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SUBROUTINE IB01RD( JOB, N, M, L, NSMP, A, LDA, B, LDB, C, LDC, D,
$ LDD, U, LDU, Y, LDY, X0, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To estimate the initial state of a linear time-invariant (LTI)
C discrete-time system, given the system matrices (A,B,C,D) and
C the input and output trajectories of the system. The model
C structure is :
C
C x(k+1) = Ax(k) + Bu(k), k >= 0,
C y(k) = Cx(k) + Du(k),
C
C where x(k) is the n-dimensional state vector (at time k),
C u(k) is the m-dimensional input vector,
C y(k) is the l-dimensional output vector,
C and A, B, C, and D are real matrices of appropriate dimensions.
C Matrix A is assumed to be in a real Schur form.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOB CHARACTER*1
C Specifies whether or not the matrix D is zero, as follows:
C = 'Z': the matrix D is zero;
C = 'N': the matrix D is not zero.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the system. N >= 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 NSMP (input) INTEGER
C The number of rows of matrices U and Y (number of
C samples used, t). NSMP >= N.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C system state matrix A in a real Schur form.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= MAX(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C system input matrix B (corresponding to the real Schur
C form of A).
C If N = 0 or M = 0, this array is not referenced.
C
C LDB INTEGER
C The leading dimension of the array B.
C LDB >= N, if N > 0 and M > 0;
C LDB >= 1, if N = 0 or M = 0.
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading L-by-N part of this array must contain the
C system output matrix C (corresponding to the real Schur
C form of A).
C
C LDC INTEGER
C The leading dimension of the array C. LDC >= L.
C
C D (input) DOUBLE PRECISION array, dimension (LDD,M)
C The leading L-by-M part of this array must contain the
C system input-output matrix.
C If M = 0 or JOB = 'Z', this array is not referenced.
C
C LDD INTEGER
C The leading dimension of the array D.
C LDD >= L, if M > 0 and JOB = 'N';
C LDD >= 1, if M = 0 or JOB = 'Z'.
C
C U (input) DOUBLE PRECISION array, dimension (LDU,M)
C If M > 0, the leading NSMP-by-M part of this array must
C contain the t-by-m input-data sequence matrix U,
C U = [u_1 u_2 ... u_m]. Column j of U contains the
C NSMP values of the j-th input component for consecutive
C time increments.
C If M = 0, this array is not referenced.
C
C LDU INTEGER
C The leading dimension of the array U.
C LDU >= MAX(1,NSMP), if M > 0;
C LDU >= 1, if M = 0.
C
C Y (input) DOUBLE PRECISION array, dimension (LDY,L)
C The leading NSMP-by-L part of this array must contain the
C t-by-l output-data sequence matrix Y,
C Y = [y_1 y_2 ... y_l]. Column j of Y contains the
C NSMP values of the j-th output component for consecutive
C time increments.
C
C LDY INTEGER
C The leading dimension of the array Y. LDY >= MAX(1,NSMP).
C
C X0 (output) DOUBLE PRECISION array, dimension (N)
C The estimated initial state of the system, x(0).
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; a matrix whose estimated condition
C number is less than 1/TOL is considered to be of full
C rank. If the user sets TOL <= 0, then EPS is used
C instead, where EPS is the relative machine precision
C (see LAPACK Library routine DLAMCH). TOL <= 1.
C
C Workspace
C
C IWORK INTEGER array, dimension (N)
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK and DWORK(2) contains the reciprocal condition
C number of the triangular factor of the QR factorization of
C the matrix Gamma (see METHOD).
C On exit, if INFO = -22, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= max( 2, min( LDW1, LDW2 ) ), where
C LDW1 = t*L*(N + 1) + 2*N + max( 2*N*N, 4*N ),
C LDW2 = N*(N + 1) + 2*N +
C max( q*(N + 1) + 2*N*N + L*N, 4*N ),
C q = N*L.
C For good performance, LDWORK should be larger.
C If LDWORK >= LDW1, then standard QR factorization of
C the matrix Gamma (see METHOD) is used. Otherwise, the
C QR factorization is computed sequentially by performing
C NCYCLE cycles, each cycle (except possibly the last one)
C processing s samples, where s is chosen by equating
C LDWORK to LDW2, for q replaced by s*L.
C The computational effort may increase and the accuracy may
C decrease with the decrease of s. Recommended value is
C LDRWRK = LDW1, assuming a large enough cache size, to
C also accommodate A, B, C, D, U, and Y.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 4: the least squares problem to be solved has a
C rank-deficient 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 An extension and refinement of the method in [1] is used.
C Specifically, the output y0(k) of the system for zero initial
C state is computed for k = 0, 1, ..., t-1 using the given model.
C Then the following least squares problem is solved for x(0)
C
C ( C ) ( y(0) - y0(0) )
C ( C*A ) ( y(1) - y0(1) )
C Gamma * x(0) = ( : ) * x(0) = ( : ).
C ( : ) ( : )
C ( C*A^(t-1) ) ( y(t-1) - y0(t-1) )
C
C The coefficient matrix Gamma is evaluated using powers of A with
C exponents 2^k. The QR decomposition of this matrix is computed.
C If its triangular factor R is too ill conditioned, then singular
C value decomposition of R is used.
C
C If the coefficient matrix cannot be stored in the workspace (i.e.,
C LDWORK < LDW1), the QR decomposition is computed sequentially.
C
C REFERENCES
C
C [1] Verhaegen M., and Varga, A.
C Some Experience with the MOESP Class of Subspace Model
C Identification Methods in Identifying the BO105 Helicopter.
C Report TR R165-94, DLR Oberpfaffenhofen, 1994.
C
C NUMERICAL ASPECTS
C
C The implemented method is numerically stable.
C
C CONTRIBUTOR
C
C V. Sima, Research Institute for Informatics, Bucharest, Apr. 2000.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Feb. 2004.
C
C KEYWORDS
C
C Identification methods; least squares solutions; multivariable
C systems; QR decomposition; 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 IBLOCK is a threshold value for switching to a block algorithm
C for U (to avoid row by row passing through U).
INTEGER IBLOCK
PARAMETER ( IBLOCK = 16384 )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDU,
$ LDWORK, LDY, M, N, NSMP
CHARACTER JOB
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), U(LDU, *), X0(*), Y(LDY, *)
INTEGER IWORK(*)
C .. Local Scalars ..
DOUBLE PRECISION RCOND, TOLL
INTEGER I2, IA, IAS, IC, ICYCLE, IE, IERR, IEXPON,
$ IG, INIGAM, INIH, INIR, INIT, IQ, IREM, IRHS,
$ ISIZE, ISV, ITAU, IU, IUPNT, IUT, IUTRAN, IX,
$ IXINIT, IY, IYPNT, J, JWORK, K, LDDW, LDR,
$ LDW1, LDW2, MAXWRK, MINSMP, MINWLS, MINWRK, NC,
$ NCP1, NCYCLE, NN, NOBS, NRBL, NROW, NSMPL, RANK
LOGICAL BLOCK, FIRST, NCYC, POWER2, SWITCH, WITHD
C .. Local Arrays ..
DOUBLE PRECISION DUM(1)
C .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, ILAENV, LSAME
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGELSS, DGEMV, DGEQRF, DLACPY,
$ DLASET, DORMQR, DTRCON, DTRMM, DTRMV, DTRSV,
$ MA02AD, MB01TD, MB04OD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, LOG, MAX, MIN, MOD
C .. Executable Statements ..
C
C Check the input parameters.
C
WITHD = LSAME( JOB, 'N' )
IWARN = 0
INFO = 0
NN = N*N
MINSMP = N
C
IF( .NOT.( LSAME( JOB, 'Z' ) .OR. WITHD ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( L.LE.0 ) THEN
INFO = -4
ELSE IF( NSMP.LT.MINSMP ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.1 .OR. ( LDB.LT.N .AND. M.GT.0 ) ) THEN
INFO = -9
ELSE IF( LDC.LT.L ) THEN
INFO = -11
ELSE IF( LDD.LT.1 .OR. ( WITHD .AND. LDD.LT.L .AND. M.GT.0 ) )
$ THEN
INFO = -13
ELSE IF( LDU.LT.1 .OR. ( M.GT.0 .AND. LDU.LT.NSMP ) ) THEN
INFO = -15
ELSE IF( LDY.LT.MAX( 1, NSMP ) ) THEN
INFO = -17
ELSE IF( TOL.GT.ONE ) THEN
INFO = -19
END IF
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
NSMPL = NSMP*L
IQ = MINSMP*L
NCP1 = N + 1
ISIZE = NSMPL*NCP1
IC = 2*NN
MINWLS = MINSMP*NCP1
ITAU = IC + L*N
LDW1 = ISIZE + 2*N + MAX( IC, 4*N )
LDW2 = MINWLS + 2*N + MAX( IQ*NCP1 + ITAU, 4*N )
MINWRK = MAX( MIN( LDW1, LDW2 ), 2 )
IF ( INFO.EQ.0 .AND. LDWORK.GE.MINWRK ) THEN
MAXWRK = ISIZE + 2*N + MAX( N*ILAENV( 1, 'DGEQRF', ' ', NSMPL,
$ N, -1, -1 ),
$ ILAENV( 1, 'DORMQR', 'LT', NSMPL,
$ 1, N, -1 ) )
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
C
IF ( INFO.EQ.0 .AND. LDWORK.LT.MINWRK ) THEN
INFO = -22
DWORK(1) = MINWRK
END IF
C
C Return if there are illegal arguments.
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'IB01RD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( N.EQ.0 ) THEN
DWORK(1) = TWO
DWORK(2) = ONE
RETURN
END IF
C
C Set up the least squares problem, either directly, if enough
C workspace, or sequentially, otherwise.
C
IYPNT = 1
IUPNT = 1
INIR = 1
IF ( LDWORK.GE.LDW1 ) THEN
C
C Enough workspace for solving the problem directly.
C
NCYCLE = 1
NOBS = NSMP
LDDW = NSMPL
INIGAM = 1
ELSE
C
C NCYCLE > 1 cycles are needed for solving the problem
C sequentially, taking NOBS samples in each cycle (or the
C remaining samples in the last cycle).
C
JWORK = LDWORK - MINWLS - 2*N - ITAU
LDDW = JWORK/NCP1
NOBS = LDDW/L
LDDW = L*NOBS
NCYCLE = NSMP/NOBS
IF ( MOD( NSMP, NOBS ).NE.0 )
$ NCYCLE = NCYCLE + 1
INIH = INIR + NN
INIGAM = INIH + N
END IF
C
NCYC = NCYCLE.GT.1
IRHS = INIGAM + LDDW*N
IXINIT = IRHS + LDDW
IC = IXINIT + N
IF( NCYC ) THEN
IA = IC + L*N
LDR = N
IE = INIGAM
ELSE
INIH = IRHS
IA = IC
LDR = LDDW
IE = IXINIT
END IF
IUTRAN = IA
IAS = IA + NN
ITAU = IA
DUM(1) = ZERO
C
C Set block parameters for passing through the array U.
C
BLOCK = M.GT.1 .AND. NSMP*M.GE.IBLOCK
IF ( BLOCK ) THEN
NRBL = ( LDWORK - IUTRAN + 1 )/M
NC = NOBS/NRBL
IF ( MOD( NOBS, NRBL ).NE.0 )
$ NC = NC + 1
INIT = ( NC - 1 )*NRBL
BLOCK = BLOCK .AND. NRBL.GT.1
END IF
C
C Perform direct of sequential compression of the matrix Gamma.
C
DO 150 ICYCLE = 1, NCYCLE
FIRST = ICYCLE.EQ.1
IF ( .NOT.FIRST ) THEN
IF ( ICYCLE.EQ.NCYCLE ) THEN
NOBS = NSMP - ( NCYCLE - 1 )*NOBS
LDDW = L*NOBS
IF ( BLOCK ) THEN
NC = NOBS/NRBL
IF ( MOD( NOBS, NRBL ).NE.0 )
$ NC = NC + 1
INIT = ( NC - 1 )*NRBL
END IF
END IF
END IF
C
C Compute the extended observability matrix Gamma.
C Workspace: need s*L*(N + 1) + 2*N*N + 2*N + a + w,
C where s = NOBS,
C a = 0, w = 0, if NCYCLE = 1,
C a = L*N, w = N*(N + 1), if NCYCLE > 1;
C prefer as above, with s = t, a = w = 0.
C
JWORK = IAS + NN
IEXPON = INT( LOG( DBLE( NOBS ) )/LOG( TWO ) )
IREM = L*( NOBS - 2**IEXPON )
POWER2 = IREM.EQ.0
IF ( .NOT.POWER2 )
$ IEXPON = IEXPON + 1
C
IF ( FIRST ) THEN
CALL DLACPY( 'Full', L, N, C, LDC, DWORK(INIGAM), LDDW )
ELSE
CALL DLACPY( 'Full', L, N, DWORK(IC), L, DWORK(INIGAM),
$ LDDW )
END IF
C p
C Use powers of the matrix A: A , p = 2**(J-1).
C
CALL DLACPY( 'Upper', N, N, A, LDA, DWORK(IA), N )
IF ( N.GT.1 )
$ CALL DCOPY( N-1, A(2,1), LDA+1, DWORK(IA+1), N+1 )
I2 = L
NROW = 0
C
DO 20 J = 1, IEXPON
IG = INIGAM
IF ( J.LT.IEXPON .OR. POWER2 ) THEN
NROW = I2
ELSE
NROW = IREM
END IF
C
CALL DLACPY( 'Full', NROW, N, DWORK(IG), LDDW, DWORK(IG+I2),
$ LDDW )
CALL DTRMM( 'Right', 'Upper', 'No Transpose', 'Non Unit',
$ NROW, N, ONE, DWORK(IA), N, DWORK(IG+I2),
$ LDDW )
C p
C Compute the contribution of the subdiagonal of A to the
C product.
C
DO 10 IX = 1, N - 1
CALL DAXPY( NROW, DWORK(IA+(IX-1)*N+IX), DWORK(IG+LDDW),
$ 1, DWORK(IG+I2), 1 )
IG = IG + LDDW
10 CONTINUE
C
IF ( J.LT.IEXPON ) THEN
CALL DLACPY( 'Upper', N, N, DWORK(IA), N, DWORK(IAS), N )
CALL DCOPY( N-1, DWORK(IA+1), N+1, DWORK(IAS+1), N+1 )
CALL MB01TD( N, DWORK(IAS), N, DWORK(IA), N,
$ DWORK(JWORK), IERR )
I2 = I2*2
END IF
20 CONTINUE
C
IF ( NCYC ) THEN
IG = INIGAM + I2 + NROW - L
CALL DLACPY( 'Full', L, N, DWORK(IG), LDDW, DWORK(IC), L )
CALL DTRMM( 'Right', 'Upper', 'No Transpose', 'Non Unit', L,
$ N, ONE, A, LDA, DWORK(IC), L )
C
C Compute the contribution of the subdiagonal of A to the
C product.
C
DO 30 IX = 1, N - 1
CALL DAXPY( L, A(IX+1,IX), DWORK(IG+LDDW), 1,
$ DWORK(IC+(IX-1)*L), 1 )
IG = IG + LDDW
30 CONTINUE
C
END IF
C
C Setup (part of) the right hand side of the least squares
C problem starting from DWORK(IRHS); use the estimated output
C trajectory for zero initial state, or for the saved final state
C value of the previous cycle.
C A specialization of SLICOT Library routine TF01ND is used.
C For large input sets (NSMP*M >= IBLOCK), chunks of U are
C transposed, to reduce the number of row-wise passes.
C Workspace: need s*L*(N + 1) + N + w;
C prefer as above, with s = t, w = 0.
C
IF ( FIRST )
$ CALL DCOPY( N, DUM, 0, DWORK(IXINIT), 1 )
CALL DCOPY( N, DWORK(IXINIT), 1, X0, 1 )
IY = IRHS
C
DO 40 J = 1, L
CALL DCOPY( NOBS, Y(IYPNT,J), 1, DWORK(IY), L )
IY = IY + 1
40 CONTINUE
C
IY = IRHS
IU = IUPNT
IF ( M.GT.0 ) THEN
IF ( WITHD ) THEN
C
IF ( BLOCK ) THEN
SWITCH = .TRUE.
NROW = NRBL
C
DO 60 K = 1, NOBS
IF ( MOD( K-1, NROW ).EQ.0 .AND. SWITCH ) THEN
IUT = IUTRAN
IF ( K.GT.INIT ) THEN
NROW = NOBS - INIT
SWITCH = .FALSE.
END IF
CALL MA02AD( 'Full', NROW, M, U(IU,1), LDU,
$ DWORK(IUT), M )
IU = IU + NROW
END IF
CALL DGEMV( 'No transpose', L, N, -ONE, C, LDC, X0,
$ 1, ONE, DWORK(IY), 1 )
CALL DGEMV( 'No transpose', L, M, -ONE, D, LDD,
$ DWORK(IUT), 1, ONE, DWORK(IY), 1 )
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', N,
$ A, LDA, X0, 1 )
C
DO 50 IX = 2, N
X0(IX) = X0(IX) + A(IX,IX-1)*DWORK(IXINIT+IX-2)
50 CONTINUE
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ DWORK(IUT), 1, ONE, X0, 1 )
CALL DCOPY( N, X0, 1, DWORK(IXINIT), 1 )
IY = IY + L
IUT = IUT + M
60 CONTINUE
C
ELSE
C
DO 80 K = 1, NOBS
CALL DGEMV( 'No transpose', L, N, -ONE, C, LDC, X0,
$ 1, ONE, DWORK(IY), 1 )
CALL DGEMV( 'No transpose', L, M, -ONE, D, LDD,
$ U(IU,1), LDU, ONE, DWORK(IY), 1 )
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', N,
$ A, LDA, X0, 1 )
C
DO 70 IX = 2, N
X0(IX) = X0(IX) + A(IX,IX-1)*DWORK(IXINIT+IX-2)
70 CONTINUE
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ U(IU,1), LDU, ONE, X0, 1 )
CALL DCOPY( N, X0, 1, DWORK(IXINIT), 1 )
IY = IY + L
IU = IU + 1
80 CONTINUE
C
END IF
C
ELSE
C
IF ( BLOCK ) THEN
SWITCH = .TRUE.
NROW = NRBL
C
DO 100 K = 1, NOBS
IF ( MOD( K-1, NROW ).EQ.0 .AND. SWITCH ) THEN
IUT = IUTRAN
IF ( K.GT.INIT ) THEN
NROW = NOBS - INIT
SWITCH = .FALSE.
END IF
CALL MA02AD( 'Full', NROW, M, U(IU,1), LDU,
$ DWORK(IUT), M )
IU = IU + NROW
END IF
CALL DGEMV( 'No transpose', L, N, -ONE, C, LDC, X0,
$ 1, ONE, DWORK(IY), 1 )
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', N,
$ A, LDA, X0, 1 )
C
DO 90 IX = 2, N
X0(IX) = X0(IX) + A(IX,IX-1)*DWORK(IXINIT+IX-2)
90 CONTINUE
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ DWORK(IUT), 1, ONE, X0, 1 )
CALL DCOPY( N, X0, 1, DWORK(IXINIT), 1 )
IY = IY + L
IUT = IUT + M
100 CONTINUE
C
ELSE
C
DO 120 K = 1, NOBS
CALL DGEMV( 'No transpose', L, N, -ONE, C, LDC, X0,
$ 1, ONE, DWORK(IY), 1 )
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', N,
$ A, LDA, X0, 1 )
C
DO 110 IX = 2, N
X0(IX) = X0(IX) + A(IX,IX-1)*DWORK(IXINIT+IX-2)
110 CONTINUE
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ U(IU,1), LDU, ONE, X0, 1 )
CALL DCOPY( N, X0, 1, DWORK(IXINIT), 1 )
IY = IY + L
IU = IU + 1
120 CONTINUE
C
END IF
C
END IF
C
ELSE
C
DO 140 K = 1, NOBS
CALL DGEMV( 'No transpose', L, N, -ONE, C, LDC, X0, 1,
$ ONE, DWORK(IY), 1 )
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', N, A,
$ LDA, X0, 1 )
C
DO 130 IX = 2, N
X0(IX) = X0(IX) + A(IX,IX-1)*DWORK(IXINIT+IX-2)
130 CONTINUE
C
CALL DCOPY( N, X0, 1, DWORK(IXINIT), 1 )
IY = IY + L
140 CONTINUE
C
END IF
C
C Compress the data using (sequential) QR factorization.
C Workspace: need v + 2*N;
C where v = s*L*(N + 1) + N + a + w.
C
JWORK = ITAU + N
IF ( FIRST ) THEN
C
C Compress the first data segment of Gamma.
C Workspace: need v + 2*N,
C prefer v + N + N*NB.
C
CALL DGEQRF( LDDW, N, DWORK(INIGAM), LDDW, DWORK(ITAU),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Apply the transformation to the right hand side part.
C Workspace: need v + N + 1,
C prefer v + N + NB.
C
CALL DORMQR( 'Left', 'Transpose', LDDW, 1, N, DWORK(INIGAM),
$ LDDW, DWORK(ITAU), DWORK(IRHS), LDDW,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
IF ( NCYC ) THEN
C
C Save the triangular factor of Gamma and the
C corresponding right hand side.
C
CALL DLACPY( 'Upper', N, NCP1, DWORK(INIGAM), LDDW,
$ DWORK(INIR), LDR )
END IF
ELSE
C
C Compress the current (but not the first) data segment of
C Gamma.
C Workspace: need v + N - 1.
C
CALL MB04OD( 'Full', N, 1, LDDW, DWORK(INIR), LDR,
$ DWORK(INIGAM), LDDW, DWORK(INIH), LDR,
$ DWORK(IRHS), LDDW, DWORK(ITAU), DWORK(JWORK) )
END IF
C
IUPNT = IUPNT + NOBS
IYPNT = IYPNT + NOBS
150 CONTINUE
C
C Estimate the reciprocal condition number of the triangular factor
C of the QR decomposition.
C Workspace: need u + 3*N, where
C u = t*L*(N + 1), if NCYCLE = 1;
C u = w, if NCYCLE > 1.
C
CALL DTRCON( '1-norm', 'Upper', 'No Transpose', N, DWORK(INIR),
$ LDR, RCOND, DWORK(IE), IWORK, IERR )
C
TOLL = TOL
IF ( TOLL.LE.ZERO )
$ TOLL = DLAMCH( 'Precision' )
IF ( RCOND.LE.TOLL**( TWO/THREE ) ) THEN
IWARN = 4
C
C The least squares problem is ill-conditioned.
C Use SVD to solve it.
C Workspace: need u + 6*N;
C prefer larger.
C
CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, DWORK(INIR+1),
$ LDR )
ISV = IE
JWORK = ISV + N
CALL DGELSS( N, N, 1, DWORK(INIR), LDR, DWORK(INIH), LDR,
$ DWORK(ISV), TOLL, RANK, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
IF ( IERR.GT.0 ) THEN
C
C Return if SVD algorithm did not converge.
C
INFO = 2
RETURN
END IF
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) - JWORK + 1 )
ELSE
C
C Find the least squares solution using QR decomposition only.
C
CALL DTRSV( 'Upper', 'No Transpose', 'Non Unit', N,
$ DWORK(INIR), LDR, DWORK(INIH), 1 )
END IF
C
C Return the estimated initial state of the system x0.
C
CALL DCOPY( N, DWORK(INIH), 1, X0, 1 )
C
DWORK(1) = MAXWRK
DWORK(2) = RCOND
C
RETURN
C
C *** End of IB01RD ***
END
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