Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( H ) and H = ( H ). (1)
( E A ) ( G -B )
This routine computes the eigenvalues using an embedding to a real
skew-Hamiltonian/skew-Hamiltonian pencil aB_S - bB_T, defined as
( Re(A) -Im(A) | Re(D) -Im(D) )
( | )
( Im(A) Re(A) | Im(D) Re(D) )
( | )
B_S = (-----------------+-----------------) , and
( | T T )
( Re(E) -Im(E) | Re(A ) Im(A ) )
( | T T )
( Im(E) Re(E) | -Im(A ) Re(A ) )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H.
( | T T )
( -Im(G) -Re(G) | -Im(B ) Re(B ) )
( | T T )
( Re(G) -Im(G) | -Re(B ) -Im(B ) )
Optionally, if JOB = 'T', the pencil aB_S - bB_H (B_H = -i*B_T) is
transformed by a unitary matrix Q to the structured Schur form
( BA BD ) ( BB BF )
B_Sout = ( H ) and B_Hout = ( H ), (3)
( 0 BA ) ( 0 -BB )
where BA and BB are upper triangular, BD is skew-Hermitian, and
BF is Hermitian. The embedding doubles the multiplicities of the
eigenvalues of the pencil aS - bH. Optionally, if COMPQ = 'C', the
unitary matrix Q is computed.
Specification
SUBROUTINE MB04BZ( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ DWORK, LDWORK, ZWORK, LZWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LZWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
$ FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (3).
= 'T': put S and H into the forms in (3) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the unitary transformation
matrix Q, as follows:
= 'N': Q is not computed;
= 'C': compute the unitary transformation matrix Q.
Input/Output Parameters
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, K)
where K = N/2, if JOB = 'E', and K = N, if JOB = 'T'.
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed, but it is preserved.
If JOB = 'E', this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, K).
DE (input/output) COMPLEX*16 array, dimension
(LDDE, MIN(K+1,N))
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian 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-Hermitian matrix D.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the skew-Hermitian matrix BD in (3) (see
also METHOD). The strictly lower triangular part of the
input matrix is preserved.
If JOB = 'E', this array is unchanged on exit.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, K).
B (input/output) COMPLEX*16 array, dimension (LDB, K)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; the elements below the first subdiagonal of the
input matrix are preserved.
If JOB = 'E', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, K).
FG (input/output) COMPLEX*16 array, dimension
(LDFG, MIN(K+1,N))
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian 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
Hermitian matrix F.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts.
If JOB = 'E', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, K).
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading 2*N-by-2*N part of
this array contains the unitary transformation matrix Q
that reduced the matrices B_S and B_H to the form in (3).
However, if JOB = 'E', the reduction was possibly not
completed: the matrix B_H may have 2-by-2 diagonal blocks,
and the array Q returns the orthogonal matrix that
performed the partial reduction.
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)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
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)
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.
Workspace
IWORK INTEGER array, dimension (2*N+3)
On exit, IWORK(1) contains the number, q, of unreliable,
possibly inaccurate (pairs of) eigenvalues, and the
absolute values in IWORK(2), ..., IWORK(q+1) are their
indices, as well as of the corresponding 1-by-1 and 2-by-2
diagonal blocks of the arrays B and A on exit, if
JOB = 'T'. Specifically, a positive value is an index of
a real or purely imaginary eigenvalue, corresponding to a
1-by-1 block, while the absolute value of a negative entry
in IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. Moreover, IWORK(q+2),..., IWORK(2*q+1)
contain pointers to the starting elements in DWORK where
each block pair is stored. Specifically, if IWORK(i+1) > 0
then DWORK(r) and DWORK(r+1) store corresponding diagonal
elements of T11 and S11, respectively, and if
IWORK(i+1) < 0, then DWORK(r:r+3) and DWORK(r+4:r+7) store
the elements of the block in T11 and S11, respectively
(see Section METHOD), where r = IWORK(q+1+i). Moreover,
IWORK(2*q+2) contains the number s of the 1-by-1 blocks,
and IWORK(2*q+3) contains the number t of the 2-by-2
blocks, corresponding to unreliable eigenvalues.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2) and DWORK(3) contain the
Frobenius norms of the matrices B_S and B_T. These norms
are used in the tests to decide that some eigenvalues are
considered as numerically unreliable. Moreover, DWORK(4),
..., DWORK(3+2*s) contain the s pairs of values of the
1-by-1 diagonal elements of T11 and S11 corresponding to
unreliable eigenvalues. (Such an eigenvalue is obtained
from -i*sqrt(T11(i,i)/S11(i,i)).) Similarly, DWORK(4+2*s),
..., DWORK(3+2*s+8*t) contain the t groups of pairs of
2-by-2 diagonal submatrices of T11 and S11 corresponding
to pairs of unreliable eigenvalues. (Such an eigenvalue
pair is obtained from -i*sqrt( ev ), where ev are the
eigenvalues of (T11(i:i+1,i:i+1),S11(i:i+1,i:i+1)).)
On exit, if INFO = -19, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK. If COMPQ = 'N',
LDWORK >= MAX( 3, 4*N*N + 3*N ), if JOB = 'E';
LDWORK >= MAX( 3, 5*N*N + 3*N ), if JOB = 'T';
LDWORK >= MAX( 3, 11*N*N + 2*N ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then 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 related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -21, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if JOB = 'E'; otherwise,
LZWORK >= 6*N + 4, if COMPQ = 'N';
LZWORK >= 8*N + 4, if COMPQ = 'C'.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if JOB = 'E';
LBWORK >= N, if JOB = 'T'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the SLICOT Library routine
MB04FD (QZ iteration did not converge or computation
of the shifts failed);
= 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
trying to triangularize the 2-by-2 blocks;
= 3: warning: the pencil is numerically singular.
Method
First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04FD is applied to compute the structured Schur
form, i.e., the factorizations
~ T T ( S11 S12 )
B_S = J Q J B_S Q = ( T ) and
( 0 S11 )
~ T T ( T11 T12 )
B_T = J Q J B_T Q = ( T ),
( 0 T11 )
where Q is real orthogonal, S11 is upper triangular, and T11 is
upper quasi-triangular. If JOB = 'T', then the matrices above are
~
further transformed so that the 2-by-2 blocks in i*B_T are split
into 1-by-1 blocks. If COMPQ = 'C', the transformations are
accumulated in the unitary matrix Q.
See also page 22 in [1] for more details.
References
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) complex floating point operations.Further Comments
The returned eigenvalues are those of the pencil (-i*T11,S11), where i is the purely imaginary unit. If JOB = 'E', the returned matrix T11 is quasi-triangular. Note that the off-diagonal elements of the 2-by-2 blocks of S11 are zero by construction. If JOB = 'T', the returned eigenvalues correspond to the diagonal elements of BB and BA. This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB04BZ 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, LZWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDDE = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX,
$ LDWORK = 11*NMAX*NMAX + 2*NMAX,
$ LZWORK = 8*NMAX + 4 )
*
* .. Local Scalars ..
CHARACTER COMPQ, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
COMPLEX*16 A( LDA, NMAX ), B( LDB, NMAX ),
$ DE( LDDE, NMAX ), FG( LDFG, NMAX ),
$ Q( LDQ, 2*NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ),
$ BETA( NMAX ), DWORK( LDWORK )
INTEGER IWORK( 2*NMAX+3 )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04BZ
*
* .. Intrinsic Functions ..
INTRINSIC MOD
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) 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 of a complex skew-Hamiltonian/
* Hamiltonian pencil.
CALL MB04BZ( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG, LDFG,
$ Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
40 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, 2*N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, 2*N )
50 CONTINUE
END IF
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99987 )
WRITE( NOUT, FMT = 99989 ) ( BETA( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT ( 'MB04BZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04BZ = ', I2 )
99996 FORMAT (/'The matrix A on exit is ' )
99995 FORMAT (20( 1X, F9.4, SP, F9.4, S, 'i ') )
99994 FORMAT (/'The matrix D on exit is ' )
99993 FORMAT (/'The matrix B on exit is ' )
99992 FORMAT (/'The matrix F on exit is ' )
99991 FORMAT (/'The matrix Q is ' )
99990 FORMAT (/'The vector ALPHAR is ' )
99989 FORMAT ( 50( 1X, F8.4 ) )
99988 FORMAT (/'The vector ALPHAI is ' )
99987 FORMAT (/'The vector BETA is ' )
END
Program Data
MB04BZ EXAMPLE PROGRAM DATA
T C 4
(0.0604,0.6568) (0.5268,0.2919)
(0.3992,0.6279) (0.4167,0.4316)
(0,0.4896) (0,0.9516) (0.3724,0.0526)
(0.9840,0.3394) (0,0.9203) (0,0.7378)
(0.2691,0.4177) (0.5478,0.3014)
(0.4228,0.9830) (0.9427,0.7010)
0.6663 0.6981 (0.1781,0.8818)
(0.5391,0.1711) 0.6665 0.1280
Program Results
MB04BZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
0.7430 +0.0000i 0.0389 -0.4330i -0.1155 -0.1366i -0.6586 -0.3210i
0.3992 +0.6279i 0.7548 +0.0000i 0.6099 -0.2308i 0.2140 +0.1260i
0.0000 +0.0000i 0.0000 +0.0000i 1.4085 +0.0000i 0.0848 +0.4972i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 1.4725 +0.0000i
The matrix D on exit is
0.0000 -0.6858i 0.1839 -0.0474i -0.4428 -0.1290i 0.4759 +0.0380i
0.9840 +0.3394i 0.0000 +0.6858i -0.6339 +0.1358i 0.4204 -0.2140i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 -0.2110i -0.0159 -0.0338i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.2110i
The matrix B on exit is
-1.5832 +0.5069i -0.0097 +0.0866i 0.1032 -0.1431i -0.0426 +0.7942i
0.0000 +0.0000i 1.6085 +0.5150i -0.1342 -0.8180i 0.5143 +0.0178i
0.0000 +0.0000i 0.0000 +0.0000i -0.0842 -0.1642i 0.0246 -0.0264i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0880 -0.1716i
The matrix F on exit is
0.3382 0.0000i 0.0234 +0.0907i -0.1619 +0.9033i -0.8227 +0.0204i
0.5391 +0.1711i -0.3382 +0.0000i -0.6525 +0.2455i -0.3532 -0.6409i
0.0000 +0.0000i 0.0000 +0.0000i 0.0120 0.0000i 0.0019 -0.0009i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.0120 +0.0000i
The matrix Q is
0.1422 +0.5446i -0.3877 -0.1273i -0.4363 +0.1705i 0.0348 -0.5440i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.1594 -0.2382i 0.1967 -0.2467i -0.1376 -0.0961i -0.1070 -0.2058i -0.1273 +0.0585i -0.0852 +0.1020i 0.6125 -0.1059i -0.0172 +0.5589i
-0.3659 -0.0211i -0.0291 +0.4967i -0.0729 +0.4236i 0.3169 -0.0008i 0.2947 -0.1080i 0.1614 -0.2342i 0.2867 -0.0578i -0.0170 +0.2603i
0.1846 +0.4089i -0.2815 -0.2018i 0.3220 -0.1600i -0.0526 +0.3937i 0.2747 -0.0655i 0.1045 -0.2159i 0.2085 -0.3104i -0.3052 +0.1463i
-0.0201 -0.2898i 0.2131 -0.0081i -0.2165 -0.1055i -0.1324 -0.3133i 0.1660 -0.1635i 0.2250 -0.1390i -0.1590 -0.4634i -0.5310 -0.2239i
0.1342 -0.1295i 0.1128 -0.1990i -0.0712 -0.1686i -0.1490 -0.1336i 0.6198 +0.0113i 0.0281 -0.4762i -0.0462 +0.3244i 0.3464 +0.0086i
0.2305 -0.1358i 0.1292 -0.3311i -0.0106 +0.4992i 0.3906 +0.0997i 0.1429 +0.3376i -0.4310 -0.0866i -0.0894 -0.1336i -0.1601 -0.1055i
-0.2601 +0.0835i -0.0940 +0.3652i -0.0213 -0.3116i -0.2502 -0.0995i 0.1361 +0.4589i -0.5898 -0.0730i 0.0294 -0.1192i -0.1253 +0.0085i
The vector ALPHAR is
-1.5832 1.5832 -0.0842 0.0842
The vector ALPHAI is
0.5069 0.5069 -0.1642 -0.1642
The vector BETA is
0.7430 0.7430 1.4085 1.4085