sgglse()

subroutine sgglse	(	integer	m,
		integer	n,
		integer	p,
		real, dimension( lda, * )	a,
		integer	lda,
		real, dimension( ldb, * )	b,
		integer	ldb,
		real, dimension( * )	c,
		real, dimension( * )	d,
		real, dimension( * )	x,
		real, dimension( * )	work,
		integer	lwork,
		integer	info )

SGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Download SGGLSE + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> SGGLSE 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.
!>
!> Callers of this subroutine should note that the singularity/rank-deficiency checks
!> implemented in this subroutine are rudimentary. The STRTRS subroutine called by this
!> subroutine only signals a failure due to singularity if the problem is exactly singular.
!>
!> It is conceivable for one (or more) of the factors involved in the generalized RQ
!> factorization of the pair (B, A) to be subnormally close to singularity without this
!> subroutine signalling an error. The solutions computed for such almost-rank-deficient
!> problems may be less accurate due to a loss of numerical precision.
!> 
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !>
[in]	P	!> P is INTEGER !> The number of rows of the matrix B. 0 <= P <= N <= M+P. !>
[in,out]	A	!> A is REAL 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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[in,out]	B	!> B is REAL 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. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !>
[in,out]	C	!> C is REAL 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. !>
[in,out]	D	!> D is REAL array, dimension (P) !> On entry, D contains the right hand side vector for the !> constrained equation. !> On exit, D is destroyed. !>
[out]	X	!> X is REAL array, dimension (N) !> On exit, X is the solution of the LSE problem. !>
[out]	WORK	!> WORK is REAL array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is 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 !> SGEQRF, SGERQF, SORMQR 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. !>
[out]	INFO	!> INFO is 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 exactly !> 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 exactly singular, so that !> rank( (A) ) < N; the least squares solution could not !> ( (B) ) !> be computed. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 186 of file sgglse.f.

*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), C( * ), D( * ),
     $                   WORK( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      parameter( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
     $                   NB4, NR
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, scopy, sgemv, sggrqf, sormqr,
     $                   sormrq,
     $                   strmv, strtrs, xerbla
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      REAL               SROUNDUP_LWORK
      EXTERNAL           ilaenv, sroundup_lwork
*     ..
*     .. 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, 'SGEQRF', ' ', m, n, -1, -1 )
            nb2 = ilaenv( 1, 'SGERQF', ' ', m, n, -1, -1 )
            nb3 = ilaenv( 1, 'SORMQR', ' ', m, n, p, -1 )
            nb4 = ilaenv( 1, 'SORMRQ', ' ', 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 ) = sroundup_lwork(lwkopt)
*
         IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
            info = -12
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGGLSE', -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**T = (  0  T12 ) P   Z**T*A*Q**T = ( 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 sggrqf( p, m, n, b, ldb, work, a, lda, work( p+1 ),
     $             work( p+mn+1 ), lwork-p-mn, info )
      lopt = int( work( p+mn+1 ) )
*
*     Update c = Z**T *c = ( c1 ) N-P
*                          ( c2 ) M+P-N
*
      CALL sormqr( '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 strtrs( '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 scopy( p, d, 1, x( n-p+1 ), 1 )
*
*        Update c1
*
         CALL sgemv( '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 strtrs( '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 scopy( 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 sgemv( '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 strmv( 'Upper', 'No transpose', 'Non unit', nr,
     $               a( n-p+1, n-p+1 ), lda, d, 1 )
         CALL saxpy( nr, -one, d, 1, c( n-p+1 ), 1 )
      END IF
*
*     Backward transformation x = Q**T*x
*
      CALL sormrq( 'Left', 'Transpose', n, 1, p, b, ldb, work( 1 ),
     $             x,
     $             n, work( p+mn+1 ), lwork-p-mn, info )
      work( 1 ) = real( p + mn + max( lopt, int( work( p+mn+1 ) ) ) )
*
      RETURN
*
*     End of SGGLSE
*

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