*DECK DLSEI
SUBROUTINE DLSEI (W, MDW, ME, MA, MG, N, PRGOPT, X, RNORME,
+ RNORML, MODE, WS, IP)
C***BEGIN PROLOGUE DLSEI
C***PURPOSE Solve a linearly constrained least squares problem with
C equality and inequality constraints, and optionally compute
C a covariance matrix.
C***LIBRARY SLATEC
C***CATEGORY K1A2A, D9
C***TYPE DOUBLE PRECISION (LSEI-S, DLSEI-D)
C***KEYWORDS CONSTRAINED LEAST SQUARES, CURVE FITTING, DATA FITTING,
C EQUALITY CONSTRAINTS, INEQUALITY CONSTRAINTS,
C QUADRATIC PROGRAMMING
C***AUTHOR Hanson, R. J., (SNLA)
C Haskell, K. H., (SNLA)
C***DESCRIPTION
C
C Abstract
C
C This subprogram solves a linearly constrained least squares
C problem with both equality and inequality constraints, and, if the
C user requests, obtains a covariance matrix of the solution
C parameters.
C
C Suppose there are given matrices E, A and G of respective
C dimensions ME by N, MA by N and MG by N, and vectors F, B and H of
C respective lengths ME, MA and MG. This subroutine solves the
C linearly constrained least squares problem
C
C EX = F, (E ME by N) (equations to be exactly
C satisfied)
C AX = B, (A MA by N) (equations to be
C approximately satisfied,
C least squares sense)
C GX .GE. H,(G MG by N) (inequality constraints)
C
C The inequalities GX .GE. H mean that every component of the
C product GX must be .GE. the corresponding component of H.
C
C In case the equality constraints cannot be satisfied, a
C generalized inverse solution residual vector length is obtained
C for F-EX. This is the minimal length possible for F-EX.
C
C Any values ME .GE. 0, MA .GE. 0, or MG .GE. 0 are permitted. The
C rank of the matrix E is estimated during the computation. We call
C this value KRANKE. It is an output parameter in IP(1) defined
C below. Using a generalized inverse solution of EX=F, a reduced
C least squares problem with inequality constraints is obtained.
C The tolerances used in these tests for determining the rank
C of E and the rank of the reduced least squares problem are
C given in Sandia Tech. Rept. SAND-78-1290. They can be
C modified by the user if new values are provided in
C the option list of the array PRGOPT(*).
C
C The user must dimension all arrays appearing in the call list..
C W(MDW,N+1),PRGOPT(*),X(N),WS(2*(ME+N)+K+(MG+2)*(N+7)),IP(MG+2*N+2)
C where K=MAX(MA+MG,N). This allows for a solution of a range of
C problems in the given working space. The dimension of WS(*)
C given is a necessary overestimate. Once a particular problem
C has been run, the output parameter IP(3) gives the actual
C dimension required for that problem.
C
C The parameters for DLSEI( ) are
C
C Input.. All TYPE REAL variables are DOUBLE PRECISION
C
C W(*,*),MDW, The array W(*,*) is doubly subscripted with
C ME,MA,MG,N first dimensioning parameter equal to MDW.
C For this discussion let us call M = ME+MA+MG. Then
C MDW must satisfy MDW .GE. M. The condition
C MDW .LT. M is an error.
C
C The array W(*,*) contains the matrices and vectors
C
C (E F)
C (A B)
C (G H)
C
C in rows and columns 1,...,M and 1,...,N+1
C respectively.
C
C The integers ME, MA, and MG are the
C respective matrix row dimensions
C of E, A and G. Each matrix has N columns.
C
C PRGOPT(*) This real-valued array is the option vector.
C If the user is satisfied with the nominal
C subprogram features set
C
C PRGOPT(1)=1 (or PRGOPT(1)=1.0)
C
C Otherwise PRGOPT(*) is a linked list consisting of
C groups of data of the following form
C
C LINK
C KEY
C DATA SET
C
C The parameters LINK and KEY are each one word.
C The DATA SET can be comprised of several words.
C The number of items depends on the value of KEY.
C The value of LINK points to the first
C entry of the next group of data within
C PRGOPT(*). The exception is when there are
C no more options to change. In that
C case, LINK=1 and the values KEY and DATA SET
C are not referenced. The general layout of
C PRGOPT(*) is as follows.
C
C ...PRGOPT(1) = LINK1 (link to first entry of next group)
C . PRGOPT(2) = KEY1 (key to the option change)
C . PRGOPT(3) = data value (data value for this change)
C . .
C . .
C . .
C ...PRGOPT(LINK1) = LINK2 (link to the first entry of
C . next group)
C . PRGOPT(LINK1+1) = KEY2 (key to the option change)
C . PRGOPT(LINK1+2) = data value
C ... .
C . .
C . .
C ...PRGOPT(LINK) = 1 (no more options to change)
C
C Values of LINK that are nonpositive are errors.
C A value of LINK .GT. NLINK=100000 is also an error.
C This helps prevent using invalid but positive
C values of LINK that will probably extend
C beyond the program limits of PRGOPT(*).
C Unrecognized values of KEY are ignored. The
C order of the options is arbitrary and any number
C of options can be changed with the following
C restriction. To prevent cycling in the
C processing of the option array, a count of the
C number of options changed is maintained.
C Whenever this count exceeds NOPT=1000, an error
C message is printed and the subprogram returns.
C
C Options..
C
C KEY=1
C Compute in W(*,*) the N by N
C covariance matrix of the solution variables
C as an output parameter. Nominally the
C covariance matrix will not be computed.
C (This requires no user input.)
C The data set for this option is a single value.
C It must be nonzero when the covariance matrix
C is desired. If it is zero, the covariance
C matrix is not computed. When the covariance matrix
C is computed, the first dimensioning parameter
C of the array W(*,*) must satisfy MDW .GE. MAX(M,N).
C
C KEY=10
C Suppress scaling of the inverse of the
C normal matrix by the scale factor RNORM**2/
C MAX(1, no. of degrees of freedom). This option
C only applies when the option for computing the
C covariance matrix (KEY=1) is used. With KEY=1 and
C KEY=10 used as options the unscaled inverse of the
C normal matrix is returned in W(*,*).
C The data set for this option is a single value.
C When it is nonzero no scaling is done. When it is
C zero scaling is done. The nominal case is to do
C scaling so if option (KEY=1) is used alone, the
C matrix will be scaled on output.
C
C KEY=2
C Scale the nonzero columns of the
C entire data matrix.
C (E)
C (A)
C (G)
C
C to have length one. The data set for this
C option is a single value. It must be
C nonzero if unit length column scaling
C is desired.
C
C KEY=3
C Scale columns of the entire data matrix
C (E)
C (A)
C (G)
C
C with a user-provided diagonal matrix.
C The data set for this option consists
C of the N diagonal scaling factors, one for
C each matrix column.
C
C KEY=4
C Change the rank determination tolerance for
C the equality constraint equations from
C the nominal value of SQRT(DRELPR). This quantity can
C be no smaller than DRELPR, the arithmetic-
C storage precision. The quantity DRELPR is the
C largest positive number such that T=1.+DRELPR
C satisfies T .EQ. 1. The quantity used
C here is internally restricted to be at
C least DRELPR. The data set for this option
C is the new tolerance.
C
C KEY=5
C Change the rank determination tolerance for
C the reduced least squares equations from
C the nominal value of SQRT(DRELPR). This quantity can
C be no smaller than DRELPR, the arithmetic-
C storage precision. The quantity used
C here is internally restricted to be at
C least DRELPR. The data set for this option
C is the new tolerance.
C
C For example, suppose we want to change
C the tolerance for the reduced least squares
C problem, compute the covariance matrix of
C the solution parameters, and provide
C column scaling for the data matrix. For
C these options the dimension of PRGOPT(*)
C must be at least N+9. The Fortran statements
C defining these options would be as follows:
C
C PRGOPT(1)=4 (link to entry 4 in PRGOPT(*))
C PRGOPT(2)=1 (covariance matrix key)
C PRGOPT(3)=1 (covariance matrix wanted)
C
C PRGOPT(4)=7 (link to entry 7 in PRGOPT(*))
C PRGOPT(5)=5 (least squares equas. tolerance key)
C PRGOPT(6)=... (new value of the tolerance)
C
C PRGOPT(7)=N+9 (link to entry N+9 in PRGOPT(*))
C PRGOPT(8)=3 (user-provided column scaling key)
C
C CALL DCOPY (N, D, 1, PRGOPT(9), 1) (Copy the N
C scaling factors from the user array D(*)
C to PRGOPT(9)-PRGOPT(N+8))
C
C PRGOPT(N+9)=1 (no more options to change)
C
C The contents of PRGOPT(*) are not modified
C by the subprogram.
C The options for WNNLS( ) can also be included
C in this array. The values of KEY recognized
C by WNNLS( ) are 6, 7 and 8. Their functions
C are documented in the usage instructions for
C subroutine WNNLS( ). Normally these options
C do not need to be modified when using DLSEI( ).
C
C IP(1), The amounts of working storage actually
C IP(2) allocated for the working arrays WS(*) and
C IP(*), respectively. These quantities are
C compared with the actual amounts of storage
C needed by DLSEI( ). Insufficient storage
C allocated for either WS(*) or IP(*) is an
C error. This feature was included in DLSEI( )
C because miscalculating the storage formulas
C for WS(*) and IP(*) might very well lead to
C subtle and hard-to-find execution errors.
C
C The length of WS(*) must be at least
C
C LW = 2*(ME+N)+K+(MG+2)*(N+7)
C
C where K = max(MA+MG,N)
C This test will not be made if IP(1).LE.0.
C
C The length of IP(*) must be at least
C
C LIP = MG+2*N+2
C This test will not be made if IP(2).LE.0.
C
C Output.. All TYPE REAL variables are DOUBLE PRECISION
C
C X(*),RNORME, The array X(*) contains the solution parameters
C RNORML if the integer output flag MODE = 0 or 1.
C The definition of MODE is given directly below.
C When MODE = 0 or 1, RNORME and RNORML
C respectively contain the residual vector
C Euclidean lengths of F - EX and B - AX. When
C MODE=1 the equality constraint equations EX=F
C are contradictory, so RNORME .NE. 0. The residual
C vector F-EX has minimal Euclidean length. For
C MODE .GE. 2, none of these parameters is defined.
C
C MODE Integer flag that indicates the subprogram
C status after completion. If MODE .GE. 2, no
C solution has been computed.
C
C MODE =
C
C 0 Both equality and inequality constraints
C are compatible and have been satisfied.
C
C 1 Equality constraints are contradictory.
C A generalized inverse solution of EX=F was used
C to minimize the residual vector length F-EX.
C In this sense, the solution is still meaningful.
C
C 2 Inequality constraints are contradictory.
C
C 3 Both equality and inequality constraints
C are contradictory.
C
C The following interpretation of
C MODE=1,2 or 3 must be made. The
C sets consisting of all solutions
C of the equality constraints EX=F
C and all vectors satisfying GX .GE. H
C have no points in common. (In
C particular this does not say that
C each individual set has no points
C at all, although this could be the
C case.)
C
C 4 Usage error occurred. The value
C of MDW is .LT. ME+MA+MG, MDW is
C .LT. N and a covariance matrix is
C requested, or the option vector
C PRGOPT(*) is not properly defined,
C or the lengths of the working arrays
C WS(*) and IP(*), when specified in
C IP(1) and IP(2) respectively, are not
C long enough.
C
C W(*,*) The array W(*,*) contains the N by N symmetric
C covariance matrix of the solution parameters,
C provided this was requested on input with
C the option vector PRGOPT(*) and the output
C flag is returned with MODE = 0 or 1.
C
C IP(*) The integer working array has three entries
C that provide rank and working array length
C information after completion.
C
C IP(1) = rank of equality constraint
C matrix. Define this quantity
C as KRANKE.
C
C IP(2) = rank of reduced least squares
C problem.
C
C IP(3) = the amount of storage in the
C working array WS(*) that was
C actually used by the subprogram.
C The formula given above for the length
C of WS(*) is a necessary overestimate.
C If exactly the same problem matrices
C are used in subsequent executions,
C the declared dimension of WS(*) can
C be reduced to this output value.
C User Designated
C Working Arrays..
C
C WS(*),IP(*) These are respectively type real
C and type integer working arrays.
C Their required minimal lengths are
C given above.
C
C***REFERENCES K. H. Haskell and R. J. Hanson, An algorithm for
C linear least squares problems with equality and
C nonnegativity constraints, Report SAND77-0552, Sandia
C Laboratories, June 1978.
C K. H. Haskell and R. J. Hanson, Selected algorithms for
C the linearly constrained least squares problem - a
C users guide, Report SAND78-1290, Sandia Laboratories,
C August 1979.
C K. H. Haskell and R. J. Hanson, An algorithm for
C linear least squares problems with equality and
C nonnegativity constraints, Mathematical Programming
C 21 (1981), pp. 98-118.
C R. J. Hanson and K. H. Haskell, Two algorithms for the
C linearly constrained least squares problem, ACM
C Transactions on Mathematical Software, September 1982.
C***ROUTINES CALLED D1MACH, DASUM, DAXPY, DCOPY, DDOT, DH12, DLSI,
C DNRM2, DSCAL, DSWAP, XERMSG
C***REVISION HISTORY (YYMMDD)
C 790701 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890618 Completely restructured and extensively revised (WRB & RWC)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 900510 Convert XERRWV calls to XERMSG calls. (RWC)
C 900604 DP version created from SP version. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DLSEI
INTEGER IP(3), MA, MDW, ME, MG, MODE, N
DOUBLE PRECISION PRGOPT(*), RNORME, RNORML, W(MDW,*), WS(*), X(*)
C
EXTERNAL D1MACH, DASUM, DAXPY, DCOPY, DDOT, DH12, DLSI, DNRM2,
* DSCAL, DSWAP, XERMSG
DOUBLE PRECISION D1MACH, DASUM, DDOT, DNRM2
C
DOUBLE PRECISION DRELPR, ENORM, FNORM, GAM, RB, RN, RNMAX, SIZE,
* SN, SNMAX, T, TAU, UJ, UP, VJ, XNORM, XNRME
INTEGER I, IMAX, J, JP1, K, KEY, KRANKE, LAST, LCHK, LINK, M,
* MAPKE1, MDEQC, MEND, MEP1, N1, N2, NEXT, NLINK, NOPT, NP1,
* NTIMES
LOGICAL COV, FIRST
CHARACTER*8 XERN1, XERN2, XERN3, XERN4
SAVE FIRST, DRELPR
C
DATA FIRST /.TRUE./
C***FIRST EXECUTABLE STATEMENT DLSEI
C
C Set the nominal tolerance used in the code for the equality
C constraint equations.
C
IF (FIRST) DRELPR = D1MACH(4)
FIRST = .FALSE.
TAU = SQRT(DRELPR)
C
C Check that enough storage was allocated in WS(*) and IP(*).
C
MODE = 4
IF (MIN(N,ME,MA,MG) .LT. 0) THEN
WRITE (XERN1, '(I8)') N
WRITE (XERN2, '(I8)') ME
WRITE (XERN3, '(I8)') MA
WRITE (XERN4, '(I8)') MG
CALL XERMSG ('SLATEC', 'LSEI', 'ALL OF THE VARIABLES N, ME,' //
* ' MA, MG MUST BE .GE. 0$$ENTERED ROUTINE WITH' //
* '$$N = ' // XERN1 //
* '$$ME = ' // XERN2 //
* '$$MA = ' // XERN3 //
* '$$MG = ' // XERN4, 2, 1)
RETURN
ENDIF
C
IF (IP(1).GT.0) THEN
LCHK = 2*(ME+N) + MAX(MA+MG,N) + (MG+2)*(N+7)
IF (IP(1).LT.LCHK) THEN
WRITE (XERN1, '(I8)') LCHK
CALL XERMSG ('SLATEC', 'DLSEI', 'INSUFFICIENT STORAGE ' //
* 'ALLOCATED FOR WS(*), NEED LW = ' // XERN1, 2, 1)
RETURN
ENDIF
ENDIF
C
IF (IP(2).GT.0) THEN
LCHK = MG + 2*N + 2
IF (IP(2).LT.LCHK) THEN
WRITE (XERN1, '(I8)') LCHK
CALL XERMSG ('SLATEC', 'DLSEI', 'INSUFFICIENT STORAGE ' //
* 'ALLOCATED FOR IP(*), NEED LIP = ' // XERN1, 2, 1)
RETURN
ENDIF
ENDIF
C
C Compute number of possible right multiplying Householder
C transformations.
C
M = ME + MA + MG
IF (N.LE.0 .OR. M.LE.0) THEN
MODE = 0
RNORME = 0
RNORML = 0
RETURN
ENDIF
C
IF (MDW.LT.M) THEN
CALL XERMSG ('SLATEC', 'DLSEI', 'MDW.LT.ME+MA+MG IS AN ERROR',
+ 2, 1)
RETURN
ENDIF
C
NP1 = N + 1
KRANKE = MIN(ME,N)
N1 = 2*KRANKE + 1
N2 = N1 + N
C
C Set nominal values.
C
C The nominal column scaling used in the code is
C the identity scaling.
C
CALL DCOPY (N, 1.D0, 0, WS(N1), 1)
C
C No covariance matrix is nominally computed.
C
COV = .FALSE.
C
C Process option vector.
C Define bound for number of options to change.
C
NOPT = 1000
NTIMES = 0
C
C Define bound for positive values of LINK.
C
NLINK = 100000
LAST = 1
LINK = PRGOPT(1)
IF (LINK.EQ.0 .OR. LINK.GT.NLINK) THEN
CALL XERMSG ('SLATEC', 'DLSEI',
+ 'THE OPTION VECTOR IS UNDEFINED', 2, 1)
RETURN
ENDIF
C
100 IF (LINK.GT.1) THEN
NTIMES = NTIMES + 1
IF (NTIMES.GT.NOPT) THEN
CALL XERMSG ('SLATEC', 'DLSEI',
+ 'THE LINKS IN THE OPTION VECTOR ARE CYCLING.', 2, 1)
RETURN
ENDIF
C
KEY = PRGOPT(LAST+1)
IF (KEY.EQ.1) THEN
COV = PRGOPT(LAST+2) .NE. 0.D0
ELSEIF (KEY.EQ.2 .AND. PRGOPT(LAST+2).NE.0.D0) THEN
DO 110 J = 1,N
T = DNRM2(M,W(1,J),1)
IF (T.NE.0.D0) T = 1.D0/T
WS(J+N1-1) = T
110 CONTINUE
ELSEIF (KEY.EQ.3) THEN
CALL DCOPY (N, PRGOPT(LAST+2), 1, WS(N1), 1)
ELSEIF (KEY.EQ.4) THEN
TAU = MAX(DRELPR,PRGOPT(LAST+2))
ENDIF
C
NEXT = PRGOPT(LINK)
IF (NEXT.LE.0 .OR. NEXT.GT.NLINK) THEN
CALL XERMSG ('SLATEC', 'DLSEI',
+ 'THE OPTION VECTOR IS UNDEFINED', 2, 1)
RETURN
ENDIF
C
LAST = LINK
LINK = NEXT
GO TO 100
ENDIF
C
DO 120 J = 1,N
CALL DSCAL (M, WS(N1+J-1), W(1,J), 1)
120 CONTINUE
C
IF (COV .AND. MDW.LT.N) THEN
CALL XERMSG ('SLATEC', 'DLSEI',
+ 'MDW .LT. N WHEN COV MATRIX NEEDED, IS AN ERROR', 2, 1)
RETURN
ENDIF
C
C Problem definition and option vector OK.
C
MODE = 0
C
C Compute norm of equality constraint matrix and right side.
C
ENORM = 0.D0
DO 130 J = 1,N
ENORM = MAX(ENORM,DASUM(ME,W(1,J),1))
130 CONTINUE
C
FNORM = DASUM(ME,W(1,NP1),1)
SNMAX = 0.D0
RNMAX = 0.D0
DO 150 I = 1,KRANKE
C
C Compute maximum ratio of vector lengths. Partition is at
C column I.
C
DO 140 K = I,ME
SN = DDOT(N-I+1,W(K,I),MDW,W(K,I),MDW)
RN = DDOT(I-1,W(K,1),MDW,W(K,1),MDW)
IF (RN.EQ.0.D0 .AND. SN.GT.SNMAX) THEN
SNMAX = SN
IMAX = K
ELSEIF (K.EQ.I .OR. SN*RNMAX.GT.RN*SNMAX) THEN
SNMAX = SN
RNMAX = RN
IMAX = K
ENDIF
140 CONTINUE
C
C Interchange rows if necessary.
C
IF (I.NE.IMAX) CALL DSWAP (NP1, W(I,1), MDW, W(IMAX,1), MDW)
IF (SNMAX.GT.RNMAX*TAU**2) THEN
C
C Eliminate elements I+1,...,N in row I.
C
CALL DH12 (1, I, I+1, N, W(I,1), MDW, WS(I), W(I+1,1), MDW,
+ 1, M-I)
ELSE
KRANKE = I - 1
GO TO 160
ENDIF
150 CONTINUE
C
C Save diagonal terms of lower trapezoidal matrix.
C
160 CALL DCOPY (KRANKE, W, MDW+1, WS(KRANKE+1), 1)
C
C Use Householder transformation from left to achieve
C KRANKE by KRANKE upper triangular form.
C
IF (KRANKE.LT.ME) THEN
DO 170 K = KRANKE,1,-1
C
C Apply transformation to matrix cols. 1,...,K-1.
C
CALL DH12 (1, K, KRANKE+1, ME, W(1,K), 1, UP, W, 1, MDW,
* K-1)
C
C Apply to rt side vector.
C
CALL DH12 (2, K, KRANKE+1, ME, W(1,K), 1, UP, W(1,NP1), 1,
+ 1, 1)
170 CONTINUE
ENDIF
C
C Solve for variables 1,...,KRANKE in new coordinates.
C
CALL DCOPY (KRANKE, W(1, NP1), 1, X, 1)
DO 180 I = 1,KRANKE
X(I) = (X(I)-DDOT(I-1,W(I,1),MDW,X,1))/W(I,I)
180 CONTINUE
C
C Compute residuals for reduced problem.
C
MEP1 = ME + 1
RNORML = 0.D0
DO 190 I = MEP1,M
W(I,NP1) = W(I,NP1) - DDOT(KRANKE,W(I,1),MDW,X,1)
SN = DDOT(KRANKE,W(I,1),MDW,W(I,1),MDW)
RN = DDOT(N-KRANKE,W(I,KRANKE+1),MDW,W(I,KRANKE+1),MDW)
IF (RN.LE.SN*TAU**2 .AND. KRANKE.LT.N)
* CALL DCOPY (N-KRANKE, 0.D0, 0, W(I,KRANKE+1), MDW)
190 CONTINUE
C
C Compute equality constraint equations residual length.
C
RNORME = DNRM2(ME-KRANKE,W(KRANKE+1,NP1),1)
C
C Move reduced problem data upward if KRANKE.LT.ME.
C
IF (KRANKE.LT.ME) THEN
DO 200 J = 1,NP1
CALL DCOPY (M-ME, W(ME+1,J), 1, W(KRANKE+1,J), 1)
200 CONTINUE
ENDIF
C
C Compute solution of reduced problem.
C
CALL DLSI(W(KRANKE+1, KRANKE+1), MDW, MA, MG, N-KRANKE, PRGOPT,
+ X(KRANKE+1), RNORML, MODE, WS(N2), IP(2))
C
C Test for consistency of equality constraints.
C
IF (ME.GT.0) THEN
MDEQC = 0
XNRME = DASUM(KRANKE,W(1,NP1),1)
IF (RNORME.GT.TAU*(ENORM*XNRME+FNORM)) MDEQC = 1
MODE = MODE + MDEQC
C
C Check if solution to equality constraints satisfies inequality
C constraints when there are no degrees of freedom left.
C
IF (KRANKE.EQ.N .AND. MG.GT.0) THEN
XNORM = DASUM(N,X,1)
MAPKE1 = MA + KRANKE + 1
MEND = MA + KRANKE + MG
DO 210 I = MAPKE1,MEND
SIZE = DASUM(N,W(I,1),MDW)*XNORM + ABS(W(I,NP1))
IF (W(I,NP1).GT.TAU*SIZE) THEN
MODE = MODE + 2
GO TO 290
ENDIF
210 CONTINUE
ENDIF
ENDIF
C
C Replace diagonal terms of lower trapezoidal matrix.
C
IF (KRANKE.GT.0) THEN
CALL DCOPY (KRANKE, WS(KRANKE+1), 1, W, MDW+1)
C
C Reapply transformation to put solution in original coordinates.
C
DO 220 I = KRANKE,1,-1
CALL DH12 (2, I, I+1, N, W(I,1), MDW, WS(I), X, 1, 1, 1)
220 CONTINUE
C
C Compute covariance matrix of equality constrained problem.
C
IF (COV) THEN
DO 270 J = MIN(KRANKE,N-1),1,-1
RB = WS(J)*W(J,J)
IF (RB.NE.0.D0) RB = 1.D0/RB
JP1 = J + 1
DO 230 I = JP1,N
W(I,J) = RB*DDOT(N-J,W(I,JP1),MDW,W(J,JP1),MDW)
230 CONTINUE
C
GAM = 0.5D0*RB*DDOT(N-J,W(JP1,J),1,W(J,JP1),MDW)
CALL DAXPY (N-J, GAM, W(J,JP1), MDW, W(JP1,J), 1)
DO 250 I = JP1,N
DO 240 K = I,N
W(I,K) = W(I,K) + W(J,I)*W(K,J) + W(I,J)*W(J,K)
W(K,I) = W(I,K)
240 CONTINUE
250 CONTINUE
UJ = WS(J)
VJ = GAM*UJ
W(J,J) = UJ*VJ + UJ*VJ
DO 260 I = JP1,N
W(J,I) = UJ*W(I,J) + VJ*W(J,I)
260 CONTINUE
CALL DCOPY (N-J, W(J, JP1), MDW, W(JP1,J), 1)
270 CONTINUE
ENDIF
ENDIF
C
C Apply the scaling to the covariance matrix.
C
IF (COV) THEN
DO 280 I = 1,N
CALL DSCAL (N, WS(I+N1-1), W(I,1), MDW)
CALL DSCAL (N, WS(I+N1-1), W(1,I), 1)
280 CONTINUE
ENDIF
C
C Rescale solution vector.
C
290 IF (MODE.LE.1) THEN
DO 300 J = 1,N
X(J) = X(J)*WS(N1+J-1)
300 CONTINUE
ENDIF
C
IP(1) = KRANKE
IP(3) = IP(3) + 2*KRANKE + N
RETURN
END