# sdrgsx.f - Man Page

TESTING/EIG/sdrgsx.f

## Synopsis

### Functions/Subroutines

subroutine sdrgsx (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi, z, q, alphar, alphai, beta, c, ldc, s, work, lwork, iwork, liwork, bwork, info)
SDRGSX

## Function/Subroutine Documentation

### subroutine sdrgsx (integer nsize, integer ncmax, 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( lda, * ) z, real, dimension( lda, * ) q, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) s, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

SDRGSX

Purpose:

``` SDRGSX checks the nonsymmetric generalized eigenvalue (Schur form)
problem expert driver SGGESX.

SGGESX 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(1),beta(1)), ...,
(alpha(n),beta(n)). Thus, w(j) = alpha(j)/beta(j) is a root of the
characteristic equation

det( A - w(j) B ) = 0

Optionally it also reorders the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms; computes a reciprocal condition number for the average
of the selected eigenvalues; and computes a reciprocal condition
number for the right and left deflating subspaces corresponding to
the selected eigenvalues.

When SDRGSX is called with NSIZE > 0, five (5) types of built-in
matrix pairs are used to test the routine SGGESX.

When SDRGSX is called with NSIZE = 0, it reads in test matrix data
to test SGGESX.

For each matrix pair, the following tests will be performed and
compared with the threshold THRESH except for the tests (7) and (9):

(1)   | A - Q S Z' | / ( |A| n ulp )

(2)   | B - Q T Z' | / ( |B| n ulp )

(3)   | I - QQ' | / ( n ulp )

(4)   | I - ZZ' | / ( n ulp )

(5)   if A is in Schur form (i.e. quasi-triangular form)

(6)   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.

(7)   if sorting worked and SDIM is the number of eigenvalues
which were selected.

(8)   the estimated value DIF does not differ from the true values of
Difu and Difl more than a factor 10*THRESH. If the estimate DIF
equals zero the corresponding true values of Difu and Difl
should be less than EPS*norm(A, B). If the true value of Difu
and Difl equal zero, the estimate DIF should be less than
EPS*norm(A, B).

(9)   If INFO = N+3 is returned by SGGESX, the reordering 'failed'
and we check that DIF = PL = PR = 0 and that the true value of
Difu and Difl is < EPS*norm(A, B). We count the events when
INFO=N+3.

For read-in test matrices, the above tests are run except that the
exact value for DIF (and PL) is input data.  Additionally, there is
one more test run for read-in test matrices:

(10)  the estimated value PL does not differ from the true value of
PLTRU more than a factor THRESH. If the estimate PL equals
zero the corresponding true value of PLTRU should be less than
EPS*norm(A, B). If the true value of PLTRU equal zero, the
estimate PL should be less than EPS*norm(A, B).

Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1)
matrix pairs are generated and tested. NSIZE should be kept small.

SVD (routine SGESVD) is used for computing the true value of DIF_u
and DIF_l when testing the built-in test problems.

Built-in Test Matrices
======================

All built-in test matrices are the 2 by 2 block of triangular
matrices

A = [ A11 A12 ]    and      B = [ B11 B12 ]
[     A22 ]                 [     B22 ]

where for different type of A11 and A22 are given as the following.
A12 and B12 are chosen so that the generalized Sylvester equation

A11*R - L*A22 = -A12
B11*R - L*B22 = -B12

have prescribed solution R and L.

Type 1:  A11 = J_m(1,-1) and A_22 = J_k(1-a,1).
B11 = I_m, B22 = I_k
where J_k(a,b) is the k-by-k Jordan block with “a” on
diagonal and “b” on superdiagonal.

Type 2:  A11 = (a_ij) = ( 2(.5-sin(i)) ) and
B11 = (b_ij) = ( 2(.5-sin(ij)) ) for i=1,...,m, j=i,...,m
A22 = (a_ij) = ( 2(.5-sin(i+j)) ) and
B22 = (b_ij) = ( 2(.5-sin(ij)) ) for i=m+1,...,k, j=i,...,k

Type 3:  A11, A22 and B11, B22 are chosen as for Type 2, but each
second diagonal block in A_11 and each third diagonal block
in A_22 are made as 2 by 2 blocks.

Type 4:  A11 = ( 20(.5 - sin(ij)) ) and B22 = ( 2(.5 - sin(i+j)) )
for i=1,...,m,  j=1,...,m and
A22 = ( 20(.5 - sin(i+j)) ) and B22 = ( 2(.5 - sin(ij)) )
for i=m+1,...,k,  j=m+1,...,k

Type 5:  (A,B) and have potentially close or common eigenvalues and
very large departure from block diagonality A_11 is chosen
as the m x m leading submatrix of A_1:
|  1  b                            |
| -b  1                            |
|        1+d  b                    |
|         -b 1+d                   |
A_1 =  |                  d  1            |
|                 -1  d            |
|                        -d  1     |
|                        -1 -d     |
|                               1  |
and A_22 is chosen as the k x k leading submatrix of A_2:
| -1  b                            |
| -b -1                            |
|       1-d  b                     |
|       -b  1-d                    |
A_2 =  |                 d 1+b            |
|               -1-b d             |
|                       -d  1+b    |
|                      -1+b  -d    |
|                              1-d |
and matrix B are chosen as identity matrices (see SLATM5).```
Parameters

NSIZE

```          NSIZE is INTEGER
The maximum size of the matrices to use. NSIZE >= 0.
If NSIZE = 0, no built-in tests matrices are used, but
read-in test matrices are used to test SGGESX.```

NCMAX

```          NCMAX is INTEGER
Maximum allowable NMAX for generating Kroneker matrix
in call to SLAKF2```

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.```

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 store 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, Z and Q,
LDA >= max( 1, NSIZE ). For the read-in test,
LDA >= max( 1, N ), N is the size of the test matrices.```

B

```          B is REAL array, dimension (LDA, NSIZE)
Used to store 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 SGGESX.```

BI

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

Z

```          Z is REAL array, dimension (LDA, NSIZE)
Z holds the left Schur vectors computed by SGGESX.```

Q

```          Q is REAL array, dimension (LDA, NSIZE)
Q holds the right Schur vectors computed by SGGESX.```

ALPHAR

`          ALPHAR is REAL array, dimension (NSIZE)`

ALPHAI

`          ALPHAI is REAL array, dimension (NSIZE)`

BETA

```          BETA is REAL array, dimension (NSIZE)
\verbatim
On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.```

C

```          C is REAL array, dimension (LDC, LDC)
Store the matrix generated by subroutine SLAKF2, this is the
matrix formed by Kronecker products used for estimating
DIF.```

LDC

```          LDC is INTEGER
The leading dimension of C. LDC >= max(1, LDA*LDA/2 ).```

S

```          S is REAL array, dimension (LDC)
Singular values of C```

WORK

`          WORK is REAL array, dimension (LWORK)`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >= MAX( 5*NSIZE*NSIZE/2 - 2, 10*(NSIZE+1) )```

IWORK

`          IWORK is INTEGER array, dimension (LIWORK)`

LIWORK

```          LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= NSIZE + 6.```

BWORK

`          BWORK is LOGICAL array, dimension (LDA)`

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

NAG Ltd.

Definition at line 356 of file sdrgsx.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

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

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