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HYBRJ1.


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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 calculates
         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 functions
         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 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 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 as
       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 from
       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.


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       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))


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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