc) is usually the third step in the solution of the complex generalized eigenvalue problem

A x = λ B x .

The (optional) first step balances the two matrices using nag_zggbal (f08wvc). In the second step, matrix

B

is reduced to upper triangular form using the

Q R

factorization function nag_zgeqrf (f08asc) and this unitary transformation

Q

is applied to matrix

A

by calling nag_zunmqr (f08auc). The driver, nag_zggev3 (f08wqc), solves the complex generalized eigenvalue problem by combining all the required steps including those just listed.

nag_zgghd3 (f08wtc) reduces a pair of complex matrices

(A, B)

, where

B

is triangular, to the generalized upper Hessenberg form using unitary transformations. This two-sided transformation is of the form

\begin{matrix} Q^{H} A Z = H, \\ Q^{H} B Z = T \end{matrix}

where

H

is an upper Hessenberg matrix,

T

is an upper triangular matrix and

Q

and

Z

are unitary matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices

Q_{1}

and

Z_{1}

, so that

\begin{matrix} Q_{1} A Z_{1}^{H} = (Q_{1} Q) H {(Z_{1} Z)}^{H}, \\ Q_{1} B Z_{1}^{H} = (Q_{1} Q) T {(Z_{1} Z)}^{H} . \end{matrix}

4

References

Golub G H and Van Loan C F (2012) Matrix Computations (4th 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

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 3.3.1.3 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: $compq$ – Nag_ComputeQTypeInput

On entry: specifies the form of the computed unitary matrix

Q

$compq = Nag_NotQ$: Do not compute $Q$ .
$compq = Nag_InitQ$: The unitary matrix $Q$ is returned.
$compq = Nag_UpdateSchur$: q must contain a unitary matrix $Q_{1}$ , and the product $Q_{1} Q$ is returned.

Constraint:

compq = Nag_NotQ

Nag_InitQ

Nag_UpdateSchur

3: $compz$ – Nag_ComputeZTypeInput

On entry: specifies the form of the computed unitary matrix

Z

$compz = Nag_NotZ$: Do not compute $Z$ .
$compz = Nag_UpdateZ$: z must contain a unitary matrix $Z_{1}$ , and the product $Z_{1} Z$ is returned.
$compz = Nag_InitZ$: The unitary matrix $Z$ is returned.

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

4: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ilo$ – IntegerInput

6: $ihi$ – IntegerInput

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to nag_zggbal (f08wvc). Otherwise, they should be set to

1

and

n

, respectively.

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

7: $a [\dim]$ – ComplexInput/Output

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

\max (1, 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

of the matrix pair

(A, B)

. Usually, this is the matrix

A

returned by nag_zunmqr (f08auc).

On exit: a is overwritten by the upper Hessenberg matrix

H

8: $pda$ – 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)

9: $b [\dim]$ – ComplexInput/Output

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

\max (1, 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 upper triangular matrix

B

of the matrix pair

(A, B)

. Usually, this is the matrix

B

returned by the

Q R

factorization function nag_zgeqrf (f08asc).

On exit: b is overwritten by the upper triangular matrix

T

10: $pdb$ – 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)

11: $q [\dim]$ – ComplexInput/Output

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

$\max (1, pdq \times n)$ when $compq = Nag_InitQ$ or $Nag_UpdateSchur$ ;
$1$ when $compq = Nag_NotQ$ .

The

(i, j)

th element of the matrix

Q

is stored in

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

On entry: if

compq = Nag_UpdateSchur

, q must contain a unitary matrix

Q_{1}

compq = Nag_NotQ

, q is not referenced.

On exit: if

compq = Nag_InitQ

, q contains the unitary matrix

Q

Iif

compq = Nag_UpdateSchur

, q is overwritten by

Q_{1} Q

12: $pdq$ – IntegerInput

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

Constraints:

if $compq = Nag_InitQ$ or $Nag_UpdateSchur$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

13: $z [\dim]$ – ComplexInput/Output

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

$\max (1, pdz \times n)$ when $compz = Nag_UpdateZ$ or $Nag_InitZ$ ;
$1$ when $compz = Nag_NotZ$ .

The

(i, j)

th element of the matrix

Z

is stored in

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

On entry: if

compz = Nag_UpdateZ

, z must contain a unitary matrix

Z_{1}

compz = Nag_NotZ

, z is not referenced.

On exit: if

compz = Nag_InitZ

, z contains the unitary matrix

Z

compz = Nag_UpdateZ

, z is overwritten by

Z_{1} Z

14: $pdz$ – IntegerInput

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

Constraints:

if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .

15: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 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_ENUM_INT_2: On entry, $compq = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compq = Nag_InitQ$ or $Nag_UpdateSchur$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .
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, $pdq = 〈value〉$ .
Constraint: $pdq > 0$ .

On entry, $pdz = 〈value〉$ .
Constraint: $pdz > 0$ .
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_INT_3: On entry, $n = 〈value〉$ , $ilo = 〈value〉$ and $ihi = 〈value〉$ .
Constraint: $ilo \geq 1$ and
if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .
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 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

The reduction to the generalized Hessenberg form is implemented using unitary transformations which are backward stable.

8

Parallelism and Performance

nag_zgghd3 (f08wtc) 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

This function is usually followed by nag_zhgeqz (f08xsc) which implements the

Q Z

algorithm for computing generalized eigenvalues of a reduced pair of matrices.

The real analogue of this function is nag_dgghd3 (f08wfc).

10

Example

See Section 10 in nag_zhgeqz (f08xsc) and nag_ztgevc (f08yxc).

nag_zgghd3 (f08wtc) (PDF version)

f08 Chapter Contents

f08 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_zgghd3 (f08wtc)

▸▿ 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

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