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SUBROUTINE IB01ND( METH, JOBD, NOBR, M, L, R, LDR, SV, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C
C RELEASE 4.0, WGS COPYRIGHT 2000.
C
C PURPOSE
C
C To find the singular value decomposition (SVD) giving the system
C order, using the triangular factor of the concatenated block
C Hankel matrices. Related preliminary calculations needed for
C computing the system matrices are also performed.
C
C ARGUMENTS
C
C Mode Parameters
C
C METH CHARACTER*1
C Specifies the subspace identification method to be used,
C as follows:
C = 'M': MOESP algorithm with past inputs and outputs;
C = 'N': N4SID algorithm.
C
C JOBD CHARACTER*1
C Specifies whether or not the matrices B and D should later
C be computed using the MOESP approach, as follows:
C = 'M': the matrices B and D should later be computed
C using the MOESP approach;
C = 'N': the matrices B and D should not be computed using
C the MOESP approach.
C This parameter is not relevant for METH = 'N'.
C
C Input/Output Parameters
C
C NOBR (input) INTEGER
C The number of block rows, s, in the input and output
C block Hankel matrices. NOBR > 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C L (input) INTEGER
C The number of system outputs. L > 0.
C
C R (input/output) DOUBLE PRECISION array, dimension
C ( LDR,2*(M+L)*NOBR )
C On entry, the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper
C triangular part of this array must contain the upper
C triangular factor R from the QR factorization of the
C concatenated block Hankel matrices. Denote R_ij,
C i,j = 1:4, the ij submatrix of R, partitioned by
C M*NOBR, M*NOBR, L*NOBR, and L*NOBR rows and columns.
C On exit, if INFO = 0, the leading
C 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this
C array contains the matrix S, the processed upper
C triangular factor R, as required by other subroutines.
C Specifically, let S_ij, i,j = 1:4, be the ij submatrix
C of S, partitioned by M*NOBR, L*NOBR, M*NOBR, and
C L*NOBR rows and columns. The submatrix S_22 contains
C the matrix of left singular vectors needed subsequently.
C Useful information is stored in S_11 and in the
C block-column S_14 : S_44. For METH = 'M' and JOBD = 'M',
C the upper triangular part of S_31 contains the upper
C triangular factor in the QR factorization of the matrix
C R_1c = [ R_12' R_22' R_11' ]', and S_12 contains the
C corresponding leading part of the transformed matrix
C R_2c = [ R_13' R_23' R_14' ]'. For METH = 'N', the
C subarray S_41 : S_43 contains the transpose of the
C matrix contained in S_14 : S_34.
C
C LDR INTEGER
C The leading dimension of the array R.
C LDR >= MAX( 2*(M+L)*NOBR, 3*M*NOBR ),
C for METH = 'M' and JOBD = 'M';
C LDR >= 2*(M+L)*NOBR, for METH = 'M' and JOBD = 'N' or
C for METH = 'N'.
C
C SV (output) DOUBLE PRECISION array, dimension ( L*NOBR )
C The singular values of the relevant part of the triangular
C factor from the QR factorization of the concatenated block
C Hankel matrices.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used for estimating the rank of
C matrices. If the user sets TOL > 0, then the given value
C of TOL is used as a lower bound for the reciprocal
C condition number; an m-by-n matrix whose estimated
C condition number is less than 1/TOL is considered to
C be of full rank. If the user sets TOL <= 0, then an
C implicitly computed, default tolerance, defined by
C TOLDEF = m*n*EPS, is used instead, where EPS is the
C relative machine precision (see LAPACK Library routine
C DLAMCH).
C This parameter is not used for METH = 'M'.
C
C Workspace
C
C IWORK INTEGER array, dimension ((M+L)*NOBR)
C This parameter is not referenced for METH = 'M'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK, and, for METH = 'N', DWORK(2) and DWORK(3)
C contain the reciprocal condition numbers of the
C triangular factors of the matrices U_f and r_1 [6].
C On exit, if INFO = -12, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= max( (2*M-1)*NOBR, (M+L)*NOBR, 5*L*NOBR ),
C if METH = 'M' and JOBD = 'M';
C LDWORK >= 5*L*NOBR, if METH = 'M' and JOBD = 'N';
C LDWORK >= 5*(M+L)*NOBR, if METH = 'N'.
C For good performance, LDWORK should be larger.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 4: the least squares problems with coefficient matrix
C U_f, used for computing the weighted oblique
C projection (for METH = 'N'), have a rank-deficient
C coefficient matrix;
C = 5: the least squares problem with coefficient matrix
C r_1 [6], used for computing the weighted oblique
C projection (for METH = 'N'), has a rank-deficient
C coefficient matrix.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 2: the singular value decomposition (SVD) algorithm did
C not converge.
C
C METHOD
C
C A singular value decomposition (SVD) of a certain matrix is
C computed, which reveals the order n of the system as the number
C of "non-zero" singular values. For the MOESP approach, this matrix
C is [ R_24' R_34' ]' := R(ms+1:(2m+l)s,(2m+l)s+1:2(m+l)s),
C where R is the upper triangular factor R constructed by SLICOT
C Library routine IB01MD. For the N4SID approach, a weighted
C oblique projection is computed from the upper triangular factor R
C and its SVD is then found.
C
C REFERENCES
C
C [1] Verhaegen M., and Dewilde, P.
C Subspace Model Identification. Part 1: The output-error
C state-space model identification class of algorithms.
C Int. J. Control, 56, pp. 1187-1210, 1992.
C
C [2] Verhaegen M.
C Subspace Model Identification. Part 3: Analysis of the
C ordinary output-error state-space model identification
C algorithm.
C Int. J. Control, 58, pp. 555-586, 1993.
C
C [3] Verhaegen M.
C Identification of the deterministic part of MIMO state space
C models given in innovations form from input-output data.
C Automatica, Vol.30, No.1, pp.61-74, 1994.
C
C [4] Van Overschee, P., and De Moor, B.
C N4SID: Subspace Algorithms for the Identification of
C Combined Deterministic-Stochastic Systems.
C Automatica, Vol.30, No.1, pp. 75-93, 1994.
C
C [5] Van Overschee, P., and De Moor, B.
C Subspace Identification for Linear Systems: Theory -
C Implementation - Applications.
C Kluwer Academic Publishers, Boston/London/Dordrecht, 1996.
C
C [6] Sima, V.
C Subspace-based Algorithms for Multivariable System
C Identification.
C Studies in Informatics and Control, 5, pp. 335-344, 1996.
C
C NUMERICAL ASPECTS
C
C The implemented method is numerically stable.
C 3
C The algorithm requires 0(((m+l)s) ) floating point operations.
C
C CONTRIBUTOR
C
C V. Sima, Research Institute for Informatics, Bucharest, Aug. 1999.
C
C REVISIONS
C
C Feb. 2000, Feb. 2001.
C
C KEYWORDS
C
C Identification methods, multivariable systems, QR decomposition,
C singular value decomposition.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDR, LDWORK, M, NOBR
CHARACTER JOBD, METH
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), R(LDR, *), SV(*)
INTEGER IWORK(*)
C .. Local Scalars ..
DOUBLE PRECISION EPS, RCOND1, RCOND2, SVLMAX, THRESH, TOLL
INTEGER I, IERR, ITAU, ITAU2, ITAU3, JWORK, LLMNOB,
$ LLNOBR, LMMNOB, LMNOBR, LNOBR, MAXWRK, MINWRK,
$ MMNOBR, MNOBR, NR, NR2, NR3, NR4, NRSAVE, RANK,
$ RANK1
LOGICAL JOBDM, MOESP, N4SID
C .. Local Arrays ..
DOUBLE PRECISION DUM(1), SVAL(3)
C .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, ILAENV, LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DGEQRF, DLACPY, DLASET, DORMQR, DSWAP,
$ DTRCON, MA02AD, MB03OD, MB03UD, MB04ID, MB04IY,
$ MB04OD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC INT, MAX
C ..
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
MOESP = LSAME( METH, 'M' )
N4SID = LSAME( METH, 'N' )
JOBDM = LSAME( JOBD, 'M' )
MNOBR = M*NOBR
LNOBR = L*NOBR
LLNOBR = LNOBR + LNOBR
LMNOBR = LNOBR + MNOBR
MMNOBR = MNOBR + MNOBR
LMMNOB = MMNOBR + LNOBR
NR = LMNOBR + LMNOBR
IWARN = 0
INFO = 0
C
C Check the scalar input parameters.
C
IF( .NOT.( MOESP .OR. N4SID ) ) THEN
INFO = -1
ELSE IF( MOESP .AND. .NOT.( JOBDM .OR. LSAME( JOBD, 'N' ) ) ) THEN
INFO = -2
ELSE IF( NOBR.LE.0 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( L.LE.0 ) THEN
INFO = -5
ELSE IF( LDR.LT.NR .OR. ( MOESP .AND. JOBDM .AND.
$ LDR.LT.3*MNOBR ) ) THEN
INFO = -7
ELSE
C
C Compute workspace.
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of workspace needed at that point in the code,
C as well as the preferred amount for good performance.
C NB refers to the optimal block size for the immediately
C following subroutine, as returned by ILAENV.)
C
MINWRK = 1
IF ( LDWORK.GE.1 ) THEN
IF ( MOESP ) THEN
MINWRK = 5*LNOBR
IF ( JOBDM )
$ MINWRK = MAX( MMNOBR - NOBR, LMNOBR, MINWRK )
MAXWRK = LNOBR + LNOBR*ILAENV( 1, 'DGEQRF', ' ', LMNOBR,
$ LNOBR, -1, -1 )
ELSE
C
MINWRK = MAX( MINWRK, 5*LMNOBR )
MAXWRK = MAX( MNOBR + MNOBR*ILAENV( 1, 'DGEQRF', ' ',
$ MMNOBR, MNOBR, -1, -1 ),
$ MNOBR + LLNOBR*ILAENV( 1, 'DORMQR', 'LT',
$ MMNOBR, LLNOBR, MNOBR, -1 ) )
MAXWRK = MAX( MAXWRK, MNOBR + LNOBR*ILAENV( 1, 'DORMQR',
$ 'LN', MMNOBR, LNOBR, MNOBR,
$ -1 ) )
MAXWRK = MAX( MAXWRK, LNOBR + LNOBR*ILAENV( 1, 'DGEQRF',
$ ' ', LMMNOB, LNOBR, -1, -1 ) )
END IF
MAXWRK = MAX( MINWRK, MAXWRK )
END IF
C
IF( LDWORK.LT.MINWRK ) THEN
INFO = -12
DWORK( 1 ) = MINWRK
END IF
END IF
C
C Return if there are illegal arguments.
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'IB01ND', -INFO )
RETURN
END IF
C
C Compute pointers to the needed blocks of R.
C
NR2 = MNOBR + 1
NR3 = MMNOBR + 1
NR4 = LMMNOB + 1
ITAU = 1
JWORK = ITAU + MNOBR
C
IF( MOESP ) THEN
C
C MOESP approach.
C
IF( M.GT.0 .AND. JOBDM ) THEN
C
C Rearrange the blocks of R:
C Copy the (1,1) block into the position (3,2) and
C copy the (1,4) block into (3,3).
C
CALL DLACPY( 'Upper', MNOBR, MNOBR, R, LDR, R(NR3,NR2),
$ LDR )
CALL DLACPY( 'Full', MNOBR, LNOBR, R(1,NR4), LDR,
$ R(NR3,NR3), LDR )
C
C Using structure, triangularize the matrix
C R_1c = [ R_12' R_22' R_11' ]'
C and then apply the transformations to the matrix
c R_2c = [ R_13' R_23' R_14' ]'.
C Workspace: need M*NOBR + MAX(M-1,L)*NOBR.
C
CALL MB04OD( 'Upper', MNOBR, LNOBR, MNOBR, R(NR2,NR2), LDR,
$ R(NR3,NR2), LDR, R(NR2,NR3), LDR, R(NR3,NR3),
$ LDR, DWORK(ITAU), DWORK(JWORK) )
CALL MB04ID( MMNOBR, MNOBR, MNOBR-1, LNOBR, R(1,NR2), LDR,
$ R(1,NR3), LDR, DWORK(ITAU), DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Copy the leading M*NOBR x M*NOBR and M*NOBR x L*NOBR
C submatrices of R_1c and R_2c, respectively, into their
C final positions, required by SLICOT Library routine IB01PD.
C
CALL DLACPY( 'Upper', MNOBR, MNOBR, R(1,NR2), LDR,
$ R(LMNOBR+1,1), LDR )
CALL DLACPY( 'Full', MNOBR, LNOBR, R(1,NR3), LDR, R(1,NR2),
$ LDR )
END IF
C
C Copy [ R_24' R_34' ]' in [ R_22' R_32' ]'.
C
CALL DLACPY( 'Full', LMNOBR, LNOBR, R(NR2,NR4), LDR,
$ R(NR2,NR2), LDR )
C
C Triangularize the matrix in [ R_22' R_32' ]'.
C Workspace: need 2*L*NOBR; prefer L*NOBR + L*NOBR*NB.
C
JWORK = ITAU + LNOBR
CALL DGEQRF( LMNOBR, LNOBR, R(NR2,NR2), LDR, DWORK(ITAU),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
ELSE
C
C N4SID approach.
C
DUM(1) = ZERO
LLMNOB = LLNOBR + MNOBR
C
C Set the precision parameters. A threshold value EPS**(2/3) is
C used for deciding to use pivoting or not, where EPS is the
C relative machine precision (see LAPACK Library routine DLAMCH).
C
TOLL = TOL
EPS = DLAMCH( 'Precision' )
THRESH = EPS**( TWO/THREE )
C
IF( M.GT.0 ) THEN
C
C For efficiency of later calculations, interchange the first
C two block-columns. The corresponding submatrices are
C redefined according to their new position.
C
DO 10 I = 1, MNOBR
CALL DSWAP( I, R(1,I), 1, R(1,MNOBR+I), 1 )
CALL DCOPY( MNOBR, R(I+1,MNOBR+I), 1, R(I+1,I), 1 )
CALL DCOPY( MMNOBR-I, DUM, 0, R(I+1,MNOBR+I), 1 )
10 CONTINUE
C
C Now,
C
C U_f = [ R_11' R_21' 0 0 ]',
C U_p = [ R_12' 0 0 0 ]',
C Y_p = [ R_13' R_23' R_33' 0 ]', and
C Y_f = [ R_14' R_24' R_34' R_44' ]',
C
C where R_21, R_12, R_33, and R_44 are upper triangular.
C Define W_p := [ U_p Y_p ].
C
C Prepare the computation of residuals of the two least
C squares problems giving the weighted oblique projection P:
C
C r_1 = W_p - U_f X_1, X_1 = arg min || U_f X - W_p ||,
C r_2 = Y_f - U_f X_2, X_2 = arg min || U_f X - Y_f ||,
C
C P = (arg min || r_1 X - r_2 ||)' r_1'. (1)
C
C Alternately, P' is given by the projection
C P' = Q_1 (Q_1)' r_2,
C where Q_1 contains the first k columns of the orthogonal
C matrix in the QR factorization of r_1, k := rank(r_1).
C
C Triangularize the matrix U_f = q r (using structure), and
C apply the transformation q' to the corresponding part of
C the matrices W_p, and Y_f.
C Workspace: need 2*(M+L)*NOBR.
C
CALL MB04ID( MMNOBR, MNOBR, MNOBR-1, LLMNOB, R, LDR,
$ R(1,NR2), LDR, DWORK(ITAU), DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Save updated Y_f (transposed) in the last block-row of R.
C
CALL MA02AD( 'Full', LMMNOB, LNOBR, R(1,NR4), LDR, R(NR4,1),
$ LDR )
C
C Check the condition of the triangular factor r and decide
C to use pivoting or not.
C Workspace: need 4*M*NOBR.
C
CALL DTRCON( '1-norm', 'Upper', 'NonUnit', MNOBR, R, LDR,
$ RCOND1, DWORK(JWORK), IWORK, IERR )
C
IF( TOLL.LE.ZERO )
$ TOLL = MNOBR*MNOBR*EPS
IF ( RCOND1.GT.MAX( TOLL, THRESH ) ) THEN
C
C U_f is considered full rank and no pivoting is used.
C
CALL DLASET( 'Full', MNOBR, LLMNOB, ZERO, ZERO, R(1,NR2),
$ LDR )
ELSE
C
C Save information about q in the (2,1) block of R.
C Use QR factorization with column pivoting, r P = Q R.
C Information on Q is stored in the strict lower triangle
C of R_11 and in DWORK(ITAU2).
C
DO 20 I = 1, MNOBR - 1
CALL DCOPY( MNOBR, R(I+1,I), -1, R(NR2,I), -1 )
CALL DCOPY( MNOBR-I, DUM, 0, R(I+1,I), 1 )
IWORK(I) = 0
20 CONTINUE
C
IWORK(MNOBR) = 0
C
C Workspace: need 5*M*NOBR.
C
ITAU2 = JWORK
JWORK = ITAU2 + MNOBR
SVLMAX = ZERO
CALL MB03OD( 'QR', MNOBR, MNOBR, R, LDR, IWORK, TOLL,
$ SVLMAX, DWORK(ITAU2), RANK, SVAL,
$ DWORK(JWORK), IERR )
C
C Workspace: need 2*M*NOBR + (M+2*L)*NOBR;
C prefer 2*M*NOBR + (M+2*L)*NOBR*NB.
C
CALL DORMQR( 'Left', 'Transpose', MNOBR, LLMNOB, MNOBR,
$ R, LDR, DWORK(ITAU2), R(1,NR2), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
IF ( RANK.LT.MNOBR ) THEN
C
C The least squares problem is rank-deficient.
C
IWARN = 4
END IF
C
C Determine residuals r_1 and r_2: premultiply by Q and
C then by q.
C Workspace: need 2*M*NOBR + (M+2*L)*NOBR);
C prefer 2*M*NOBR + (M+2*L)*NOBR*NB.
C
CALL DLASET( 'Full', RANK, LLMNOB, ZERO, ZERO, R(1,NR2),
$ LDR )
CALL DORMQR( 'Left', 'NoTranspose', MNOBR, LLMNOB, MNOBR,
$ R, LDR, DWORK(ITAU2), R(1,NR2), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
JWORK = ITAU2
C
C Restore the transformation q.
C
DO 30 I = 1, MNOBR - 1
CALL DCOPY( MNOBR, R(NR2,I), 1, R(I+1,I), 1 )
30 CONTINUE
C
END IF
C
C Premultiply by the transformation q (apply transformations
C in backward order).
C Workspace: need M*NOBR + (M+2*L)*NOBR;
C prefer larger.
C
CALL MB04IY( 'Left', 'NoTranspose', MMNOBR, LLMNOB, MNOBR,
$ MNOBR-1, R, LDR, DWORK(ITAU), R(1,NR2), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
ELSE
C
C Save Y_f (transposed) in the last block-row of R.
C
CALL MA02AD( 'Full', LMMNOB, LNOBR, R(1,NR4), LDR, R(NR4,1),
$ LDR )
RCOND1 = ONE
END IF
C
C Triangularize the matrix r_1 for determining the oblique
C projection P in least squares problem in (1). Exploit the
C fact that the third block-row of r_1 has the structure
C [ 0 T ], where T is an upper triangular matrix. Then apply
C the corresponding transformations Q' to the matrix r_2.
C Workspace: need 2*M*NOBR;
C prefer M*NOBR + M*NOBR*NB.
C
CALL DGEQRF( MMNOBR, MNOBR, R(1,NR2), LDR, DWORK(ITAU),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Workspace: need M*NOBR + 2*L*NOBR;
C prefer M*NOBR + 2*L*NOBR*NB.
C
CALL DORMQR( 'Left', 'Transpose', MMNOBR, LLNOBR, MNOBR,
$ R(1,NR2), LDR, DWORK(ITAU), R(1,NR3), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
NRSAVE = NR2
C
ITAU2 = JWORK
JWORK = ITAU2 + LNOBR
CALL MB04ID( LMNOBR, LNOBR, LNOBR-1, LNOBR, R(NR2,NR3), LDR,
$ R(NR2,NR4), LDR, DWORK(ITAU2), DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Check the condition of the triangular matrix of order (m+l)*s
C just determined, and decide to use pivoting or not.
C Workspace: need 4*(M+L)*NOBR.
C
CALL DTRCON( '1-norm', 'Upper', 'NonUnit', LMNOBR, R(1,NR2),
$ LDR, RCOND2, DWORK(JWORK), IWORK, IERR )
C
IF( TOL.LE.ZERO )
$ TOLL = LMNOBR*LMNOBR*EPS
IF ( RCOND2.LE.MAX( TOLL, THRESH ) ) THEN
IF ( M.GT.0 ) THEN
C
C Save information about Q in R_11 (in the strict lower
C triangle), R_21 and R_31 (transposed information).
C
CALL DLACPY( 'Lower', MMNOBR-1, MNOBR, R(2,NR2), LDR,
$ R(2,1), LDR )
NRSAVE = 1
C
DO 40 I = NR2, LMNOBR
CALL DCOPY( MNOBR, R(I+1,MNOBR+I), 1, R(MNOBR+I,1),
$ LDR )
40 CONTINUE
C
END IF
C
CALL DLASET( 'Lower', LMNOBR-1, LMNOBR-1, ZERO, ZERO,
$ R(2,NR2), LDR )
C
C Use QR factorization with column pivoting.
C Workspace: need 5*(M+L)*NOBR.
C
DO 50 I = 1, LMNOBR
IWORK(I) = 0
50 CONTINUE
C
ITAU3 = JWORK
JWORK = ITAU3 + LMNOBR
SVLMAX = ZERO
CALL MB03OD( 'QR', LMNOBR, LMNOBR, R(1,NR2), LDR, IWORK,
$ TOLL, SVLMAX, DWORK(ITAU3), RANK1, SVAL,
$ DWORK(JWORK), IERR )
C
C Workspace: need 2*(M+L)*NOBR + L*NOBR;
C prefer 2*(M+L)*NOBR + L*NOBR*NB.
C
CALL DORMQR( 'Left', 'Transpose', LMNOBR, LNOBR, LMNOBR,
$ R(1,NR2), LDR, DWORK(ITAU3), R(1,NR4), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
IF ( RANK1.LT.LMNOBR ) THEN
C
C The least squares problem is rank-deficient.
C
IWARN = 5
END IF
C
C Apply the orthogonal transformations, in backward order, to
C [r_2(1:rank(r_1),:)' 0]', to obtain P'.
C Workspace: need 2*(M+L)*NOBR + L*NOBR;
C prefer 2*(M+L)*NOBR + L*NOBR*NB.
C
CALL DLASET( 'Full', LMNOBR-RANK1, LNOBR, ZERO, ZERO,
$ R(RANK1+1,NR4), LDR )
CALL DORMQR( 'Left', 'NoTranspose', LMNOBR, LNOBR, LMNOBR,
$ R(1,NR2), LDR, DWORK(ITAU3), R(1,NR4), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
JWORK = ITAU3
C
IF ( M.GT.0 ) THEN
C
C Restore the saved transpose matrix from R_31.
C
DO 60 I = NR2, LMNOBR
CALL DCOPY( MNOBR, R(MNOBR+I,1), LDR, R(I+1,MNOBR+I),
$ 1 )
60 CONTINUE
C
END IF
C
END IF
C
C Workspace: need M*NOBR + L*NOBR;
C prefer larger.
C
CALL MB04IY( 'Left', 'NoTranspose', LMNOBR, LNOBR, LNOBR,
$ LNOBR-1, R(NR2,NR3), LDR, DWORK(ITAU2),
$ R(NR2,NR4), LDR, DWORK(JWORK), LDWORK-JWORK+1,
$ IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Workspace: need M*NOBR + L*NOBR;
C prefer M*NOBR + L*NOBR*NB.
C
JWORK = ITAU2
CALL DORMQR( 'Left', 'NoTranspose', MMNOBR, LNOBR, MNOBR,
$ R(1,NRSAVE), LDR, DWORK(ITAU), R(1,NR4), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Now, the matrix P' is available in R_14 : R_34.
C Triangularize the matrix P'.
C Workspace: need 2*L*NOBR;
C prefer L*NOBR + L*NOBR*NB.
C
JWORK = ITAU + LNOBR
CALL DGEQRF( LMMNOB, LNOBR, R(1,NR4), LDR, DWORK(ITAU),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Copy the triangular factor to its final position, R_22.
C
CALL DLACPY( 'Upper', LNOBR, LNOBR, R(1,NR4), LDR, R(NR2,NR2),
$ LDR )
C
C Restore Y_f.
C
CALL MA02AD( 'Full', LNOBR, LMMNOB, R(NR4,1), LDR, R(1,NR4),
$ LDR )
END IF
C
C Find the singular value decomposition of R_22.
C Workspace: need 5*L*NOBR.
C
CALL MB03UD( 'NoVectors', 'Vectors', LNOBR, R(NR2,NR2), LDR,
$ DUM, 1, SV, DWORK, LDWORK, IERR )
IF ( IERR.NE.0 ) THEN
INFO = 2
RETURN
END IF
MAXWRK = MAX( MAXWRK, INT( DWORK(1) ) )
C
C Transpose R(m*s+1:(m+L)*s,m*s+1:(m+L)*s) in-situ; its
C columns will then be the singular vectors needed subsequently.
C
DO 70 I = NR2+1, LMNOBR
CALL DSWAP( LMNOBR-I+1, R(I,I-1), 1, R(I-1,I), LDR )
70 CONTINUE
C
C Return optimal workspace in DWORK(1) and reciprocal condition
C numbers, if METH = 'N'.
C
DWORK(1) = MAXWRK
IF ( N4SID ) THEN
DWORK(2) = RCOND1
DWORK(3) = RCOND2
END IF
RETURN
C
C *** Last line of IB01ND ***
END
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