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SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
$ INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ),
$ WORK( * ), X( * )
* ..
*
* Purpose
* =======
*
* DGGLSE 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 generalized RQ factorization of the
* matrices (B, A) given by
*
* B = (0 R)*Q, A = Z*T*Q.
*
* 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) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the elements on and above the diagonal of the array
* contain the min(M,N)-by-N upper trapezoidal matrix T.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
* On entry, the P-by-N matrix B.
* On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
* contains the P-by-P upper triangular matrix R.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* C (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (P)
* On entry, D contains the right hand side vector for the
* constrained equation.
* On exit, D is destroyed.
*
* X (output) DOUBLE PRECISION array, dimension (N)
* On exit, X is the solution of the LSE problem.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,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
* DGEQRF, SGERQF, DORMQR and SORMRQ.
*
* 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.
* = 1: the upper triangular factor R associated with B in the
* generalized RQ factorization of the pair (B, A) is
* singular, so that rank(B) < P; the least squares
* solution could not be computed.
* = 2: the (N-P) by (N-P) part of the upper trapezoidal factor
* T associated with A in the generalized RQ factorization
* of the pair (B, A) is singular, so that
* rank( (A) ) < N; the least squares solution could not
* ( (B) )
* be computed.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
$ NB4, NR
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGGRQF, DORMQR, DORMRQ,
$ DTRMV, DTRTRS, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
MN = MIN( M, N )
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
END IF
*
* Calculate workspace
*
IF( INFO.EQ.0) THEN
IF( N.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, P, -1 )
NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKMIN = M + N + P
LWKOPT = P + MN + MAX( M, N )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGLSE', -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
* orthogonal.
*
CALL DGGRQF( 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 DORMQR( 'Left', '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
*
IF( P.GT.0 ) THEN
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', P, 1,
$ B( 1, N-P+1 ), LDB, D, P, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 1
RETURN
END IF
*
* Put the solution in X
*
CALL DCOPY( P, D, 1, X( N-P+1 ), 1 )
*
* Update c1
*
CALL DGEMV( 'No transpose', N-P, P, -ONE, A( 1, N-P+1 ), LDA,
$ D, 1, ONE, C, 1 )
END IF
*
* Solve R11*x1 = c1 for x1
*
IF( N.GT.P ) THEN
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N-P, 1,
$ A, LDA, C, N-P, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 2
RETURN
END IF
*
* Put the solutions in X
*
CALL DCOPY( N-P, C, 1, X, 1 )
END IF
*
* Compute the residual vector:
*
IF( M.LT.N ) THEN
NR = M + P - N
IF( NR.GT.0 )
$ CALL DGEMV( 'No transpose', NR, N-M, -ONE, A( N-P+1, M+1 ),
$ LDA, D( NR+1 ), 1, ONE, C( N-P+1 ), 1 )
ELSE
NR = P
END IF
IF( NR.GT.0 ) THEN
CALL DTRMV( 'Upper', 'No transpose', 'Non unit', NR,
$ A( N-P+1, N-P+1 ), LDA, D, 1 )
CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 )
END IF
*
* Backward transformation x = Q'*x
*
CALL DORMRQ( 'Left', '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 DGGLSE
*
END
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