NAG Library Function Document

nag_dhseqr (f08pec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_dhseqr (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>

#include <nagf08.h>

void	nag_dhseqr (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)

3 Description

nag_dhseqr (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 8 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 nag_dgehrd (f08nec) has been called to reduce

A

to Hessenberg form, nag_dorghr (f08nfc) must be called to form

Q

explicitly;

Q

is then passed to nag_dhseqr (f08pec), which must be called with

compz = Nag_UpdateZ

The function can also take advantage of a previous call to nag_dgebal (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 nag_dhseqr (f08pec) directly. Also, nag_dgebak (f08njc) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If nag_dgebal (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, nag_dgebal (f08nhc) must not be called with

job = Nag_Schur

Nag_DoBoth

, because the balancing transformation is not orthogonal.

nag_dhseqr (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_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: job – Nag_JobTypeInput

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_ComputeZTypeInput

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).
$compz = Nag_InitZ$: The Schur vectors of $H$ are computed (and the array z is initialized by the function).
$compz = Nag_UpdateZ$: The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

4: n – IntegerInput

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

5: ilo – IntegerInput

6: ihi – IntegerInput

On entry: if the matrix

A

has been balanced by nag_dgebal (f08nhc), then 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[ $\dim$ ] – doubleInput/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

n

upper Hessenberg matrix

H

, as returned by nag_dgehrd (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 – IntegerInput

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[ $\dim$ ] – doubleOutput

10: wi[ $\dim$ ] – doubleOutput

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 8 for details.

11: z[ $\dim$ ] – doubleInput/Output

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

$\max (1, pdz \times n)$ when $compz = Nag_UpdateZ$ or $Nag_InitZ$ and $order = Nag_ColMajor$ ;
$\max (1, \times pdz)$ when $compz = Nag_UpdateZ$ or $Nag_InitZ$ and $order = Nag_RowMajor$ ;
$1$ 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 – 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 $compz = Nag_InitZ$ or $Nag_UpdateZ$ , $pdz \geq \max (1, n)$ ;
- if $compz = Nag_NotZ$ , $pdz \geq 1$ ;
if order=Nag_RowMajor,
- 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
NE_ENUM_INT_2: On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compz = Nag_InitZ$ or $Nag_UpdateZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .; On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ , $n = 〈value〉$ .
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, $pdh = 〈value〉$ .
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.

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 nag_dtrsna (f08qlc).

8 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

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 nag_zhseqr (f08psc).

9 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 Section 9 in nag_dorghr (f08nfc), which illustrates the use of this function to compute the Schur factorization of a general matrix.

NAG Library Function Documentnag_dhseqr (f08pec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dhseqr (f08pec)