**Purpose**

To compute the eigenvalues of a real N-by-N skew-Hamiltonian/ skew-Hamiltonian pencil aS - bT with ( A D ) ( B F ) S = ( ) and T = ( ). (1) ( E A' ) ( G B' ) Optionally, if JOB = 'T', the pencil aS - bT will be transformed to the structured Schur form: an orthogonal transformation matrix Q is computed such that ( Aout Dout ) J Q' J' S Q = ( ), and ( 0 Aout' ) (2) ( Bout Fout ) ( 0 I ) J Q' J' T Q = ( ), where J = ( ), ( 0 Bout' ) ( -I 0 ) Aout is upper triangular, and Bout is upper quasi-triangular. The notation M' denotes the transpose of the matrix M. Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal transformation matrix Q will be computed.

SUBROUTINE MB04FD( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG, $ LDFG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK, $ DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER COMPQ, JOB INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK, N C .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), $ B( LDB, * ), BETA( * ), DE( LDDE, * ), $ DWORK( * ), FG( LDFG, * ), Q( LDQ, * ) INTEGER IWORK( * )

**Mode Parameters**

JOB CHARACTER*1 Specifies the computation to be performed, as follows: = 'E': compute the eigenvalues only; S and T will not necessarily be put into skew-Hamiltonian triangular form (2); = 'T': put S and T into skew-Hamiltonian triangular form (2), and return the eigenvalues in ALPHAR, ALPHAI and BETA. COMPQ CHARACTER*1 Specifies whether to compute the orthogonal transformation matrix Q as follows: = 'N': Q is not computed; = 'I': the array Q is initialized internally to the unit matrix, and the orthogonal matrix Q is returned; = 'U': the array Q contains an orthogonal matrix Q0 on entry, and the product Q0*Q is returned, where Q is the product of the orthogonal transformations that are applied to the pencil aS - bT to reduce S and T to the forms in (2), for COMPQ = 'I'.

N (input) INTEGER The order of the pencil aS - bT. 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 JOB = 'T', the leading N/2-by-N/2 part of this array contains the matrix Aout; otherwise, it contains meaningless elements, except for the diagonal blocks, which are correctly set. 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 strictly lower triangular part of this array must contain the strictly lower triangular part of the skew-symmetric matrix E, and the N/2-by-N/2 strictly upper triangular part of the submatrix in the columns 2 to N/2+1 of this array must contain the strictly upper triangular part of the skew-symmetric matrix D. The entries on the diagonal and the first superdiagonal of this array are not referenced, but are assumed to be zero. On exit, if JOB = 'T', the leading N/2-by-N/2 strictly upper triangular part of the submatrix in the columns 2 to N/2+1 of this array contains the strictly upper triangular part of the skew-symmetric matrix Dout. If JOB = 'E', the leading N/2-by-N/2 strictly upper triangular part of the submatrix in the columns 2 to N/2+1 of this array contains the strictly upper triangular part of the skew-symmetric matrix D just before the application of the QZ algorithm. The remaining entries are meaningless. 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 JOB = 'T', the leading N/2-by-N/2 part of this array contains the matrix Bout; otherwise, it contains meaningless elements, except for the diagonal 1-by-1 and 2-by-2 blocks, which are correctly set. 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 strictly lower triangular part of this array must contain the strictly lower triangular part of the skew-symmetric matrix G, and the N/2-by-N/2 strictly upper triangular part of the submatrix in the columns 2 to N/2+1 of this array must contain the strictly upper triangular part of the skew-symmetric matrix F. The entries on the diagonal and the first superdiagonal of this array are not referenced, but are assumed to be zero. On exit, if JOB = 'T', the leading N/2-by-N/2 strictly upper triangular part of the submatrix in the columns 2 to N/2+1 of this array contains the strictly upper triangular part of the skew-symmetric matrix Fout. If JOB = 'E', the leading N/2-by-N/2 strictly upper triangular part of the submatrix in the columns 2 to N/2+1 of this array contains the strictly upper triangular part of the skew-symmetric matrix F just before the application of the QZ algorithm. The remaining entries are meaningless. LDFG INTEGER The leading dimension of the array FG. LDFG >= MAX(1, N/2). Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) On entry, if COMPQ = 'U', then the leading N-by-N part of this array must contain a given matrix Q0, and on exit, the leading N-by-N part of this array contains the product of the input matrix Q0 and the transformation matrix Q used to transform the matrices S and T. On exit, if COMPQ = 'I', then the leading N-by-N part of this array contains the orthogonal transformation matrix Q. 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, N), if COMPQ = 'I' or COMPQ = 'U'. ALPHAR (output) DOUBLE PRECISION array, dimension (N/2) The real parts of each scalar alpha defining an eigenvalue of the pencil aS - bT. ALPHAI (output) DOUBLE PRECISION array, dimension (N/2) The imaginary parts of each scalar alpha defining an eigenvalue of the pencil aS - bT. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair. BETA (output) DOUBLE PRECISION array, dimension (N/2) The scalars beta that define the eigenvalues of the pencil aS - bT. Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta = BETA(j) represent the j-th eigenvalue of the pencil aS - bT, in the form lambda = alpha/beta. Since lambda may overflow, the ratios should not, in general, be computed. Due to the skew-Hamiltonian/skew-Hamiltonian structure of the pencil, every eigenvalue occurs twice and thus it has only to be saved once in ALPHAR, ALPHAI and BETA.

IWORK INTEGER array, dimension (N/2+1) On exit, IWORK(1) contains the number of (pairs of) possibly inaccurate eigenvalues, q <= N/2, and the absolute values in IWORK(2), ..., IWORK(q+1) are their indices, as well as of the corresponding diagonal blocks. Specifically, a positive value is an index of a real eigenvalue, corresponding to a 1-by-1 block pair, 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 pair. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK; DWORK(2) and DWORK(3) contain the Frobenius norms of the matrices S and T on entry. These norms are used in the tests to decide that some eigenvalues are considered as unreliable. On exit, if INFO = -19, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The dimension of the array DWORK. LDWORK >= MAX(3,N/2), if JOB = 'E' and COMPQ = 'N'; LDWORK >= MAX(3,N**2/4+N/2), if JOB = 'T' and COMPQ = 'N'; LDWORK >= MAX(1,3*N**2/4), if COMPQ<> 'N'. For good performance LDWORK should generally be 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.

INFO INTEGER = 0: succesful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: QZ iteration failed in the LAPACK Library routine DHGEQZ. (QZ iteration did not converge or computation of the shifts failed.) = 2: warning: the pencil is numerically singular.

The algorithm uses Givens rotations and Householder reflections to annihilate elements in S and T such that S is in skew-Hamiltonian triangular form and T is in skew-Hamiltonian Hessenberg form: ( A1 D1 ) ( B1 F1 ) S = ( ), T = ( ), ( 0 A1' ) ( 0 B1' ) where A1 is upper triangular and B1 is upper Hessenberg. Subsequently, the QZ algorithm is applied to the pencil aA1 - bB1 to determine orthogonal matrices Q1 and Q2 such that Q2' A1 Q1 is upper triangular and Q2' B1 Q1 is upper quasi- triangular. See also page 40 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 ) real floating point operations.

None

**Program Text**

* MB04FD 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 PARAMETER ( LDA = NMAX/2, LDB = NMAX/2, LDDE = NMAX/2, $ LDFG = NMAX/2, LDQ = NMAX, LDWORK = $ 3*NMAX*NMAX/4 ) * * .. Local Scalars .. CHARACTER COMPQ, JOB INTEGER I, INFO, J, N * * .. Local Arrays .. 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, NMAX ) INTEGER IWORK( NMAX/2+1 ) * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * * .. External Subroutines .. EXTERNAL MB04FD * * .. Intrinsic Functions .. INTRINSIC MOD * * .. Executable statements .. * WRITE( NOUT, FMT = 99999 ) * * Skip first line in data file. * READ( NIN, FMT = * ) READ( NIN, FMT = * ) JOB, COMPQ, 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( LSAME( COMPQ, 'U' ) ) $ READ( NIN, FMT = * ) ( ( Q( I, J ), J = 1, N ), I = 1, N ) IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN WRITE( NOUT, FMT = 99998 ) N ELSE * * Test of MB04FD. * CALL MB04FD( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG, LDFG, $ Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK, DWORK, $ LDWORK, 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/2 WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N/2 ) 10 CONTINUE END IF WRITE( NOUT, FMT = 99994 ) DO 20 I = 1, N/2 WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N/2+1 ) 20 CONTINUE IF( LSAME( JOB, 'T' ) ) THEN WRITE( NOUT, FMT = 99993 ) DO 30 I = 1, N/2 WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N/2 ) 30 CONTINUE END IF WRITE( NOUT, FMT = 99992 ) DO 40 I = 1, N/2 WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N/2+1 ) 40 CONTINUE IF( .NOT.LSAME( COMPQ, 'N' ) ) THEN WRITE( NOUT, FMT = 99991 ) DO 50 I = 1, N WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, N ) 50 CONTINUE END IF WRITE( NOUT, FMT = 99990 ) WRITE( NOUT, FMT = 99995 ) ( ALPHAR(I), I = 1, N/2 ) WRITE( NOUT, FMT = 99995 ) ( ALPHAI(I), I = 1, N/2 ) WRITE( NOUT, FMT = 99995 ) ( BETA(I), I = 1, N/2 ) END IF END IF STOP 99999 FORMAT ( 'MB04FD EXAMPLE PROGRAM RESULTS', 1X ) 99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 ) 99997 FORMAT ( 'INFO on exit from MB04FD = ', I2 ) 99996 FORMAT (/' The transformed matrix A is' ) 99995 FORMAT ( 51( 1X, F8.4 ) ) 99994 FORMAT (/' The transformed matrix DE is' ) 99993 FORMAT (/' The transformed matrix B is' ) 99992 FORMAT (/' The transformed matrix FG is' ) 99991 FORMAT (/' The transformed matrix Q is ' ) 99990 FORMAT (/' The real, imaginary, and beta parts of eigenvalues are' $ ) END

MB04FD EXAMPLE PROGRAM DATA T I 8 0.8147 0.6323 0.9575 0.9571 0.9057 0.0975 0.9648 0.4853 0.1269 0.2784 0.1576 0.8002 0.9133 0.5468 0.9705 0.1418 0.4217 0.6557 0.6787 0.6554 0.2769 0.9157 0.0357 0.7577 0.1711 0.0461 0.7922 0.8491 0.7431 0.7060 0.0971 0.9594 0.9339 0.3922 0.0318 0.8234 0.6948 0.4387 0.1868 0.7093 0.3170 0.3815 0.4897 0.7546 0.9502 0.7655 0.4455 0.2760 0.0344 0.7951 0.6463 0.6797 0.6550 0.9597 0.7512 0.8909 0.1492 0.1626 0.3403 0.2550 0.9592 0.2575 0.1189 0.5852 0.5059 0.5472 0.8407 0.4983 0.2238 0.6990 0.1386 0.2542

MB04FD EXAMPLE PROGRAM RESULTS The transformed matrix A is 0.0550 -0.3064 0.1543 0.2170 0.0000 1.2189 0.3267 -1.3622 0.0000 0.0000 0.7734 -0.6215 0.0000 0.0000 0.0000 2.5172 The transformed matrix DE is 0.4217 0.6557 0.4277 -0.2877 -0.3980 -1.5448 0.0357 0.7577 -0.2582 -0.4775 0.3220 1.1004 0.7431 0.7060 -1.1102 0.3899 0.3463 0.4843 0.0318 0.8234 The transformed matrix B is 0.8482 -0.4425 -0.3643 0.8333 0.0000 -0.5919 -0.0987 -0.7923 0.0000 0.0000 1.1021 0.1926 0.0000 0.0000 0.0000 1.9788 The transformed matrix FG is 0.6550 0.9597 0.3266 -0.6059 -0.9618 -0.3151 0.3403 0.2550 -0.4771 -0.3005 0.1987 -0.0141 0.5059 0.5472 -0.4172 0.7914 0.0371 0.0000 0.1386 0.2542 The transformed matrix Q is 0.0762 0.7824 -0.2771 -0.5526 0.0000 0.0000 0.0000 0.0000 0.1786 -0.2450 0.4261 -0.5360 0.1843 -0.5857 0.2437 -0.0522 -0.2452 0.1425 0.4831 -0.0742 -0.1834 -0.0856 -0.6529 0.4620 -0.2123 -0.3870 -0.6431 -0.2548 0.3061 -0.1791 -0.2518 0.3707 -0.4085 0.0154 0.0784 -0.0738 0.3908 0.0160 -0.3950 -0.7157 0.6418 -0.2868 0.0258 -0.3305 -0.0286 0.4768 -0.3901 -0.1253 -0.0077 -0.1340 -0.2740 -0.0534 -0.7708 -0.3910 -0.1981 -0.3432 -0.5275 -0.2394 0.1131 -0.4684 -0.3018 0.4869 0.3216 0.0252 The real, imaginary, and beta parts of eigenvalues are 0.8482 -0.5919 1.1021 1.9788 0.0000 0.0000 0.0000 0.0000 0.0550 1.2189 0.7734 2.5172