void	f08qkc (Nag_OrderType order, Nag_SideType side, Nag_HowManyType how_many, Nag_Boolean select[], Integer n, const double t[], Integer pdt, double vl[], Integer pdvl, double vr[], Integer pdvr, Integer mm, Integer m, NagError fail)

The function may be called by the names: f08qkc, nag_lapackeig_dtrevc or nag_dtrevc.

3 Description

f08qkc computes left and/or right eigenvectors of a real upper quasi-triangular matrix

T

in canonical Schur form. Such a matrix arises from the Schur factorization of a real general matrix, as computed by f08pec, 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^{T} y = \bar{λ} y) .

Note that even though

T

is real,

λ

x

and

y

may be complex. If

x

is an eigenvector corresponding to a complex eigenvalue

λ

, then the complex conjugate vector

\bar{x}

is the eigenvector corresponding to the complex conjugate eigenvalue

\bar{λ}

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 an orthogonal matrix from the Schur factorization of a matrix

A

A = Q T Q^{T}

; 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, for a real eigenvector

x

\max (| x_{i} |) = 1

, and for a complex eigenvector,

\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_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: $side$ – Nag_SideType Input

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_HowManyType Input

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: $select [\dim]$ – Nag_Boolean Input/Output

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 real eigenvector corresponding to the real eigenvalue

λ_{j}

select [j - 1]

must be set Nag_TRUE. To select the complex eigenvector corresponding to a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

select [j - 1]

and/or

select [j]

must be set Nag_TRUE; the eigenvector corresponding to the first eigenvalue in the pair is computed.

On exit: if a complex eigenvector was selected as specified above,

select [j - 1]

is set to Nag_TRUE and

select [j]

to Nag_FALSE.

how_many = Nag_ComputeAll

Nag_BackTransform

, select is not referenced and may be NULL.

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

6: $t [\dim]$ – const double Input

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 \times n

upper quasi-triangular matrix

T

in canonical Schur form, as returned by f08pec.

7: $pdt$ – Integer Input

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

Constraint:

pdt \geq \max (1, n)

8: $vl [\dim]$ – double Input/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 \times n

matrix

Q

(usually the matrix of Schur vectors returned by f08pec).

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 of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one row or column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two rows or columns; the first row or column holds the real part and the second row or column holds the imaginary part.

side = Nag_RightSide

, vl is not referenced and may be NULL.

9: $pdvl$ – Integer Input

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: $vr [\dim]$ – double Input/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 \times n

matrix

Q

(usually the matrix of Schur vectors returned by f08pec).

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 of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one row or column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two rows or columns; the first row or column holds the real part and the second row or column holds the imaginary part.

side = Nag_LeftSide

, vr is not referenced and may be NULL.

11: $pdvr$ – Integer Input

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: $mm$ – Integer Input

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

m

, is

n

how_many = Nag_ComputeAll

Nag_BackTransform

; if

how_many = Nag_ComputeSelected

m

is obtained by counting

1

for each selected real eigenvector and

2

for each selected complex eigenvector (see select), in which case

0 \leq m \leq n

Constraints:

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

13: $m$ – Integer * Output

On exit:

m

, the number of rows or columns of vl and/or vr actually used to store the computed eigenvectors. If

how_many = Nag_ComputeAll

Nag_BackTransform

, m is set to

n

14: $fail$ – 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, $how_many = ⟨ value ⟩$ , $mm = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $how_many = Nag_ComputeAll$ or $Nag_BackTransform$ , $mm \geq n$ ;
otherwise $mm \geq m$ .

On entry, $side = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
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, $side = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
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: $n \geq 0$ .

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: $pdt \geq \max (1, n)$ .
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

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 f08qlc.

8 Parallelism and Performance

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

f08qkc 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.