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SUBROUTINE MB04WU( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2,
$ CS, TAU, DWORK, LDWORK, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To generate a matrix Q with orthogonal columns (spanning an
C isotropic subspace), which is defined as the first n columns
C of a product of symplectic reflectors and Givens rotators,
C
C Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
C diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
C ....
C diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
C
C The matrix Q is returned in terms of its first 2*M rows
C
C [ op( Q1 ) op( Q2 ) ]
C Q = [ ].
C [ -op( Q2 ) op( Q1 ) ]
C
C ARGUMENTS
C
C Mode Parameters
C
C TRANQ1 CHARACTER*1
C Specifies the form of op( Q1 ) as follows:
C = 'N': op( Q1 ) = Q1;
C = 'T': op( Q1 ) = Q1';
C = 'C': op( Q1 ) = Q1'.
C
C TRANQ2 CHARACTER*1
C Specifies the form of op( Q2 ) as follows:
C = 'N': op( Q2 ) = Q2;
C = 'T': op( Q2 ) = Q2';
C = 'C': op( Q2 ) = Q2'.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of rows of the matrices Q1 and Q2. M >= 0.
C
C N (input) INTEGER
C The number of columns of the matrices Q1 and Q2.
C M >= N >= 0.
C
C K (input) INTEGER
C The number of symplectic Givens rotators whose product
C partly defines the matrix Q. N >= K >= 0.
C
C Q1 (input/output) DOUBLE PRECISION array, dimension
C (LDQ1,N) if TRANQ1 = 'N',
C (LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C'
C On entry with TRANQ1 = 'N', the leading M-by-K part of
C this array must contain in its i-th column the vector
C which defines the elementary reflector F(i).
C On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
C K-by-M part of this array must contain in its i-th row
C the vector which defines the elementary reflector F(i).
C On exit with TRANQ1 = 'N', the leading M-by-N part of this
C array contains the matrix Q1.
C On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
C N-by-M part of this array contains the matrix Q1'.
C
C LDQ1 INTEGER
C The leading dimension of the array Q1.
C LDQ1 >= MAX(1,M), if TRANQ1 = 'N';
C LDQ1 >= MAX(1,N), if TRANQ1 = 'T' or TRANQ1 = 'C'.
C
C Q2 (input/output) DOUBLE PRECISION array, dimension
C (LDQ2,N) if TRANQ2 = 'N',
C (LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C'
C On entry with TRANQ2 = 'N', the leading M-by-K part of
C this array must contain in its i-th column the vector
C which defines the elementary reflector H(i) and, on the
C diagonal, the scalar factor of H(i).
C On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
C K-by-M part of this array must contain in its i-th row the
C vector which defines the elementary reflector H(i) and, on
C the diagonal, the scalar factor of H(i).
C On exit with TRANQ2 = 'N', the leading M-by-N part of this
C array contains the matrix Q2.
C On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
C N-by-M part of this array contains the matrix Q2'.
C
C LDQ2 INTEGER
C The leading dimension of the array Q2.
C LDQ2 >= MAX(1,M), if TRANQ2 = 'N';
C LDQ2 >= MAX(1,N), if TRANQ2 = 'T' or TRANQ2 = 'C'.
C
C CS (input) DOUBLE PRECISION array, dimension (2*K)
C On entry, the first 2*K elements of this array must
C contain the cosines and sines of the symplectic Givens
C rotators G(i).
C
C TAU (input) DOUBLE PRECISION array, dimension (K)
C On entry, the first K elements of this array must
C contain the scalar factors of the elementary reflectors
C F(i).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal
C value of LDWORK.
C On exit, if INFO = -13, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK. LDWORK >= MAX(1,M+N).
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C REFERENCES
C
C [1] Bunse-Gerstner, A.
C Matrix factorizations for symplectic QR-like methods.
C Linear Algebra Appl., 83, pp. 49-77, 1986.
C
C CONTRIBUTORS
C
C D. Kressner, Technical Univ. Berlin, Germany, and
C P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
C
C REVISIONS
C
C V. Sima, June 2008 (SLICOT version of the HAPACK routine DOSGSQ).
C
C KEYWORDS
C
C Elementary matrix operations, orthogonal symplectic matrix.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
C .. Scalar Arguments ..
CHARACTER TRANQ1, TRANQ2
INTEGER INFO, K, LDQ1, LDQ2, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)
C .. Local Scalars ..
LOGICAL LTRQ1, LTRQ2
INTEGER I, J
DOUBLE PRECISION NU
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DLARF, DLASET, DROT, DSCAL, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, MAX
C
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
INFO = 0
LTRQ1 = LSAME( TRANQ1,'T' ) .OR. LSAME( TRANQ1,'C' )
LTRQ2 = LSAME( TRANQ2,'T' ) .OR. LSAME( TRANQ2,'C' )
C
C Check the scalar input parameters.
C
IF ( .NOT.( LTRQ1 .OR. LSAME( TRANQ1, 'N' ) ) ) THEN
INFO = -1
ELSE IF ( .NOT.( LTRQ2 .OR. LSAME( TRANQ2, 'N' ) ) ) THEN
INFO = -2
ELSE IF ( M.LT.0 ) THEN
INFO = -3
ELSE IF ( N.LT.0 .OR. N.GT.M ) THEN
INFO = -4
ELSE IF ( K.LT.0 .OR. K.GT.N ) THEN
INFO = -5
ELSE IF ( ( LTRQ1 .AND. LDQ1.LT.MAX( 1, N ) ) .OR.
$ ( .NOT.LTRQ1 .AND. LDQ1.LT.MAX( 1, M ) ) ) THEN
INFO = -7
ELSE IF ( ( LTRQ2 .AND. LDQ2.LT.MAX( 1, N ) ) .OR.
$ ( .NOT.LTRQ2 .AND. LDQ2.LT.MAX( 1, M ) ) ) THEN
INFO = -9
ELSE IF ( LDWORK.LT.MAX( 1,M + N ) ) THEN
DWORK(1) = DBLE( MAX( 1,M + N ) )
INFO = -13
END IF
C
C Return if there were illegal values.
C
IF ( INFO.NE.0 ) THEN
CALL XERBLA( 'MB04WU', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 ) THEN
DWORK(1) = ONE
RETURN
END IF
C
C Initialize columns K+1:N to columns of the unit matrix.
C
DO 20 J = K + 1, N
DO 10 I = 1, M
Q1(I,J) = ZERO
10 CONTINUE
Q1(J,J) = ONE
20 CONTINUE
CALL DLASET( 'All', M, N-K, ZERO, ZERO, Q2(1,K+1), LDQ2 )
C
IF ( LTRQ1.AND.LTRQ2 ) THEN
DO 50 I = K, 1, -1
C
C Apply F(I) to Q1(I+1:N,I:M) and Q2(I+1:N,I:M) from the
C right.
C
CALL DCOPY( M-I+1, Q2(I,I), LDQ2, DWORK, 1 )
IF ( I.LT.N ) THEN
Q1(I,I) = ONE
CALL DLARF( 'Right', N-I, M-I+1, Q1(I,I), LDQ1, TAU(I),
$ Q1(I+1,I), LDQ1, DWORK(M+1) )
CALL DLARF( 'Right', N-I, M-I+1, Q1(I,I), LDQ1, TAU(I),
$ Q2(I+1,I), LDQ2, DWORK(M+1) )
END IF
IF ( I.LT.M )
$ CALL DSCAL( M-I, -TAU(I), Q1(I,I+1), LDQ1 )
Q1(I,I) = ONE - TAU(I)
C
C Set Q1(I,1:I-1) and Q2(I,1:M) to zero.
C
DO 30 J = 1, I - 1
Q1(I,J) = ZERO
30 CONTINUE
DO 40 J = 1, M
Q2(I,J) = ZERO
40 CONTINUE
C
C Apply G(I) to Q1(I:N,I) and Q2(I:N,I) from the right.
C
CALL DROT( N-I+1, Q1(I,I), 1, Q2(I,I), 1, CS(2*I-1),
$ CS(2*I) )
C
C Apply H(I) to Q1(I:N,I:M) and Q2(I:N,I:M) from the right.
C
NU = DWORK(1)
DWORK(1) = ONE
CALL DLARF( 'Right', N-I+1, M-I+1, DWORK, 1, NU, Q1(I,I),
$ LDQ1, DWORK(M+1) )
CALL DLARF( 'Right', N-I+1, M-I+1, DWORK, 1, NU, Q2(I,I),
$ LDQ2, DWORK(M+1) )
50 CONTINUE
ELSE IF ( LTRQ1 ) THEN
DO 80 I = K, 1, -1
C
C Apply F(I) to Q1(I+1:N,I:M) from the right and to
C Q2(I:M,I+1:N) from the left.
C
CALL DCOPY( M-I+1, Q2(I,I), 1, DWORK, 1 )
IF ( I.LT.N ) THEN
Q1(I,I) = ONE
CALL DLARF( 'Right', N-I, M-I+1, Q1(I,I), LDQ1, TAU(I),
$ Q1(I+1,I), LDQ1, DWORK(M+1) )
CALL DLARF( 'Left', M-I+1, N-I, Q1(I,I), LDQ1, TAU(I),
$ Q2(I,I+1), LDQ2, DWORK(M+1) )
END IF
IF ( I.LT.M )
$ CALL DSCAL( M-I, -TAU(I), Q1(I,I+1), LDQ1 )
Q1(I,I) = ONE - TAU(I)
C
C Set Q1(I,1:I-1) and Q2(1:M,I) to zero.
C
DO 60 J = 1, I - 1
Q1(I,J) = ZERO
60 CONTINUE
DO 70 J = 1, M
Q2(J,I) = ZERO
70 CONTINUE
C
C Apply G(I) to Q1(I:N,I) from the right and to Q2(I,I:N)
C from the left.
C
CALL DROT( N-I+1, Q1(I,I), 1, Q2(I,I), LDQ2, CS(2*I-1),
$ CS(2*I) )
C
C Apply H(I) to Q1(I:N,I:M) from the right and to Q2(I:M,I:N)
C from the left.
C
NU = DWORK(1)
DWORK(1) = ONE
CALL DLARF( 'Right', N-I+1, M-I+1, DWORK, 1, NU, Q1(I,I),
$ LDQ1, DWORK(M+1) )
CALL DLARF( 'Left', M-I+1, N-I+1, DWORK, 1, NU, Q2(I,I),
$ LDQ2, DWORK(M+1) )
80 CONTINUE
ELSE IF ( LTRQ2 ) THEN
DO 110 I = K, 1, -1
C
C Apply F(I) to Q1(I:M,I+1:N) from the left and to
C Q2(I+1:N,I:M) from the right.
C
CALL DCOPY( M-I+1, Q2(I,I), LDQ2, DWORK, 1 )
IF ( I.LT.N ) THEN
Q1(I,I) = ONE
CALL DLARF( 'Left', M-I+1, N-I, Q1(I,I), 1, TAU(I),
$ Q1(I,I+1), LDQ1, DWORK(M+1) )
CALL DLARF( 'Right', N-I, M-I+1, Q1(I,I), 1, TAU(I),
$ Q2(I+1,I), LDQ2, DWORK(M+1) )
END IF
IF ( I.LT.M )
$ CALL DSCAL( M-I, -TAU(I), Q1(I+1,I), 1 )
Q1(I,I) = ONE - TAU(I)
C
C Set Q1(1:I-1,I) and Q2(I,1:M) to zero.
C
DO 90 J = 1, I - 1
Q1(J,I) = ZERO
90 CONTINUE
DO 100 J = 1, M
Q2(I,J) = ZERO
100 CONTINUE
C
C Apply G(I) to Q1(I,I:N) from the left and to Q2(I:N,I)
C from the right.
C
CALL DROT( N-I+1, Q1(I,I), LDQ1, Q2(I,I), 1, CS(2*I-1),
$ CS(2*I) )
C
C Apply H(I) to Q1(I:M,I:N) from the left and to Q2(I:N,I:M)
C from the left.
C
NU = DWORK(1)
DWORK(1) = ONE
CALL DLARF( 'Left', M-I+1, N-I+1, DWORK, 1, NU, Q1(I,I),
$ LDQ1, DWORK(M+1) )
CALL DLARF( 'Right', N-I+1, M-I+1, DWORK, 1, NU, Q2(I,I),
$ LDQ2, DWORK(M+1) )
110 CONTINUE
ELSE
DO 140 I = K, 1, -1
C
C Apply F(I) to Q1(I:M,I+1:N) and Q2(I:M,I+1:N) from the left.
C
CALL DCOPY( M-I+1, Q2(I,I), 1, DWORK, 1 )
IF ( I.LT.N ) THEN
Q1(I,I) = ONE
CALL DLARF( 'Left', M-I+1, N-I, Q1(I,I), 1, TAU(I),
$ Q1(I,I+1), LDQ1, DWORK(M+1) )
CALL DLARF( 'Left', M-I+1, N-I, Q1(I,I), 1, TAU(I),
$ Q2(I,I+1), LDQ2, DWORK(M+1) )
END IF
IF ( I.LT.M )
$ CALL DSCAL( M-I, -TAU(I), Q1(I+1,I), 1 )
Q1(I,I) = ONE - TAU(I)
C
C Set Q1(1:I-1,I) and Q2(1:M,I) to zero.
C
DO 120 J = 1, I - 1
Q1(J,I) = ZERO
120 CONTINUE
DO 130 J = 1, M
Q2(J,I) = ZERO
130 CONTINUE
C
C Apply G(I) to Q1(I,I:N) and Q2(I,I:N) from the left.
C
CALL DROT( N-I+1, Q1(I,I), LDQ1, Q2(I,I), LDQ2, CS(2*I-1),
$ CS(2*I) )
C
C Apply H(I) to Q1(I:M,I:N) and Q2(I:M,I:N) from the left.
C
NU = DWORK(1)
DWORK(1) = ONE
CALL DLARF( 'Left', M-I+1, N-I+1, DWORK, 1, NU, Q1(I,I),
$ LDQ1, DWORK(M+1) )
CALL DLARF( 'Left', M-I+1, N-I+1, DWORK, 1, NU, Q2(I,I),
$ LDQ2, DWORK(M+1) )
140 CONTINUE
END IF
DWORK(1) = DBLE( MAX( 1, M+N ) )
C *** Last line of MB04WU ***
END
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