void	f08wtc (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)

The function may be called by the names: f08wtc, nag_lapackeig_zgghd3 or nag_zgghd3.

3 Description

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 f08wvc. In the second step, matrix

B

is reduced to upper triangular form using the

Q R

factorization function f08asc and this unitary transformation

Q

is applied to matrix

A

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

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_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: $compq$ – Nag_ComputeQType Input

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_ComputeZType Input

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: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ilo$ – Integer Input

6: $ihi$ – Integer Input

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to 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: $a [\dim]$ – Complex Input/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 f08auc.

On exit: a is overwritten by the upper Hessenberg matrix

H

8: $pda$ – 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)

9: $b [\dim]$ – Complex Input/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 f08asc.

On exit: b is overwritten by the upper triangular matrix

T

10: $pdb$ – 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)

11: $q [\dim]$ – Complex Input/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: $pdq$ – Integer Input

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: $z [\dim]$ – Complex Input/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: $pdz$ – Integer Input

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 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_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: 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 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

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

8 Parallelism and Performance

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

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.