Previous: hybrj Up: ../minpack.html Next: lmder

Page 1 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 Page 2 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 Page 3 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 Page 4 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. Page 5 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)) Page 6 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