*DECK SNSQE
SUBROUTINE SNSQE (FCN, JAC, IOPT, N, X, FVEC, TOL, NPRINT, INFO,
+ WA, LWA)
C***BEGIN PROLOGUE SNSQE
C***PURPOSE An easy-to-use code to find a zero of a system of N
C nonlinear functions in N variables by a modification of
C the Powell hybrid method.
C***LIBRARY SLATEC
C***CATEGORY F2A
C***TYPE SINGLE PRECISION (SNSQE-S, DNSQE-D)
C***KEYWORDS EASY-TO-USE, NONLINEAR SQUARE SYSTEM,
C POWELL HYBRID METHOD, ZEROS
C***AUTHOR Hiebert, K. L., (SNLA)
C***DESCRIPTION
C
C 1. Purpose.
C
C
C The purpose of SNSQE is to find a zero of a system of N non-
C linear functions in N variables by a modification of the Powell
C hybrid method. This is done by using the more general nonlinear
C equation solver SNSQ. The user must provide a subroutine which
C calculates the functions. The user has the option of either to
C provide a subroutine which calculates the Jacobian or to let the
C code calculate it by a forward-difference approximation. This
C code is the combination of the MINPACK codes (Argonne) HYBRD1
C and HYBRJ1.
C
C
C 2. Subroutine and Type Statements.
C
C SUBROUTINE SNSQE(FCN,JAC,IOPT,N,X,FVEC,TOL,NPRINT,INFO,
C * WA,LWA)
C INTEGER IOPT,N,NPRINT,INFO,LWA
C REAL TOL
C REAL X(N),FVEC(N),WA(LWA)
C EXTERNAL FCN,JAC
C
C
C 3. Parameters.
C
C Parameters designated as input parameters must be specified on
C entry to SNSQE and are not changed on exit, while parameters
C designated as output parameters need not be specified on entry
C and are set to appropriate values on exit from SNSQE.
C
C FCN is the name of the user-supplied subroutine which calculates
C the functions. FCN must be declared in an EXTERNAL statement
C in the user calling program, and should be written as follows.
C
C SUBROUTINE FCN(N,X,FVEC,IFLAG)
C INTEGER N,IFLAG
C REAL X(N),FVEC(N)
C ----------
C Calculate the functions at X and
C return this vector in FVEC.
C ----------
C RETURN
C END
C
C The value of IFLAG should not be changed by FCN unless the
C user wants to terminate execution of SNSQE. In this case, set
C IFLAG to a negative integer.
C
C JAC is the name of the user-supplied subroutine which calculates
C the Jacobian. If IOPT=1, then JAC must be declared in an
C EXTERNAL statement in the user calling program, and should be
C written as follows.
C
C SUBROUTINE JAC(N,X,FVEC,FJAC,LDFJAC,IFLAG)
C INTEGER N,LDFJAC,IFLAG
C REAL X(N),FVEC(N),FJAC(LDFJAC,N)
C ----------
C Calculate the Jacobian at X and return this
C matrix in FJAC. FVEC contains the function
C values at X and should not be altered.
C ----------
C RETURN
C END
C
C The value of IFLAG should not be changed by JAC unless the
C user wants to terminate execution of SNSQE. In this case, set
C IFLAG to a negative integer.
C
C If IOPT=2, JAC can be ignored (treat it as a dummy argument).
C
C IOPT is an input variable which specifies how the Jacobian will
C be calculated. If IOPT=1, then the user must supply the
C Jacobian through the subroutine JAC. If IOPT=2, then the
C code will approximate the Jacobian by forward-differencing.
C
C N is a positive integer input variable set to the number of
C functions and variables.
C
C X is an array of length N. On input, X must contain an initial
C estimate of the solution vector. On output, X contains the
C final estimate of the solution vector.
C
C FVEC is an output array of length N which contains the functions
C evaluated at the output X.
C
C TOL is a non-negative input variable. Termination occurs when
C the algorithm estimates that the relative error between X and
C the solution is at most TOL. Section 4 contains more details
C about TOL.
C
C NPRINT is an integer input variable that enables controlled
C printing of iterates if it is positive. In this case, FCN is
C called with IFLAG = 0 at the beginning of the first iteration
C and every NPRINT iteration thereafter and immediately prior
C to return, with X and FVEC available for printing. Appropriate
C print statements must be added to FCN (see example). If NPRINT
C is not positive, no special calls of FCN with IFLAG = 0 are
C made.
C
C INFO is an integer output variable. If the user has terminated
C execution, INFO is set to the (negative) value of IFLAG. See
C description of FCN and JAC. Otherwise, INFO is set as follows.
C
C INFO = 0 improper input parameters.
C
C INFO = 1 algorithm estimates that the relative error between
C X and the solution is at most TOL.
C
C INFO = 2 number of calls to FCN has reached or exceeded
C 100*(N+1) for IOPT=1 or 200*(N+1) for IOPT=2.
C
C INFO = 3 TOL is too small. No further improvement in the
C approximate solution X is possible.
C
C INFO = 4 iteration is not making good progress.
C
C Sections 4 and 5 contain more details about INFO.
C
C WA is a work array of length LWA.
C
C LWA is a positive integer input variable not less than
C (3*N**2+13*N))/2.
C
C
C 4. Successful Completion.
C
C The accuracy of SNSQE is controlled by the convergence parame-
C ter TOL. This parameter is used in a test which makes a compar-
C ison between the approximation X and a solution XSOL. SNSQE
C terminates when the test is satisfied. If TOL is less than the
C machine precision (as defined by the function R1MACH(4)), then
C SNSQE attempts only to satisfy the test defined by the machine
C precision. Further progress is not usually possible. Unless
C high precision solutions are required, the recommended value
C for TOL is the square root of the machine precision.
C
C The test assumes that the functions are reasonably well behaved,
C and, if the Jacobian is supplied by the user, that the functions
C and the Jacobian coded consistently. If these conditions
C are not satisfied, SNSQE may incorrectly indicate convergence.
C The coding of the Jacobian can be checked by the subroutine
C CHKDER. If the Jacobian is coded correctly or IOPT=2, then
C the validity of the answer can be checked, for example, by
C rerunning SNSQE with a tighter tolerance.
C
C Convergence Test. If ENORM(Z) denotes the Euclidean norm of a
C vector Z, then this test attempts to guarantee that
C
C ENORM(X-XSOL) .LE. TOL*ENORM(XSOL).
C
C If this condition is satisfied with TOL = 10**(-K), then the
C larger components of X have K significant decimal digits and
C INFO is set to 1. There is a danger that the smaller compo-
C nents of X may have large relative errors, but the fast rate
C of convergence of SNSQE usually avoids this possibility.
C
C
C 5. Unsuccessful Completion.
C
C Unsuccessful termination of SNSQE can be due to improper input
C parameters, arithmetic interrupts, an excessive number of func-
C tion evaluations, errors in the functions, or lack of good prog-
C ress.
C
C Improper Input Parameters. INFO is set to 0 if IOPT .LT. 1, or
C IOPT .GT. 2, or N .LE. 0, or TOL .LT. 0.E0, or
C LWA .LT. (3*N**2+13*N)/2.
C
C Arithmetic Interrupts. If these interrupts occur in the FCN
C subroutine during an early stage of the computation, they may
C be caused by an unacceptable choice of X by SNSQE. In this
C case, it may be possible to remedy the situation by not evalu-
C ating the functions here, but instead setting the components
C of FVEC to numbers that exceed those in the initial FVEC.
C
C Excessive Number of Function Evaluations. If the number of
C calls to FCN reaches 100*(N+1) for IOPT=1 or 200*(N+1) for
C IOPT=2, then this indicates that the routine is converging
C very slowly as measured by the progress of FVEC, and INFO is
C set to 2. This situation should be unusual because, as
C indicated below, lack of good progress is usually diagnosed
C earlier by SNSQE, causing termination with INFO = 4.
C
C Errors in the Functions. When IOPT=2, the choice of step length
C in the forward-difference approximation to the Jacobian
C assumes that the relative errors in the functions are of the
C order of the machine precision. If this is not the case,
C SNSQE may fail (usually with INFO = 4). The user should
C then either use SNSQ and set the step length or use IOPT=1
C and supply the Jacobian.
C
C Lack of Good Progress. SNSQE searches for a zero of the system
C by minimizing the sum of the squares of the functions. In so
C doing, it can become trapped in a region where the minimum
C does not correspond to a zero of the system and, in this situ-
C ation, the iteration eventually fails to make good progress.
C In particular, this will happen if the system does not have a
C zero. If the system has a zero, rerunning SNSQE from a dif-
C ferent starting point may be helpful.
C
C
C 6. Characteristics of the Algorithm.
C
C SNSQE is a modification of the Powell hybrid method. Two of
C its main characteristics involve the choice of the correction as
C a convex combination of the Newton and scaled gradient direc-
C tions, and the updating of the Jacobian by the rank-1 method of
C Broyden. The choice of the correction guarantees (under reason-
C able conditions) global convergence for starting points far from
C the solution and a fast rate of convergence. The Jacobian is
C calculated at the starting point by either the user-supplied
C subroutine or a forward-difference approximation, but it is not
C recalculated until the rank-1 method fails to produce satis-
C factory progress.
C
C Timing. The time required by SNSQE to solve a given problem
C depends on N, the behavior of the functions, the accuracy
C requested, and the starting point. The number of arithmetic
C operations needed by SNSQE is about 11.5*(N**2) to process
C each evaluation of the functions (call to FCN) and 1.3*(N**3)
C to process each evaluation of the Jacobian (call to JAC,
C if IOPT = 1). Unless FCN and JAC can be evaluated quickly,
C the timing of SNSQE will be strongly influenced by the time
C spent in FCN and JAC.
C
C Storage. SNSQE requires (3*N**2 + 17*N)/2 single precision
C storage locations, in addition to the storage required by the
C program. There are no internally declared storage arrays.
C
C
C 7. Example.
C
C The problem is to determine the values of X(1), X(2), ..., X(9),
C which solve the system of tridiagonal equations
C
C (3-2*X(1))*X(1) -2*X(2) = -1
C -X(I-1) + (3-2*X(I))*X(I) -2*X(I+1) = -1, I=2-8
C -X(8) + (3-2*X(9))*X(9) = -1
C
C **********
C
C PROGRAM TEST
C C
C C Driver for SNSQE example.
C C
C INTEGER J,N,IOPT,NPRINT,INFO,LWA,NWRITE
C REAL TOL,FNORM
C REAL X(9),FVEC(9),WA(180)
C REAL ENORM,R1MACH
C EXTERNAL FCN
C DATA NWRITE /6/
C C
C IOPT = 2
C N = 9
C C
C C The following starting values provide a rough solution.
C C
C DO 10 J = 1, 9
C X(J) = -1.E0
C 10 CONTINUE
C
C LWA = 180
C NPRINT = 0
C C
C C Set TOL to the square root of the machine precision.
C C Unless high precision solutions are required,
C C this is the recommended setting.
C C
C TOL = SQRT(R1MACH(4))
C C
C CALL SNSQE(FCN,JAC,IOPT,N,X,FVEC,TOL,NPRINT,INFO,WA,LWA)
C FNORM = ENORM(N,FVEC)
C WRITE (NWRITE,1000) FNORM,INFO,(X(J),J=1,N)
C STOP
C 1000 FORMAT (5X,' FINAL L2 NORM OF THE RESIDUALS',E15.7 //
C * 5X,' EXIT PARAMETER',16X,I10 //
C * 5X,' FINAL APPROXIMATE SOLUTION' // (5X,3E15.7))
C END
C SUBROUTINE FCN(N,X,FVEC,IFLAG)
C INTEGER N,IFLAG
C REAL X(N),FVEC(N)
C INTEGER K
C REAL ONE,TEMP,TEMP1,TEMP2,THREE,TWO,ZERO
C DATA ZERO,ONE,TWO,THREE /0.E0,1.E0,2.E0,3.E0/
C C
C DO 10 K = 1, N
C TEMP = (THREE - TWO*X(K))*X(K)
C TEMP1 = ZERO
C IF (K .NE. 1) TEMP1 = X(K-1)
C TEMP2 = ZERO
C IF (K .NE. N) TEMP2 = X(K+1)
C FVEC(K) = TEMP - TEMP1 - TWO*TEMP2 + ONE
C 10 CONTINUE
C RETURN
C END
C
C Results obtained with different compilers or machines
C may be slightly different.
C
C FINAL L2 NORM OF THE RESIDUALS 0.1192636E-07
C
C EXIT PARAMETER 1
C
C FINAL APPROXIMATE SOLUTION
C
C -0.5706545E+00 -0.6816283E+00 -0.7017325E+00
C -0.7042129E+00 -0.7013690E+00 -0.6918656E+00
C -0.6657920E+00 -0.5960342E+00 -0.4164121E+00
C
C***REFERENCES M. J. D. Powell, A hybrid method for nonlinear equa-
C tions. In Numerical Methods for Nonlinear Algebraic
C Equations, P. Rabinowitz, Editor. Gordon and Breach,
C 1988.
C***ROUTINES CALLED SNSQ, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800301 DATE WRITTEN
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE SNSQE
INTEGER IOPT,N,NPRINT,INFO,LWA
REAL TOL
REAL X(*),FVEC(*),WA(LWA)
EXTERNAL FCN, JAC
INTEGER INDEX,J,LR,MAXFEV,ML,MODE,MU,NFEV,NJEV
REAL EPSFCN,FACTOR,ONE,XTOL,ZERO
SAVE FACTOR, ONE, ZERO
DATA FACTOR,ONE,ZERO /1.0E2,1.0E0,0.0E0/
C***FIRST EXECUTABLE STATEMENT SNSQE
INFO = 0
C
C CHECK THE INPUT PARAMETERS FOR ERRORS.
C
IF (IOPT .LT. 1 .OR. IOPT .GT. 2 .OR. N .LE. 0
1 .OR. TOL .LT. ZERO .OR. LWA .LT. (3*N**2 +13*N)/2)
2 GO TO 20
C
C CALL SNSQ.
C
MAXFEV = 100*(N + 1)
IF (IOPT .EQ. 2) MAXFEV = 2*MAXFEV
XTOL = TOL
ML = N - 1
MU = N - 1
EPSFCN = ZERO
MODE = 2
DO 10 J = 1, N
WA(J) = ONE
10 CONTINUE
LR = (N*(N + 1))/2
INDEX=6*N+LR
CALL SNSQ(FCN,JAC,IOPT,N,X,FVEC,WA(INDEX+1),N,XTOL,MAXFEV,ML,MU,
1 EPSFCN,WA(1),MODE,FACTOR,NPRINT,INFO,NFEV,NJEV,
2 WA(6*N+1),LR,WA(N+1),WA(2*N+1),WA(3*N+1),WA(4*N+1),
3 WA(5*N+1))
IF (INFO .EQ. 5) INFO = 4
20 CONTINUE
IF (INFO .EQ. 0) CALL XERMSG ('SLATEC', 'SNSQE',
+ 'INVALID INPUT PARAMETER.', 2, 1)
RETURN
C
C LAST CARD OF SUBROUTINE SNSQE.
C
END