f08pec computes all the eigenvalues and, optionally, the Schur factorization of a real Hessenberg matrix or a real general matrix which has been reduced to Hessenberg form.

2 Specification

#include <nag.h>

void	f08pec (Nag_OrderType order, Nag_JobType job, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, double h[], Integer pdh, double wr[], double wi[], double z[], Integer pdz, NagError *fail)

The function may be called by the names: f08pec, nag_lapackeig_dhseqr or nag_dhseqr.

3 Description

f08pec computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix

H

H = Z T Z^{T},

where

T

is an upper quasi-triangular matrix (the Schur form of

H

), and

Z

is the orthogonal matrix whose columns are the Schur vectors

z_{i}

. See Section 9 for details of the structure of

T

The function may also be used to compute the Schur factorization of a real general matrix

A

which has been reduced to upper Hessenberg form

H

\begin{array}{l} A & = & Q H Q^{T}, where ​ Q ​ is orthogonal, \\ = & (Q Z) T {(Q Z)}^{T} . \end{array}

In this case, after f08nec has been called to reduce

A

to Hessenberg form, f08nfc must be called to form

Q

explicitly;

Q

is then passed to f08pec, which must be called with

compz = Nag_UpdateZ

The function can also take advantage of a previous call to f08nhc which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix

H

has the structure:

(\begin{matrix} H_{11} & H_{12} & H_{13} \\ H_{22} & H_{23} \\ H_{33} \end{matrix})

where

H_{11}

and

H_{33}

are upper triangular. If so, only the central diagonal block

H_{22}

(in rows and columns

i_{lo}

i_{hi}

) needs to be further reduced to Schur form (the blocks

H_{12}

and

H_{23}

are also affected). Therefore, the values of

i_{lo}

and

i_{hi}

can be supplied to f08pec directly. Also, f08njc must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nhc has not been called however, then

i_{lo}

must be set to

1

and

i_{hi}

n

. Note that if the Schur factorization of

A

is required, f08nhc must not be called with

job = Nag_Schur

Nag_DoBoth

, because the balancing transformation is not orthogonal.

f08pec uses a multishift form of the upper Hessenberg

Q R

algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that

{‖ z_{i} ‖}_{2} = 1

, but are determined only to within a factor

\pm 1

4 References

Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift

Q R

iteration Internat. J. High Speed Comput. 1 97–112

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

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: $job$ – Nag_JobType Input

On entry: indicates whether eigenvalues only or the Schur form

T

is required.

$job = Nag_EigVals$: Eigenvalues only are required.
$job = Nag_Schur$: The Schur form $T$ is required.

Constraint:

job = Nag_EigVals

Nag_Schur

3: $compz$ – Nag_ComputeZType Input

On entry: indicates whether the Schur vectors are to be computed.

$compz = Nag_NotZ$: No Schur vectors are computed (and the array z is not referenced).
$compz = Nag_UpdateZ$: The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz = Nag_InitZ$: The Schur vectors of $H$ are computed (and the array z is initialized by the function).

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

5: $ilo$ – Integer Input

6: $ihi$ – Integer Input

On entry: if the matrix

A

has been balanced by f08nhc, ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to

1

and ihi to n.

Constraint:

ilo \geq 1

and

\min (ilo, n) \leq ihi \leq n

7: $h [\dim]$ – double Input/Output

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

\max (1, pdh \times n)

where

H (i, j)

appears in this document, it refers to the array element

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

On entry: the

n \times n

upper Hessenberg matrix

H

, as returned by f08nec.

On exit: if

job = Nag_EigVals

, the array contains no useful information.

job = Nag_Schur

, h is overwritten by the upper quasi-triangular matrix

T

from the Schur decomposition (the Schur form) unless

fail . code =

NE_CONVERGENCE.

8: $pdh$ – Integer Input

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

Constraint:

pdh \geq \max (1, n)

9: $wr [\dim]$ – double Output

10: $wi [\dim]$ – double Output

Note: the dimension, dim, of the arrays wr and wi must be at least

\max (1, n)

On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless

fail . code =

NE_CONVERGENCE (in which case see Section 6). Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first. The eigenvalues are stored in the same order as on the diagonal of the Schur form

T

(if computed); see Section 9 for details.

11: $z [\dim]$ – double 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 the orthogonal matrix

Q

from the reduction to Hessenberg form.

compz = Nag_InitZ

, z need not be set.

On exit: if

compz = Nag_UpdateZ

Nag_InitZ

, z contains the orthogonal matrix of the required Schur vectors, unless

fail . code =

NE_CONVERGENCE.

compz = Nag_NotZ

, z is not referenced.

12: $pdz$ – 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 $compz = Nag_NotZ$ , $pdz \geq 1$ .

13: $fail$ – 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_CONVERGENCE

The algorithm has failed to find all the eigenvalues after a total of

30 (ihi - ilo + 1)

iterations.

fail . errnum = i

, elements

1, 2, \dots, ilo - 1

and

i + 1, i + 2, \dots, n

of wr and wi contain the real and imaginary parts of contain the eigenvalues which have been found.

job = Nag_EigVals

, then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix

\hat{H}

, formed from

H (ilo : fail . errnum, ilo : fail . errnum)

, i.e., the ilo through

fail . errnum

rows and columns of the final output matrix

H

job = Nag_Schur

, then on exit

(*) H_{i} U = U \tilde{H}

for some matrix

U

, where

H_{i}

is the input upper Hessenberg matrix and

\tilde{H}

is an upper Hessenberg matrix formed from

H (fail . errnum + 1 : ihi, fail . errnum + 1 : ihi)

compz = Nag_UpdateZ

, then on exit

Z_{out} = Z_{in} U

where

U

is defined in

(*)

(regardless of the value of job).

compz = Nag_InitZ

, then on exit

Z_{out} = U

where

U

is defined in

(*)

(regardless of the value of job).

fail . code =

NE_CONVERGENCE and

compz = Nag_NotZ

, z is not accessed.

NE_ENUM_INT_2

On entry,

compz = ⟨ value ⟩

pdz = ⟨ value ⟩

and

n = ⟨ value ⟩

.
Constraint: if

compz = Nag_UpdateZ

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,

pdh = ⟨ value ⟩

.
Constraint:

pdh > 0

On entry,

pdz = ⟨ value ⟩

.
Constraint:

pdz > 0

NE_INT_2

On entry,

pdh = ⟨ value ⟩

and

n = ⟨ value ⟩

.
Constraint:

pdh \geq \max (1, n)

NE_INT_3

On entry,

n = ⟨ value ⟩

ilo = ⟨ value ⟩

and

ihi = ⟨ value ⟩

.
Constraint:

ilo \geq 1

and

\min (ilo, n) \leq ihi \leq 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

The computed Schur factorization is the exact factorization of a nearby matrix

(H + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ H ‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue, and

{\tilde{λ}}_{i}

is the corresponding computed value, then

| {\tilde{λ}}_{i} - λ_{i} | \leq \frac{c (n) ε {‖ H ‖}_{2}}{s_{i}},

where

c (n)

is a modestly increasing function of

n

, and

s_{i}

is the reciprocal condition number of

λ_{i}

. The condition numbers

s_{i}

may be computed by calling f08qlc.

8 Parallelism and Performance

f08pec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08pec 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

The total number of floating-point operations depends on how rapidly the algorithm converges, but is typically about:

$7 n^{3}$ if only eigenvalues are computed;
$10 n^{3}$ if the Schur form is computed;
$20 n^{3}$ if the full Schur factorization is computed.

The Schur form

T

has the following structure (referred to as canonical Schur form).

If all the computed eigenvalues are real,

T

is upper triangular, and the diagonal elements of

T

are the eigenvalues;

wr [i - 1] = t_{i i}

, for

i = 1, 2, \dots, n

, and

wi [i - 1] = 0.0

If some of the computed eigenvalues form complex conjugate pairs, then

T

has

2 \times 2

diagonal blocks. Each diagonal block has the form

(\begin{matrix} t_{i i} & t_{i, i + 1} \\ t_{i + 1, i} & t_{i + 1, i + 1} \end{matrix}) = (\begin{matrix} α & β \\ γ & α \end{matrix})

where

β γ < 0

. The corresponding eigenvalues are

α \pm \sqrt{β γ}

;

wr [i - 1] = wr [i] = α

;

wi [i - 1] = + \sqrt{| β γ |}

;

wi [i] = - wi [i - 1]

The complex analogue of this function is f08psc.

10 Example

This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix

H

, where

H = (\begin{array}{r} 0.3500 & - 0.1160 & - 0.3886 & - 0.2942 \\ - 0.5140 & 0.1225 & 0.1004 & 0.1126 \\ 0.0000 & 0.6443 & - 0.1357 & - 0.0977 \\ 0.0000 & 0.0000 & 0.4262 & 0.1632 \end{array}) .

See also f08nfc, which illustrates the use of this function to compute the Schur factorization of a general matrix.

f08pe: FL CL CPP AD

NAG CL Interfacef08pec (dhseqr)

▸▿ 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
f08pec (dhseqr)