# sdrgvx.f - Man Page

TESTING/EIG/sdrgvx.f

## Synopsis

### Functions/Subroutines

subroutine sdrgvx (nsize, thresh, nin, nout, a, lda, b, ai, bi, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, s, stru, dif, diftru, work, lwork, iwork, liwork, result, bwork, info)
SDRGVX

## Function/Subroutine Documentation

### subroutine sdrgvx (integer nsize, real thresh, integer nin, integer nout, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) ai, real, dimension( lda, * ) bi, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( lda, * ) vl, real, dimension( lda, * ) vr, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real, dimension( * ) s, real, dimension( * ) stru, real, dimension( * ) dif, real, dimension( * ) diftru, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, real, dimension( 4 ) result, logical, dimension( * ) bwork, integer info)

SDRGVX

Purpose:

``` SDRGVX checks the nonsymmetric generalized eigenvalue problem
expert driver SGGEVX.

SGGEVX computes the generalized eigenvalues, (optionally) the left
and/or right eigenvectors, (optionally) computes a balancing
transformation to improve the conditioning, and (optionally)
reciprocal condition numbers for the eigenvalues and eigenvectors.

When SDRGVX is called with NSIZE > 0, two types of test matrix pairs
are generated by the subroutine SLATM6 and test the driver SGGEVX.
The test matrices have the known exact condition numbers for
eigenvalues. For the condition numbers of the eigenvectors
corresponding the first and last eigenvalues are also know
“exactly” (see SLATM6).

For each matrix pair, the following tests will be performed and
compared with the threshold THRESH.

(1) max over all left eigenvalue/-vector pairs (beta/alpha,l) of

| l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) )

where l**H is the conjugate transpose of l.

(2) max over all right eigenvalue/-vector pairs (beta/alpha,r) of

| (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )

(3) The condition number S(i) of eigenvalues computed by SGGEVX
differs less than a factor THRESH from the exact S(i) (see
SLATM6).

(4) DIF(i) computed by STGSNA differs less than a factor 10*THRESH
from the exact value (for the 1st and 5th vectors only).

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

Two kinds of test matrix pairs

(A, B) = inverse(YH) * (Da, Db) * inverse(X)

are used in the tests:

1: Da = 1+a   0    0    0    0    Db = 1   0   0   0   0
0   2+a   0    0    0         0   1   0   0   0
0    0   3+a   0    0         0   0   1   0   0
0    0    0   4+a   0         0   0   0   1   0
0    0    0    0   5+a ,      0   0   0   0   1 , and

2: Da =  1   -1    0    0    0    Db = 1   0   0   0   0
1    1    0    0    0         0   1   0   0   0
0    0    1    0    0         0   0   1   0   0
0    0    0   1+a  1+b        0   0   0   1   0
0    0    0  -1-b  1+a ,      0   0   0   0   1 .

In both cases the same inverse(YH) and inverse(X) are used to compute
(A, B), giving the exact eigenvectors to (A,B) as (YH, X):

YH:  =  1    0   -y    y   -y    X =  1   0  -x  -x   x
0    1   -y    y   -y         0   1   x  -x  -x
0    0    1    0    0         0   0   1   0   0
0    0    0    1    0         0   0   0   1   0
0    0    0    0    1,        0   0   0   0   1 , where

a, b, x and y will have all values independently of each other from
{ sqrt(sqrt(ULP)),  0.1,  1,  10,  1/sqrt(sqrt(ULP)) }.```
Parameters

NSIZE

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

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.  It must be at least zero.```

NIN

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

NOUT

```          NOUT 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, NSIZE)
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, B, AI, BI, Ao, and Bo.
It must be at least 1 and at least NSIZE.```

B

```          B is REAL array, dimension (LDA, NSIZE)
Used to hold the matrix whose eigenvalues are to be
computed.  On exit, B contains the last matrix actually used.```

AI

```          AI is REAL array, dimension (LDA, NSIZE)
Copy of A, modified by SGGEVX.```

BI

```          BI is REAL array, dimension (LDA, NSIZE)
Copy of B, modified by SGGEVX.```

ALPHAR

`          ALPHAR is REAL array, dimension (NSIZE)`

ALPHAI

`          ALPHAI is REAL array, dimension (NSIZE)`

BETA

```          BETA is REAL array, dimension (NSIZE)

On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.```

VL

```          VL is REAL array, dimension (LDA, NSIZE)
VL holds the left eigenvectors computed by SGGEVX.```

VR

```          VR is REAL array, dimension (LDA, NSIZE)
VR holds the right eigenvectors computed by SGGEVX.```

ILO

`                ILO is INTEGER`

IHI

`                IHI is INTEGER`

LSCALE

`                LSCALE is REAL array, dimension (N)`

RSCALE

`                RSCALE is REAL array, dimension (N)`

S

`                S is REAL array, dimension (N)`

STRU

`                STRU is REAL array, dimension (N)`

DIF

`                DIF is REAL array, dimension (N)`

DIFTRU

`                DIFTRU is REAL array, dimension (N)`

WORK

`          WORK is REAL array, dimension (LWORK)`

LWORK

```          LWORK is INTEGER
Leading dimension of WORK.  LWORK >= 2*N*N+12*N+16.```

IWORK

`          IWORK is INTEGER array, dimension (LIWORK)`

LIWORK

```          LIWORK is INTEGER
Leading dimension of IWORK.  Must be at least N+6.```

RESULT

`                RESULT is REAL array, dimension (4)`

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.```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 297 of file sdrgvx.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

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

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