Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( ) and H = ( ). (1)
( E A' ) ( G -B' )
The structured Schur form of the embedded 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
( | )
( Re(E) -Im(E) | Re(A') Im(A') )
( | )
( 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,
( | )
( -Im(G) -Re(G) | -Im(B') Re(B') )
( | )
( Re(G) -Im(G) | -Re(B') -Im(B') )
is determined and used to compute the eigenvalues. The notation M'
denotes the conjugate transpose of the matrix M. Optionally,
if COMPQ = 'C', an orthonormal basis of the right deflating
subspace of the pencil aS - bH, corresponding to the eigenvalues
with strictly negative real part, is computed. Namely, after
transforming aB_S - bB_H by unitary matrices, we have
( BA BD ) ( BB BF )
B_Sout = ( ) and B_Hout = ( ), (3)
( 0 BA' ) ( 0 -BB' )
and the eigenvalues with strictly negative real part of the
complex pencil aB_Sout - bB_Hout are moved to the top. The
embedding doubles the multiplicities of the eigenvalues of the
pencil aS - bH.
Specification
SUBROUTINE MB03LZ( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
$ IWORK, DWORK, LDWORK, ZWORK, LZWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, ORTH
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LZWORK, N, NEIG
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
COMPQ CHARACTER*1
Specifies whether to compute the deflating subspace
corresponding to the eigenvalues of aS - bH with strictly
negative real part.
= 'N': do not compute the deflating subspace; compute the
eigenvalues only;
= 'C': compute the deflating subspace and store it in the
leading subarray of Q.
ORTH CHARACTER*1
If COMPQ = 'C', specifies the technique for computing an
orthonormal 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) COMPLEX*16 array, dimension (LDA, N)
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-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; it is preserved in the leading N/2-by-N/2 part.
If COMPQ = 'N', this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N).
DE (input/output) COMPLEX*16 array, dimension (LDDE, 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 COMPQ = 'C', 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 COMPQ = 'N', this array is unchanged on exit.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
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-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 COMPQ = 'N', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N).
FG (input/output) COMPLEX*16 array, dimension (LDFG, 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 COMPQ = 'C', 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 COMPQ = 'N', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, N).
NEIG (output) INTEGER
If COMPQ = 'C', the number of eigenvalues in aS - bH with
strictly negative real part.
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading N-by-NEIG part of
this array contains an orthonormal basis of the right
deflating subspace corresponding to the eigenvalues of the
pencil aS - bH with strictly negative real part.
The remaining entries are meaningless.
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 (N+1)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= MAX( 4*N*N + 2*N + MAX(3,N) ), if COMPQ = 'N';
LDWORK >= MAX( 1, 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 = -22, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if COMPQ = 'N';
LZWORK >= 8*N + 4, if COMPQ = 'C'.
For good performance LZWORK should be generally larger.
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 COMPQ = 'N';
LBWORK >= N, if COMPQ = 'C'.
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: the singular value decomposition failed in the LAPACK
routine ZGESVD (for ORTH = 'S');
= 4: 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
~ ( S11 S12 )
B_S = J Q' J' B_S Q = ( ) and
( 0 S11' )
~ ( T11 T12 ) ( 0 I )
B_T = J Q' J' B_T Q = ( ), with J = ( ),
( 0 T11' ) ( -I 0 )
where Q is real orthogonal, S11 is upper triangular, and T11 is
upper quasi-triangular.
Second, the SLICOT Library routine MB03JZ is applied, to compute a
~
unitary matrix Q, such that
~ ~
~ ~ ~ ( S11 S12 )
J Q' J' B_S Q = ( ~ ) =: B_Sout,
( 0 S11' )
~ ~ ~ ( H11 H12 )
J Q' J'(-i*B_T) Q = ( ) =: B_Hout,
( 0 -H11' )
~ ~ ~
with S11, H11 upper triangular, and such that Spec_-(B_S, -i*B_T)
is contained in the spectrum of the 2*NEIG-by-2*NEIG leading
~
principal subpencil aS11 - bH11.
Finally, the right deflating subspace is computed.
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
This routine does not perform any scaling of the matrices. Scaling might sometimes be useful, and it should be done externally.Example
Program Text
* MB03LZ 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*1 COMPQ, ORTH
INTEGER I, INFO, J, N, NEIG
*
* .. 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( NMAX + 1 )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03LZ
*
* .. Intrinsic Functions ..
INTRINSIC MOD
*
* .. Executable statements ..
*
WRITE( NOUT, FMT = 99999 )
*
* Skip first line in data file.
*
READ( NIN, FMT = * )
READ( NIN, FMT = * ) COMPQ, ORTH, N
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, N/2 ), I = 1, N/2 )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
* Compute the eigenvalues and an orthogonal basis of the right
* deflating subspace of a complex skew-Hamiltonian/Hamiltonian
* pencil, corresponding to the eigenvalues with strictly negative
* real part.
CALL MB03LZ( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, NEIG, 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( COMPQ, 'C' ) ) 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
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99990 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99989 )
WRITE( NOUT, FMT = 99990 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99990 ) ( BETA( I ), I = 1, N )
IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
WRITE( NOUT, FMT = 99987 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
50 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) )
$ WRITE( NOUT, FMT = 99986 ) NEIG
END IF
END IF
STOP
99999 FORMAT ( 'MB03LZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB03LZ = ', 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 vector ALPHAR is ' )
99990 FORMAT ( 50( 1X, F8.4 ) )
99989 FORMAT (/'The vector ALPHAI is ' )
99988 FORMAT (/'The vector BETA is ' )
99987 FORMAT (/'The deflating subspace corresponding to the ',
$ 'eigenvalues with negative real part is ' )
99986 FORMAT (/'The number of eigenvalues in the initial pencil with ',
$ 'negative real part is ', I2 )
END
Program Data
MB03LZ EXAMPLE PROGRAM DATA
C P 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
MB03LZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
0.7430 +0.0000i -0.1431 -0.1304i -0.4169 -0.0495i 0.0650 -0.0262i
0.3992 +0.6279i 0.7398 -1.2647i -0.0861 -0.1075i 0.2826 +0.7725i
0.0000 +0.0000i 0.0000 +0.0000i 1.4799 +0.1442i -0.1094 -0.1061i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.6816 +0.2278i
The matrix D on exit is
0.0000 -0.6858i -0.3122 -0.1018i -0.7813 -0.4163i -0.1343 +0.3259i
0.9840 +0.3394i 0.0000 +0.1465i -0.1678 +0.2971i -0.0728 -0.6524i
0.0000 +0.0000i 0.0000 +0.0000i -0.0000 +0.2979i -0.0728 +0.3971i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.2414i
The matrix B on exit is
-1.5832 +0.5069i -0.0819 -0.1073i 0.7749 -0.0519i 0.0635 -0.0052i
0.0000 +0.0000i -0.1916 -0.0106i -0.0074 +0.0165i -0.1546 -0.6817i
0.0000 +0.0000i 0.0000 +0.0000i -0.0716 -0.1811i 0.3146 +0.1558i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -1.6078 -0.0203i
The matrix F on exit is
0.3382 0.0000i -0.0622 +0.8488i 0.0042 +0.9053i -0.1584 +0.0726i
0.5391 +0.1711i -0.5888 +0.0000i 0.4089 +0.2018i -0.6913 -0.5011i
0.0000 +0.0000i 0.0000 +0.0000i -0.2712 +0.0000i 0.5114 +0.3726i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.5218 +0.0000i
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
The deflating subspace corresponding to the eigenvalues with negative real part is
-0.0793 -0.1949i 0.4845 -0.5472i
0.4349 +0.1710i -0.2878 +0.0952i
-0.1266 +0.1505i 0.1364 -0.4776i
-0.5035 +0.6671i 0.1628 +0.3174i
The number of eigenvalues in the initial pencil with negative real part is 2