Nag_RCondType sense, Integer n, double a[], Integer pda, Integer *sdim, double wr[], double wi[], double vs[], Integer pdvs, double *rconde, double *rcondv, NagError *fail)

3

Description

The real Schur factorization of

A

is given by

A = Z T Z^{T},

where

Z

, the matrix of Schur vectors, is orthogonal and

T

is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with

1

1

and

2

2

blocks.

2

2

blocks will be standardized in the form

[\begin{array}{c} a & b \\ c & a \end{array}]

where

b c < 0

. The eigenvalues of such a block are

a \pm \sqrt{b c}

Optionally, nag_dgeesx (f08pbc) also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of

Z

form an orthonormal basis for this invariant subspace.

For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8 of Anderson et al. (1999) (where these quantities are called

s

and

sep

respectively).

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

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 3.3.1.3 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: $jobvs$ – Nag_JobTypeInput

On entry: if

jobvs = Nag_DoNothing

, Schur vectors are not computed.

jobvs = Nag_Schur

, Schur vectors are computed.

Constraint:

jobvs = Nag_DoNothing

Nag_Schur

3: $sort$ – Nag_SortEigValsTypeInput

On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.

$sort = Nag_NoSortEigVals$: Eigenvalues are not ordered.
$sort = Nag_SortEigVals$: Eigenvalues are ordered (see select).

Constraint:

sort = Nag_NoSortEigVals

Nag_SortEigVals

4: $select$ – function, supplied by the userExternal Function

sort = Nag_SortEigVals

, select is used to select eigenvalues to sort to the top left of the Schur form.

sort = Nag_NoSortEigVals

, select is not referenced and nag_dgeesx (f08pbc) may be specified as NULLFN.

An eigenvalue

wr [j - 1] + \sqrt{- 1} \times wi [j - 1]

is selected if

select (wr [j - 1], wi [j - 1])

is Nag_TRUE. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy

select (wr [j - 1], wi [j - 1]) = Nag_TRUE

after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case

fail . errnum

is set to

n + 2

The specification of select is:

Nag_Boolean

select (double wr, double wi)

1: $wr$ – doubleInput
2: $wi$ – doubleInput: On entry: the real and imaginary parts of the eigenvalue.

5: $sense$ – Nag_RCondTypeInput

On entry: determines which reciprocal condition numbers are computed.

$sense = Nag_NotRCond$: None are computed.
$sense = Nag_RCondEigVals$: Computed for average of selected eigenvalues only.
$sense = Nag_RCondEigVecs$: Computed for selected right invariant subspace only.
$sense = Nag_RCondBoth$: Computed for both.

sense = Nag_RCondEigVals

Nag_RCondEigVecs

Nag_RCondBoth

sort = Nag_SortEigVals

Constraint:

sense = Nag_NotRCond

Nag_RCondEigVals

Nag_RCondEigVecs

Nag_RCondBoth

6: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

7: $a [\dim]$ – doubleInput/Output

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

\max (1, 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

n

n

matrix

A

On exit: a is overwritten by its real Schur form

T

8: $pda$ – 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)

9: $sdim$ – Integer *Output

On exit: if

sort = Nag_NoSortEigVals

sdim = 0

sort = Nag_SortEigVals

sdim =

number of eigenvalues (after sorting) for which select is Nag_TRUE. (Complex conjugate pairs for which select is Nag_TRUE for either eigenvalue count as

2

10: $wr [\dim]$ – doubleOutput

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

\max (1, n)

On exit: see the description of wi.

11: $wi [\dim]$ – doubleOutput

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

\max (1, n)

On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form

T

. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

12: $vs [\dim]$ – doubleOutput

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

$\max (1, pdvs \times n)$ when $jobvs = Nag_Schur$ ;
$1$ otherwise.

The

i

th element of the

j

th vector is stored in

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

On exit: if

jobvs = Nag_Schur

, vs contains the orthogonal matrix

Z

of Schur vectors.

jobvs = Nag_DoNothing

, vs is not referenced.

13: $pdvs$ – IntegerInput

On entry: the stride used in the array vs.

Constraints:

if $jobvs = Nag_Schur$ , $pdvs \geq \max (1, n)$ ;
otherwise $pdvs \geq 1$ .

14: $rconde$ – double *Output

On exit: if

sense = Nag_RCondEigVals

Nag_RCondBoth

, contains the reciprocal condition number for the average of the selected eigenvalues.

sense = Nag_NotRCond

Nag_RCondEigVecs

, rconde is not referenced.

15: $rcondv$ – double *Output

On exit: if

sense = Nag_RCondEigVecs

Nag_RCondBoth

, rcondv contains the reciprocal condition number for the selected right invariant subspace.

sense = Nag_NotRCond

Nag_RCondEigVals

, rcondv is not referenced.

16: $fail$ – 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_CONVERGENCE: The $Q R$ algorithm failed to compute all the eigenvalues.
NE_ENUM_INT_2: On entry, $jobvs = 〈value〉$ , $pdvs = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $jobvs = Nag_Schur$ , $pdvs \geq \max (1, n)$ ;
otherwise $pdvs \geq 1$ .
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdvs = 〈value〉$ .
Constraint: $pdvs > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .
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.
NE_SCHUR_REORDER: The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT: After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy $select = Nag_TRUE$ . This could also be caused by underflow due to scaling.

7

Accuracy

The computed Schur factorization satisfies

A + E = Z T Z^{T},

where

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

nag_dgeesx (f08pbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgeesx (f08pbc) 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

The total number of floating-point operations is proportional to

n^{3}

The complex analogue of this function is nag_zgeesx (f08ppc).

10

Example

This example finds the Schur factorization of the matrix

A = (\begin{array}{r} 0.35 & 0.45 & - 0.14 & - 0.17 \\ 0.09 & 0.07 & - 0.54 & 0.35 \\ - 0.44 & - 0.33 & - 0.03 & 0.17 \\ 0.25 & - 0.32 & - 0.13 & 0.11 \end{array}),

such that the real positive eigenvalues of

A

are the top left diagonal elements of the Schur form,

T

. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding invariant subspace are also returned.

NAG Library Function Document

nag_dgeesx (f08pbc)

▸▿ 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