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

       *        5X,27H FINAL APPROXIMATE SOLUTION // (5X,3D15.7))
  C
  C     LAST CARD OF DRIVER FOR HYBRD EXAMPLE.
  C
        END
        SUBROUTINE FCN(N,X,FVEC,IFLAG)
        INTEGER N,IFLAG
        DOUBLE PRECISION X(N),FVEC(N)
  C
  C     SUBROUTINE FCN FOR HYBRD EXAMPLE.
  C
        INTEGER K
        DOUBLE PRECISION ONE,TEMP,TEMP1,TEMP2,THREE,TWO,ZERO
        DATA ZERO,ONE,TWO,THREE /0.D0,1.D0,2.D0,3.D0/
  C
        IF (IFLAG .NE. 0) GO TO 5
  C
  C     INSERT PRINT STATEMENTS HERE WHEN NPRINT IS POSITIVE.
  C
        RETURN
      5 CONTINUE
        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
        RETURN
  C
  C     LAST CARD OF SUBROUTINE FCN.
  C
        END
        Results obtained with different compilers or machines
        may be slightly different.
        FINAL L2 NORM OF THE RESIDUALS  0.1192636D-07
        NUMBER OF FUNCTION EVALUATIONS        14
 
                                                                  Page
        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 HYBRJ1
                         Double precision version
                       Argonne National Laboratory
          Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
                                March 1980
 
  1. Purpose.
        The purpose of HYBRJ1 is to find a zero of a system of N non-
        linear functions in N variables by a modification of the Powell
        hybrid method.  This is done by using the more general nonlinear
        equation solver HYBRJ.  The user must provide a subroutine which
        calculates the functions and the Jacobian.
 
  2. Subroutine and type statements.
        SUBROUTINE HYBRJ1(FCN,N,X,FVEC,FJAC,LDFJAC,TOL,INFO,WA,LWA)
        INTEGER N,LDFJAC,INFO,LWA
        DOUBLE PRECISION TOL
        DOUBLE PRECISION X(N),FVEC(N),FJAC(LDFJAC,N),WA(LWA)
        EXTERNAL FCN
 
  3. Parameters.
        Parameters designated as input parameters must be specified on
        entry to HYBRJ1 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 HYBRJ1.
        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(N,X,FVEC,FJAC,LDFJAC,IFLAG)
          INTEGER N,LDFJAC,IFLAG
          DOUBLE PRECISION X(N),FVEC(N),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
          The value of IFLAG should not be changed by FCN unless the
 
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          user wants to terminate execution of HYBRJ1.  In this case set
          IFLAG to a negative integer.
        N is a positive integer input variable set to the number of
          functions and variables.
        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 N which contains the function
          evaluated at the output X.
        FJAC is an output N by N array which contains the orthogonal
          matrix Q produced by the QR factorization of the final approx-
          imate Jacobian.  Section 6 contains more details about the
          approximation to the Jacobian.
        LDFJAC is a positive integer input variable not less than N
          which specifies the leading dimension of the array FJAC.
        TOL is a nonnegative input variable.  Termination occurs when
          the algorithm estimates that the relative error between X and
          the solution is at most TOL.  Section 4 contains 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 between
                    X and the solution is at most TOL.
          INFO = 2  Number of calls to FCN with IFLAG = 1 has reached
                    100*(N+1).
          INFO = 3  TOL is too small.  No further improvement in the
                    approximate solution X is possible.
          INFO = 4  Iteration is not making good progress.
          Sections 4 and 5 contain more details about INFO.
        WA is a work array of length LWA.
        LWA is a positive integer input variable not less than
          (N*(N+13))/2.
 
  4. Successful completion.
        The accuracy of HYBRJ1 is controlled by the convergence
 
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        parameter TOL.  This parameter is used in a test which makes a
        comparison between the approximation X and a solution XSOL.
        HYBRJ1 terminates when the test is satisfied.  If TOL is less
        than the machine precision (as defined by the MINPACK function
        DPMPAR(1)), then HYBRJ1 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 test assumes that the functions and the Jacobian are coded
        consistently, and that the functions are reasonably well
        behaved.  If these conditions are not satisfied, then HYBRJ1 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 HYBRJ1 with a tighter toler-
        ance.
        Convergence test.  If ENORM(Z) denotes the Euclidean norm of a
          vector Z, then this test attempts to guarantee that
                ENORM(X-XSOL) .LE. TOL*ENORM(XSOL).
          If this condition is satisfied with TOL = 10**(-K), then the
          larger components of X have K significant decimal digits and
          INFO is set to 1.  There is a danger that the smaller compo-
          nents of X may have large relative errors, but the fast rate
          of convergence of HYBRJ1 usually avoids this possibility.
 
  5. Unsuccessful completion.
        Unsuccessful termination of HYBRJ1 can be due to improper input
        parameters, arithmetic interrupts, an excessive number of func-
        tion evaluations, or lack of good progress.
        Improper input parameters.  INFO is set to 0 if N .LE. 0, or
          LDFJAC .LT. N, or TOL .LT. 0.D0, or LWA .LT. (N*(N+13))/2.
        Arithmetic interrupts.  If these interrupts occur in the FCN
          subroutine during an early stage of the computation, they may
          be caused by an unacceptable choice of X by HYBRJ1.  In this
          case, it may be possible to remedy the situation by not evalu
          ating the functions here, but instead setting the components
          of FVEC to numbers that exceed those in the initial FVEC,
          thereby indirectly reducing the step length.  The step length
          can be more directly controlled by using instead HYBRJ, which
          includes in its calling sequence the step-length- governing
          parameter FACTOR.
        Excessive number of function evaluations.  If the number of
          calls to FCN with IFLAG = 1 reaches 100*(N+1), then this indi-
          cates that the routine is converging very slowly as measured
 
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          by the progress of FVEC, and INFO is set to 2.  This situation
          should be unusual because, as indicated below, lack of good
          progress is usually diagnosed earlier by HYBRJ1, causing ter-
          mination with INFO = 4.
        Lack of good progress.  HYBRJ1 searches for a zero of the system
          by minimizing the sum of the squares of the functions.  In so
          doing, it can become trapped in a region where the minimum
          does not correspond to a zero of the system and, in this situ-
          ation, the iteration eventually fails to make good progress.
          In particular, this will happen if the system does not have a
          zero.  If the system has a zero, rerunning HYBRJ1 from a dif-
          ferent starting point may be helpful.
 
  6. Characteristics of the algorithm.
        HYBRJ1 is a modification of the Powell hybrid method.  Two of
        its main characteristics involve the choice of the correction a
        a convex combination of the Newton and scaled gradient direc-
        tions, and the updating of the Jacobian by the rank-1 method of
        Broyden.  The choice of the correction guarantees (under reason
        able conditions) global convergence for starting points far fro
        the solution and a fast rate of convergence.  The Jacobian is
        calculated at the starting point, but it is not recalculated
        until the rank-1 method fails to produce satisfactory progress.
        Timing.  The time required by HYBRJ1 to solve a given problem
          depends on N, the behavior of the functions, the accuracy
          requested, and the starting point.  The number of arithmetic
          operations needed by HYBRJ1 is about 11.5*(N**2) to process
          each evaluation of the functions (call to FCN with IFLAG = 1)
          and 1.3*(N**3) to process each evaluation of the Jacobian
          (call to FCN with IFLAG = 2).  Unless FCN can be evaluated
          quickly, the timing of HYBRJ1 will be strongly influenced by
          the time spent in FCN.
        Storage.  HYBRJ1 requires (3*N**2 + 17*N)/2 double precision
          storage locations, in addition to the storage required by the
          program.  There are no internally declared storage arrays.
 
  7. Subprograms required.
        USER-supplied ...... FCN
        MINPACK-supplied ... DOGLEG,DPMPAR,ENORM,HYBRJ,
                             QFORM,QRFAC,R1MPYQ,R1UPDT
        FORTRAN-supplied ... DABS,DMAX1,DMIN1,DSQRT,MIN0,MOD
 
  8. References.
 
                                                                  Page
        M. J. D. Powell, A Hybrid Method for Nonlinear Equations.
        Numerical Methods for Nonlinear Algebraic Equations,
        P. Rabinowitz, editor. Gordon and Breach, 1970.
 
  9. Example.
        The problem is to determine the values of x(1), x(2), ..., x(9)
        which solve the system of tridiagonal equations
        (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 HYBRJ1 EXAMPLE.
  C     DOUBLE PRECISION VERSION
  C
  C     **********
        INTEGER J,N,LDFJAC,INFO,LWA,NWRITE
        DOUBLE PRECISION TOL,FNORM
        DOUBLE PRECISION X(9),FVEC(9),FJAC(9,9),WA(99)
        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
        LWA = 99
  C
  C     SET TOL TO THE SQUARE ROOT OF THE MACHINE PRECISION.
  C     UNLESS HIGH PRECISION SOLUTIONS ARE REQUIRED,
  C     THIS IS THE RECOMMENDED SETTING.
  C
        TOL = DSQRT(DPMPAR(1))
  C
        CALL HYBRJ1(FCN,N,X,FVEC,FJAC,LDFJAC,TOL,INFO,WA,LWA)
        FNORM = ENORM(N,FVEC)
        WRITE (NWRITE,1000) FNORM,INFO,(X(J),J=1,N)
        STOP
   1000 FORMAT (5X,31H FINAL L2 NORM OF THE RESIDUALS,D15.7 //
       *        5X,15H EXIT PARAMETER,16X,I10 //
       *        5X,27H FINAL APPROXIMATE SOLUTION // (5X,3D15.7))