# zdrvsx.f - Man Page

TESTING/EIG/zdrvsx.f

## Synopsis

### Functions/Subroutines

subroutine zdrvsx (nsizes, nn, ntypes, dotype, iseed, thresh, niunit, nounit, a, lda, h, ht, w, wt, wtmp, vs, ldvs, vs1, result, work, lwork, rwork, bwork, info)
ZDRVSX

## Function/Subroutine Documentation

### subroutine zdrvsx (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, double precision thresh, integer niunit, integer nounit, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) h, complex*16, dimension( lda, * ) ht, complex*16, dimension( * ) w, complex*16, dimension( * ) wt, complex*16, dimension( * ) wtmp, complex*16, dimension( ldvs, * ) vs, integer ldvs, complex*16, dimension( ldvs, * ) vs1, double precision, dimension( 17 ) result, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

ZDRVSX

Purpose:

```    ZDRVSX checks the nonsymmetric eigenvalue (Schur form) problem
expert driver ZGEESX.

ZDRVSX uses both test matrices generated randomly depending on
data supplied in the calling sequence, as well as on data
read from an input file and including precomputed condition
numbers to which it compares the ones it computes.

When ZDRVSX 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, one matrix will be generated and used
to test the nonsymmetric eigenroutines.  For each matrix, 15
tests will be performed:

(1)     0 if T is in Schur form, 1/ulp otherwise
(no sorting of eigenvalues)

(2)     | A - VS T VS' | / ( n |A| ulp )

Here VS is the matrix of Schur eigenvectors, and T is in Schur
form  (no sorting of eigenvalues).

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

(4)     0     if W are eigenvalues of T
1/ulp otherwise
(no sorting of eigenvalues)

(5)     0     if T(with VS) = T(without VS),
1/ulp otherwise
(no sorting of eigenvalues)

(6)     0     if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(no sorting of eigenvalues)

(7)     0 if T is in Schur form, 1/ulp otherwise
(with sorting of eigenvalues)

(8)     | A - VS T VS' | / ( n |A| ulp )

Here VS is the matrix of Schur eigenvectors, and T is in Schur
form  (with sorting of eigenvalues).

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

(10)    0     if W are eigenvalues of T
1/ulp otherwise
If workspace sufficient, also compare W with and
without reciprocal condition numbers
(with sorting of eigenvalues)

(11)    0     if T(with VS) = T(without VS),
1/ulp otherwise
If workspace sufficient, also compare T with and without
reciprocal condition numbers
(with sorting of eigenvalues)

(12)    0     if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
If workspace sufficient, also compare VS with and without
reciprocal condition numbers
(with sorting of eigenvalues)

(13)    if sorting worked and SDIM is the number of
eigenvalues which were SELECTed
If workspace sufficient, also compare SDIM with and
without reciprocal condition numbers

(14)    if RCONDE the same no matter if VS and/or RCONDV computed

(15)    if RCONDV the same no matter if VS and/or RCONDE computed

The 'sizes' 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)  The zero matrix.
(2)  The identity matrix.
(3)  A (transposed) Jordan block, with 1's on the diagonal.

(4)  A diagonal matrix with evenly spaced entries
1, ..., ULP  and random complex angles.
(ULP = (first number larger than 1) - 1 )
(5)  A diagonal matrix with geometrically spaced entries
1, ..., ULP  and random complex angles.
(6)  A diagonal matrix with 'clustered' entries 1, ULP, ..., ULP
and random complex angles.

(7)  Same as (4), but multiplied by a constant near
the overflow threshold
(8)  Same as (4), but multiplied by a constant near
the underflow threshold

(9)  A matrix of the form  U' T U, where U is unitary and
T has evenly spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.

(10) A matrix of the form  U' T U, where U is unitary and
T has geometrically spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.

(11) A matrix of the form  U' T U, where U is orthogonal and
T has 'clustered' entries 1, ULP,..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.

(12) A matrix of the form  U' T U, where U is unitary and
T has complex eigenvalues randomly chosen from
ULP < |z| < 1   and random O(1) entries in the upper
triangle.

(13) A matrix of the form  X' T X, where X has condition
SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.

(14) A matrix of the form  X' T X, where X has condition
SQRT( ULP ) and T has geometrically spaced entries
1, ..., ULP with random complex angles on the diagonal
and random O(1) entries in the upper triangle.

(15) A matrix of the form  X' T X, where X has condition
SQRT( ULP ) and T has 'clustered' entries 1, ULP,..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.

(16) A matrix of the form  X' T X, where X has condition
SQRT( ULP ) and T has complex eigenvalues randomly chosen
from ULP < |z| < 1 and random O(1) entries in the upper
triangle.

(17) Same as (16), but multiplied by a constant
near the overflow threshold
(18) Same as (16), but multiplied by a constant
near the underflow threshold

(19) Nonsymmetric matrix with random entries chosen from (-1,1).
If N is at least 4, all entries in first two rows and last
row, and first column and last two columns are zero.
(20) Same as (19), but multiplied by a constant
near the overflow threshold
(21) Same as (19), but multiplied by a constant
near the underflow threshold

In addition, an input file will be read from logical unit number
NIUNIT. The file contains matrices along with precomputed
eigenvalues and reciprocal condition numbers for the eigenvalue
average and right invariant subspace. For these matrices, in
addition to tests (1) to (15) we will compute the following two
tests:

(16)  |RCONDE - RCDEIN| / cond(RCONDE)

RCONDE is the reciprocal average eigenvalue condition number
computed by ZGEESX and RCDEIN (the precomputed true value)
is supplied as input.  cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.

(17)  |RCONDV - RCDVIN| / cond(RCONDV)

RCONDV is the reciprocal right invariant subspace condition
number computed by ZGEESX and RCDVIN (the precomputed true
value) is supplied as input. cond(RCONDV) is the condition
number of RCONDV, and takes errors in computing RCONDV into
account, so that the resulting quantity should be O(ULP).
cond(RCONDV) is essentially given by norm(A)/RCONDE.```
Parameters

NSIZES

```          NSIZES is INTEGER
The number of sizes of matrices to use.  NSIZES must be at
least zero. If it is zero, no randomly generated matrices
are tested, but any test matrices read from NIUNIT will be
tested.```

NN

```          NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped.  The values must be at least
zero.```

NTYPES

```          NTYPES is INTEGER
The number of elements in DOTYPE. NTYPES must be at least
zero. If it is zero, no randomly generated test matrices
are tested, but and test matrices read from NIUNIT will be
tested. 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.  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 ZDRVSX to continue the same random number
sequence.```

THRESH

```          THRESH is DOUBLE PRECISION
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.  It must be at least zero.```

NIUNIT

```          NIUNIT is INTEGER
The FORTRAN unit number for reading in the data file of
problems to solve.```

NOUNIT

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

A

```          A is COMPLEX*16 array, dimension (LDA, max(NN))
Used to hold the matrix whose eigenvalues are to be
computed.  On exit, A contains the last matrix actually used.```

LDA

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

H

```          H is COMPLEX*16 array, dimension (LDA, max(NN))
Another copy of the test matrix A, modified by ZGEESX.```

HT

```          HT is COMPLEX*16 array, dimension (LDA, max(NN))
Yet another copy of the test matrix A, modified by ZGEESX.```

W

```          W is COMPLEX*16 array, dimension (max(NN))
The computed eigenvalues of A.```

WT

```          WT is COMPLEX*16 array, dimension (max(NN))
Like W, this array contains the eigenvalues of A,
but those computed when ZGEESX only computes a partial
eigendecomposition, i.e. not Schur vectors```

WTMP

```          WTMP is COMPLEX*16 array, dimension (max(NN))
More temporary storage for eigenvalues.```

VS

```          VS is COMPLEX*16 array, dimension (LDVS, max(NN))
VS holds the computed Schur vectors.```

LDVS

```          LDVS is INTEGER
Leading dimension of VS. Must be at least max(1,max(NN)).```

VS1

```          VS1 is COMPLEX*16 array, dimension (LDVS, max(NN))
VS1 holds another copy of the computed Schur vectors.```

RESULT

```          RESULT is DOUBLE PRECISION array, dimension (17)
The values computed by the 17 tests described above.
The values are currently limited to 1/ulp, to avoid overflow.```

WORK

`          WORK is COMPLEX*16 array, dimension (LWORK)`

LWORK

```          LWORK is INTEGER
The number of entries in WORK.  This must be at least
max(1,2*NN(j)**2) for all j.```

RWORK

`          RWORK is DOUBLE PRECISION array, dimension (max(NN))`

BWORK

`          BWORK is LOGICAL array, dimension (max(NN))`

INFO

```          INFO is INTEGER
If 0,  successful exit.
<0,  input parameter -INFO is incorrect
>0,  ZLATMR, CLATMS, CLATME or ZGET24 returned an error
code and INFO is its absolute value

-----------------------------------------------------------------------

Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE       Real 0 and 1.
MAXTYP          The number of types defined.
NMAX            Largest value in NN.
NERRS           The number of tests which have exceeded THRESH
COND, CONDS,
IMODE           Values to be passed to the matrix generators.
ANORM           Norm of A; passed to matrix generators.

OVFL, UNFL      Overflow and underflow thresholds.
ULP, ULPINV     Finest relative precision and its inverse.
RTULP, RTULPI   Square roots of the previous 4 values.
The following four arrays decode JTYPE:
KTYPE(j)        The general type (1-10) for type 'j'.
KMODE(j)        The MODE value to be passed to the matrix
generator for type 'j'.
KMAGN(j)        The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
KCONDS(j)       Selectw whether CONDS is to be 1 or
1/sqrt(ulp).  (0 means irrelevant.)```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 431 of file zdrvsx.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

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

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