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SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
$ WORK, 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, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine DGELSY.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' [ inv(T11)*Q1'*B ]
* [ 0 ]
* where Q1 consists of the first RANK columns of Q.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of
* columns of matrices B and X. NRHS >= 0.
*
* A (input/output) 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) 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.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,M,N).
*
* JPVT (input/output) 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.
*
* RCOND (input) 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.
*
* RANK (output) 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.
*
* WORK (workspace) DOUBLE PRECISION array, dimension
* (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. 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' * 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' * 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
|
