f08yyc (Nag_OrderType order, Nag_JobType job, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const Complex a[], Integer pda, const Complex b[], Integer pdb, const Complex vl[], Integer pdvl, const Complex vr[], Integer pdvr, double s[], double dif[], Integer mm, Integer *m, NagError *fail)

The function may be called by the names: f08yyc, nag_lapackeig_ztgsna or nag_ztgsna.

3 Description

f08yyc estimates condition numbers for specified eigenvalues and/or right eigenvectors of an

n \times n

matrix pair

(S, T)

in generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.

The pair

(S, T)

are in generalized Schur form if

S

and

T

are upper triangular as returned, for example, by f08xpc or f08xqc, or f08xsc with

job = Nag_Schur

. The diagonal elements define the generalized eigenvalues

(α_{i}, β_{i})

, for

i = 1, 2, \dots, n

, of the pair

(S, T)

and the eigenvalues are given by

λ_{i} = α_{i} / β_{i},

so that

β_{i} S x_{i} = α_{i} T x_{i} or S x_{i} = λ_{i} T x_{i},

where

x_{i}

is the corresponding (right) eigenvector.

S

and

T

are the result of a generalized Schur factorization of a matrix pair

(A, B)

A = Q S Z^{H}, B = Q T Z^{H}

then the eigenvalues and condition numbers of the pair

(S, T)

are the same as those of the pair

(A, B)

Let

(α, β) \neq (0, 0)

be a simple generalized eigenvalue of

(A, B)

. Then the reciprocal of the condition number of the eigenvalue

λ = α / β

is defined as

s (λ) = \frac{{({| y^{H} A x |}^{2} + {| y^{H} B x |}^{2})}^{1 / 2}}{({‖ x ‖}_{2} {‖ y ‖}_{2})},

where

x

and

y

are the right and left eigenvectors of

(A, B)

corresponding to

λ

. If both

α

and

β

are zero, then

(A, B)

is singular and

s (λ) = −1

is returned.

U

and

V

are unitary transformations such that

U^{H} (A, B) V = (S, T) = (\begin{matrix} α & * \\ 0 & S_{22} \end{matrix}) (\begin{matrix} β & * \\ 0 & T_{22} \end{matrix}),

where

S_{22}

and

T_{22}

are

(n - 1) \times (n - 1)

matrices, then the reciprocal condition number is given by

Dif (x) \equiv Dif (y) = Dif ((α, β), (S_{22}, T_{22})) = σ_{\min} (Z),

where

σ_{\min} (Z)

denotes the smallest singular value of the

2 (n - 1) \times 2 (n - 1)

matrix

Z = (\begin{matrix} α \otimes I & −1 \otimes S_{22} \\ β \otimes I & −1 \otimes T_{22} \end{matrix})

and

\otimes

is the Kronecker product.

See Sections 2.4.8 and 4.11 of Anderson et al. (1999) and Kågström and Poromaa (1996) for further details and information.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103

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 eigenvalue

λ_{j}

select [j - 1]

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 pair

(S, T)

Constraint:

n \geq 0

6: $a [\dim]$ – const Complex Input

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

pda \times n

The

(i, j)

th element of the matrix

A

is stored in

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

On entry: the upper triangular matrix

S

7: $pda$ – Integer Input

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

Constraint:

pda \geq n

8: $b [\dim]$ – const Complex Input

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

pdb \times n

The

(i, j)

th element of the matrix

B

is stored in

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

On entry: the upper triangular matrix

T

9: $pdb$ – Integer Input

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

Constraint:

pdb \geq n

10: $vl [\dim]$ – const Complex 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.

i

th element of the

j

th vector 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 left eigenvectors of

(S, T)

, corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive columns of vl, as returned by f08wqc or f08yxc.

job = Nag_EigVecs

, vl is not referenced and may be NULL.

11: $pdvl$ – Integer Input

On entry: the stride used in the array vl.

Constraints:

if $order = Nag_ColMajor$ ,
- if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvl \geq n$ ;
- otherwise $pdvl \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvl \geq mm$ ;
- otherwise vl may be NULL.

12: $vr [\dim]$ – const Complex 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.

i

th element of the

j

th vector 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 right eigenvectors of

(S, T)

, corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive columns of vr, as returned by f08wqc or f08yxc.

job = Nag_EigVecs

, vr is not referenced and may be NULL.

13: $pdvr$ – Integer Input

On entry: the stride used in the array vr.

Constraints:

if $order = Nag_ColMajor$ ,
- if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvr \geq n$ ;
- otherwise $pdvr \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $job = Nag_EigVals$ or $Nag_DoBoth$ , $pdvr \geq mm$ ;
- otherwise vr may be NULL.

14: $s [\dim]$ – 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: if

job = Nag_EigVals

Nag_DoBoth

, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array.

job = Nag_EigVecs

, s is not referenced and may be NULL.

15: $dif [\dim]$ – double Output

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

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

On exit: if

job = Nag_EigVecs

Nag_DoBoth

, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute

dif [j - 1]

dif [j - 1]

is set to

0

; this can only occur when the true value would be very small anyway.

job = Nag_EigVals

, dif is not referenced and may be NULL.

16: $mm$ – Integer Input

On entry: the number of elements in the arrays s and dif.

Constraints:

if $how_many = Nag_ComputeAll$ , $mm \geq n$ ;
otherwise $mm \geq the number of selected eigenvalues$ .

17: $m$ – Integer * Output

On exit: the number of elements of the arrays s and dif used to store the specified condition numbers; for each selected eigenvalue one element is used.

how_many = Nag_ComputeAll

, m is set to n.

18: $fail$ – 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, $how_many = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $mm = ⟨ value ⟩$ .
Constraint: if $how_many = Nag_ComputeAll$ , $mm \geq n$ ;
otherwise $mm \geq the number of selected eigenvalues$ .

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: $n > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .

On entry, $pdvl = ⟨ value ⟩$ .
Constraint: $pdvl > 0$ .

On entry, $pdvr = ⟨ value ⟩$ .
Constraint: $pdvr > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq n$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq n$ .
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.
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

Not applicable.

8 Parallelism and Performance

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

f08yyc 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

An approximate asymptotic error bound on the chordal distance between the computed eigenvalue

\tilde{λ}

and the corresponding exact eigenvalue

λ

χ (\tilde{λ}, λ) \leq ε {‖ (A, B) ‖}_{F} / S (λ)

where

ε

is the machine precision.

An approximate asymptotic error bound for the right or left computed eigenvectors

\tilde{x}

\tilde{y}

corresponding to the right and left eigenvectors

x

and

y

is given by

θ (\tilde{z}, z) \leq ε {‖ (A, B) ‖}_{F} / Dif .

The real analogue of this function is f08ylc.

10 Example

This example estimates condition numbers and approximate error estimates for all the eigenvalues and right eigenvectors of the pair

(S, T)

given by

S = (\begin{array}{r} 4.0 + 4.0 i & 1.0 + 1.0 i & 1.0 + 1.0 i & 2.0 - 1.0 i \\ 0 & 2.0 + 1.0 i & 1.0 + 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 2.0 - 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 0 & 6.0 - 2.0 i \end{array})

and

T = (\begin{array}{r} 2.0 & 1.0 + 1.0 i & 1.0 + 1.0 i & 3.0 - 1.0 i \\ 0 & 1.0 & 2.0 + 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 1.0 & 1.0 + 1.0 i \\ 0 & 0 & 0 & 2.0 \end{array}) .

The eigenvalues and eigenvectors are computed by calling f08yxc.

f08yy: FL CL CPP AD PY MB

NAG CL Interfacef08yyc (ztgsna)

▸▿ 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
f08yyc (ztgsna)