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

 
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  C
  C     LAST CARD OF DRIVER FOR HYBRJ1 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 HYBRJ1 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 .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.
        FINAL L2 NORM OF THE RESIDUALS  0.1192636D-07
        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 HYBRJ
                         Double precision version
                       Argonne National Laboratory
          Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
                                March 1980
 
  1. Purpose.
        The purpose of HYBRJ is to find a zero of a system of N non-
        linear functions in N variables by a modification of the Powell
        hybrid method.  The user must provide a subroutine which calcu-
        lates the functions and the Jacobian.
 
  2. Subroutine and type statements.
        SUBROUTINE HYBRJ(FCN,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,DIAG,
       *                 MODE,FACTOR,NPRINT,INFO,NFEV,NJEV,R,LR,QTF,
       *                 WA1,WA2,WA3,WA4)
        INTEGER N,LDFJAC,MAXFEV,MODE,NPRINT,INFO,NFEV,NJEV,LR
        DOUBLE PRECISION XTOL,FACTOR
        DOUBLE PRECISION X(N),FVEC(N),FJAC(LDFJAC,N),DIAG(N),R(LR),QTF(
       *                 WA1(N),WA2(N),WA3(N),WA4(N)
 
  3. Parameters.
        Parameters designated as input parameters must be specified on
        entry to HYBRJ 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 HYBRJ.
        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
 
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          The value of IFLAG should not be changed by FCN unless the
          user wants to terminate execution of HYBRJ.  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.
        XTOL is a nonnegative input variable.  Termination occurs when
          the relative error between two consecutive iterates is at most
          XTOL.  Therefore, XTOL measures the relative error desired in
          the approximate solution.  Section 4 contains more details
          about XTOL.
        MAXFEV is a positive integer input variable.  Termination occur
          when the number of calls to FCN with IFLAG = 1 has reached
          MAXFEV.
        DIAG is an array of length N.  If MODE = 1 (see below), DIAG is
          internally set.  If MODE = 2, DIAG must contain positive
          entries that serve as multiplicative scale factors for the
          variables.
        MODE is an integer input variable.  If MODE = 1, the variables
          will be scaled internally.  If MODE = 2, the scaling is speci-
          fied by the input DIAG.  Other values of MODE are equivalent
          to MODE = 1.
        FACTOR is a positive input variable used in determining the ini-
          tial step bound.  This bound is set to the product of FACTOR
          and the Euclidean norm of DIAG*X if nonzero, or else to FACTO
          itself.  In most cases FACTOR should lie in the interval
          (.1,100.).  100. is a generally recommended value.
        NPRINT is an integer input variable that enables controlled
          printing of iterates if it is positive.  In this case, FCN is
          called with IFLAG = 0 at the beginning of the first iteration
          and every NPRINT iterations thereafter and immediately prior
          to return, with X and FVEC available for printing.  FVEC and
          FJAC should not be altered.  If NPRINT is not positive, no
 
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          special calls of FCN with IFLAG = 0 are made.
        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  Relative error between two consecutive iterates is
                    at most XTOL.
          INFO = 2  Number of calls to FCN with IFLAG = 1 has reached
                    MAXFEV.
          INFO = 3  XTOL is too small.  No further improvement in the
                    approximate solution X is possible.
          INFO = 4  Iteration is not making good progress, as measured
                    by the improvement from the last five Jacobian eval-
                    uations.
          INFO = 5  Iteration is not making good progress, as measured
                    by the improvement from the last ten iterations.
          Sections 4 and 5 contain more details about INFO.
        NFEV is an integer output variable set to the number of calls t
          FCN with IFLAG = 1.
        NJEV is an integer output variable set to the number of calls t
          FCN with IFLAG = 2.
        R is an output array of length LR which contains the upper
          triangular matrix produced by the QR factorization of the
          final approximate Jacobian, stored rowwise.
        LR is a positive integer input variable not less than
          (N*(N+1))/2.
        QTF is an output array of length N which contains the vector
          (Q transpose)*FVEC.
        WA1, WA2, WA3, and WA4 are work arrays of length N.
 
  4. Successful completion.
        The accuracy of HYBRJ is controlled by the convergence parameter
        XTOL.  This parameter is used in a test which makes a comparison
        between the approximation X and a solution XSOL.  HYBRJ termi-
        nates when the test is satisfied.  If the convergence parameter
        is less than the machine precision (as defined by the MINPACK
        function DPMPAR(1)), then HYBRJ only attempts to satisfy the
        test defined by the machine precision.  Further progress is not
 
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        usually possible.
        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 HYBRJ may
        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 HYBRJ with a tighter toler-
        ance.
        Convergence test.  If ENORM(Z) denotes the Euclidean norm of a
          vector Z and D is the diagonal matrix whose entries are
          defined by the array DIAG, then this test attempts to guaran-
          tee that
                ENORM(D*(X-XSOL)) .LE. XTOL*ENORM(D*XSOL).
          If this condition is satisfied with XTOL = 10**(-K), then the
          larger components of D*X have K significant decimal digits an
          INFO is set to 1.  There is a danger that the smaller compo-
          nents of D*X may have large relative errors, but the fast rat
          of convergence of HYBRJ usually avoids this possibility.
          Unless high precision solutions are required, the recommended
          value for XTOL is the square root of the machine precision.
 
  5. Unsuccessful completion.
        Unsuccessful termination of HYBRJ 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 XTOL .LT. 0.D0, or MAXFEV .LE. 0, or
          FACTOR .LE. 0.D0, or LR .LT. (N*(N+1))/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 HYBRJ.  In this
          case, it may be possible to remedy the situation by rerunning
          HYBRJ with a smaller value of FACTOR.
        Excessive number of function evaluations.  A reasonable value
          for MAXFEV is 100*(N+1).  If the number of calls to FCN with
          IFLAG = 1 reaches MAXFEV, then this indicates that the routine
          is converging very slowly as measured 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 HYBRJ, causing termination with INFO = 4
          or INFO = 5.
        Lack of good progress.  HYBRJ searches for a zero of the system
          by minimizing the sum of the squares of the functions.  In so
 
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          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 HYBRJ from a dif-
          ferent starting point may be helpful.
 
  6. Characteristics of the algorithm.
        HYBRJ is a modification of the Powell hybrid method.  Two of it
        main characteristics involve the choice of the correction as a
        convex combination of the Newton and scaled gradient directions
        and the updating of the Jacobian by the rank-1 method of Broy-
        den.  The choice of the correction guarantees (under reasonable
        conditions) global convergence for starting points far from the
        solution and a fast rate of convergence.  The Jacobian is calcu
        lated at the starting point, but it is not recalculated until
        the rank-1 method fails to produce satisfactory progress.
        Timing.  The time required by HYBRJ 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 HYBRJ 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 HYBRJ will be strongly influenced by
          the time spent in FCN.
        Storage.  HYBRJ 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,
                             QFORM,QRFAC,R1MPYQ,R1UPDT
        FORTRAN-supplied ... DABS,DMAX1,DMIN1,DSQRT,MIN0,MOD
 
  8. References.
        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.
 
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        The problem is to determine the values of x(1), x(2), ..., x(9)
        which solve the system of tridiagonal equations