# sdrges3.f - Man Page

TESTING/EIG/sdrges3.f

## Synopsis

### Functions/Subroutines

subroutine sdrges3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, alphar, alphai, beta, work, lwork, result, bwork, info)
SDRGES3

## Function/Subroutine Documentation

### subroutine sdrges3 (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) s, real, dimension( lda, * ) t, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldq, * ) z, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( * ) work, integer lwork, real, dimension( 13 ) result, logical, dimension( * ) bwork, integer info)

SDRGES3

Purpose:

``` SDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form)
problem driver SGGES3.

SGGES3 factors A and B as Q S Z'  and Q T Z' , where ' means
transpose, T is upper triangular, S is in generalized Schur form
(block upper triangular, with 1x1 and 2x2 blocks on the diagonal,
the 2x2 blocks corresponding to complex conjugate pairs of
generalized eigenvalues), and Q and Z are orthogonal. It also
computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n,
Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic
equation
det( A - w(j) B ) = 0
Optionally it also reorder the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms.

When SDRGES3 is called, a number of matrix 'sizes' ('N's') and a
number of matrix 'TYPES' are specified.  For each size ('N')
and each TYPE of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following 13 tests
will be performed and compared with the threshold THRESH except
the tests (5), (11) and (13).

(1)   | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)

(2)   | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)

(3)   | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)

(4)   | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)

(5)   if A is in Schur form (i.e. quasi-triangular form)
(no sorting of eigenvalues)

(6)   if eigenvalues = diagonal blocks of the Schur form (S, T),
i.e., test the maximum over j of D(j)  where:

if alpha(j) is real:
|alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

if alpha(j) is complex:
| det( s S - w T ) |
D(j) = ---------------------------------------------------
ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

and S and T are here the 2 x 2 diagonal blocks of S and T
corresponding to the j-th and j+1-th eigenvalues.
(no sorting of eigenvalues)

(7)   | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp )
(with sorting of eigenvalues).

(8)   | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).

(9)   | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).

(10)  if A is in Schur form (i.e. quasi-triangular form)
(with sorting of eigenvalues).

(11)  if eigenvalues = diagonal blocks of the Schur form (S, T),
i.e. test the maximum over j of D(j)  where:

if alpha(j) is real:
|alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

if alpha(j) is complex:
| det( s S - w T ) |
D(j) = ---------------------------------------------------
ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

and S and T are here the 2 x 2 diagonal blocks of S and T
corresponding to the j-th and j+1-th eigenvalues.
(with sorting of eigenvalues).

(12)  if sorting worked and SDIM is the number of eigenvalues
which were SELECTed.

Test Matrices
=============

The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The 'types' are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type 'j' will be generated.
Currently, the list of possible types is:

(1)  ( 0, 0 )         (a pair of zero matrices)

(2)  ( I, 0 )         (an identity and a zero matrix)

(3)  ( 0, I )         (an identity and a zero matrix)

(4)  ( I, I )         (a pair of identity matrices)

t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

t                ( I   0  )
(6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
( 0   I  )          ( 0   J  )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2

(7)  ( D, I )         where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8)  ( I, D )

(9)  ( big*D, small*I ) where 'big' is near overflow and small=1/big

(10) ( small*D, big*I )

(11) ( big*I, small*D )

(12) ( small*I, big*D )

(13) ( big*D, big*I )

(14) ( small*D, small*I )

(15) ( D1, D2 )        where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t   t
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.

(17) Q ( T1, T2 ) Z    where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )

(18) Q ( T1, T2 ) Z    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.

(19) Q ( T1, T2 ) Z    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )

N-5
(20) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

(21) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.

(22) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(23) Q ( small*T1, big*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(24) Q ( small*T1, small*T2 ) Z  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(25) Q ( big*T1, big*T2 ) Z      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(26) Q ( T1, T2 ) Z     where T1 and T2 are random upper-triangular
matrices.```
Parameters

NSIZES

```          NSIZES is INTEGER
The number of sizes of matrices to use.  If it is zero,
SDRGES3 does nothing.  NSIZES >= 0.```

NN

```          NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped.  NN >= 0.```

NTYPES

```          NTYPES is INTEGER
The number of elements in DOTYPE.   If it is zero, SDRGES3
does nothing.  It must be at least zero.  If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A on input.
This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .```

DOTYPE

```          DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.```

ISEED

```          ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd.  The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to SDRGES3 to continue the same random number
sequence.```

THRESH

```          THRESH is REAL
A test will count as 'failed' if the 'error', computed as
described above, exceeds THRESH.  Note that the error is
scaled to be O(1), so THRESH should be a reasonably small
multiple of 1, e.g., 10 or 100.  In particular, it should
not depend on the precision (single vs. double) or the size
of the matrix.  THRESH >= 0.```

NOUNIT

```          NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)```

A

```          A is REAL array,
dimension(LDA, max(NN))
Used to hold the original A matrix.  Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.```

LDA

```          LDA is INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ).```

B

```          B is REAL array,
dimension(LDA, max(NN))
Used to hold the original B matrix.  Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.```

S

```          S is REAL array, dimension (LDA, max(NN))
The Schur form matrix computed from A by SGGES3.  On exit, S
contains the Schur form matrix corresponding to the matrix
in A.```

T

```          T is REAL array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by SGGES3.```

Q

```          Q is REAL array, dimension (LDQ, max(NN))
The (left) orthogonal matrix computed by SGGES3.```

LDQ

```          LDQ is INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ).```

Z

```          Z is REAL array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by SGGES3.```

ALPHAR

`          ALPHAR is REAL array, dimension (max(NN))`

ALPHAI

`          ALPHAI is REAL array, dimension (max(NN))`

BETA

```          BETA is REAL array, dimension (max(NN))

The generalized eigenvalues of (A,B) computed by SGGES3.
( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
generalized eigenvalue of A and B.```

WORK

`          WORK is REAL array, dimension (LWORK)`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest
matrix dimension.```

RESULT

```          RESULT is REAL array, dimension (15)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow.```

BWORK

`          BWORK is LOGICAL array, dimension (N)`

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  A routine returned an error code.  INFO is the
absolute value of the INFO value returned.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 399 of file sdrges3.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sdrges3(3) is an alias of sdrges3.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK