**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.

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, * )

**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 leading subarray of Q. 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.

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.

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)

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.

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 is computed. See also page 12 in [1] for more details.

[1] 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.

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.

**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

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

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