minpack-documentation.txt

该程序实现了非线性最小二乘问题和非线性方程组的解法

        (3-2*x(1))*x(1)           -2*x(2)                   = -1
                -x(i-1) + (3-2*x(i))*x(i)         -2*x(i+1) = -1, i=2-8
                                    -x(8) + (3-2*x(9))*x(9) = -1
  C     **********
  C
  C     DRIVER FOR HYBRJ EXAMPLE.
  C     DOUBLE PRECISION VERSION
  C
  C     **********
        INTEGER J,N,LDFJAC,MAXFEV,MODE,NPRINT,INFO,NFEV,NJEV,LR,NWRITE
        DOUBLE PRECISION XTOL,FACTOR,FNORM
        DOUBLE PRECISION X(9),FVEC(9),FJAC(9,9),DIAG(9),R(45),QTF(9),
       *                 WA1(9),WA2(9),WA3(9),WA4(9)
        DOUBLE PRECISION ENORM,DPMPAR
        EXTERNAL FCN
  C
  C     LOGICAL OUTPUT UNIT IS ASSUMED TO BE NUMBER 6.
  C
        DATA NWRITE /6/
  C
        N = 9
  C
  C     THE FOLLOWING STARTING VALUES PROVIDE A ROUGH SOLUTION.
  C
        DO 10 J = 1, 9
           X(J) = -1.D0
     10    CONTINUE
  C
        LDFJAC = 9
        LR = 45
  C
  C     SET XTOL TO THE SQUARE ROOT OF THE MACHINE PRECISION.
  C     UNLESS HIGH PRECISION SOLUTIONS ARE REQUIRED,
  C     THIS IS THE RECOMMENDED SETTING.
  C
        XTOL = DSQRT(DPMPAR(1))
  C
        MAXFEV = 1000
        MODE = 2
        DO 20 J = 1, 9
           DIAG(J) = 1.D0
     20    CONTINUE
        FACTOR = 1.D2
        NPRINT = 0
  C
        CALL HYBRJ(FCN,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,DIAG,
       *           MODE,FACTOR,NPRINT,INFO,NFEV,NJEV,R,LR,QTF,
       *           WA1,WA2,WA3,WA4)
        FNORM = ENORM(N,FVEC)
        WRITE (NWRITE,1000) FNORM,NFEV,NJEV,INFO,(X(J),J=1,N)
 
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        STOP
   1000 FORMAT (5X,31H FINAL L2 NORM OF THE RESIDUALS,D15.7 //
       *        5X,31H NUMBER OF FUNCTION EVALUATIONS,I10 //
       *        5X,31H NUMBER OF JACOBIAN EVALUATIONS,I10 //
       *        5X,15H EXIT PARAMETER,16X,I10 //
       *        5X,27H FINAL APPROXIMATE SOLUTION // (5X,3D15.7))
  C
  C     LAST CARD OF DRIVER FOR HYBRJ EXAMPLE.
  C
        END
        SUBROUTINE FCN(N,X,FVEC,FJAC,LDFJAC,IFLAG)
        INTEGER N,LDFJAC,IFLAG
        DOUBLE PRECISION X(N),FVEC(N),FJAC(LDFJAC,N)
  C
  C     SUBROUTINE FCN FOR HYBRJ EXAMPLE.
  C
        INTEGER J,K
        DOUBLE PRECISION ONE,TEMP,TEMP1,TEMP2,THREE,TWO,ZERO
        DATA ZERO,ONE,TWO,THREE,FOUR /0.D0,1.D0,2.D0,3.D0,4.D0/
  C
        IF (IFLAG .NE. 0) GO TO 5
  C
  C     INSERT PRINT STATEMENTS HERE WHEN NPRINT IS POSITIVE.
  C
        RETURN
      5 CONTINUE
        IF (IFLAG .EQ. 2) GO TO 20
        DO 10 K = 1, N
           TEMP = (THREE - TWO*X(K))*X(K)
           TEMP1 = ZERO
           IF (K .NE. 1) TEMP1 = X(K-1)
           TEMP2 = ZERO
           IF (K .NE. N) TEMP2 = X(K+1)
           FVEC(K) = TEMP - TEMP1 - TWO*TEMP2 + ONE
     10    CONTINUE
        GO TO 50
     20 CONTINUE
        DO 40 K = 1, N
           DO 30 J = 1, N
              FJAC(K,J) = ZERO
     30       CONTINUE
           FJAC(K,K) = THREE - FOUR*X(K)
           IF (K .NE. 1) FJAC(K,K-1) = -ONE
           IF (K .NE. N) FJAC(K,K+1) = -TWO
     40    CONTINUE
     50 CONTINUE
        RETURN
  C
  C     LAST CARD OF SUBROUTINE FCN.
  C
        END
        Results obtained with different compilers or machines
        may be slightly different.
 
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        FINAL L2 NORM OF THE RESIDUALS  0.1192636D-07
        NUMBER OF FUNCTION EVALUATIONS        11
        NUMBER OF JACOBIAN EVALUATIONS         1
        EXIT PARAMETER                         1
        FINAL APPROXIMATE SOLUTION
        -0.5706545D+00 -0.6816283D+00 -0.7017325D+00
        -0.7042129D+00 -0.7013690D+00 -0.6918656D+00
        -0.6657920D+00 -0.5960342D+00 -0.4164121D+00
 
 
 
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               Documentation for MINPACK subroutine LMDER1
                         Double precision version
                       Argonne National Laboratory
          Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
                                March 1980
 
  1. Purpose.
        The purpose of LMDER1 is to minimize the sum of the squares of
        nonlinear functions in N variables by a modification of the
        Levenberg-Marquardt algorithm.  This is done by using the more
        general least-squares solver LMDER.  The user must provide a
        subroutine which calculates the functions and the Jacobian.
 
  2. Subroutine and type statements.
        SUBROUTINE LMDER1(FCN,M,N,X,FVEC,FJAC,LDFJAC,TOL,
       *                  INFO,IPVT,WA,LWA)
        INTEGER M,N,LDFJAC,INFO,LWA
        INTEGER IPVT(N)
        DOUBLE PRECISION TOL
        DOUBLE PRECISION X(N),FVEC(M),FJAC(LDFJAC,N),WA(LWA)
        EXTERNAL FCN
 
  3. Parameters.
        Parameters designated as input parameters must be specified on
        entry to LMDER1 and are not changed on exit, while parameters
        designated as output parameters need not be specified on entry
        and are set to appropriate values on exit from LMDER1.
        FCN is the name of the user-supplied subroutine which calculate
          the functions and the Jacobian.  FCN must be declared in an
          EXTERNAL statement in the user calling program, and should be
          written as follows.
          SUBROUTINE FCN(M,N,X,FVEC,FJAC,LDFJAC,IFLAG)
          INTEGER M,N,LDFJAC,IFLAG
          DOUBLE PRECISION X(N),FVEC(M),FJAC(LDFJAC,N)
          ----------
          IF IFLAG = 1 CALCULATE THE FUNCTIONS AT X AND
          RETURN THIS VECTOR IN FVEC.  DO NOT ALTER FJAC.
          IF IFLAG = 2 CALCULATE THE JACOBIAN AT X AND
          RETURN THIS MATRIX IN FJAC.  DO NOT ALTER FVEC.
          ----------
          RETURN
          END
 
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          The value of IFLAG should not be changed by FCN unless the
          user wants to terminate execution of LMDER1.  In this case set
          IFLAG to a negative integer.
        M is a positive integer input variable set to the number of
          functions.
        N is a positive integer input variable set to the number of
          variables.  N must not exceed M.
        X is an array of length N.  On input X must contain an initial
          estimate of the solution vector.  On output X contains the
          final estimate of the solution vector.
        FVEC is an output array of length M which contains the function
          evaluated at the output X.
        FJAC is an output M by N array.  The upper N by N submatrix of
          FJAC contains an upper triangular matrix R with diagonal ele-
          ments of nonincreasing magnitude such that
                 T     T           T
                P *(JAC *JAC)*P = R *R,
          where P is a permutation matrix and JAC is the final calcu-
          lated Jacobian.  Column j of P is column IPVT(j) (see below)
          of the identity matrix.  The lower trapezoidal part of FJAC
          contains information generated during the computation of R.
        LDFJAC is a positive integer input variable not less than M
          which specifies the leading dimension of the array FJAC.
        TOL is a nonnegative input variable.  Termination occurs when
          the algorithm estimates either that the relative error in the
          sum of squares is at most TOL or that the relative error
          between X and the solution is at most TOL.  Section 4 contain
          more details about TOL.
        INFO is an integer output variable.  If the user has terminated
          execution, INFO is set to the (negative) value of IFLAG.  See
          description of FCN.  Otherwise, INFO is set as follows.
          INFO = 0  Improper input parameters.
          INFO = 1  Algorithm estimates that the relative error in the
                    sum of squares is at most TOL.
          INFO = 2  Algorithm estimates that the relative error between
                    X and the solution is at most TOL.
          INFO = 3  Conditions for INFO = 1 and INFO = 2 both hold.
          INFO = 4  FVEC is orthogonal to the columns of the Jacobian t
                    machine precision.
 
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          INFO = 5  Number of calls to FCN with IFLAG = 1 has reached
                    100*(N+1).
          INFO = 6  TOL is too small.  No further reduction in the sum
                    of squares is possible.
          INFO = 7  TOL is too small.  No further improvement in the
                    approximate solution X is possible.
          Sections 4 and 5 contain more details about INFO.
        IPVT is an integer output array of length N.  IPVT defines a
          permutation matrix P such that JAC*P = Q*R, where JAC is the
          final calculated Jacobian, Q is orthogonal (not stored), and
          is upper triangular with diagonal elements of nonincreasing
          magnitude.  Column j of P is column IPVT(j) of the identity
          matrix.
        WA is a work array of length LWA.
        LWA is a positive integer input variable not less than 5*N+M.
 
  4. Successful completion.
        The accuracy of LMDER1 is controlled by the convergence parame-
        ter TOL.  This parameter is used in tests which make three type
        of comparisons between the approximation X and a solution XSOL.
        LMDER1 terminates when any of the tests is satisfied.  If TOL i
        less than the machine precision (as defined by the MINPACK func-
        tion DPMPAR(1)), then LMDER1 only attempts to satisfy the test
        defined by the machine precision.  Further progress is not usu-
        ally possible.  Unless high precision solutions are required,
        the recommended value for TOL is the square root of the machine
        precision.
        The tests assume that the functions and the Jacobian are coded
        consistently, and that the functions are reasonably well
        behaved.  If these conditions are not satisfied, then LMDER1 ma
        incorrectly indicate convergence.  The coding of the Jacobian
        can be checked by the MINPACK subroutine CHKDER.  If the Jaco-
        bian is coded correctly, then the validity of the answer can be
        checked, for example, by rerunning LMDER1 with a tighter toler-
        ance.
        First convergence test.  If ENORM(Z) denotes the Euclidean norm
          of a vector Z, then this test attempts to guarantee that
                ENORM(FVEC) .LE. (1+TOL)*ENORM(FVECS),
          where FVECS denotes the functions evaluated at XSOL.  If this
          condition is satisfied with TOL = 10**(-K), then the final
          residual norm ENORM(FVEC) has K significant decimal digits an
          INFO is set to 1 (or to 3 if the second test is also
 
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          satisfied).
        Second convergence test.  If D is a diagonal matrix (implicitly
          generated by LMDER1) whose entries contain scale factors for
          the variables, then this test attempts to guarantee that
                ENORM(D*(X-XSOL)) .LE. TOL*ENORM(D*XSOL).
          If this condition is satisfied with TOL = 10**(-K), then the
          larger components of D*X have K significant decimal digits an
          INFO is set to 2 (or to 3 if the first test is also satis-
          fied).  There is a danger that the smaller components of D*X
          may have large relative errors, but the choice of D is such
          that the accuracy of the components of X is usually related t
          their sensitivity.
        Third convergence test.  This test is satisfied when FVEC is
          orthogonal to the columns of the Jacobian to machine preci-
          sion.  There is no clear relationship between this test and
          the accuracy of LMDER1, and furthermore, the test is equally
          well satisfied at other critical points, namely maximizers an
          saddle points.  Therefore, termination caused by this test
          (INFO = 4) should be examined carefully.
 
  5. Unsuccessful completion.
        Unsuccessful termination of LMDER1 can be due to improper in