*> \brief \b SGGGLM
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE SGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
* $ X( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGGGLM solves a general Gauss-Markov linear model (GLM) problem:
*>
*> minimize || y ||_2 subject to d = A*x + B*y
*> x
*>
*> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
*> given N-vector. It is assumed that M <= N <= M+P, and
*>
*> rank(A) = M and rank( A B ) = N.
*>
*> Under these assumptions, the constrained equation is always
*> consistent, and there is a unique solution x and a minimal 2-norm
*> solution y, which is obtained using a generalized QR factorization
*> of the matrices (A, B) given by
*>
*> A = Q*(R), B = Q*T*Z.
*> (0)
*>
*> In particular, if matrix B is square nonsingular, then the problem
*> GLM is equivalent to the following weighted linear least squares
*> problem
*>
*> minimize || inv(B)*(d-A*x) ||_2
*> x
*>
*> where inv(B) denotes the inverse of B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix A. 0 <= M <= N.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of columns of the matrix B. P >= N-M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,M)
*> On entry, the N-by-M matrix A.
*> On exit, the upper triangular part of the array A contains
*> the M-by-M upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is REAL array, dimension (LDB,P)
*> On entry, the N-by-P matrix B.
*> On exit, if N <= P, the upper triangle of the subarray
*> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
*> if N > P, the elements on and above the (N-P)th subdiagonal
*> contain the N-by-P upper trapezoidal matrix T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> On entry, D is the left hand side of the GLM equation.
*> On exit, D is destroyed.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is REAL array, dimension (M)
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is REAL array, dimension (P)
*>
*> On exit, X and Y are the solutions of the GLM problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N+M+P).
*> For optimum performance, LWORK >= M+min(N,P)+max(N,P)*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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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 A in the
*> generalized QR factorization of the pair (A, B) is
*> singular, so that rank(A) < M; the least squares
*> solution could not be computed.
*> = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal
*> factor T associated with B in the generalized QR
*> factorization of the pair (A, B) is singular, so that
*> rank( A B ) < N; the least squares solution could not
*> be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup realOTHEReigen
*
* =====================================================================
SUBROUTINE SGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
$ INFO )
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
$ X( * ), Y( * )
* ..
*
* ===================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
$ NB4, NP
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGEMV, SGGQRF, SORMQR, SORMRQ, STRTRS,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NP = MIN( N, P )
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
INFO = -2
ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) 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', ' ', N, M, -1, -1 )
NB2 = ILAENV( 1, 'SGERQF', ' ', N, M, -1, -1 )
NB3 = ILAENV( 1, 'SORMQR', ' ', N, M, P, -1 )
NB4 = ILAENV( 1, 'SORMRQ', ' ', N, M, P, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKMIN = M + N + P
LWKOPT = M + NP + MAX( N, P )*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( 'SGGGLM', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Compute the GQR factorization of matrices A and B:
*
* Q**T*A = ( R11 ) M, Q**T*B*Z**T = ( T11 T12 ) M
* ( 0 ) N-M ( 0 T22 ) N-M
* M M+P-N N-M
*
* where R11 and T22 are upper triangular, and Q and Z are
* orthogonal.
*
CALL SGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
$ WORK( M+NP+1 ), LWORK-M-NP, INFO )
LOPT = WORK( M+NP+1 )
*
* Update left-hand-side vector d = Q**T*d = ( d1 ) M
* ( d2 ) N-M
*
CALL SORMQR( 'Left', 'Transpose', N, 1, M, A, LDA, WORK, D,
$ MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
*
* Solve T22*y2 = d2 for y2
*
IF( N.GT.M ) THEN
CALL STRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
$ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 1
RETURN
END IF
*
CALL SCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
END IF
*
* Set y1 = 0
*
DO 10 I = 1, M + P - N
Y( I ) = ZERO
10 CONTINUE
*
* Update d1 = d1 - T12*y2
*
CALL SGEMV( 'No transpose', M, N-M, -ONE, B( 1, M+P-N+1 ), LDB,
$ Y( M+P-N+1 ), 1, ONE, D, 1 )
*
* Solve triangular system: R11*x = d1
*
IF( M.GT.0 ) THEN
CALL STRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA,
$ D, M, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 2
RETURN
END IF
*
* Copy D to X
*
CALL SCOPY( M, D, 1, X, 1 )
END IF
*
* Backward transformation y = Z**T *y
*
CALL SORMRQ( 'Left', 'Transpose', P, 1, NP,
$ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
$ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
*
RETURN
*
* End of SGGGLM
*
END