void f08qlc (Nag_OrderType order, Nag_JobType job, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const double t[], Integer pdt, const double vl[], Integer pdvl, const double vr[], Integer pdvr, double s[], double sep[], Integer mm, Integer *m, NagError *fail)

The function may be called by the names: f08qlc, nag_lapackeig_dtrsna or nag_dtrsna.

3 Description

f08qlc estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix

T

in canonical Schur form. These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix

A = Z T Z^{T}

(with orthogonal

Z

), from which

T

may have been derived.

f08qlc computes the reciprocal of the condition number of an eigenvalue

λ_{i}

s_{i} = \frac{| v^{H} u |}{{‖ u ‖}_{E} {‖ v ‖}_{E}},

where

u

and

v

are the right and left eigenvectors of

T

, respectively, corresponding to

λ_{i}

. This reciprocal condition number always lies between zero (i.e., ill-conditioned) and one (i.e., well-conditioned).

An approximate error estimate for a computed eigenvalue

λ_{i}

is then given by

\frac{ε ‖ T ‖}{s_{i}},

where

ε

is the machine precision.

To estimate the reciprocal of the condition number of the right eigenvector corresponding to

λ_{i}

, the function first calls f08qfc to reorder the eigenvalues so that

λ_{i}

is in the leading position:

T = Q (\begin{matrix} λ_{i} & c^{T} \\ 0 & T_{22} \end{matrix}) Q^{T} .

The reciprocal condition number of the eigenvector is then estimated as

{sep}_{i}

, the smallest singular value of the matrix

(T_{22} - λ_{i} I)

. This number ranges from zero (i.e., ill-conditioned) to very large (i.e., well-conditioned).

An approximate error estimate for a computed right eigenvector

u

corresponding to

λ_{i}

is then given by

\frac{ε ‖ T ‖}{{sep}_{i}} .

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: $job$ – Nag_JobType Input

On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.

$job = Nag_EigVals$: Condition numbers for eigenvalues only are computed.
$job = Nag_EigVecs$: Condition numbers for eigenvectors only are computed.
$job = Nag_DoBoth$: Condition numbers for both eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_EigVals

Nag_EigVecs

Nag_DoBoth

3: $how_many$ – Nag_HowManyType Input

On entry: indicates how many condition numbers are to be computed.

$how_many = Nag_ComputeAll$: Condition numbers for all eigenpairs are computed.
$how_many = Nag_ComputeSelected$: Condition numbers for selected eigenpairs (as specified by select) are computed.

Constraint:

how_many = Nag_ComputeAll

Nag_ComputeSelected

4: $select [\dim]$ – const Nag_Boolean Input

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 the eigenpairs for which condition numbers are to be computed if

how_many = Nag_ComputeSelected

. To select condition numbers for the eigenpair corresponding to the real eigenvalue

λ_{j}

select [j - 1]

must be set Nag_TRUE. To select condition numbers corresponding to a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

select [j - 1]

and/or

select [j]

must be set to Nag_TRUE.

how_many = Nag_ComputeAll

, 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: $vl [\dim]$ – const double Input

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

$pdvl \times mm$ when $job = Nag_EigVals$ or $Nag_DoBoth$ and $order = Nag_ColMajor$ ;
$n \times pdvl$ when $job = Nag_EigVals$ or $Nag_DoBoth$ 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

job = Nag_EigVals

Nag_DoBoth

, vl must contain the left eigenvectors of

T

(or of any matrix

Q T Q^{T}

with

Q

orthogonal) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns of vl, as returned by f08pkc or f08qkc.

job = Nag_EigVecs

, 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 $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvl \geq n$ ;
- if $job = Nag_EigVecs$ , vl may be NULL;
if $order = Nag_RowMajor$ ,
- if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvl \geq mm$ ;
- if $job = Nag_EigVecs$ , vl may be NULL.

10: $vr [\dim]$ – const double Input

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

$pdvr \times mm$ when $job = Nag_EigVals$ or $Nag_DoBoth$ and $order = Nag_ColMajor$ ;
$n \times pdvr$ when $job = Nag_EigVals$ or $Nag_DoBoth$ 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

job = Nag_EigVals

Nag_DoBoth

, vr must contain the right eigenvectors of

T

(or of any matrix

Q T Q^{T}

with

Q

orthogonal) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns of vr, as returned by f08pkc or f08qkc.

job = Nag_EigVecs

, 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 $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvr \geq n$ ;
- if $job = Nag_EigVecs$ , vr may be NULL;
if $order = Nag_RowMajor$ ,
- if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvr \geq mm$ ;
- if $job = Nag_EigVecs$ , vr may be NULL.

12: $s [\dim]$ – double Output

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

$mm$ when $job = Nag_EigVals$ or $Nag_DoBoth$ ;
otherwise s may be NULL.

On exit: the reciprocal condition numbers of the selected eigenvalues if

job = Nag_EigVals

Nag_DoBoth

, stored in consecutive elements of the array. Thus

s [j - 1]

sep [j - 1]

and the

j

th rows or columns of vl and vr all correspond to the same eigenpair (but not in general the

j

th eigenpair unless all eigenpairs have been selected). For a complex conjugate pair of eigenvalues, two consecutive elements of s are set to the same value.

job = Nag_EigVecs

, s is not referenced and may be NULL.

13: $sep [\dim]$ – double Output

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

$mm$ when $job = Nag_EigVecs$ or $Nag_DoBoth$ ;
otherwise sep may be NULL.

On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if

job = Nag_EigVecs

Nag_DoBoth

, stored in consecutive elements of the array. For a complex eigenvector, two consecutive elements of sep are set to the same value. If the eigenvalues cannot be reordered to compute

sep [j]

sep [j]

is set to zero; this can only occur when the true value would be very small anyway.

job = Nag_EigVals

, sep is not referenced and may be NULL.

14: $mm$ – Integer Input

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

m

, is

n

how_many = Nag_ComputeAll

; if

how_many = Nag_ComputeSelected

m

is obtained by counting

1

for each selected real eigenvalue, and

2

for each selected complex conjugate pair of eigenvalues (see select), in which case

0 \leq m \leq n

Constraint:

mm \geq m

Constraint: if

how_many = Nag_ComputeAll

mm \geq n

15: $m$ – Integer * Output

On exit:

m

, the number of elements of s and/or sep actually used to store the estimated condition numbers. If

how_many = Nag_ComputeAll

, m is set to

n

16: $fail$ – 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 $how_many = Nag_ComputeAll$ , $mm \geq n$ .

On entry, $job = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvl \geq mm$ .

On entry, $job = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvl \geq n$ .

On entry, $job = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvr \geq mm$ .

On entry, $job = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvr \geq n$ .
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, $mm = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $mm \geq m$ .

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

The computed values

{sep}_{i}

may over estimate the true value, but seldom by a factor of more than

3

8 Parallelism and Performance

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

f08qlc 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

For a description of canonical Schur form, see the document for f08pec.

The complex analogue of this function is f08qyc.

10 Example

This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix

T

, where

T = (\begin{matrix} 0.7995 & - 0.1144 & 0.0060 & 0.0336 \\ 0.0000 & - 0.0994 & 0.2478 & 0.3474 \\ 0.0000 & - 0.6483 & - 0.0994 & 0.2026 \\ 0.0000 & 0.0000 & 0.0000 & - 0.1007 \end{matrix}) .

f08ql: FL CL CPP AD PY MB

NAG CL Interfacef08qlc (dtrsna)

▸▿ 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
f08qlc (dtrsna)