void	nag_dhgeqz (Nag_OrderType order, Nag_JobType job, Nag_ComputeQType compq, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double q[], Integer pdq, double z[], Integer pdz, NagError *fail)

3

Description

nag_dhgeqz (f08xec) implements a single-double-shift version of the

Q Z

method for finding the generalized eigenvalues of the real matrix pair

(A, B)

which is in the generalized upper Hessenberg form. If the matrix pair

(A, B)

is not in the generalized upper Hessenberg form, then the function nag_dgghrd (f08wec) should be called before invoking nag_dhgeqz (f08xec).

This problem is mathematically equivalent to solving the equation

\det (A - λ B) = 0 .

Note that, to avoid underflow, overflow and other arithmetic problems, the generalized eigenvalues

λ_{j}

are never computed explicitly by this function but defined as ratios between two computed values,

α_{j}

and

β_{j}

λ_{j} = α_{j} / β_{j} .

The arguments

α_{j}

, in general, are finite complex values and

β_{j}

are finite real non-negative values.

If desired, the matrix pair

(A, B)

may be reduced to generalized Schur form. That is, the transformed matrix

B

is upper triangular and the transformed matrix

A

is block upper triangular, where the diagonal blocks are either

1

1

2

2

. The

1

1

blocks provide generalized eigenvalues which are real and the

2

2

blocks give complex generalized eigenvalues.

The argument job specifies two options. If

job = Nag_Schur

then the matrix pair

(A, B)

is simultaneously reduced to Schur form by applying one orthogonal transformation (usually called

Q

) on the left and another (usually called

Z

) on the right. That is,

\begin{matrix} A \leftarrow Q^{T} A Z \\ B \leftarrow Q^{T} B Z \end{matrix}

The

2

2

upper-triangular diagonal blocks of

B

corresponding to

2

2

blocks of a will be reduced to non-negative diagonal matrices. That is, if

A (j + 1, j)

is nonzero, then

B (j + 1, j) = B (j, j + 1) = 0

and

B (j, j)

and

B (j + 1, j + 1)

will be non-negative.

job = Nag_EigVals

, then at each iteration the same transformations are computed but they are only applied to those parts of

A

and

B

which are needed to compute

α

and

β

. This option could be used if generalized eigenvalues are required but not generalized eigenvectors.

job = Nag_Schur

and

compq = Nag_AccumulateQ

Nag_InitQ

, and

compz = Nag_AccumulateZ

Nag_InitZ

, then the orthogonal transformations used to reduce the pair

(A, B)

are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to

job = Nag_Schur

and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to

compq = Nag_AccumulateQ

Nag_InitQ

and not

compq \neq Nag_NotQ

compq = Nag_InitQ

, then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix

Q

. However, if

compq = Nag_AccumulateQ

then the transformations are accumulated on the user-supplied matrix

Q_{0}

in array q on entry and thus on exit q contains the matrix product

Q Q_{0}

. A similar convention is used for compz.

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 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: $job$ – Nag_JobTypeInput

On entry: specifies the operations to be performed on

(A, B)

$job = Nag_EigVals$: The matrix pair $(A, B)$ on exit might not be in the generalized Schur form.
$job = Nag_Schur$: The matrix pair $(A, B)$ on exit will be in the generalized Schur form.

Constraint:

job = Nag_EigVals

Nag_Schur

3: $compq$ – Nag_ComputeQTypeInput

On entry: specifies the operations to be performed on

Q

$compq = Nag_NotQ$: The array q is unchanged.
$compq = Nag_AccumulateQ$: The left transformation $Q$ is accumulated on the array q.
$compq = Nag_InitQ$: The array q is initialized to the identity matrix before the left transformation $Q$ is accumulated in q.

Constraint:

compq = Nag_NotQ

Nag_AccumulateQ

Nag_InitQ

4: $compz$ – Nag_ComputeZTypeInput

On entry: specifies the operations to be performed on

Z

$compz = Nag_NotZ$: The array z is unchanged.
$compz = Nag_AccumulateZ$: The right transformation $Z$ is accumulated on the array z.
$compz = Nag_InitZ$: The array z is initialized to the identity matrix before the right transformation $Z$ is accumulated in z.

Constraint:

compz = Nag_NotZ

Nag_AccumulateZ

Nag_InitZ

5: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

B

Q

and

Z

Constraint:

n \geq 0

6: $ilo$ – IntegerInput

7: $ihi$ – IntegerInput

On entry: the indices

i_{lo}

and

i_{hi}

, respectively which define the upper triangular parts of

A

. The submatrices

A (1 : i_{lo} - 1, 1 : i_{lo} - 1)

and

A (i_{hi} + 1 : n, i_{hi} + 1 : n)

are then upper triangular. These arguments are provided by nag_dggbal (f08whc) if the matrix pair was previously balanced; otherwise,

ilo = 1

and

ihi = n

Constraints:

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

8: $a [\dim]$ – doubleInput/Output

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

\max (1, pda \times n)

Where

A (i, j)

appears in this document, it refers to the array element

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

On entry: the

n

n

upper Hessenberg matrix

A

. The elements below the first subdiagonal must be set to zero.

On exit: if

job = Nag_Schur

, the matrix pair

(A, B)

will be simultaneously reduced to generalized Schur form.

job = Nag_EigVals

, the

1

1

and

2

2

diagonal blocks of the matrix pair

(A, B)

will give generalized eigenvalues but the remaining elements will be irrelevant.

9: $pda$ – 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)

10: $b [\dim]$ – doubleInput/Output

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

\max (1, pdb \times n)

Where

B (i, j)

appears in this document, it refers to the array element

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

On entry: the

n

n

upper triangular matrix

B

. The elements below the diagonal must be zero.

On exit: if

job = Nag_Schur

, the matrix pair

(A, B)

will be simultaneously reduced to generalized Schur form.

job = Nag_EigVals

, the

1

1

and

2

2

diagonal blocks of the matrix pair

(A, B)

will give generalized eigenvalues but the remaining elements will be irrelevant.

11: $pdb$ – 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)

12: $alphar [n]$ – doubleOutput

On exit: the real parts of

α_{j}

, for

j = 1, 2, \dots, n

13: $alphai [n]$ – doubleOutput

On exit: the imaginary parts of

α_{j}

, for

j = 1, 2, \dots, n

14: $beta [n]$ – doubleOutput

On exit:

β_{j}

, for

j = 1, 2, \dots, n

15: $q [\dim]$ – doubleInput/Output

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

$\max (1, pdq \times n)$ when $compq = Nag_AccumulateQ$ or $Nag_InitQ$ ;
$1$ 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_AccumulateQ

, the matrix

Q_{0}

. The matrix

Q_{0}

is usually the matrix

Q

returned by nag_dgghrd (f08wec).

compq = Nag_NotQ

, q is not referenced.

On exit: if

compq = Nag_AccumulateQ

, q contains the matrix product

Q Q_{0}

compq = Nag_InitQ

, q contains the transformation matrix

Q

16: $pdq$ – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $compq = Nag_AccumulateQ$ or $Nag_InitQ$ , $pdq \geq n$ ;
- if $compq = Nag_NotQ$ , $pdq \geq 1$ ;
if order=Nag_RowMajor,
- if $compq = Nag_AccumulateQ$ or $Nag_InitQ$ , $pdq \geq \max (1, n)$ ;
- if $compq = Nag_NotQ$ , $pdq \geq 1$ .

17: $z [\dim]$ – doubleInput/Output

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

$\max (1, pdz \times n)$ when $compz = Nag_AccumulateZ$ 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_AccumulateZ

, the matrix

Z_{0}

. The matrix

Z_{0}

is usually the matrix

Z

returned by nag_dgghrd (f08wec).

compz = Nag_NotZ

, z is not referenced.

On exit: if

compz = Nag_AccumulateZ

, z contains the matrix product

Z Z_{0}

compz = Nag_InitZ

, z contains the transformation matrix

Z

18: $pdz$ – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $compz = Nag_AccumulateZ$ or $Nag_InitZ$ , $pdz \geq n$ ;
- if $compz = Nag_NotZ$ , $pdz \geq 1$ ;
if order=Nag_RowMajor,
- if $compz = Nag_AccumulateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
- if $compz = Nag_NotZ$ , $pdz \geq 1$ .

19: $fail$ – 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, $compq = 〈value〉$ , $pdq = 〈value〉$ , $n = 〈value〉$ .
Constraint: if $compq = Nag_AccumulateQ$ or $Nag_InitQ$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

On entry, $compq = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compq = Nag_AccumulateQ$ or $Nag_InitQ$ , $pdq \geq n$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

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

On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compz = Nag_AccumulateZ$ or $Nag_InitZ$ , $pdz \geq n$ ;
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.

An unexpected Library error has occurred.
NE_ITERATION_QZ: The $Q Z$ iteration did not converge and the matrix pair $(A, B)$ is not in the generalized Schur form. The computed $α_{i}$ and $β_{i}$ should be correct for $i = 〈value〉, \dots, 〈value〉$ .
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.
NE_SCHUR: The computation of shifts failed and the matrix pair $(A, B)$ is not in the generalized Schur form. The computed $α_{i}$ and $β_{i}$ should be correct for $i = 〈value〉, \dots, 〈value〉$ .

7

Accuracy

Please consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990), for more information.

8

Parallelism and Performance

nag_dhgeqz (f08xec) 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_dhgeqz (f08xec) is the fifth step in the solution of the real generalized eigenvalue problem and is called after nag_dgghrd (f08wec).

The complex analogue of this function is nag_zhgeqz (f08xsc).

10

Example

This example computes the

α

and

β

arguments, which defines the generalized eigenvalues, of the matrix pair

(A, B)

given by

A = (\begin{array}{r} 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ 2.0 & 4.0 & 8.0 & 16.0 & 32.0 \\ 3.0 & 9.0 & 27.0 & 81.0 & 243.0 \\ 4.0 & 16.0 & 64.0 & 256.0 & 1024.0 \\ 5.0 & 25.0 & 125.0 & 625.0 & 3125.0 \end{array})

B = (\begin{array}{r} 1.0 & 2.0 & 3.0 & 4.0 & 5.0 \\ 1.0 & 4.0 & 9.0 & 16.0 & 25.0 \\ 1.0 & 8.0 & 27.0 & 64.0 & 125.0 \\ 1.0 & 16.0 & 81.0 & 256.0 & 625.0 \\ 1.0 & 32.0 & 243.0 & 1024.0 & 3125.0 \end{array}) .

This requires calls to five functions: nag_dggbal (f08whc) to balance the matrix, nag_dgeqrf (f08aec) to perform the

Q R

factorization of

B

, nag_dormqr (f08agc) to apply

Q

A

, nag_dgghrd (f08wec) to reduce the matrix pair to the generalized Hessenberg form and nag_dhgeqz (f08xec) to compute the eigenvalues using the

Q Z

algorithm.

NAG Library Function Document

nag_dhgeqz (f08xec)

▸▿ 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

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