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267 lines
8.3 KiB
Fortran
267 lines
8.3 KiB
Fortran
*DECK LMPAR
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SUBROUTINE LMPAR (N, R, LDR, IPVT, DIAG, QTB, DELTA, PAR, X,
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+ SIGMA, WA1, WA2)
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C***BEGIN PROLOGUE LMPAR
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C***SUBSIDIARY
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C***PURPOSE Subsidiary to SNLS1 and SNLS1E
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C***LIBRARY SLATEC
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C***TYPE SINGLE PRECISION (LMPAR-S, DMPAR-D)
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C***AUTHOR (UNKNOWN)
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C***DESCRIPTION
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C
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C Given an M by N matrix A, an N by N nonsingular DIAGONAL
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C matrix D, an M-vector B, and a positive number DELTA,
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C the problem is to determine a value for the parameter
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C PAR such that if X solves the system
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C
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C A*X = B , SQRT(PAR)*D*X = 0 ,
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C
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C in the least squares sense, and DXNORM is the Euclidean
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C norm of D*X, then either PAR is zero and
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C
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C (DXNORM-DELTA) .LE. 0.1*DELTA ,
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C
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C or PAR is positive and
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C
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C ABS(DXNORM-DELTA) .LE. 0.1*DELTA .
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C
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C This subroutine completes the solution of the problem
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C if it is provided with the necessary information from the
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C QR factorization, with column pivoting, of A. That is, if
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C A*P = Q*R, where P is a permutation matrix, Q has orthogonal
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C columns, and R is an upper triangular matrix with diagonal
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C elements of nonincreasing magnitude, then LMPAR expects
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C the full upper triangle of R, the permutation matrix P,
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C and the first N components of (Q TRANSPOSE)*B. On output
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C LMPAR also provides an upper triangular matrix S such that
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C
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C T T T
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C P *(A *A + PAR*D*D)*P = S *S .
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C
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C S is employed within LMPAR and may be of separate interest.
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C
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C Only a few iterations are generally needed for convergence
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C of the algorithm. If, however, the limit of 10 iterations
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C is reached, then the output PAR will contain the best
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C value obtained so far.
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C
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C The subroutine statement is
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C
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C SUBROUTINE LMPAR(N,R,LDR,IPVT,DIAG,QTB,DELTA,PAR,X,SIGMA,
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C WA1,WA2)
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C
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C where
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C
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C N is a positive integer input variable set to the order of R.
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C
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C R is an N by N array. On input the full upper triangle
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C must contain the full upper triangle of the matrix R.
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C On output the full upper triangle is unaltered, and the
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C strict lower triangle contains the strict upper triangle
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C (transposed) of the upper triangular matrix S.
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C
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C LDR is a positive integer input variable not less than N
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C which specifies the leading dimension of the array R.
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C
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C IPVT is an integer input array of length N which defines the
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C permutation matrix P such that A*P = Q*R. Column J of P
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C is column IPVT(J) of the identity matrix.
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C
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C DIAG is an input array of length N which must contain the
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C diagonal elements of the matrix D.
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C
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C QTB is an input array of length N which must contain the first
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C N elements of the vector (Q TRANSPOSE)*B.
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C
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C DELTA is a positive input variable which specifies an upper
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C bound on the Euclidean norm of D*X.
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C
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C PAR is a nonnegative variable. On input PAR contains an
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C initial estimate of the Levenberg-Marquardt parameter.
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C On output PAR contains the final estimate.
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C
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C X is an output array of length N which contains the least
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C squares solution of the system A*X = B, SQRT(PAR)*D*X = 0,
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C for the output PAR.
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C
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C SIGMA is an output array of length N which contains the
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C diagonal elements of the upper triangular matrix S.
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C
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C WA1 and WA2 are work arrays of length N.
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C
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C***SEE ALSO SNLS1, SNLS1E
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C***ROUTINES CALLED ENORM, QRSOLV, R1MACH
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C***REVISION HISTORY (YYMMDD)
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C 800301 DATE WRITTEN
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C 890531 Changed all specific intrinsics to generic. (WRB)
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C 890831 Modified array declarations. (WRB)
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 900326 Removed duplicate information from DESCRIPTION section.
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C (WRB)
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C 900328 Added TYPE section. (WRB)
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C***END PROLOGUE LMPAR
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INTEGER N,LDR
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INTEGER IPVT(*)
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REAL DELTA,PAR
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REAL R(LDR,*),DIAG(*),QTB(*),X(*),SIGMA(*),WA1(*),WA2(*)
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INTEGER I,ITER,J,JM1,JP1,K,L,NSING
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REAL DXNORM,DWARF,FP,GNORM,PARC,PARL,PARU,P1,P001,SUM,TEMP,ZERO
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REAL R1MACH,ENORM
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SAVE P1, P001, ZERO
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DATA P1,P001,ZERO /1.0E-1,1.0E-3,0.0E0/
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C***FIRST EXECUTABLE STATEMENT LMPAR
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DWARF = R1MACH(1)
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C
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C COMPUTE AND STORE IN X THE GAUSS-NEWTON DIRECTION. IF THE
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C JACOBIAN IS RANK-DEFICIENT, OBTAIN A LEAST SQUARES SOLUTION.
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C
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NSING = N
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DO 10 J = 1, N
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WA1(J) = QTB(J)
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IF (R(J,J) .EQ. ZERO .AND. NSING .EQ. N) NSING = J - 1
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IF (NSING .LT. N) WA1(J) = ZERO
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10 CONTINUE
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IF (NSING .LT. 1) GO TO 50
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DO 40 K = 1, NSING
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J = NSING - K + 1
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WA1(J) = WA1(J)/R(J,J)
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TEMP = WA1(J)
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JM1 = J - 1
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IF (JM1 .LT. 1) GO TO 30
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DO 20 I = 1, JM1
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WA1(I) = WA1(I) - R(I,J)*TEMP
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20 CONTINUE
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30 CONTINUE
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40 CONTINUE
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50 CONTINUE
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DO 60 J = 1, N
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L = IPVT(J)
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X(L) = WA1(J)
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60 CONTINUE
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C
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C INITIALIZE THE ITERATION COUNTER.
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C EVALUATE THE FUNCTION AT THE ORIGIN, AND TEST
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C FOR ACCEPTANCE OF THE GAUSS-NEWTON DIRECTION.
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C
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ITER = 0
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DO 70 J = 1, N
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WA2(J) = DIAG(J)*X(J)
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70 CONTINUE
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DXNORM = ENORM(N,WA2)
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FP = DXNORM - DELTA
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IF (FP .LE. P1*DELTA) GO TO 220
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C
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C IF THE JACOBIAN IS NOT RANK DEFICIENT, THE NEWTON
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C STEP PROVIDES A LOWER BOUND, PARL, FOR THE ZERO OF
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C THE FUNCTION. OTHERWISE SET THIS BOUND TO ZERO.
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C
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PARL = ZERO
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IF (NSING .LT. N) GO TO 120
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DO 80 J = 1, N
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L = IPVT(J)
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WA1(J) = DIAG(L)*(WA2(L)/DXNORM)
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80 CONTINUE
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DO 110 J = 1, N
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SUM = ZERO
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JM1 = J - 1
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IF (JM1 .LT. 1) GO TO 100
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DO 90 I = 1, JM1
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SUM = SUM + R(I,J)*WA1(I)
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90 CONTINUE
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100 CONTINUE
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WA1(J) = (WA1(J) - SUM)/R(J,J)
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110 CONTINUE
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TEMP = ENORM(N,WA1)
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PARL = ((FP/DELTA)/TEMP)/TEMP
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120 CONTINUE
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C
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C CALCULATE AN UPPER BOUND, PARU, FOR THE ZERO OF THE FUNCTION.
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C
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DO 140 J = 1, N
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SUM = ZERO
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DO 130 I = 1, J
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SUM = SUM + R(I,J)*QTB(I)
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130 CONTINUE
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L = IPVT(J)
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WA1(J) = SUM/DIAG(L)
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140 CONTINUE
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GNORM = ENORM(N,WA1)
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PARU = GNORM/DELTA
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IF (PARU .EQ. ZERO) PARU = DWARF/MIN(DELTA,P1)
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C
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C IF THE INPUT PAR LIES OUTSIDE OF THE INTERVAL (PARL,PARU),
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C SET PAR TO THE CLOSER ENDPOINT.
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C
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PAR = MAX(PAR,PARL)
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PAR = MIN(PAR,PARU)
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IF (PAR .EQ. ZERO) PAR = GNORM/DXNORM
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C
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C BEGINNING OF AN ITERATION.
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C
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150 CONTINUE
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ITER = ITER + 1
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C
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C EVALUATE THE FUNCTION AT THE CURRENT VALUE OF PAR.
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C
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IF (PAR .EQ. ZERO) PAR = MAX(DWARF,P001*PARU)
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TEMP = SQRT(PAR)
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DO 160 J = 1, N
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WA1(J) = TEMP*DIAG(J)
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160 CONTINUE
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CALL QRSOLV(N,R,LDR,IPVT,WA1,QTB,X,SIGMA,WA2)
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DO 170 J = 1, N
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WA2(J) = DIAG(J)*X(J)
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170 CONTINUE
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DXNORM = ENORM(N,WA2)
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TEMP = FP
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FP = DXNORM - DELTA
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C
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C IF THE FUNCTION IS SMALL ENOUGH, ACCEPT THE CURRENT VALUE
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C OF PAR. ALSO TEST FOR THE EXCEPTIONAL CASES WHERE PARL
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C IS ZERO OR THE NUMBER OF ITERATIONS HAS REACHED 10.
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C
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IF (ABS(FP) .LE. P1*DELTA
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1 .OR. PARL .EQ. ZERO .AND. FP .LE. TEMP
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2 .AND. TEMP .LT. ZERO .OR. ITER .EQ. 10) GO TO 220
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C
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C COMPUTE THE NEWTON CORRECTION.
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C
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DO 180 J = 1, N
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L = IPVT(J)
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WA1(J) = DIAG(L)*(WA2(L)/DXNORM)
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180 CONTINUE
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DO 210 J = 1, N
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WA1(J) = WA1(J)/SIGMA(J)
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TEMP = WA1(J)
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JP1 = J + 1
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IF (N .LT. JP1) GO TO 200
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DO 190 I = JP1, N
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WA1(I) = WA1(I) - R(I,J)*TEMP
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190 CONTINUE
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200 CONTINUE
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210 CONTINUE
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TEMP = ENORM(N,WA1)
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PARC = ((FP/DELTA)/TEMP)/TEMP
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C
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C DEPENDING ON THE SIGN OF THE FUNCTION, UPDATE PARL OR PARU.
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C
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IF (FP .GT. ZERO) PARL = MAX(PARL,PAR)
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IF (FP .LT. ZERO) PARU = MIN(PARU,PAR)
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C
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C COMPUTE AN IMPROVED ESTIMATE FOR PAR.
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C
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PAR = MAX(PARL,PAR+PARC)
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C
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C END OF AN ITERATION.
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C
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GO TO 150
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220 CONTINUE
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C
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C TERMINATION.
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C
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IF (ITER .EQ. 0) PAR = ZERO
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RETURN
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C
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C LAST CARD OF SUBROUTINE LMPAR.
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C
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END
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