NAG Library Function Document

nag_ztrevc (f08qxc)

+− 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

nag_ztrevc (f08qxc) computes selected left and/or right eigenvectors of a complex upper triangular matrix.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_ztrevc (Nag_OrderType order, Nag_SideType side, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, Complex t[], Integer pdt, Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, Integer mm, Integer m, NagError fail)

3 Description

nag_ztrevc (f08qxc) computes left and/or right eigenvectors of a complex upper triangular matrix

T

. Such a matrix arises from the Schur factorization of a complex general matrix, as computed by nag_zhseqr (f08psc), for example.

The right eigenvector

x

, and the left eigenvector

y

, corresponding to an eigenvalue

λ

, are defined by:

T x = λ x and y^{H} T = λ y^{H} (or ​ T^{H} y = \bar{λ} y) .

The function can compute the eigenvectors corresponding to selected eigenvalues, or it can compute all the eigenvectors. In the latter case the eigenvectors may optionally be pre-multiplied by an input matrix

Q

. Normally

Q

is a unitary matrix from the Schur factorization of a matrix

A

A = Q T Q^{H}

; if

x

is a (left or right) eigenvector of

T

, then

Q x

is an eigenvector of

A

The eigenvectors are computed by forward or backward substitution. They are scaled so that

\max (|Re (x_{i})| + |Im x_{i}|) = 1

4 References

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

Nag_ColMajor

2: side – Nag_SideTypeInput

On entry: indicates whether left and/or right eigenvectors are to be computed.

$side = Nag_RightSide$: Only right eigenvectors are computed.
$side = Nag_LeftSide$: Only left eigenvectors are computed.
$side = Nag_BothSides$: Both left and right eigenvectors are computed.

Constraint:

side = Nag_RightSide

Nag_LeftSide

Nag_BothSides

3: how_many – Nag_HowManyTypeInput

On entry: indicates how many eigenvectors are to be computed.

$how_many = Nag_ComputeAll$: All eigenvectors (as specified by side) are computed.
$how_many = Nag_BackTransform$: All eigenvectors (as specified by side) are computed and then pre-multiplied by the matrix $Q$ (which is overwritten).
$how_many = Nag_ComputeSelected$: Selected eigenvectors (as specified by side and select) are computed.

Constraint:

how_many = Nag_ComputeAll

Nag_BackTransform

Nag_ComputeSelected

4: select[ $\dim$ ] – const Nag_BooleanInput

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

$n$ when $how_many = Nag_ComputeSelected$ ;
otherwise select may be NULL.

On entry: specifies which eigenvectors are to be computed if

how_many = Nag_ComputeSelected

. To obtain the eigenvector corresponding to the eigenvalue

λ_{j}

select [j - 1]

must be set Nag_TRUE.

how_many = Nag_ComputeAll

Nag_BackTransform

, select is not referenced and may be NULL.

5: n – IntegerInput

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

6: t[ $\dim$ ] – ComplexInput/Output

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

pdt \times n

The

(i, j)

th element of the matrix

T

is stored in

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

On entry: the

n

n

upper triangular matrix

T

, as returned by nag_zhseqr (f08psc).

On exit: is used as internal workspace prior to being restored and hence is unchanged.

7: pdt – IntegerInput

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

Constraints:

if $order = Nag_ColMajor$ , $pdt \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdt \geq n$ .

8: vl[ $\dim$ ] – ComplexInput/Output

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

$pdvl \times mm$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$n \times pdvl$ when $side = Nag_LeftSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
otherwise vl may be NULL.

The

(i, j)

th element of the matrix is stored in

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

On entry: if

how_many = Nag_BackTransform

and

side = Nag_LeftSide

Nag_BothSides

, vl must contain an

n

n

matrix

Q

(usually the matrix of Schur vectors returned by nag_zhseqr (f08psc)).

how_many = Nag_ComputeAll

Nag_ComputeSelected

, vl need not be set.

On exit: if

side = Nag_LeftSide

Nag_BothSides

, vl contains the computed left eigenvectors (as specified by how_many and select). The eigenvectors are stored consecutively in the rows or columns (depending on the value of order) of the array, in the same order as their eigenvalues.

side = Nag_RightSide

, vl is not referenced and may be NULL.

9: pdvl – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq n$ ;
- if $side = Nag_RightSide$ , vl may be NULL;
if order=Nag_RowMajor,
- if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq mm$ ;
- if $side = Nag_RightSide$ , vl may be NULL.

10: vr[ $\dim$ ] – ComplexInput/Output

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

$pdvr \times mm$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_ColMajor$ ;
$n \times pdvr$ when $side = Nag_RightSide$ or $Nag_BothSides$ and $order = Nag_RowMajor$ ;
otherwise vr may be NULL.

The

(i, j)

th element of the matrix is stored in

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

On entry: if

how_many = Nag_BackTransform

and

side = Nag_RightSide

Nag_BothSides

, vr must contain an

n

n

matrix

Q

(usually the matrix of Schur vectors returned by nag_zhseqr (f08psc)).

how_many = Nag_ComputeAll

Nag_ComputeSelected

, vr need not be set.

On exit: if

side = Nag_RightSide

Nag_BothSides

, vr contains the computed right eigenvectors (as specified by how_many and select). The eigenvectors are stored consecutively in the rows or columns (depending on the value of order) of the array, in the same order as their eigenvalues.

side = Nag_LeftSide

, vr is not referenced and may be NULL.

11: pdvr – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq n$ ;
- if $side = Nag_LeftSide$ , vr may be NULL;
if order=Nag_RowMajor,
- if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq mm$ ;
- if $side = Nag_LeftSide$ , vr may be NULL.

12: mm – IntegerInput

On entry: the number of rows or columns (depending on the value of order) in the arrays vl and/or vr. The precise number of rows or columns required,

required_rowcol

, is

n

how_many = Nag_ComputeAll

Nag_BackTransform

; if

how_many = Nag_ComputeSelected

required_rowcol

is the number of selected eigenvectors (see select), in which case

0 \leq required_rowcol \leq n

Constraints:

if $how_many = Nag_ComputeAll$ or $Nag_BackTransform$ , $mm \geq n$ ;
otherwise $mm \geq required_rowcol$ .

13: m – Integer *Output

On exit:

required_rowcol

, the number of selected eigenvectors. If

how_many = Nag_ComputeAll

Nag_BackTransform

, m is set to

n

14: 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_ENUM_INT: On entry, $side = ⟨value⟩$ and $mm = ⟨value⟩$ .
Constraint: $mm > 0$ .
NE_ENUM_INT_2: On entry, $mm = ⟨value⟩$ , $n = ⟨value⟩$ and $how_many = ⟨value⟩$ .
Constraint: if $how_many = Nag_ComputeAll$ or $Nag_BackTransform$ , $mm \geq n$ ;
otherwise $mm \geq required_rowcol$ .
On entry, $side = ⟨value⟩$ , $pdvl = ⟨value⟩$ , $mm = ⟨value⟩$ .
Constraint: if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq mm$ .
On entry, $side = ⟨value⟩$ , $pdvl = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $side = Nag_LeftSide$ or $Nag_BothSides$ , $pdvl \geq n$ .
On entry, $side = ⟨value⟩$ , $pdvr = ⟨value⟩$ , $mm = ⟨value⟩$ .
Constraint: if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq mm$ .
On entry, $side = ⟨value⟩$ , $pdvr = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $side = Nag_RightSide$ or $Nag_BothSides$ , $pdvr \geq n$ .
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pdt = ⟨value⟩$ .
Constraint: $pdt > 0$ .
On entry, $pdvl = ⟨value⟩$ .
Constraint: $pdvl > 0$ .
On entry, $pdvr = ⟨value⟩$ .
Constraint: $pdvr > 0$ .
NE_INT_2: On entry, $pdt = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdt \geq \max (1, n)$ .
On entry, $pdt = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdt \geq n$ .
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

x_{i}

is an exact right eigenvector, and

{\tilde{x}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{x}}_{i}, x_{i})

between them is bounded as follows:

θ ({\tilde{x}}_{i}, x_{i}) \leq \frac{c (n) ε {‖T‖}_{2}}{{sep}_{i}}

where

{sep}_{i}

is the reciprocal condition number of

x_{i}

The condition number

{sep}_{i}

may be computed by calling nag_ztrsna (f08qyc).

8 Parallelism and Performance

nag_ztrevc (f08qxc) is not threaded by NAG in any implementation.

nag_ztrevc (f08qxc) 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 Users' Note for your implementation for any additional implementation-specific information.