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

3

Description

nag_dtgsna (f08ylc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an

n

n

matrix pair

(S, T)

in real 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 real generalized Schur form if

S

is block upper triangular with

1

1

and

2

2

diagonal blocks and

T

is upper triangular as returned, for example, by nag_dgges (f08xac) or nag_dggesx (f08xbc), or nag_dhgeqz (f08xec) with

job = Nag_Schur

. The diagonal elements, or blocks, 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^{T}, B = Q T Z^{T}

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^{T} A x|}^{2} + {|y^{T} 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.

The definition of the reciprocal of the estimated condition number of the right eigenvector

x

and the left eigenvector

y

corresponding to the simple eigenvalue

λ

depends upon whether

λ

is a real eigenvalue, or one of a complex conjugate pair.

If the eigenvalue

λ

is real and

U

and

V

are orthogonal transformations such that

U^{T} (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)

(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)

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.

λ

is part of a complex conjugate pair and

U

and

V

are orthogonal transformations such that

U^{T} (A, B) V = (S, T) = (\begin{matrix} S_{11} & * \\ 0 & S_{22} \end{matrix}) (\begin{matrix} T_{11} & * \\ 0 & T_{22} \end{matrix}),

where

S_{11}

and

T_{11}

are two by two matrices,

S_{22}

and

T_{22}

are

(n - 2)

(n - 2)

matrices, and

(S_{11}, T_{11})

corresponds to the complex conjugate eigenvalue pair

λ

\bar{λ}

, then there exist unitary matrices

U_{1}

and

V_{1}

such that

U_{1}^{H} S_{11} V_{1} = (\begin{matrix} s_{11} & s_{12} \\ 0 & s_{22} \end{matrix}) and U_{1}^{H} T_{11} V_{1} = (\begin{matrix} t_{11} & t_{12} \\ 0 & t_{22} \end{matrix}) .

The eigenvalues are given by

λ = s_{11} / t_{11}

and

\bar{λ} = s_{22} / t_{22}

. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by

Dif (x) \equiv Dif (y) \leq \min (d_{1}, \max (1, |Re (s_{11}) / Re (s_{22})|), d_{2})

where

Re (z)

denotes the real part of

z

d_{1} = Dif ((s_{11}, t_{11}), (s_{22}, t_{22})) = σ_{\min} (Z_{1})

Z_{1}

is the complex two by two matrix

Z_{1} = (\begin{matrix} s_{11} & - s_{22} \\ t_{11} & - t_{22} \end{matrix}),

and

d_{2}

is an upper bound on

Dif ((S_{11}, T_{11}), (S_{22}, T_{22}))

; i.e., an upper bound on

σ_{\min} (Z_{2})

, where

Z_{2}

is the

(2 n - 2)

(2 n - 2)

matrix

Z_{2} = (\begin{matrix} S_{11}^{T} \otimes I & - I \otimes S_{22} \\ T_{11}^{T} \otimes I & - I \otimes T_{22} \end{matrix}) .

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 http://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_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $job$ – Nag_JobTypeInput

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: $howmny$ – Nag_HowManyTypeInput

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

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

Constraint:

howmny = Nag_ComputeAll

Nag_ComputeSelected

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

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

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

On entry: specifies the eigenpairs for which condition numbers are to be computed if

howmny = 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.

howmny = Nag_ComputeAll

, select is not referenced and may be NULL.

5: $n$ – IntegerInput

On entry:

n

, the order of the matrix pair

(S, T)

Constraint:

n \geq 0

6: $a [\dim]$ – const doubleInput

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 quasi-triangular matrix

S

7: $pda$ – IntegerInput

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

Constraint:

pda \geq \max (1, n)

8: $b [\dim]$ – const doubleInput

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: $pdb$ – IntegerInput

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

Constraint:

pdb \geq \max (1, n)

10: $vl [\dim]$ – const doubleInput

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 left eigenvectors of

(S, T)

, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by nag_dggev (f08wac) or nag_dtgevc (f08ykc).

job = Nag_EigVecs

, vl is not referenced and may be NULL.

11: $pdvl$ – 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 $job = Nag_EigVals$ 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 doubleInput

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 right eigenvectors of

(S, T)

, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by nag_dggev (f08wac) or nag_dtgevc (f08ykc).

job = Nag_EigVecs

, vr is not referenced and may be NULL.

13: $pdvr$ – 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 $job = Nag_EigVals$ 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: $s [\dim]$ – doubleOutput

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. For a complex conjugate pair of eigenvalues two consecutive elements of s are set to the same value. Thus

s [j - 1]

dif [j - 1]

, and the

j

th columns of

VL

and

VR

all correspond to the same eigenpair (but not in general the

j

th eigenpair, unless all eigenpairs are selected).

job = Nag_EigVecs

, s is not referenced and may be NULL.

15: $dif [\dim]$ – doubleOutput

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. For a complex eigenvector two consecutive elements of dif are set to the same value. 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: $mm$ – IntegerInput

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

Constraints:

if $howmny = Nag_ComputeAll$ , $mm \geq n$ ;
otherwise $mm \geq m$ .

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 real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If

howmny = Nag_ComputeAll

, m is set to n.

18: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_ENUM_INT_2: On entry, $job = 〈value〉$ , $pdvl = 〈value〉$ , $mm = 〈value〉$ .
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, $job = 〈value〉$ , $pdvr = 〈value〉$ , $mm = 〈value〉$ .
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_ENUM_INT_3: On entry, $howmny = 〈value〉$ , $n = 〈value〉$ , $mm = 〈value〉$ and $m = 〈value〉$ .
Constraint: if $howmny = Nag_ComputeAll$ , $mm \geq n$ ;
otherwise $mm \geq m$ .
NE_INT: 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: $pda \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

None.

8

Parallelism and Performance

nag_dtgsna (f08ylc) 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 complex analogue of this function is nag_ztgsna (f08yyc).

10

Example

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

(S, T)

given by

S = (\begin{array}{r} 4.0 & 1.0 & 1.0 & 2.0 \\ 0 & 3.0 & - 1.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 6.0 \end{array}) and T = (\begin{array}{r} 2.0 & 1.0 & 1.0 & 3.0 \\ 0 & 1.0 & 0.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 2.0 \end{array}) .

The eigenvalues and eigenvectors are computed by calling nag_dtgevc (f08ykc).

NAG Library Function Document

nag_dtgsna (f08ylc)

▸▿ 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

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