Purpose
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
H T ( B F ) ( Z11 Z12 )
S = J Z J Z and H = ( H ), Z =: ( ). (1)
( G -B ) ( Z21 Z22 )
The structured Schur form of the embedded real skew-Hamiltonian/
H T
skew-Hamiltonian pencil, aB_S - bB_T, with B_S = J B_Z J B_Z,
( Re(Z11) -Im(Z11) | Re(Z12) -Im(Z12) )
( | )
( Im(Z11) Re(Z11) | Im(Z12) Re(Z12) )
( | )
B_Z = (---------------------+---------------------) ,
( | )
( Re(Z21) -Im(Z21) | Re(Z22) -Im(Z22) )
( | )
( Im(Z21) Re(Z21) | Im(Z22) Re(Z22) )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H,
( | T T )
( -Im(G) -Re(G) | -Im(B ) Re(B ) )
( | T T )
( Re(G) -Im(G) | -Re(B ) -Im(B ) )
is determined and used to compute the eigenvalues. Optionally,
if JOB = 'T', the pencil aB_S - bB_H is transformed by a unitary
matrix Q and a unitary symplectic matrix U to the structured Schur
H T
form aB_Sout - bB_Hout, with B_Sout = J B_Zout J B_Zout,
( BA BD ) ( BB BF )
B_Zout = ( ) and B_Hout = ( H ), (3)
( 0 BC ) ( 0 -BB )
where BA and BB are upper triangular, BC is lower triangular,
and BF is Hermitian. The embedding doubles the multiplicities of
the eigenvalues of the pencil aS - bH.
Optionally, if COMPQ = 'C', the unitary matrix Q is computed.
Optionally, if COMPU = 'C', the unitary symplectic matrix U is
computed.
Specification
SUBROUTINE MB04AZ( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG,
$ LDFG, D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR,
$ ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK,
$ ZWORK, LZWORK, BWORK, INFO )C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU, JOB
INTEGER INFO, LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK,
$ LDZ, LIWORK, LZWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 B( LDB, * ), C( LDC, * ), D( LDD, * ),
$ FG( LDFG, * ), Q( LDQ, * ), U( LDU, * ),
$ Z( LDZ, * ), ZWORK( * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (3).
= 'T': put S and H into the forms in (3) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the unitary transformation
matrix Q, as follows:
= 'N': do not compute the unitary matrix Q;
= 'C': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned.
COMPU CHARACTER*1
Specifies whether to compute the unitary symplectic
transformation matrix U, as follows:
= 'N': do not compute the unitary symplectic matrix U;
= 'C': the array U is initialized internally to the unit
matrix, and the unitary symplectic matrix U is
returned.
Input/Output Parameters
N (input) INTEGER
Order of the pencil aS - bH. N >= 0, even.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the non-trivial factor Z in the factorization
H T
S = J Z J Z of the skew-Hamiltonian matrix S.
On exit, if JOB = 'T', 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. The submatrix in the rows N/2+1 to N and the
first N/2 columns is unchanged, except possibly for the
entry (N/2+1,N/2), which might be set to zero.
If JOB = 'E', this array is unchanged on exit.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1, N).
B (input/output) COMPLEX*16 array, dimension (LDB, K), where
K = N, if JOB = 'T', and K = M, if JOB = 'E'.
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-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.
If JOB = 'E', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B.
LDB >= MAX(1, M), if JOB = 'E';
LDB >= MAX(1, N), if JOB = 'T'.
FG (input/output) COMPLEX*16 array, dimension (LDFG, P),
where P = MAX(M+1,N), if JOB = 'T', and
P = M+1, if JOB = 'E'.
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 JOB = 'T', 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 JOB = 'E', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, M), if JOB = 'E';
LDFG >= MAX(1, N), if JOB = 'T'.
D (output) COMPLEX*16 array, dimension (LDD, N)
If JOB = 'T', the leading N-by-N part of this array
contains the matrix BD in (3) (see also METHOD).
If JOB = 'E', this array is not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= 1, if JOB = 'E';
LDD >= MAX(1, N), if JOB = 'T'.
C (output) COMPLEX*16 array, dimension (LDC, N)
If JOB = 'T', the leading N-by-N part of this array
contains the lower triangular matrix BC in (3) (see also
METHOD).
If JOB = 'E', this array is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= 1, if JOB = 'E';
LDC >= MAX(1, N), if JOB = 'T'.
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C' and JOB = 'T', then the leading
2*N-by-2*N part of this array contains the unitary
transformation matrix Q.
If COMPQ = 'C' and JOB = 'E', this array contains the
orthogonal transformation which reduced B_Z and B_T
in the first step of the algorithm (see METHOD).
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'.
U (output) COMPLEX*16 array, dimension (LDU, 2*N)
On exit, if COMPU = 'C' and JOB = 'T', then the leading
N-by-2*N part of this array contains the leading N-by-2*N
part of the unitary symplectic transformation matrix U.
If COMPU = 'C' and JOB = 'E', this array contains the
first N rows of the transformation U which reduced B_Z
and B_T in the first step of the algorithm (see METHOD).
If COMPU = 'N', this array is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if COMPU = 'N';
LDU >= MAX(1, N), if COMPU = '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 (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= 2*N+9.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -25, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= c*N**2 + N + MAX(2*N, 24) + 3, where
c = 18, if COMPU = 'C';
c = 16, if COMPQ = 'C' and COMPU = 'N';
c = 13, if COMPQ = 'N' and COMPU = 'N'.
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 = -27, ZWORK(1) returns the minimum
value of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 8*N + 28, if JOB = 'T' and COMPQ = 'C';
LZWORK >= 6*N + 28, if JOB = 'T' and COMPQ = 'N';
LZWORK >= 1, if JOB = 'E'.
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 JOB = 'E';
LBWORK >= N, if JOB = 'T'.
Error Indicator
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the algorithm was not able to reveal information
about the eigenvalues from the 2-by-2 blocks in the
SLICOT Library routine MB03BD (called by MB04ED);
= 2: periodic QZ iteration failed in the SLICOT Library
routines MB03BD or MB03BZ when trying to
triangularize the 2-by-2 blocks.
Method
First T = i*H is set. Then, the embeddings, B_Z and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04ED is applied to compute the structured Schur
form, i.e., the factorizations
~ T ( BZ11 BZ12 )
B_Z = U B_Z Q = ( ) and
( 0 BZ22 )
~ T T ( T11 T12 )
B_T = J Q J B_T Q = ( ),
( 0 T11' )
where Q is real orthogonal, U is real orthogonal symplectic, BZ11,
BZ22' are upper triangular and T11 is upper quasi-triangular.
The notation M' denotes the transpose of the matrix M.
If JOB = 'T', the 2-by-2 blocks are triangularized using the
periodic QZ algorithm.
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
* MB04AZ 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 LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK, LDZ,
$ LIWORK, LZWORK
PARAMETER ( LDB = NMAX, LDC = NMAX, LDD = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX, LDU = NMAX,
$ LDWORK = 18*NMAX*NMAX + NMAX + MAX( 2*NMAX,
$ 24 ) + 3,
$ LDZ = NMAX, LIWORK = 2*NMAX + 9,
$ LZWORK = 8*NMAX + 28 )
*
* .. Local Scalars ..
CHARACTER COMPQ, COMPU, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
COMPLEX*16 B( LDB, NMAX ), C( LDC, NMAX ),
$ D( LDD, NMAX ), FG( LDFG, NMAX ),
$ Q( LDQ, 2*NMAX ), U( LDU, 2*NMAX ),
$ Z( LDZ, NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ),
$ BETA( NMAX ), DWORK(LDWORK )
INTEGER IWORK( LIWORK )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04AZ
*
* .. Intrinsic Functions ..
INTRINSIC MAX, MOD
*
* .. 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, COMPQ, COMPU, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I=1, N )
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 )
* Compute the eigenvalues of a complex skew-Hamiltonian/
* Hamiltonian pencil (factored version).
CALL MB04AZ( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG, LDFG,
$ D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR, ALPHAI,
$ BETA, IWORK, LIWORK, DWORK, LDWORK, ZWORK, LZWORK,
$ BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
M = N/2
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( D( I, J ), J = 1, N )
40 CONTINUE
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( C( I, J ), J = 1, N )
50 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99990 )
DO 60 I = 1, 2*N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, 2*N )
60 CONTINUE
END IF
IF( LSAME( COMPU, 'C' ) ) THEN
WRITE( NOUT, FMT = 99989 )
DO 70 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, 2*N )
70 CONTINUE
END IF
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99987 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99986 )
WRITE( NOUT, FMT = 99987 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99985 )
WRITE( NOUT, FMT = 99987 ) ( BETA( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT ( 'MB04AZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04AZ = ', I2 )
99996 FORMAT (/' The transformed matrix Z is' )
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' The transformed matrix B is' )
99993 FORMAT (/' The transformed matrix FG is' )
99992 FORMAT (/' The matrix D is' )
99991 FORMAT (/' The matrix C is' )
99990 FORMAT (/' The matrix Q is' )
99989 FORMAT (/' The upper part of the matrix U is' )
99988 FORMAT (/' The vector ALPHAR is ' )
99987 FORMAT ( 50( 1X, F8.4 ) )
99986 FORMAT (/' The vector ALPHAI is ' )
99985 FORMAT (/' The vector BETA is ' )
END
Program Data
MB04AZ EXAMPLE PROGRAM DATA
T C C 4
(0.4941,0.8054) (0.8909,0.8865) (0.0305,0.9786) (0.9047,0.0596)
(0.7790,0.5767) (0.3341,0.0286) (0.7440,0.7126) (0.6098,0.6819)
(0.7150,0.1829) (0.6987,0.4899) (0.5000,0.5004) (0.6176,0.0424)
(0.9037,0.2399) (0.1978,0.1679) (0.4799,0.4710) (0.8594,0.0714)
(0.5216,0.7224) (0.8181,0.6596)
(0.0967,0.1498) (0.8175,0.5185)
0.9729 0.8003 (0.4323,0.8313)
(0.6489,0.1331) 0.4537 0.8253
Program Results
MB04AZ EXAMPLE PROGRAM RESULTS
The transformed matrix Z is
0.4545 +0.0000i 0.7904 +0.0000i -0.1601 +0.0000i -0.2691 +0.0000i
0.7790 +0.5767i 0.4273 +0.0000i 0.1459 +0.0000i 0.1298 +0.0000i
0.7150 +0.1829i 0.6987 +0.4899i 0.6715 +0.0000i -0.3001 +0.0000i
0.9037 +0.2399i 0.1978 +0.1679i 0.4799 +0.4710i 0.7924 +0.0000i
The transformed matrix B is
0.0000 -1.7219i 0.0000 +0.7762i 0.0000 +0.5342i 0.0000 +0.0845i
0.0000 +0.0000i 0.0000 +0.8862i 0.0000 +0.5186i 0.0000 -0.1429i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +1.1122i 0.0000 +0.2898i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.4889i
The transformed matrix FG is
0.0000 +0.0000i 0.0000 +0.4145i 0.0000 -0.7921i 0.0000 +0.5630i
0.6489 +0.1331i 0.0000 +0.0000i 0.0000 +1.5982i 0.0000 +0.5818i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.5819i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
The matrix D is
2.2139 +0.0000i 0.0402 +0.0000i -0.2787 +0.0000i 1.0465 +0.0000i
-0.5021 +0.0000i 0.5502 +0.0000i -0.2771 +0.0000i -0.4521 +0.0000i
-0.0398 +0.0000i 0.4046 +0.0000i 0.0149 +0.0000i 0.7577 +0.0000i
-0.1550 +0.0000i 2.0660 +0.0000i 1.6075 +0.0000i 0.5836 +0.0000i
The matrix C is
0.3159 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.7819 +0.0000i -0.7575 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
-0.3494 +0.0000i 0.8622 +0.0000i 0.7539 +0.0000i 0.0000 +0.0000i
1.1178 +0.0000i 0.3133 +0.0000i 0.4638 +0.0000i 1.2348 +0.0000i
The matrix Q is
-0.4983 +0.0000i 0.3694 +0.0000i -0.4754 +0.0000i 0.2791 +0.0000i 0.1950 +0.0000i 0.2416 +0.0000i 0.1869 +0.0000i 0.4242 +0.0000i
-0.1045 +0.0000i 0.3309 +0.0000i 0.5730 +0.0000i -0.5566 +0.0000i 0.0877 +0.0000i 0.3543 +0.0000i 0.1761 +0.0000i 0.2779 +0.0000i
-0.2586 +0.0000i -0.4457 +0.0000i -0.2838 +0.0000i -0.5436 +0.0000i 0.5524 +0.0000i -0.1064 +0.0000i -0.2040 +0.0000i 0.0205 +0.0000i
-0.0040 +0.0000i 0.3845 +0.0000i 0.2469 +0.0000i 0.2965 +0.0000i 0.5799 +0.0000i 0.0840 +0.0000i -0.4788 +0.0000i -0.3614 +0.0000i
0.7958 +0.0000i 0.1597 +0.0000i -0.3420 +0.0000i -0.1047 +0.0000i 0.3370 +0.0000i 0.1813 +0.0000i 0.2275 +0.0000i 0.1229 +0.0000i
-0.0600 +0.0000i -0.4599 +0.0000i 0.3487 +0.0000i 0.3910 +0.0000i 0.4005 +0.0000i 0.1010 +0.0000i 0.5819 +0.0000i -0.0345 +0.0000i
-0.0539 +0.0000i 0.3817 +0.0000i 0.0501 +0.0000i -0.1114 +0.0000i 0.1751 +0.0000i -0.8212 +0.0000i 0.3606 +0.0000i -0.0380 +0.0000i
0.1846 +0.0000i -0.1577 +0.0000i 0.2510 +0.0000i 0.2293 +0.0000i 0.0911 +0.0000i -0.2834 +0.0000i -0.3779 +0.0000i 0.7708 +0.0000i
The upper part of the matrix U is
-0.2544 +0.0000i -0.1844 +0.0000i 0.7632 +0.0000i 0.5646 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.4272 +0.0000i -0.0454 +0.0000i -0.2918 +0.0000i 0.5721 +0.0000i 0.1446 +0.0000i -0.1611 +0.0000i 0.3609 +0.0000i -0.4752 +0.0000i
0.6936 +0.0000i 0.3812 +0.0000i 0.2606 +0.0000i 0.0848 +0.0000i -0.3359 +0.0000i -0.1592 +0.0000i -0.3242 +0.0000i 0.2348 +0.0000i
0.2320 +0.0000i -0.4483 +0.0000i -0.1629 +0.0000i 0.1784 +0.0000i -0.2898 +0.0000i 0.7525 +0.0000i -0.0511 +0.0000i 0.1841 +0.0000i
The vector ALPHAR is
0.0000 0.0000 0.0000 0.0000
The vector ALPHAI is
-1.4991 -1.3690 1.0985 0.9993
The vector BETA is
0.1250 0.5000 0.5000 2.0000