Purpose
To compute the eigenvalues of a real N-by-N skew-Hamiltonian/ Hamiltonian pencil aS - bH with ( 0 I ) S = T Z = J Z' J' Z, where J = ( ), (1) ( -I 0 ) via generalized symplectic URV decomposition. That is, orthogonal matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2 are computed such that ( T11 T12 ) Q1' T U1 = Q1' J Z' J' U1 = ( ) = Tout, ( 0 T22 ) ( Z11 Z12 ) U2' Z Q2 = ( ) = Zout, (2) ( 0 Z22 ) ( H11 H12 ) Q1' H Q2 = ( ) = Hout, ( 0 H22 ) where T11, T22', Z11, Z22', H11 are upper triangular and H22' is upper quasi-triangular. Optionally, if COMPQ1 = 'C' the orthogonal transformation matrix Q1 will be computed. Optionally, if COMPQ2 = 'C' the orthogonal transformation matrix Q2 will be computed. Optionally, if COMPU1 = 'C' the orthogonal symplectic transformation matrix ( U11 U12 ) U1 = ( ) ( -U12 U11 ) will be computed. Optionally, if COMPU2 = 'C' the orthogonal symplectic transformation matrix ( U21 U22 ) U2 = ( ) ( -U22 U21 ) will be computed.Specification
SUBROUTINE MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ, $ H, LDH, T, LDT, Q1, LDQ1, Q2, LDQ2, U11, LDU11, $ U12, LDU12, U21, LDU21, U22, LDU22, ALPHAR, $ ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK, $ INFO ) C .. Scalar Arguments .. CHARACTER COMPQ1, COMPQ2, COMPU1, COMPU2, JOB INTEGER INFO, LDH, LDQ1, LDQ2, LDT, LDU11, LDU12, $ LDU21, LDU22, LDWORK, LDZ, LIWORK, N C .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), $ DWORK( * ), H( LDH, * ), Q1( LDQ1, * ), $ Q2( LDQ2, * ), T( LDT, * ), U11( LDU11, * ), $ U12( LDU12, * ), U21( LDU21, * ), $ U22( LDU22, * ), Z( LDZ, * )Arguments
Mode Parameters
JOB CHARACTER*1 Specifies the computation to be performed, as follows: = 'E': compute the eigenvalues only; Z, T, and H will not necessarily be put into the forms in (2); H22' is upper Hessenberg; = 'T': put Z, T, and H into the forms in (2), and return the eigenvalues in ALPHAR, ALPHAI and BETA. COMPQ1 CHARACTER*1 Specifies whether to compute the orthogonal transformation matrix Q1, as follows: = 'N': Q1 is not computed; = 'C': compute the matrix Q1 of the orthogonal transformations applied on the left to the pencil aTZ - bH to reduce its matrices to the form (2). The array Q1 is initialized internally to the identity matrix. COMPQ2 CHARACTER*1 Specifies whether to compute the orthogonal transformation matrix Q2, as follows: = 'N': Q2 is not computed; = 'C': compute the matrix Q2 of the orthogonal transformations applied on the right to the pencil aTZ - bH to reduce its matrices to the form (2). The array Q2 is initialized internally to the identity matrix. COMPU1 CHARACTER*1 Specifies whether to compute the orthogonal symplectic transformation matrix U1, as follows: = 'N': U1 is not computed; = 'C': compute the matrices U11 and U12 of the orthogonal symplectic transformations applied to the pencil aTZ - bT to reduce its matrices to the form (2). The arrays U11 and U12 are initialized internally to correspond to an identity matrix U1. COMPU2 CHARACTER*1 Specifies whether to compute the orthogonal symplectic transformation matrix U2, as follows: = 'N': U2 is not computed; = 'C': compute the matrices U21 and U22 of the orthogonal symplectic transformations applied to the pencil aTZ - bT to reduce its matrices to the form (2). The arrays U21 and U22 are initialized internally to correspond to an identity matrix U2.Input/Output Parameters
N (input) INTEGER The order of the pencil aS - bH. N >= 0, even. Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) On entry, the leading N-by-N part of this array must contain the matrix Z. On exit, if JOB = 'T', the leading N-by-N part of this array contains the matrix Zout; otherwise, it contains the matrix Z obtained just before the application of the periodic QZ algorithm. The elements of the (2,1) block, i.e., in the rows N/2+1 to N and in the columns 1 to N/2 are not set to zero, but are unchanged on exit. LDZ INTEGER The leading dimension of the array Z. LDZ >= MAX(1, N). H (input/output) DOUBLE PRECISION array, dimension (LDH, N) On entry, the leading N-by-N part of this array must contain the Hamiltonian matrix H (H22 = -H11', H12 = H12', H21 = H21'). On exit, if JOB = 'T', the leading N-by-N part of this array contains the matrix Hout; otherwise, it contains the matrix H obtained just before the application of the periodic QZ algorithm. LDH INTEGER The leading dimension of the array H. LDH >= MAX(1, N). T (output) DOUBLE PRECISION array, dimension (LDT, N) If JOB = 'T', the leading N-by-N part of this array contains the matrix Tout; otherwise, it contains the matrix T obtained just before the application of the periodic QZ algorithm. LDT INTEGER The leading dimension of the array T. LDT >= MAX(1, N). Q1 (output) DOUBLE PRECISION array, dimension (LDQ1, N) On exit, if COMPQ1 = 'C', the leading N-by-N part of this array contains the orthogonal transformation matrix Q1. If COMPQ1 = 'N', this array is not referenced. LDQ1 INTEGER The leading dimension of the array Q1. LDQ1 >= 1, if COMPQ1 = 'N'; LDQ1 >= MAX(1, N), if COMPQ1 = 'C'. Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N) On exit, if COMPQ2 = 'C', the leading N-by-N part of this array contains the orthogonal transformation matrix Q2. If COMPQ2 = 'N', this array is not referenced. LDQ2 INTEGER The leading dimension of the array Q2. LDQ2 >= 1, if COMPQ2 = 'N'; LDQ2 >= MAX(1, N), if COMPQ2 = 'C'. U11 (output) DOUBLE PRECISION array, dimension (LDU11, N/2) On exit, if COMPU1 = 'C', the leading N/2-by-N/2 part of this array contains the upper left block U11 of the orthogonal symplectic transformation matrix U1. If COMPU1 = 'N', this array is not referenced. LDU11 INTEGER The leading dimension of the array U11. LDU11 >= 1, if COMPU1 = 'N'; LDU11 >= MAX(1, N/2), if COMPU1 = 'C'. U12 (output) DOUBLE PRECISION array, dimension (LDU12, N/2) On exit, if COMPU1 = 'C', the leading N/2-by-N/2 part of this array contains the upper right block U12 of the orthogonal symplectic transformation matrix U1. If COMPU1 = 'N', this array is not referenced. LDU12 INTEGER The leading dimension of the array U12. LDU12 >= 1, if COMPU1 = 'N'; LDU12 >= MAX(1, N/2), if COMPU1 = 'C'. U21 (output) DOUBLE PRECISION array, dimension (LDU21, N/2) On exit, if COMPU2 = 'C', the leading N/2-by-N/2 part of this array contains the upper left block U21 of the orthogonal symplectic transformation matrix U2. If COMPU2 = 'N', this array is not referenced. LDU21 INTEGER The leading dimension of the array U21. LDU21 >= 1, if COMPU2 = 'N'; LDU21 >= MAX(1, N/2), if COMPU2 = 'C'. U22 (output) DOUBLE PRECISION array, dimension (LDU22, N/2) On exit, if COMPU2 = 'C', the leading N/2-by-N/2 part of this array contains the upper right block U22 of the orthogonal symplectic transformation matrix U2. If COMPU2 = 'N', this array is not referenced. LDU22 INTEGER The leading dimension of the array U22. LDU22 >= 1, if COMPU2 = 'N'; LDU22 >= MAX(1, N/2), if COMPU2 = '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 defining the eigenvalues of the pencil aS - bH. If INFO = 0, the quantities alpha = (ALPHAR(j),ALPHAI(j)), and beta = BETA(j) represent together 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, only half of the spectrum is saved in ALPHAR, ALPHAI and BETA. Specifically, the eigenvalues with positive real parts or with non-negative imaginary parts, when real parts are zero, are returned. The remaining eigenvalues have opposite signs. If INFO = 3, one or more BETA(j) is not representable, and the eigenvalues are returned as described below.Workspace
IWORK INTEGER array, dimension (LIWORK) On exit, if INFO = 3, IWORK(1), ..., IWORK(N/2) return the scaling parameters for the eigenvalues of the pencil aS - bH (see INFO = 3). LIWORK INTEGER The dimension of the array IWORK. LIWORK >= N/2+18. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK, and DWORK(2) returns the machine base, b. On exit, if INFO = -31, DWORK(1) returns the minimum value of LDWORK. LDWORK INTEGER The dimension of the array DWORK. If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N' and COMPU1 = 'N' and COMPU2 = 'N', then LDWORK >= 3/2*N**2+MAX(N, 48); else, LDWORK >= 3*N**2+MAX(N, 48). 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.Error Indicator
INFO INTEGER = 0: succesful exit. < 0: if INFO = -i, the i-th argument had an illegal value. = 1: the periodic QZ algorithm was not able to reveal information about the eigenvalues from the 2-by-2 blocks in the SLICOT Library routine MB03BD; = 2: the periodic QZ algorithm did not converge in the SLICOT Library routine MB03BD; = 3: the eigenvalues will under- or overflow if evaluated; therefore, the j-th eigenvalue is represented by the quantities alpha = (ALPHAR(j),ALPHAI(j)), beta = BETA(j), and gamma = IWORK(j) in the form lambda = (alpha/beta) * b**gamma, where b is the machine base (often 2.0). This is not an error.Method
The algorithm uses Givens rotations and Householder reflections to annihilate elements in T, Z, and H such that T11, T22', Z11, Z22' and H11 are upper triangular and H22' is upper Hessenberg. Finally the periodic QZ algorithm is applied to transform H22' to upper quasi-triangular form while T11, T22', Z11, Z22', and H11 stay in upper triangular form. See also page 17 in [1] for more details.References
[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.Numerical Aspects
3 The algorithm is numerically backward stable and needs O(N ) real floating point operations.Further Comments
NoneExample
Program Text
* MB04AD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2010 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX PARAMETER ( NMAX = 50 ) INTEGER LDH, LDQ1, LDQ2, LDT, LDU11, LDU12, LDU21, $ LDU22, LDWORK, LDZ, LIWORK PARAMETER ( LDH = NMAX, LDQ1 = NMAX, LDQ2 = NMAX, $ LDT = NMAX, LDU11 = NMAX/2, LDU12 = NMAX/2, $ LDU21 = NMAX/2, LDU22 = NMAX/2, $ LDWORK = 3*NMAX*NMAX + MAX( NMAX, 48 ), $ LDZ = NMAX, LIWORK = NMAX/2 + 18 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) * * .. Local Scalars .. CHARACTER COMPQ1, COMPQ2, COMPU1, COMPU2, JOB INTEGER I, INFO, J, M, N * * .. Local Arrays .. INTEGER IWORK( LIWORK ) DOUBLE PRECISION ALPHAI( NMAX/2 ), ALPHAR( NMAX/2 ), $ BETA( NMAX/2 ), DWORK( LDWORK ), $ H( LDH, NMAX ), Q1( LDQ1, NMAX ), $ Q2( LDQ2, NMAX ), T( LDT, NMAX ), $ U11( LDU11, NMAX/2 ), U12( LDU12, NMAX/2 ), $ U21( LDU21, NMAX/2 ), U22( LDU22, NMAX/2 ), $ Z( LDZ, NMAX ) * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * * .. External Subroutines .. EXTERNAL DLASET, MB04AD * * .. 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 = * ) JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N IF( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE( NOUT, FMT = 99998 ) N ELSE READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N ) READ( NIN, FMT = * ) ( ( H( I, J ), J = 1, N ), I = 1, N ) * Compute the eigenvalues of a real skew-Hamiltonian/Hamiltonian * pencil (factored version). CALL MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ, H, $ LDH, T, LDT, Q1, LDQ1, Q2, LDQ2, U11, LDU11, U12, $ LDU12, U21, LDU21, U22, LDU22, ALPHAR, ALPHAI, $ BETA, IWORK, LIWORK, DWORK, LDWORK, INFO ) * IF( INFO.NE.0 ) THEN WRITE( NOUT, FMT = 99997 ) INFO ELSE M = N/2 CALL DLASET( 'Full', M, M, ZERO, ZERO, Z( M+1, 1 ), LDZ ) WRITE( NOUT, FMT = 99996 ) DO 10 I = 1, N WRITE( NOUT, FMT = 99995 ) ( T( I, J ), J = 1, N ) 10 CONTINUE WRITE( NOUT, FMT = 99994 ) DO 20 I = 1, N WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N ) 20 CONTINUE WRITE( NOUT, FMT = 99993 ) DO 30 I = 1, N WRITE( NOUT, FMT = 99995 ) ( H( I, J ), J = 1, N ) 30 CONTINUE IF( LSAME( COMPQ1, 'C' ) ) THEN WRITE( NOUT, FMT = 99992 ) DO 40 I = 1, N WRITE( NOUT, FMT = 99995 ) ( Q1( I, J ), J = 1, N ) 40 CONTINUE END IF IF( LSAME( COMPQ2, 'C' ) ) THEN WRITE( NOUT, FMT = 99991 ) DO 50 I = 1, N WRITE( NOUT, FMT = 99995 ) ( Q2( I, J ), J = 1, N ) 50 CONTINUE END IF IF( LSAME( COMPU1, 'C' ) ) THEN WRITE( NOUT, FMT = 99990 ) DO 60 I = 1, M WRITE( NOUT, FMT = 99995 ) ( U11( I, J ), J = 1, M ) 60 CONTINUE WRITE( NOUT, FMT = 99989 ) DO 70 I = 1, M WRITE( NOUT, FMT = 99995 ) ( U12( I, J ), J = 1, M ) 70 CONTINUE END IF IF( LSAME( COMPU2, 'C' ) ) THEN WRITE( NOUT, FMT = 99988 ) DO 80 I = 1, M WRITE( NOUT, FMT = 99995 ) ( U21( I, J ), J = 1, M ) 80 CONTINUE WRITE( NOUT, FMT = 99987 ) DO 90 I = 1, M WRITE( NOUT, FMT = 99995 ) ( U22( I, J ), J = 1, M ) 90 CONTINUE END IF WRITE( NOUT, FMT = 99986 ) WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M ) WRITE( NOUT, FMT = 99985 ) WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M ) WRITE( NOUT, FMT = 99984 ) WRITE( NOUT, FMT = 99995 ) ( BETA( I ), I = 1, M ) END IF END IF STOP * 99999 FORMAT( 'MB04AD EXAMPLE PROGRAM RESULTS', 1X ) 99998 FORMAT( 'N is out of range.', /, 'N = ', I5 ) 99997 FORMAT( 'INFO on exit from MB04AD = ', I2 ) 99996 FORMAT( 'The matrix T on exit is ' ) 99995 FORMAT( 50( 1X, F8.4 ) ) 99994 FORMAT( 'The matrix Z on exit is ' ) 99993 FORMAT( 'The matrix H is ' ) 99992 FORMAT( 'The matrix Q1 is ' ) 99991 FORMAT( 'The matrix Q2 is ' ) 99990 FORMAT( 'The upper left block of the matrix U1 is ' ) 99989 FORMAT( 'The upper right block of the matrix U1 is ' ) 99988 FORMAT( 'The upper left block of the matrix U2 is ' ) 99987 FORMAT( 'The upper right block of the matrix U2 is ' ) 99986 FORMAT( 'The vector ALPHAR is ' ) 99985 FORMAT( 'The vector ALPHAI is ' ) 99984 FORMAT( 'The vector BETA is ' ) ENDProgram Data
MB04AD EXAMPLE PROGRAM DATA T C C C C 8 3.1472 4.5751 -0.7824 1.7874 -2.2308 -0.6126 2.0936 4.5974 4.0579 4.6489 4.1574 2.5774 -4.5383 -1.1844 2.5469 -1.5961 -3.7301 -3.4239 2.9221 2.4313 -4.0287 2.6552 -2.2397 0.8527 4.1338 4.7059 4.5949 -1.0777 3.2346 2.9520 1.7970 -2.7619 1.3236 4.5717 1.5574 1.5548 1.9483 -3.1313 1.5510 2.5127 -4.0246 -0.1462 -4.6429 -3.2881 -1.8290 -0.1024 -3.3739 -2.4490 -2.2150 3.0028 3.4913 2.0605 4.5022 -0.5441 -3.8100 0.0596 0.4688 -3.5811 4.3399 -4.6817 -4.6555 1.4631 -0.0164 1.9908 3.9090 -3.5071 3.1428 -3.0340 -1.4834 3.7401 -0.1715 0.4026 4.5929 -2.4249 -2.5648 -2.4892 3.7401 -2.1416 1.6251 2.6645 0.4722 3.4072 4.2926 1.1604 -0.1715 1.6251 -4.2415 -0.0602 -3.6138 -2.4572 -1.5002 -0.2671 0.4026 2.6645 -0.0602 -3.7009 0.6882 -1.8421 -4.1122 0.1317 -3.9090 -4.5929 -0.4722 3.6138 -1.8421 2.9428 -0.4340 1.3834 3.5071 2.4249 -3.4072 2.4572 -4.1122 -0.4340 -2.3703 0.5231 -3.1428 2.5648 -4.2926 1.5002 0.1317 1.3834 0.5231 -4.1618 3.0340 2.4892 -1.1604 0.2671Program Results
MB04AD EXAMPLE PROGRAM RESULTS The matrix T on exit is -3.9699 3.7658 5.5815 -1.7750 -0.8818 -0.0511 -4.2158 1.9054 0.0000 5.3686 -5.9166 4.9163 1.3839 0.8870 3.9458 -4.9167 0.0000 0.0000 5.9641 1.9432 -2.0680 2.4402 -1.4091 5.8512 0.0000 0.0000 0.0000 5.9983 -3.8172 4.0147 -2.0739 -1.2570 0.0000 0.0000 0.0000 0.0000 8.2005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.5732 8.0098 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6017 2.4397 5.9751 0.0000 0.0000 0.0000 0.0000 0.0000 -2.5869 0.5598 0.2544 5.2129 The matrix Z on exit is -6.4705 -2.5511 -4.0551 -1.9895 -2.7642 0.7532 -4.1047 -2.2046 0.0000 7.3589 -4.4480 -2.7491 -1.5465 -1.4345 -0.9272 1.3121 0.0000 0.0000 4.9125 -0.4968 5.3574 3.8579 5.2547 -1.7324 0.0000 0.0000 0.0000 9.0822 0.0460 -0.3382 3.9302 3.1084 0.0000 0.0000 0.0000 0.0000 6.1869 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.5573 6.6549 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.7456 -3.5789 4.3432 0.0000 0.0000 0.0000 0.0000 0.0000 0.1549 3.5335 3.1346 4.1062 The matrix H is -7.4834 0.4404 2.3558 1.6724 -0.4630 1.9533 1.5724 -2.7254 0.0000 -7.3500 3.7414 3.7466 0.2837 0.6849 0.7727 -4.2140 0.0000 0.0000 -2.3493 -3.7994 -0.6872 1.1773 -2.6901 -5.1494 0.0000 0.0000 0.0000 -3.4719 5.3322 0.4182 1.9779 1.5175 0.0000 0.0000 0.0000 0.0000 -6.1880 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -3.3324 9.0833 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -1.8703 0.0799 -2.8180 0.0000 0.0000 0.0000 0.0000 0.0000 -2.3477 3.3110 0.6561 0.7281 The matrix Q1 is -0.2489 -0.1409 0.3615 0.6458 0.0113 0.6063 -0.0470 0.0238 -0.2436 0.1294 -0.0874 -0.4103 0.3408 0.3628 -0.3267 0.6272 -0.4316 -0.2352 0.5553 -0.2811 -0.2198 -0.2880 -0.4564 -0.1773 0.1992 -0.2176 -0.5198 0.1561 -0.1523 0.1299 -0.7281 -0.2197 0.0161 0.7390 0.1125 -0.2226 -0.1003 0.3608 -0.1118 -0.4886 -0.5824 0.0984 -0.3052 0.1996 0.5889 -0.2442 0.0060 -0.3341 -0.3246 0.4661 -0.1835 0.3523 -0.5153 -0.3034 -0.0865 0.3931 -0.4559 -0.2961 -0.3790 -0.3127 -0.4356 0.3452 0.3642 -0.1467 The matrix Q2 is 0.0288 -0.1842 -0.6791 -0.2115 -0.4790 0.4212 -0.0417 -0.2253 -0.0666 -0.0787 -0.3711 0.1737 -0.0482 -0.5770 -0.6785 0.1607 0.1506 0.6328 0.0518 -0.6266 0.0652 -0.0790 -0.2854 -0.2994 -0.2900 -0.2737 -0.0076 -0.3671 -0.2017 -0.6241 0.4521 -0.2675 0.3353 0.4107 0.0326 0.1400 -0.6447 -0.2043 0.2561 0.4187 0.0905 -0.1648 -0.2363 -0.5323 0.3180 0.0286 0.1252 0.7126 -0.7246 0.0468 0.3328 -0.1794 -0.3639 0.2257 -0.2623 0.2786 0.4922 -0.5353 0.4803 -0.2501 -0.2723 0.0199 -0.3194 -0.0371 The upper left block of the matrix U1 is 0.4144 0.2249 0.6015 -0.1964 -0.0198 0.5131 -0.2823 -0.3058 -0.6620 0.1508 0.2237 0.0240 -0.0743 -0.4323 -0.0332 -0.7263 The upper right block of the matrix U1 is -0.3474 0.1306 -0.3391 -0.3530 -0.3760 0.1550 0.6087 -0.1646 0.1707 0.6553 -0.1262 -0.1177 0.3048 -0.0773 0.0767 -0.4173 The upper left block of the matrix U2 is 0.1403 -0.6447 -0.6536 -0.3707 0.7069 0.2609 -0.0091 -0.1702 -0.1218 -0.1120 0.3766 -0.5154 0.0773 0.6349 -0.5070 -0.1810 The upper right block of the matrix U2 is 0.0000 0.0000 0.0000 0.0000 0.1182 0.1587 0.1930 -0.5716 0.6051 -0.2720 0.3364 0.1089 0.2823 -0.0386 -0.1529 0.4434 The vector ALPHAR is 0.0000 0.7122 0.0000 0.7450 The vector ALPHAI is 0.7540 0.0000 0.7465 0.0000 The vector BETA is 4.0000 4.0000 8.0000 16.0000
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