## MB03LD

### Eigenvalues and right deflating subspace of a real skew-Hamiltonian/Hamiltonian pencil

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute the relevant eigenvalues of a real N-by-N skew-
Hamiltonian/Hamiltonian pencil aS - bH, with

(  A  D  )         (  B  F  )
S = (        ) and H = (        ),                           (1)
(  E  A' )         (  G -B' )

where the notation M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'C', an orthogonal basis of the right
deflating subspace of aS - bH corresponding to the eigenvalues
with strictly negative real part is computed.

```
Specification
```      SUBROUTINE MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
\$                   LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
\$                   IWORK, LIWORK, DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ, ORTH
INTEGER            INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
\$                   LIWORK, N, NEIG
C     .. Array Arguments ..
LOGICAL            BWORK( * )
INTEGER            IWORK( * )
DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
\$                   B( LDB, * ), BETA( * ), DE( LDDE, * ),
\$                   DWORK( * ), FG( LDFG, * ), Q( LDQ, * )

```
Arguments

Mode Parameters

```  COMPQ   CHARACTER*1
Specifies whether to compute the right deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N':  do not compute the deflating subspace;
= 'C':  compute the deflating subspace and store it in the

ORTH    CHARACTER*1
If COMPQ = 'C', specifies the technique for computing the
orthogonal basis of the deflating subspace, as follows:
= 'P':  QR factorization with column pivoting;
= 'S':  singular value decomposition.
If COMPQ = 'N', the ORTH value is not used.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the pencil aS - bH.  N >= 0, even.

A       (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
this array contains the upper triangular matrix Aout
(see METHOD); otherwise, it contains the upper triangular
matrix A obtained just before the application of the
periodic QZ algorithm (see SLICOT Library routine MB04BD).

LDA     INTEGER
The leading dimension of the array A.  LDA >= MAX(1, N/2).

DE      (input/output) DOUBLE PRECISION array, dimension
(LDDE, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-symmetric matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of the
skew-symmetric matrix D.
The entries on the diagonal and the first superdiagonal of
this array need not be set, but are assumed to be zero.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 lower
triangular part and the first superdiagonal contain the
transpose of the upper quasi-triangular matrix C2out (see
METHOD), and the (N/2-1)-by-(N/2-1) upper triangular part
of the submatrix in the columns 3 to N/2+1 of this array
contains the strictly upper triangular part of the
skew-symmetric matrix Dout (see METHOD), without the main
diagonal, which is zero.
On exit, if COMPQ = 'N', the leading N/2-by-N/2 lower
triangular part and the first superdiagonal contain the
transpose of the upper Hessenberg matrix C2, and the
(N/2-1)-by-(N/2-1) upper triangular part of the submatrix
in the columns 3 to N/2+1 of this array contains the
strictly upper triangular part of the skew-symmetric
matrix D (without the main diagonal) just before the
application of the periodic QZ algorithm.

LDDE    INTEGER
The leading dimension of the array DE.
LDDE >= MAX(1, N/2).

B       (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
this array contains the upper triangular matrix C1out
(see METHOD); otherwise, it contains the upper triangular
matrix C1 obtained just before the application of the
periodic QZ algorithm.

LDB     INTEGER
The leading dimension of the array B.  LDB >= MAX(1, N/2).

FG      (input/output) DOUBLE PRECISION array, dimension
(LDFG, N/2+1)
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
symmetric matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
symmetric matrix F.
On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
the submatrix in the columns 2 to N/2+1 of this array
contains the matrix Vout (see METHOD); otherwise, it
contains the matrix V obtained just before the application
of the periodic QZ algorithm.

LDFG    INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, N/2).

NEIG    (output) INTEGER
If COMPQ = 'C', the number of eigenvalues in aS - bH with
strictly negative real part.

Q       (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthogonal basis of the right
deflating subspace corresponding to the eigenvalues of
aA - bB with strictly negative real part. The remaining
part of this array is used as workspace.
If COMPQ = 'N', this array is not referenced.

LDQ     INTEGER
The leading dimension of the array Q.
LDQ >= 1,           if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.

ALPHAR  (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.

ALPHAI  (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.

BETA    (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
Due to the skew-Hamiltonian/Hamiltonian structure of the
pencil, for every eigenvalue lambda, -lambda is also an
eigenvalue, and thus it has only to be saved once in
ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)
On exit, if INFO = -19, IWORK(1) returns the minimum value
of LIWORK.

LIWORK  INTEGER
The dimension of the array IWORK.  LIWORK = 1, if N = 0,
LIWORK >= MAX( N + 12, 2*N + 3 ),     if COMPQ = 'N',
LIWORK >= MAX( 32, N + 12, 2*N + 3 ), if COMPQ = 'Q'.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum value
of LDWORK.

LDWORK  INTEGER
The dimension of the array DWORK.  LDWORK = 1, if N = 0,
LDWORK >= 3*(N/2)**2 + N**2 + MAX( MAX( N, 32 ) + 4,
4*(N+1) ), if COMPQ = 'N',
where the last term can be reduced to 4*N if N/2 is odd;
LDWORK >= 8*N**2 + MAX( 8*N + 32, N/2 + 168, 272 ),
if COMPQ = 'C'.
For good performance LDWORK should be generally larger.

If LDWORK = -1  a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.

BWORK   LOGICAL array, dimension (N/2)

```
Error Indicator
```  INFO    INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: periodic QZ iteration failed in the SLICOT Library
routines MB04BD or MB04HD (QZ iteration did not
converge or computation of the shifts failed);
= 2: standard QZ iteration failed in the SLICOT Library
routines MB04HD or MB03DD (called by MB03JD);
= 3: a numerically singular matrix was found in the SLICOT
Library routine MB03HD (called by MB03JD);
= 4: the singular value decomposition failed in the LAPACK
routine DGESVD (for ORTH = 'S');
= 5: some eigenvalues might be inaccurate. This is a
warning.

```
Method
```  First, the decompositions of S and H are computed via orthogonal
transformations Q1 and Q2 as follows:

(  Aout  Dout  )
Q1' S J Q1 J' = (              ),
(   0    Aout' )

(  Bout  Fout  )
J' Q2' J S Q2 = (              ) =: T,                       (2)
(   0    Bout' )

(  C1out  Vout  )            (  0  I  )
Q1' H Q2 = (               ), where J = (        ),
(  0     C2out' )            ( -I  0  )

and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
triangular and Dout and Fout are skew-symmetric.

Then, orthogonal matrices Q3 and Q4 are found, for the extended
matrices

(  Aout   0  )          (    0   C1out )
Se = (            ) and He = (              ),
(   0   Bout )          ( -C2out   0   )

such that S11 := Q4' Se Q3 is upper triangular and
H11 := Q4' He Q3 is upper quasi-triangular. The following matrices
are computed:

(  Dout   0  )                   (   0   Vout )
S12 := Q4' (            ) Q4 and H12 := Q4' (            ) Q4.
(   0   Fout )                   ( Vout'   0  )

Then, an orthogonal matrix Q is found such that the eigenvalues
with strictly negative real parts of the pencil

(  S11  S12  )     (  H11  H12  )
a (            ) - b (            )
(   0   S11' )     (   0  -H11' )

are moved to the top of this pencil.

Finally, an orthogonal basis of the right deflating subspace
corresponding to the eigenvalues with strictly negative real part

```
References
```   Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.

```
Numerical Aspects
```                                                            3
The algorithm is numerically backward stable and needs O(N )
floating point operations.

```
```  This routine does not perform any scaling of the matrices. Scaling
might sometimes be useful, and it should be done externally.

```
Example

Program Text

```*     MB03LD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER            NIN, NOUT
PARAMETER          ( NIN = 5, NOUT = 6 )
INTEGER            NMAX
PARAMETER          ( NMAX = 50 )
INTEGER            LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LIWORK
PARAMETER          (  LDA = NMAX/2, LDB = NMAX/2, LDDE = NMAX/2,
\$                     LDFG = NMAX/2, LDQ = 2*NMAX,
\$                     LDWORK = 8*NMAX*NMAX +
\$                              MAX( 8*NMAX + 32, NMAX/2 + 168,
\$                                   272 ),
\$                     LIWORK = MAX( 32, NMAX + 12, NMAX*2 + 3 ) )
*
*     .. Local Scalars ..
CHARACTER          COMPQ, ORTH
INTEGER            I, INFO, J, M, N, NEIG
*
*     .. Local Arrays ..
LOGICAL            BWORK( NMAX/2 )
INTEGER            IWORK( LIWORK )
DOUBLE PRECISION   A( LDA, NMAX/2 ),  ALPHAI( NMAX/2 ),
\$                   ALPHAR( NMAX/2 ),  B( LDB, NMAX/2 ),
\$                   BETA( NMAX/2 ),  DE( LDDE, NMAX/2+1 ),
\$                   DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
\$                   Q( LDQ, 2*NMAX )
*
*     .. External Subroutines ..
EXTERNAL           MB03LD
*
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*
*     .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) COMPQ, ORTH, N
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( (  A( I, J ), J = 1, M   ), I = 1, M )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
READ( NIN, FMT = * ) ( (  B( I, J ), J = 1, M   ), I = 1, M )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
*        Compute the eigenvalues and an orthogonal basis of the right
*        deflating subspace of a real skew-Hamiltonian/Hamiltonian
*        pencil, corresponding to the eigenvalues with strictly negative
*        real part.
CALL MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
\$                LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
\$                LIWORK, DWORK, LDWORK, BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, M )
10       CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, M+1 )
20       CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, M )
30       CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 2, M+1 )
40       CONTINUE
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
WRITE( NOUT, FMT = 99989 )
WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99988 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
50          CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT( 'MB03LD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03LD = ', I2 )
99996 FORMAT( 'The matrix A on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix DE on exit is ' )
99993 FORMAT( 'The matrix C1 on exit is ' )
99992 FORMAT( 'The matrix V on exit is ' )
99991 FORMAT( 'The vector ALPHAR is ' )
99990 FORMAT( 'The vector ALPHAI is ' )
99989 FORMAT( 'The vector BETA is ' )
99988 FORMAT( 'The matrix Q is ' )
END
```
Program Data
```MB03LD EXAMPLE PROGRAM DATA
C   P   8
3.1472   1.3236   4.5751   4.5717
4.0579  -4.0246   4.6489  -0.1462
-3.7301  -2.2150  -3.4239   3.0028
4.1338   0.4688   4.7059  -3.5811
0.0000   0.0000  -1.5510  -4.5974  -2.5127
3.5071   0.0000   0.0000   1.5961   2.4490
-3.1428   2.5648   0.0000   0.0000  -0.0596
3.0340   2.4892  -1.1604   0.0000   0.0000
0.6882  -3.3782  -3.3435   1.8921
-0.3061   2.9428   1.0198   2.4815
-4.8810  -1.8878  -2.3703  -0.4946
-1.6288   0.2853   1.5408  -4.1618
-2.4013  -2.7102   0.3834  -3.9335   3.1730
-3.1815  -2.3620   4.9613   4.6190   3.6869
3.6929   0.7970   0.4986  -4.9537  -4.1556
3.5303   1.2206  -1.4905   0.1325  -1.0022

```
Program Results
```MB03LD EXAMPLE PROGRAM RESULTS
The matrix A on exit is
-4.7460   4.1855   3.2696  -0.2244
0.0000   6.4157   2.8287   1.4553
0.0000   0.0000   7.4626   1.5726
0.0000   0.0000   0.0000   8.8702
The matrix DE on exit is
-5.4562   2.5550  -1.3137  -6.3615  -0.8940
-2.1348  -7.9616   0.0000   1.0704  -0.0659
4.9694   1.1516   4.8504   0.0000  -0.6922
-2.2744   3.4912   0.5046   4.4394   0.0000
The matrix C1 on exit is
6.9525  -4.9881   2.3661   4.2188
0.0000   8.5009   0.7182   5.5533
0.0000   0.0000  -4.6650  -2.8177
0.0000   0.0000   0.0000   1.5124
The matrix V on exit is
0.9136   4.1106  -0.0079   3.5789
-1.1553  -1.4785  -1.5155  -0.8018
-2.2167   4.8029   1.3645   2.5202
-1.0994  -0.6144   0.3970   2.0730
The vector ALPHAR is
0.8314  -0.8314   0.8131   0.0000
The vector ALPHAI is
0.4372   0.4372   0.0000   0.9164
The vector BETA is
0.7071   0.7071   1.4142   2.8284
The matrix Q is
-0.5844  -0.2949   0.1692
0.5470  -0.2324  -0.4524
0.0382   0.4673  -0.3092
-0.0378   0.0904  -0.4451
-0.0255   0.1497  -0.1929
-0.1286  -0.6067  -0.2275
-0.2260   0.4901   0.0951
0.5367  -0.0430   0.6123
```