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Page 1 Documentation for MINPACK subroutine HYBRD1 Double precision version Argonne National Laboratory Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More March 1980 1. Purpose. The purpose of HYBRD1 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 HYBRD. The user must provide a subroutine which calculates the functions. The Jacobian is then calculated by a forward-difference approximation. 2. Subroutine and type statements. SUBROUTINE HYBRD1(FCN,N,X,FVEC,TOL,INFO,WA,LWA) INTEGER N,INFO,LWA DOUBLE PRECISION TOL DOUBLE PRECISION X(N),FVEC(N),WA(LWA) EXTERNAL FCN 3. Parameters. Parameters designated as input parameters must be specified on entry to HYBRD1 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 HYBRD1. FCN is the name of the user-supplied subroutine which calculates the functions. FCN must be declared in an EXTERNAL statement in the user calling program, and should be written as follows. SUBROUTINE FCN(N,X,FVEC,IFLAG) INTEGER N,IFLAG DOUBLE PRECISION X(N),FVEC(N) ---------- CALCULATE THE FUNCTIONS AT X AND RETURN THIS VECTOR IN FVEC. ---------- RETURN END The value of IFLAG should not be changed by FCN unless the user wants to terminate execution of HYBRD1. In this case set IFLAG to a negative integer. Page 2 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. 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 has reached or exceeded 200*(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*(3*N+13))/2. 4. Successful completion. The accuracy of HYBRD1 is controlled by the convergence parame- ter TOL. This parameter is used in a test which makes a compar- ison between the approximation X and a solution XSOL. HYBRD1 terminates when the test is satisfied. If TOL is less than the machine precision (as defined by the MINPACK function DPMPAR(1)), then HYBRD1 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 are reasonably well behaved. Page 3 If this condition is not satisfied, then HYBRD1 may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning HYBRD1 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 HYBRD1 usually avoids this possibility. 5. Unsuccessful completion. Unsuccessful termination of HYBRD1 can be due to improper input parameters, arithmetic interrupts, an excessive number of func- tion evaluations, errors in the functions, or lack of good prog- ress. Improper input parameters. INFO is set to 0 if N .LE. 0, or TOL .LT. 0.D0, or LWA .LT. (N*(3*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 HYBRD1. 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 HYBRD, which includes in its calling sequence the step-length- governing parameter FACTOR. Excessive number of function evaluations. If the number of calls to FCN reaches 200*(N+1), 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 unu- sual because, as indicated below, lack of good progress is usually diagnosed earlier by HYBRD1, causing termination with INFO = 4. Errors in the functions. The choice of step length in the for- ward-difference approximation to the Jacobian assumes that the relative errors in the functions are of the order of the machine precision. If this is not the case, HYBRD1 may fail (usually with INFO = 4). The user should then use HYBRD instead, or one of the programs which require the analytic Jacobian (HYBRJ1 and HYBRJ). Page 4 Lack of good progress. HYBRD1 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 HYBRD1 from a dif- ferent starting point may be helpful. 6. Characteristics of the algorithm. HYBRD1 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 approximated by forward differences at the starting point, but forward differences are not used again until the rank-1 method fails to produce satisfactory progress. Timing. The time required by HYBRD1 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 HYBRD1 is about 11.5*(N**2) to process each call to FCN. Unless FCN can be evaluated quickly, the timing of HYBRD1 will be strongly influenced by the time spent in FCN. Storage. HYBRD1 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,FDJAC1,HYBRD, 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. Page 5 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 HYBRD1 EXAMPLE. C DOUBLE PRECISION VERSION C C ********** INTEGER J,N,INFO,LWA,NWRITE DOUBLE PRECISION TOL,FNORM DOUBLE PRECISION X(9),FVEC(9),WA(180) 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 LWA = 180 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 HYBRD1(FCN,N,X,FVEC,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)) C C LAST CARD OF DRIVER FOR HYBRD1 EXAMPLE. C END SUBROUTINE FCN(N,X,FVEC,IFLAG) INTEGER N,IFLAG DOUBLE PRECISION X(N),FVEC(N) C Page 6 C SUBROUTINE FCN FOR HYBRD1 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 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 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