**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.

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( * )

**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.

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.

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'.

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.

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.

[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.

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

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

**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

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

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