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Page 1 Documentation for MINPACK subroutine HYBRD Double precision version Argonne National Laboratory Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More March 1980 1. Purpose. The purpose of HYBRD 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. The Jacobian is then calculated by a for- ward-difference approximation. 2. Subroutine and type statements. SUBROUTINE HYBRD(FCN,N,X,FVEC,XTOL,MAXFEV,ML,MU,EPSFCN,DIAG, * MODE,FACTOR,NPRINT,INFO,NFEV,FJAC,LDFJAC, * R,LR,QTF,WA1,WA2,WA3,WA4) INTEGER N,MAXFEV,ML,MU,MODE,NPRINT,INFO,NFEV,LDFJAC,LR DOUBLE PRECISION XTOL,EPSFCN,FACTOR DOUBLE PRECISION X(N),FVEC(N),DIAG(N),FJAC(LDFJAC,N),R(LR),QTF(N * WA1(N),WA2(N),WA3(N),WA4(N) EXTERNAL FCN 3. Parameters. Parameters designated as input parameters must be specified on entry to HYBRD 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 HYBRD. 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 Page 2 user wants to terminate execution of HYBRD. 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. 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 occurs when the number of calls to FCN is at least MAXFEV by the end of an iteration. ML is a nonnegative integer input variable which specifies the number of subdiagonals within the band of the Jacobian matrix. If the Jacobian is not banded, set ML to at least N - 1. MU is a nonnegative integer input variable which specifies the number of superdiagonals within the band of the Jacobian matrix. If the Jacobian is not banded, set MU to at least N - 1. EPSFCN is an input variable used in determining a suitable step for the forward-difference approximation. This approximation assumes that the relative errors in the functions are of the order of EPSFCN. If EPSFCN is less than the machine preci- sion, it is assumed that the relative errors in the functions are of the order of the machine precision. 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 FACTOR itself. In most cases FACTOR should lie in the interval (.1,100.). 100. is a generally recommended value. Page 3 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. If NPRINT is not positive, no 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 has reached or exceeded 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 to FCN. 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. LDFJAC is a positive integer input variable not less than N which specifies the leading dimension of the array FJAC. 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. Page 4 4. Successful completion. The accuracy of HYBRD 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. HYBRD 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 HYBRD only attempts to satisfy the test defined by the machine precision. Further progress is not usually possible. The test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then HYBRD may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning HYBRD 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 and 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 rate of convergence of HYBRD 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 HYBRD 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 XTOL .LT. 0.D0, or MAXFEV .LE. 0, or ML .LT. 0, or MU .LT. 0, or FACTOR .LE. 0.D0, or LDFJAC .LT. N, 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 HYBRD. In this case, it may be possible to remedy the situation by rerunning HYBRD with a smaller value of FACTOR. Excessive number of function evaluations. A reasonable value for MAXFEV is 200*(N+1). If the number of calls to FCN reaches MAXFEV, then this indicates that the routine is con- verging very slowly as measured by the progress of FVEC, and Page 5 INFO is set to 2. This situation should be unusual because, as indicated below, lack of good progress is usually diagnosed earlier by HYBRD, causing termination with INFO = 4 or INFO = 5. Lack of good progress. HYBRD 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 HYBRD from a dif- ferent starting point may be helpful. 6. Characteristics of the algorithm. HYBRD 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 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 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 HYBRD 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 HYBRD is about 11.5*(N**2) to process each call to FCN. Unless FCN can be evaluated quickly, the timing of HYBRD will be strongly influenced by the time spent in FCN. Storage. HYBRD 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, QFORM,QRFAC,R1MPYQ,R1UPDT FORTRAN-supplied ... DABS,DMAX1,DMIN1,DSQRT,MIN0,MOD 8. References. M. J. D. Powell, A Hybrid Method for Nonlinear Equations. Page 6 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 HYBRD EXAMPLE. C DOUBLE PRECISION VERSION C C ********** INTEGER J,N,MAXFEV,ML,MU,MODE,NPRINT,INFO,NFEV,LDFJAC,LR,NWRITE DOUBLE PRECISION XTOL,EPSFCN,FACTOR,FNORM DOUBLE PRECISION X(9),FVEC(9),DIAG(9),FJAC(9,9),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 = 2000 ML = 1 MU = 1 EPSFCN = 0.D0 MODE = 2 DO 20 J = 1, 9 DIAG(J) = 1.D0 Page 7 20 CONTINUE FACTOR = 1.D2 NPRINT = 0 C CALL HYBRD(FCN,N,X,FVEC,XTOL,MAXFEV,ML,MU,EPSFCN,DIAG, * MODE,FACTOR,NPRINT,INFO,NFEV,FJAC,LDFJAC, * R,LR,QTF,WA1,WA2,WA3,WA4) FNORM = ENORM(N,FVEC) WRITE (NWRITE,1000) FNORM,NFEV,INFO,(X(J),J=1,N) STOP 1000 FORMAT (5X,31H FINAL L2 NORM OF THE RESIDUALS,D15.7 // * 5X,31H NUMBER OF FUNCTION EVALUATIONS,I10 // * 5X,15H EXIT PARAMETER,16X,I10 // * 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 8 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