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SUBROUTINE ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
$ INFO )
*
* -- LAPACK driver routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ),
$ WORK( * ), X( * )
* ..
*
* Purpose
* =======
*
* ZGGLSE solves the linear equality-constrained least squares (LSE)
* problem:
*
* minimize || c - A*x ||_2 subject to B*x = d
*
* where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
* M-vector, and d is a given P-vector. It is assumed that
* P <= N <= M+P, and
*
* rank(B) = P and rank( ( A ) ) = N.
* ( ( B ) )
*
* These conditions ensure that the LSE problem has a unique solution,
* which is obtained using a GRQ factorization of the matrices B and A.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. 0 <= P <= N <= M+P.
*
* A (input/output) COMPLEX*16 array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A is destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX*16 array, dimension (LDB,N)
* On entry, the P-by-N matrix B.
* On exit, B is destroyed.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* C (input/output) COMPLEX*16 array, dimension (M)
* On entry, C contains the right hand side vector for the
* least squares part of the LSE problem.
* On exit, the residual sum of squares for the solution
* is given by the sum of squares of elements N-P+1 to M of
* vector C.
*
* D (input/output) COMPLEX*16 array, dimension (P)
* On entry, D contains the right hand side vector for the
* constrained equation.
* On exit, D is destroyed.
*
* X (output) COMPLEX*16 array, dimension (N)
* On exit, X is the solution of the LSE problem.
*
* WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,M+N+P).
* For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
* where NB is an upper bound for the optimal blocksizes for
* ZGEQRF, CGERQF, ZUNMQR and CUNMRQ.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKOPT, MN, NB, NB1, NB2, NB3, NB4, NR
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGGRQF, ZTRMV,
$ ZTRSV, ZUNMQR, ZUNMRQ
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
MN = MIN( M, N )
NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, P, -1 )
NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKOPT = P + MN + MAX( M, N )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -7
ELSE IF( LWORK.LT.MAX( 1, M+N+P ) .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGGLSE', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Compute the GRQ factorization of matrices B and A:
*
* B*Q' = ( 0 T12 ) P Z'*A*Q' = ( R11 R12 ) N-P
* N-P P ( 0 R22 ) M+P-N
* N-P P
*
* where T12 and R11 are upper triangular, and Q and Z are
* unitary.
*
CALL ZGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ),
$ WORK( P+MN+1 ), LWORK-P-MN, INFO )
LOPT = WORK( P+MN+1 )
*
* Update c = Z'*c = ( c1 ) N-P
* ( c2 ) M+P-N
*
CALL ZUNMQR( 'Left', 'Conjugate Transpose', M, 1, MN, A, LDA,
$ WORK( P+1 ), C, MAX( 1, M ), WORK( P+MN+1 ),
$ LWORK-P-MN, INFO )
LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
* Solve T12*x2 = d for x2
*
CALL ZTRSV( 'Upper', 'No transpose', 'Non unit', P, B( 1, N-P+1 ),
$ LDB, D, 1 )
*
* Update c1
*
CALL ZGEMV( 'No transpose', N-P, P, -CONE, A( 1, N-P+1 ), LDA, D,
$ 1, CONE, C, 1 )
*
* Sovle R11*x1 = c1 for x1
*
CALL ZTRSV( 'Upper', 'No transpose', 'Non unit', N-P, A, LDA, C,
$ 1 )
*
* Put the solutions in X
*
CALL ZCOPY( N-P, C, 1, X, 1 )
CALL ZCOPY( P, D, 1, X( N-P+1 ), 1 )
*
* Compute the residual vector:
*
IF( M.LT.N ) THEN
NR = M + P - N
CALL ZGEMV( 'No transpose', NR, N-M, -CONE, A( N-P+1, M+1 ),
$ LDA, D( NR+1 ), 1, CONE, C( N-P+1 ), 1 )
ELSE
NR = P
END IF
CALL ZTRMV( 'Upper', 'No transpose', 'Non unit', NR,
$ A( N-P+1, N-P+1 ), LDA, D, 1 )
CALL ZAXPY( NR, -CONE, D, 1, C( N-P+1 ), 1 )
*
* Backward transformation x = Q'*x
*
CALL ZUNMRQ( 'Left', 'Conjugate Transpose', N, 1, P, B, LDB,
$ WORK( 1 ), X, N, WORK( P+MN+1 ), LWORK-P-MN, INFO )
WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
RETURN
*
* End of ZGGLSE
*
END
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