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|
SUBROUTINE MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q,
$ LDQ, R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK,
$ 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 compute a low rank QR factorization with column pivoting of a
C K*M-by-L*N block Toeplitz matrix T with blocks of size (K,L);
C specifically,
C T
C T P = Q R ,
C
C where R is lower trapezoidal, P is a block permutation matrix
C and Q^T Q = I. The number of columns in R is equivalent to the
C numerical rank of T with respect to the given tolerance TOL1.
C Note that the pivoting scheme is local, i.e., only columns
C belonging to the same block in T are permuted.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOB CHARACTER*1
C Specifies the output of the routine as follows:
C = 'Q': computes Q and R;
C = 'R': only computes R.
C
C Input/Output Parameters
C
C K (input) INTEGER
C The number of rows in one block of T. K >= 0.
C
C L (input) INTEGER
C The number of columns in one block of T. L >= 0.
C
C M (input) INTEGER
C The number of blocks in one block column of T. M >= 0.
C
C N (input) INTEGER
C The number of blocks in one block row of T. N >= 0.
C
C TC (input) DOUBLE PRECISION array, dimension (LDTC, L)
C The leading M*K-by-L part of this array must contain
C the first block column of T.
C
C LDTC INTEGER
C The leading dimension of the array TC.
C LDTC >= MAX(1,M*K).
C
C TR (input) DOUBLE PRECISION array, dimension (LDTR,(N-1)*L)
C The leading K-by-(N-1)*L part of this array must contain
C the first block row of T without the leading K-by-L
C block.
C
C LDTR INTEGER
C The leading dimension of the array TR. LDTR >= MAX(1,K).
C
C RNK (output) INTEGER
C The number of columns in R, which is equivalent to the
C numerical rank of T.
C
C Q (output) DOUBLE PRECISION array, dimension (LDQ,RNK)
C If JOB = 'Q', then the leading M*K-by-RNK part of this
C array contains the factor Q.
C If JOB = 'R', then this array is not referenced.
C
C LDQ INTEGER
C The leading dimension of the array Q.
C LDQ >= MAX(1,M*K), if JOB = 'Q';
C LDQ >= 1, if JOB = 'R'.
C
C R (output) DOUBLE PRECISION array, dimension (LDR,RNK)
C The leading N*L-by-RNK part of this array contains the
C lower trapezoidal factor R.
C
C LDR INTEGER
C The leading dimension of the array R.
C LDR >= MAX(1,N*L)
C
C JPVT (output) INTEGER array, dimension (MIN(M*K,N*L))
C This array records the column pivoting performed.
C If JPVT(j) = k, then the j-th column of T*P was
C the k-th column of T.
C
C Tolerances
C
C TOL1 DOUBLE PRECISION
C If TOL1 >= 0.0, the user supplied diagonal tolerance;
C if TOL1 < 0.0, a default diagonal tolerance is used.
C
C TOL2 DOUBLE PRECISION
C If TOL2 >= 0.0, the user supplied offdiagonal tolerance;
C if TOL2 < 0.0, a default offdiagonal tolerance is used.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK; DWORK(2) and DWORK(3) return the used values
C for TOL1 and TOL2, respectively.
C On exit, if INFO = -19, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX( 3, ( M*K + ( N - 1 )*L )*( L + 2*K ) + 9*L
C + MAX(M*K,(N-1)*L) ), if JOB = 'Q';
C LDWORK >= MAX( 3, ( N - 1 )*L*( L + 2*K + 1 ) + 9*L,
C M*K*( L + 1 ) + L ), if JOB = 'R'.
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 = 1: due to perturbations induced by roundoff errors, or
C removal of nearly linearly dependent columns of the
C generator, the Schur algorithm encountered a
C situation where a diagonal element in the negative
C generator is larger in magnitude than the
C corresponding diagonal element in the positive
C generator (modulo TOL1);
C = 2: due to perturbations induced by roundoff errors, or
C removal of nearly linearly dependent columns of the
C generator, the Schur algorithm encountered a
C situation where diagonal elements in the positive
C and negative generator are equal in magnitude
C (modulo TOL1), but the offdiagonal elements suggest
C that these columns are not linearly dependent
C (modulo TOL2*ABS(diagonal element)).
C
C METHOD
C
C Householder transformations and modified hyperbolic rotations
C are used in the Schur algorithm [1], [2].
C If, during the process, the hyperbolic norm of a row in the
C leading part of the generator is found to be less than or equal
C to TOL1, then this row is not reduced. If the difference of the
C corresponding columns has a norm less than or equal to TOL2 times
C the magnitude of the leading element, then this column is removed
C from the generator, as well as from R. Otherwise, the algorithm
C breaks down. TOL1 is set to norm(TC)*sqrt(eps) and TOL2 is set
C to N*L*sqrt(eps) by default.
C If M*K > L, the columns of T are permuted so that the diagonal
C elements in one block column of R have decreasing magnitudes.
C
C REFERENCES
C
C [1] Kailath, T. and Sayed, A.
C Fast Reliable Algorithms for Matrices with Structure.
C SIAM Publications, Philadelphia, 1999.
C
C [2] Kressner, D. and Van Dooren, P.
C Factorizations and linear system solvers for matrices with
C Toeplitz structure.
C SLICOT Working Note 2000-2, 2000.
C
C NUMERICAL ASPECTS
C
C The algorithm requires 0(K*RNK*L*M*N) floating point operations.
C
C CONTRIBUTOR
C
C D. Kressner, Technical Univ. Berlin, Germany, May 2001.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, June 2001.
C D. Kressner, Technical Univ. Berlin, Germany, July 2002.
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2004.
C
C KEYWORDS
C
C Elementary matrix operations, Householder transformation, matrix
C operations, Toeplitz matrix.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, K, L, LDQ, LDR, LDTC, LDTR, LDWORK, M, N,
$ RNK
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,*), R(LDR,*), TC(LDTC,*),
$ TR(LDTR,*)
INTEGER JPVT(*)
C .. Local Scalars ..
LOGICAL COMPQ, LAST
INTEGER CPCOL, GAP, I, IERR, J, JJ, JWORK, KK, LEN, MK,
$ NZC, PDP, PDQ, PDW, PNQ, PNR, PP, PPR, PT, RDEF,
$ RRDF, RRNK, WRKMIN, WRKOPT
DOUBLE PRECISION LTOL1, LTOL2, NRM, TEMP
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL DLAMCH, DNRM2, LSAME
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEQP3, DGEQRF, DLACPY, DLASET,
$ DORGQR, DSCAL, DSWAP, DTRMV, MA02AD, MB02CU,
$ MB02CV, MB02KD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
C
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
INFO = 0
WRKOPT = 3
MK = M*K
COMPQ = LSAME( JOB, 'Q' )
IF ( COMPQ ) THEN
WRKMIN = MAX( 3, ( MK + ( N - 1 )*L )*( L + 2*K ) + 9*L +
$ MAX( MK, ( N - 1 )*L ) )
ELSE
WRKMIN = MAX( 3, MAX ( ( N - 1 )*L*( L + 2*K + 1 ) + 9*L,
$ MK*( L + 1 ) + L ) )
END IF
C
C Check the scalar input parameters.
C
IF ( .NOT.( COMPQ .OR. LSAME( JOB, 'R' ) ) ) THEN
INFO = -1
ELSE IF ( K.LT.0 ) THEN
INFO = -2
ELSE IF ( L.LT.0 ) THEN
INFO = -3
ELSE IF ( M.LT.0 ) THEN
INFO = -4
ELSE IF ( N.LT.0 ) THEN
INFO = -5
ELSE IF ( LDTC.LT.MAX( 1, MK ) ) THEN
INFO = -7
ELSE IF ( LDTR.LT.MAX( 1, K ) ) THEN
INFO = -9
ELSE IF ( LDQ.LT.1 .OR. ( COMPQ .AND. LDQ.LT.MK ) ) THEN
INFO = -12
ELSE IF ( LDR.LT.MAX( 1, N*L ) ) THEN
INFO = -14
ELSE IF ( LDWORK.LT.WRKMIN ) THEN
DWORK(1) = DBLE( WRKMIN )
INFO = -19
END IF
C
C Return if there were illegal values.
C
IF ( INFO.NE.0 ) THEN
CALL XERBLA( 'MB02JX', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MIN( M, N, K, L ).EQ.0 ) THEN
RNK = 0
DWORK(1) = DBLE( WRKOPT )
DWORK(2) = ZERO
DWORK(3) = ZERO
RETURN
END IF
C
WRKOPT = WRKMIN
C
IF ( MK.LE.L ) THEN
C
C Catch M*K <= L.
C
CALL DLACPY( 'All', MK, L, TC, LDTC, DWORK, MK )
PDW = MK*L + 1
JWORK = PDW + MK
CALL DGEQRF( MK, L, DWORK, MK, DWORK(PDW), DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1 )
CALL MA02AD( 'Upper part', MK, L, DWORK, MK, R, LDR )
CALL DORGQR( MK, MK, MK, DWORK, MK, DWORK(PDW),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1 )
IF ( COMPQ )
$ CALL DLACPY( 'All', MK, MK, DWORK, MK, Q, LDQ )
PDW = MK*MK + 1
IF ( N.GT.1 ) THEN
CALL MB02KD( 'Row', 'Transpose', K, L, M, N-1, MK, ONE,
$ ZERO, TC, LDTC, TR, LDTR, DWORK, MK, R(L+1,1),
$ LDR, DWORK(PDW), LDWORK-PDW+1, IERR )
END IF
WRKOPT = MAX( WRKOPT, INT( DWORK(PDW) ) + PDW - 1 )
C
DO 10 I = 1, MK
JPVT(I) = I
10 CONTINUE
C
RNK = MK
DWORK(1) = DBLE( WRKOPT )
DWORK(2) = ZERO
DWORK(3) = ZERO
RETURN
END IF
C
C Compute the generator:
C
C 1st column of the generator.
C
DO 20 I = 1, L
JPVT(I) = 0
20 CONTINUE
C
LTOL1 = TOL1
LTOL2 = TOL2
C
IF ( COMPQ ) THEN
CALL DLACPY( 'All', MK, L, TC, LDTC, Q, LDQ )
CALL DGEQP3( MK, L, Q, LDQ, JPVT, DWORK, DWORK(L+1),
$ LDWORK-L, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(L+1) ) + L )
C
IF ( LTOL1.LT.ZERO ) THEN
C
C Compute default tolerance LTOL1.
C
C Estimate the 2-norm of the first block column of the
C matrix with 5 power iterations.
C
TEMP = ONE / SQRT( DBLE( L ) )
CALL DLASET( 'All', L, 1, TEMP, TEMP, DWORK(L+1), 1 )
C
DO 30 I = 1, 5
CALL DTRMV( 'Upper', 'NonTranspose', 'NonUnit', L, Q,
$ LDQ, DWORK(L+1), 1 )
CALL DTRMV( 'Upper', 'Transpose', 'NonUnit', L, Q, LDQ,
$ DWORK(L+1), 1 )
NRM = DNRM2( L, DWORK(L+1), 1 )
CALL DSCAL( L, ONE/NRM, DWORK(L+1), 1 )
30 CONTINUE
C
LTOL1 = SQRT( NRM*DLAMCH( 'Epsilon' ) )
END IF
C
I = L
C
40 CONTINUE
IF ( ABS( Q(I,I) ).LE.LTOL1 ) THEN
I = I - 1
IF ( I.GT.0 ) GO TO 40
END IF
C
RRNK = I
RRDF = L - RRNK
CALL MA02AD( 'Upper', RRNK, L, Q, LDQ, R, LDR )
IF ( RRNK.GT.1 )
$ CALL DLASET( 'Upper', L-1, RRNK-1, ZERO, ZERO, R(1,2), LDR )
CALL DORGQR( MK, L, RRNK, Q, LDQ, DWORK, DWORK(L+1), LDWORK-L,
$ IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(L+1) ) + L )
IF ( N.GT.1 ) THEN
CALL MB02KD( 'Row', 'Transpose', K, L, M, N-1, RRNK, ONE,
$ ZERO, TC, LDTC, TR, LDTR, Q, LDQ, R(L+1,1),
$ LDR, DWORK, LDWORK, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
END IF
C
ELSE
C
PDW = MK*L + 1
JWORK = PDW + L
CALL DLACPY( 'All', MK, L, TC, LDTC, DWORK, MK )
CALL DGEQP3( MK, L, DWORK, MK, JPVT, DWORK(PDW),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1 )
C
IF ( LTOL1.LT.ZERO ) THEN
C
C Compute default tolerance LTOL1.
C
C Estimate the 2-norm of the first block column of the
C matrix with 5 power iterations.
C
TEMP = ONE / SQRT( DBLE( L ) )
CALL DLASET( 'All', L, 1, TEMP, TEMP, DWORK(JWORK), 1 )
C
DO 50 I = 1, 5
CALL DTRMV( 'Upper', 'NonTranspose', 'NonUnit', L, DWORK,
$ MK, DWORK(JWORK), 1 )
CALL DTRMV( 'Upper', 'Transpose', 'NonUnit', L, DWORK,
$ MK, DWORK(JWORK), 1 )
NRM = DNRM2( L, DWORK(JWORK), 1 )
CALL DSCAL( L, ONE/NRM, DWORK(JWORK), 1 )
50 CONTINUE
C
LTOL1 = SQRT( NRM*DLAMCH( 'Epsilon' ) )
END IF
C
RRNK = L
I = ( L - 1 )*MK + L
C
60 CONTINUE
IF ( ABS( DWORK(I) ).LE.LTOL1 ) THEN
RRNK = RRNK - 1
I = I - MK - 1
IF ( I.GT.0 ) GO TO 60
END IF
C
RRDF = L - RRNK
CALL MA02AD( 'Upper part', RRNK, L, DWORK, MK, R, LDR )
IF ( RRNK.GT.1 )
$ CALL DLASET( 'Upper', L-1, RRNK-1, ZERO, ZERO, R(1,2), LDR )
CALL DORGQR( MK, L, RRNK, DWORK, MK, DWORK(PDW),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1 )
IF ( N.GT.1 ) THEN
CALL MB02KD( 'Row', 'Transpose', K, L, M, N-1, RRNK, ONE,
$ ZERO, TC, LDTC, TR, LDTR, DWORK, MK, R(L+1,1),
$ LDR, DWORK(PDW), LDWORK-PDW+1, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(PDW) ) + PDW - 1 )
END IF
END IF
C
C Quick return if N = 1.
C
IF ( N.EQ.1 ) THEN
RNK = RRNK
DWORK(1) = DBLE( WRKOPT )
DWORK(2) = LTOL1
DWORK(3) = ZERO
RETURN
END IF
C
C Compute default tolerance LTOL2.
C
IF ( LTOL2.LT.ZERO )
$ LTOL2 = DBLE( N*L )*SQRT( DLAMCH( 'Epsilon' ) )
C
DO 70 J = 1, L
CALL DCOPY( RRNK, R(J,1), LDR, R(L+JPVT(J),RRNK+1), LDR )
70 CONTINUE
C
IF ( N.GT.2 )
$ CALL DLACPY( 'All', (N-2)*L, RRNK, R(L+1,1), LDR,
$ R(2*L+1,RRNK+1), LDR )
C
C 2nd column of the generator.
C
IF ( RRDF.GT.0 )
$ CALL MA02AD( 'All', MIN( RRDF, K ), (N-1)*L, TR, LDTR,
$ R(L+1,2*RRNK+1), LDR )
IF ( K.GT.RRDF )
$ CALL MA02AD( 'All', K-RRDF, (N-1)*L, TR(RRDF+1,1), LDTR, DWORK,
$ (N-1)*L )
C
C 3rd column of the generator.
C
PNR = ( N - 1 )*L*MAX( 0, K-RRDF ) + 1
CALL DLACPY( 'All', (N-1)*L, RRNK, R(L+1,1), LDR, DWORK(PNR),
$ (N-1)*L )
C
C 4th column of the generator.
C
PDW = PNR + ( N - 1 )*L*RRNK
PT = ( M - 1 )*K + 1
C
DO 80 I = 1, MIN( M, N-1 )
CALL MA02AD( 'All', K, L, TC(PT,1), LDTC, DWORK(PDW), (N-1)*L )
PT = PT - K
PDW = PDW + L
80 CONTINUE
C
PT = 1
C
DO 90 I = M + 1, N - 1
CALL MA02AD( 'All', K, L, TR(1,PT), LDTR, DWORK(PDW), (N-1)*L )
PT = PT + L
PDW = PDW + L
90 CONTINUE
C
IF ( COMPQ ) THEN
PDQ = PNR + ( N - 1 )*L*( RRNK + K )
PNQ = PDQ + MK*MAX( 0, K-RRDF )
PDW = PNQ + MK*( RRNK + K )
CALL DLACPY( 'All', MK, RRNK, Q, LDQ, DWORK(PNQ), MK )
IF ( M.GT.1 )
$ CALL DLACPY( 'All', (M-1)*K, RRNK, Q, LDQ, Q(K+1,RRNK+1),
$ LDQ )
CALL DLASET( 'All', K, RRNK, ZERO, ZERO, Q(1,RRNK+1), LDQ )
IF ( RRDF.GT.0 )
$ CALL DLASET( 'All', MK, RRDF, ZERO, ONE, Q(1,2*RRNK+1),
$ LDQ )
CALL DLASET( 'All', RRDF, MAX( 0, K-RRDF ), ZERO, ZERO,
$ DWORK(PDQ), MK )
CALL DLASET( 'All', M*K-RRDF, MAX( 0, K-RRDF ), ZERO, ONE,
$ DWORK(PDQ+RRDF), MK )
CALL DLASET( 'All', MK, K, ZERO, ZERO, DWORK(PNQ+MK*RRNK), MK )
ELSE
PDW = PNR + ( N - 1 )*L*( RRNK + K )
END IF
PPR = 1
RNK = RRNK
RDEF = RRDF
LEN = N*L
GAP = N*L - MIN( N*L, MK )
C
C KK is the number of columns in the leading part of the
C generator. After sufficiently many rank drops or if
C M*K < N*L it may be less than L.
C
KK = MIN( L+K-RDEF, L )
KK = MIN( KK, MK-L )
C
C Generator reduction process.
C
DO 190 I = L + 1, MIN( MK, N*L ), L
IF ( I+L.LE.MIN( MK, N*L ) ) THEN
LAST = .FALSE.
ELSE
LAST = .TRUE.
END IF
PP = KK + MAX( K - RDEF, 0 )
LEN = LEN - L
CALL MB02CU( 'Deficient', KK, PP, L+K-RDEF, -1, R(I,RNK+1),
$ LDR, DWORK(PPR), (N-1)*L, DWORK(PNR), (N-1)*L,
$ RRNK, JPVT(I), DWORK(PDW), LTOL1, DWORK(PDW+5*L),
$ LDWORK-PDW-5*L+1, IERR )
IF ( IERR.NE.0 ) THEN
C
C Error return: The current generator is indefinite.
C
INFO = 1
RETURN
END IF
C
C Apply pivoting to other columns of R.
C
PDP = PDW + 6*L - I
C
DO 100 J = I, I + KK - 1
JPVT(J) = JPVT(J) + I - 1
DWORK(PDP+JPVT(J)) = DBLE(J)
100 CONTINUE
C
DO 120 J = I, I + KK - 1
TEMP = DBLE(J)
JJ = J-1
C
110 CONTINUE
JJ = JJ + 1
IF ( DWORK(PDP+JJ).NE.TEMP ) GO TO 110
C
IF ( JJ.NE.J ) THEN
DWORK(PDP+JJ) = DWORK(PDP+J)
CALL DSWAP( RNK, R(J,1), LDR, R(JJ,1), LDR )
END IF
120 CONTINUE
C
DO 130 J = I + KK, I + L - 1
JPVT(J) = J
130 CONTINUE
C
C Apply reduction to other rows of R.
C
IF ( LEN.GT.KK ) THEN
CALL MB02CV( 'Deficient', 'NoStructure', KK, LEN-KK, PP,
$ L+K-RDEF, -1, RRNK, R(I,RNK+1), LDR,
$ DWORK(PPR), (N-1)*L, DWORK(PNR), (N-1)*L,
$ R(I+KK,RNK+1), LDR, DWORK(PPR+KK), (N-1)*L,
$ DWORK(PNR+KK), (N-1)*L, DWORK(PDW),
$ DWORK(PDW+5*L), LDWORK-PDW-5*L+1, IERR )
END IF
C
C Apply reduction to Q.
C
IF ( COMPQ ) THEN
CALL MB02CV( 'Deficient', 'NoStructure', KK, MK, PP,
$ L+K-RDEF, -1, RRNK, R(I,RNK+1), LDR,
$ DWORK(PPR), (N-1)*L, DWORK(PNR), (N-1)*L,
$ Q(1,RNK+1), LDQ, DWORK(PDQ), MK, DWORK(PNQ),
$ MK, DWORK(PDW), DWORK(PDW+5*L),
$ LDWORK-PDW-5*L+1, IERR )
END IF
C
C Inspection of the rank deficient columns:
C Look for small diagonal entries.
C
NZC = 0
C
DO 140 J = KK, RRNK + 1, -1
IF ( ABS( R(I+J-1,RNK+J) ).LE.LTOL1 ) NZC = NZC + 1
140 CONTINUE
C
C The last NZC columns of the generator cannot be removed.
C Now, decide whether for the other rank deficient columns
C it is safe to remove.
C
PT = PNR
C
DO 150 J = RRNK + 1, KK - NZC
TEMP = R(I+J-1,RNK+J)
CALL DSCAL( LEN-J-GAP, TEMP, R(I+J,RNK+J), 1 )
CALL DAXPY( LEN-J-GAP, -DWORK(PT+J-1), DWORK(PT+J), 1,
$ R(I+J,RNK+J), 1 )
IF ( DNRM2( LEN-J-GAP, R(I+J,RNK+J), 1 )
$ .GT.LTOL2*ABS( TEMP ) ) THEN
C
C Unlucky case:
C It is neither advisable to remove the whole column nor
C possible to remove the diagonal entries by Hyperbolic
C rotations.
C
INFO = 2
RETURN
END IF
PT = PT + ( N - 1 )*L
150 CONTINUE
C
C Annihilate unwanted elements in the factor R.
C
RRDF = KK - RRNK
CALL DLASET( 'All', I-1, RRNK, ZERO, ZERO, R(1,RNK+1), LDR )
CALL DLASET( 'Upper', L-1, RRNK-1, ZERO, ZERO, R(I,RNK+2),
$ LDR )
C
C Construct the generator for the next step.
C
IF ( .NOT.LAST ) THEN
C
C Compute KK for the next step.
C
KK = MIN( L+K-RDEF-RRDF+NZC, L )
KK = MIN( KK, MK-I-L+1 )
C
IF ( KK.LE.0 ) THEN
RNK = RNK + RRNK
GO TO 200
END IF
C
CALL DLASET( 'All', L, RRDF, ZERO, ZERO, R(I,RNK+RRNK+1),
$ LDR )
C
C The columns with small diagonal entries form parts of the
C new positive generator.
C
IF ( ( RRDF-NZC ).GT.0 .AND. NZC.GT.0 ) THEN
CPCOL = MIN( NZC, KK )
C
DO 160 J = RNK + RRNK + 1, RNK + RRNK + CPCOL
CALL DCOPY( LEN-L, R(I+L,J+RRDF-NZC), 1,
$ R(I+L,J), 1 )
160 CONTINUE
C
END IF
C
C Construct the leading parts of the positive generator.
C
CPCOL = MIN( RRNK, KK-NZC )
IF ( CPCOL.GT.0 ) THEN
C
DO 170 J = I, I + L - 1
CALL DCOPY( CPCOL, R(J,RNK+1), LDR,
$ R(JPVT(J)+L,RNK+RRNK+NZC+1), LDR )
170 CONTINUE
C
IF ( LEN.GT.2*L ) THEN
CALL DLACPY( 'All', LEN-2*L, CPCOL, R(I+L,RNK+1), LDR,
$ R(I+2*L,RNK+RRNK+NZC+1), LDR )
END IF
END IF
PPR = PPR + L
C
C Refill the leading parts of the positive generator.
C
CPCOL = MIN( K-RDEF, KK-RRNK-NZC )
IF ( CPCOL.GT.0 ) THEN
CALL DLACPY( 'All', LEN-L, CPCOL, DWORK(PPR), (N-1)*L,
$ R(I+L,RNK+2*RRNK+NZC+1), LDR )
PPR = PPR + CPCOL*( N - 1 )*L
END IF
PNR = PNR + ( RRDF - NZC )*( N - 1 )*L + L
C
C Do the same things for Q.
C
IF ( COMPQ ) THEN
IF ( ( RRDF - NZC ).GT.0 .AND. NZC.GT.0 ) THEN
CPCOL = MIN( NZC, KK )
C
DO 180 J = RNK + RRNK + 1, RNK + RRNK + CPCOL
CALL DCOPY( MK, Q(1,J+RRDF-NZC), 1, Q(1,J), 1 )
180 CONTINUE
C
END IF
CPCOL = MIN( RRNK, KK-NZC )
IF ( CPCOL.GT.0 ) THEN
CALL DLASET( 'All', K, CPCOL, ZERO, ZERO,
$ Q(1,RNK+RRNK+NZC+1), LDQ )
IF ( M.GT.1 )
$ CALL DLACPY( 'All', (M-1)*K, CPCOL, Q(1,RNK+1),
$ LDQ, Q(K+1,RNK+RRNK+NZC+1), LDQ )
END IF
CPCOL = MIN( K-RDEF, KK-RRNK-NZC )
IF ( CPCOL.GT.0 ) THEN
CALL DLACPY( 'All', MK, CPCOL, DWORK(PDQ), MK,
$ Q(1,RNK+2*RRNK+NZC+1), LDQ )
PDQ = PDQ + CPCOL*MK
END IF
PNQ = PNQ + ( RRDF - NZC )*MK
END IF
END IF
RNK = RNK + RRNK
RDEF = RDEF + RRDF - NZC
190 CONTINUE
C
200 CONTINUE
DWORK(1) = DBLE( WRKOPT )
DWORK(2) = LTOL1
DWORK(3) = LTOL2
C
C *** Last line of MB02JX ***
END
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