*> \brief DGELSX solves overdetermined or underdetermined systems for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGELSX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, * WORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, M, N, NRHS, RANK * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * INTEGER JPVT( * ) * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> This routine is deprecated and has been replaced by routine DGELSY. *> *> DGELSX computes the minimum-norm solution to a real linear least *> squares problem: *> minimize || A * X - B || *> using a complete orthogonal factorization of A. A is an M-by-N *> matrix which may be rank-deficient. *> *> Several right hand side vectors b and solution vectors x can be *> handled in a single call; they are stored as the columns of the *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution *> matrix X. *> *> The routine first computes a QR factorization with column pivoting: *> A * P = Q * [ R11 R12 ] *> [ 0 R22 ] *> with R11 defined as the largest leading submatrix whose estimated *> condition number is less than 1/RCOND. The order of R11, RANK, *> is the effective rank of A. *> *> Then, R22 is considered to be negligible, and R12 is annihilated *> by orthogonal transformations from the right, arriving at the *> complete orthogonal factorization: *> A * P = Q * [ T11 0 ] * Z *> [ 0 0 ] *> The minimum-norm solution is then *> X = P * Z**T [ inv(T11)*Q1**T*B ] *> [ 0 ] *> where Q1 consists of the first RANK columns of Q. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of *> columns of matrices B and X. NRHS >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, A has been overwritten by details of its *> complete orthogonal factorization. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) *> On entry, the M-by-NRHS right hand side matrix B. *> On exit, the N-by-NRHS solution matrix X. *> If m >= n and RANK = n, the residual sum-of-squares for *> the solution in the i-th column is given by the sum of *> squares of elements N+1:M in that column. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,M,N). *> \endverbatim *> *> \param[in,out] JPVT *> \verbatim *> JPVT is INTEGER array, dimension (N) *> On entry, if JPVT(i) .ne. 0, the i-th column of A is an *> initial column, otherwise it is a free column. Before *> the QR factorization of A, all initial columns are *> permuted to the leading positions; only the remaining *> free columns are moved as a result of column pivoting *> during the factorization. *> On exit, if JPVT(i) = k, then the i-th column of A*P *> was the k-th column of A. *> \endverbatim *> *> \param[in] RCOND *> \verbatim *> RCOND is DOUBLE PRECISION *> RCOND is used to determine the effective rank of A, which *> is defined as the order of the largest leading triangular *> submatrix R11 in the QR factorization with pivoting of A, *> whose estimated condition number < 1/RCOND. *> \endverbatim *> *> \param[out] RANK *> \verbatim *> RANK is INTEGER *> The effective rank of A, i.e., the order of the submatrix *> R11. This is the same as the order of the submatrix T11 *> in the complete orthogonal factorization of A. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension *> (max( min(M,N)+3*N, 2*min(M,N)+NRHS )), *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup doubleGEsolve * * ===================================================================== SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, $ WORK, INFO ) * * -- LAPACK driver routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, M, N, NRHS, RANK DOUBLE PRECISION RCOND * .. * .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER IMAX, IMIN PARAMETER ( IMAX = 1, IMIN = 2 ) DOUBLE PRECISION ZERO, ONE, DONE, NTDONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, DONE = ZERO, $ NTDONE = ONE ) * .. * .. Local Scalars .. INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX, $ SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2 * .. * .. External Functions .. DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL DLAMCH, DLANGE * .. * .. External Subroutines .. EXTERNAL DGEQPF, DLAIC1, DLASCL, DLASET, DLATZM, DORM2R, $ DTRSM, DTZRQF, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. Executable Statements .. * MN = MIN( M, N ) ISMIN = MN + 1 ISMAX = 2*MN + 1 * * Test the input arguments. * INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN INFO = -7 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSX', -INFO ) RETURN END IF * * Quick return if possible * IF( MIN( M, N, NRHS ).EQ.0 ) THEN RANK = 0 RETURN END IF * * Get machine parameters * SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) * * Scale A, B if max elements outside range [SMLNUM,BIGNUM] * ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN * * Matrix all zero. Return zero solution. * CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) RANK = 0 GO TO 100 END IF * BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF * * Compute QR factorization with column pivoting of A: * A * P = Q * R * CALL DGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), INFO ) * * workspace 3*N. Details of Householder rotations stored * in WORK(1:MN). * * Determine RANK using incremental condition estimation * WORK( ISMIN ) = ONE WORK( ISMAX ) = ONE SMAX = ABS( A( 1, 1 ) ) SMIN = SMAX IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN RANK = 0 CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) GO TO 100 ELSE RANK = 1 END IF * 10 CONTINUE IF( RANK.LT.MN ) THEN I = RANK + 1 CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ), $ A( I, I ), SMINPR, S1, C1 ) CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ), $ A( I, I ), SMAXPR, S2, C2 ) * IF( SMAXPR*RCOND.LE.SMINPR ) THEN DO 20 I = 1, RANK WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 ) WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 ) 20 CONTINUE WORK( ISMIN+RANK ) = C1 WORK( ISMAX+RANK ) = C2 SMIN = SMINPR SMAX = SMAXPR RANK = RANK + 1 GO TO 10 END IF END IF * * Logically partition R = [ R11 R12 ] * [ 0 R22 ] * where R11 = R(1:RANK,1:RANK) * * [R11,R12] = [ T11, 0 ] * Y * IF( RANK.LT.N ) $ CALL DTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO ) * * Details of Householder rotations stored in WORK(MN+1:2*MN) * * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS) * CALL DORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ), $ B, LDB, WORK( 2*MN+1 ), INFO ) * * workspace NRHS * * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) * CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK, $ NRHS, ONE, A, LDA, B, LDB ) * DO 40 I = RANK + 1, N DO 30 J = 1, NRHS B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE * * B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS) * IF( RANK.LT.N ) THEN DO 50 I = 1, RANK CALL DLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA, $ WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB, $ WORK( 2*MN+1 ) ) 50 CONTINUE END IF * * workspace NRHS * * B(1:N,1:NRHS) := P * B(1:N,1:NRHS) * DO 90 J = 1, NRHS DO 60 I = 1, N WORK( 2*MN+I ) = NTDONE 60 CONTINUE DO 80 I = 1, N IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN IF( JPVT( I ).NE.I ) THEN K = I T1 = B( K, J ) T2 = B( JPVT( K ), J ) 70 CONTINUE B( JPVT( K ), J ) = T1 WORK( 2*MN+K ) = DONE T1 = T2 K = JPVT( K ) T2 = B( JPVT( K ), J ) IF( JPVT( K ).NE.I ) $ GO TO 70 B( I, J ) = T1 WORK( 2*MN+K ) = DONE END IF END IF 80 CONTINUE 90 CONTINUE * * Undo scaling * IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA, $ INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA, $ INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF * 100 CONTINUE * RETURN * * End of DGELSX * END