SLATEC Routines --- DNSQ ---
*DECK DNSQ
SUBROUTINE DNSQ (FCN, JAC, IOPT, N, X, FVEC, FJAC, LDFJAC, XTOL,
+ MAXFEV, ML, MU, EPSFCN, DIAG, MODE, FACTOR, NPRINT, INFO, NFEV,
+ NJEV, R, LR, QTF, WA1, WA2, WA3, WA4)
C***BEGIN PROLOGUE DNSQ
C***PURPOSE Find a zero of a system of a N nonlinear functions in N
C variables by a modification of the Powell hybrid method.
C***LIBRARY SLATEC
C***CATEGORY F2A
C***TYPE DOUBLE PRECISION (SNSQ-S, DNSQ-D)
C***KEYWORDS NONLINEAR SQUARE SYSTEM, POWELL HYBRID METHOD, ZEROS
C***AUTHOR Hiebert, K. L. (SNLA)
C***DESCRIPTION
C
C 1. Purpose.
C
C The purpose of DNSQ is to find a zero of a system of N nonlinear
C functions in N variables by a modification of the Powell
C hybrid method. 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.
C This code is the combination of the MINPACK codes (Argonne)
C HYBRD and HYBRDJ.
C
C 2. Subroutine and Type Statements.
C
C SUBROUTINE DNSQ(FCN,JAC,IOPT,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,
C * ML,MU,EPSFCN,DIAG,MODE,FACTOR,NPRINT,INFO,NFEV,
C * NJEV,R,LR,QTF,WA1,WA2,WA3,WA4)
C INTEGER IOPT,N,MAXFEV,ML,MU,MODE,NPRINT,INFO,NFEV,LDFJAC,NJEV,LR
C DOUBLE PRECISION XTOL,EPSFCN,FACTOR
C DOUBLE PRECISION
C X(N),FVEC(N),DIAG(N),FJAC(LDFJAC,N),R(LR),QTF(N),
C * WA1(N),WA2(N),WA3(N),WA4(N)
C EXTERNAL FCN,JAC
C
C 3. Parameters.
C
C Parameters designated as input parameters must be specified on
C entry to DNSQ 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 DNSQ.
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 DOUBLE PRECISION 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 DNSQ. 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 DOUBLE PRECISION 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 DNSQ. 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 FJAC is an output N by N array which contains the orthogonal
C matrix Q produced by the QR factorization of the final
C approximate Jacobian.
C
C LDFJAC is a positive integer input variable not less than N
C which specifies the leading dimension of the array FJAC.
C
C XTOL is a nonnegative input variable. Termination occurs when
C the relative error between two consecutive iterates is at most
C XTOL. Therefore, XTOL measures the relative error desired in
C the approximate solution. Section 4 contains more details
C about XTOL.
C
C MAXFEV is a positive integer input variable. Termination occurs
C when the number of calls to FCN is at least MAXFEV by the end
C of an iteration.
C
C ML is a nonnegative integer input variable which specifies the
C number of subdiagonals within the band of the Jacobian matrix.
C If the Jacobian is not banded or IOPT=1, set ML to at
C least N - 1.
C
C MU is a nonnegative integer input variable which specifies the
C number of superdiagonals within the band of the Jacobian
C matrix. If the Jacobian is not banded or IOPT=1, set MU to at
C least N - 1.
C
C EPSFCN is an input variable used in determining a suitable step
C for the forward-difference approximation. This approximation
C assumes that the relative errors in the functions are of the
C order of EPSFCN. If EPSFCN is less than the machine
C precision, it is assumed that the relative errors in the
C functions are of the order of the machine precision. If
C IOPT=1, then EPSFCN can be ignored (treat it as a dummy
C argument).
C
C DIAG is an array of length N. If MODE = 1 (see below), DIAG is
C internally set. If MODE = 2, DIAG must contain positive
C entries that serve as implicit (multiplicative) scale factors
C for the variables.
C
C MODE is an integer input variable. If MODE = 1, the variables
C will be scaled internally. If MODE = 2, the scaling is
C specified by the input DIAG. Other values of MODE are
C equivalent to MODE = 1.
C
C FACTOR is a positive input variable used in determining the
C initial step bound. This bound is set to the product of
C FACTOR and the Euclidean norm of DIAG*X if nonzero, or else to
C FACTOR itself. In most cases FACTOR should lie in the
C interval (.1,100.). 100. is a generally recommended value.
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 iterations 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 Relative error between two consecutive iterates is
C at most XTOL.
C
C INFO = 2 Number of calls to FCN has reached or exceeded
C MAXFEV.
C
C INFO = 3 XTOL is too small. No further improvement in the
C approximate solution X is possible.
C
C INFO = 4 Iteration is not making good progress, as measured
C by the improvement from the last five Jacobian
C evaluations.
C
C INFO = 5 Iteration is not making good progress, as measured
C by the improvement from the last ten iterations.
C
C Sections 4 and 5 contain more details about INFO.
C
C NFEV is an integer output variable set to the number of calls to
C FCN.
C
C NJEV is an integer output variable set to the number of calls to
C JAC. (If IOPT=2, then NJEV is set to zero.)
C
C R is an output array of length LR which contains the upper
C triangular matrix produced by the QR factorization of the
C final approximate Jacobian, stored rowwise.
C
C LR is a positive integer input variable not less than
C (N*(N+1))/2.
C
C QTF is an output array of length N which contains the vector
C (Q transpose)*FVEC.
C
C WA1, WA2, WA3, and WA4 are work arrays of length N.
C
C
C 4. Successful completion.
C
C The accuracy of DNSQ is controlled by the convergence parameter
C XTOL. This parameter is used in a test which makes a comparison
C between the approximation X and a solution XSOL. DNSQ
C terminates when the test is satisfied. If the convergence
C parameter is less than the machine precision (as defined by the
C function D1MACH(4)), then DNSQ only attempts to satisfy the test
C defined by the machine precision. Further progress is not
C usually possible.
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 are coded consistently. If these conditions
C are not satisfied, then DNSQ may incorrectly indicate
C convergence. The coding of the Jacobian can be checked by the
C subroutine DCKDER. If the Jacobian is coded correctly or IOPT=2,
C then the validity of the answer can be checked, for example, by
C rerunning DNSQ with a tighter tolerance.
C
C Convergence Test. If DENORM(Z) denotes the Euclidean norm of a
C vector Z and D is the diagonal matrix whose entries are
C defined by the array DIAG, then this test attempts to
C guarantee that
C
C DENORM(D*(X-XSOL)) .LE. XTOL*DENORM(D*XSOL).
C
C If this condition is satisfied with XTOL = 10**(-K), then the
C larger components of D*X have K significant decimal digits and
C INFO is set to 1. There is a danger that the smaller
C components of D*X may have large relative errors, but the fast
C rate of convergence of DNSQ usually avoids this possibility.
C Unless high precision solutions are required, the recommended
C value for XTOL is the square root of the machine precision.
C
C
C 5. Unsuccessful Completion.
C
C Unsuccessful termination of DNSQ can be due to improper input
C parameters, arithmetic interrupts, an excessive number of
C function evaluations, or lack of good progress.
C
C Improper Input Parameters. INFO is set to 0 if IOPT .LT .1,
C or IOPT .GT. 2, or N .LE. 0, or LDFJAC .LT. N, or
C XTOL .LT. 0.E0, or MAXFEV .LE. 0, or ML .LT. 0, or MU .LT. 0,
C or FACTOR .LE. 0.E0, or LR .LT. (N*(N+1))/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 DNSQ. In this
C case, it may be possible to remedy the situation by rerunning
C DNSQ with a smaller value of FACTOR.
C
C Excessive Number of Function Evaluations. A reasonable value
C for MAXFEV is 100*(N+1) for IOPT=1 and 200*(N+1) for IOPT=2.
C If the number of calls to FCN reaches MAXFEV, then this
C indicates that the routine is converging very slowly as
C measured by the progress of FVEC, and INFO is set to 2. This
C situation should be unusual because, as indicated below, lack
C of good progress is usually diagnosed earlier by DNSQ,
C causing termination with info = 4 or INFO = 5.
C
C Lack of Good Progress. DNSQ 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
C situation, the iteration eventually fails to make good
C progress. In particular, this will happen if the system does
C not have a zero. If the system has a zero, rerunning DNSQ
C from a different starting point may be helpful.
C
C
C 6. Characteristics of The Algorithm.
C
C DNSQ is a modification of the Powell Hybrid method. Two of its
C main characteristics involve the choice of the correction as a
C convex combination of the Newton and scaled gradient directions,
C and the updating of the Jacobian by the rank-1 method of
C Broyden. The choice of the correction guarantees (under
C reasonable conditions) global convergence for starting points
C far from the solution and a fast rate of convergence. The
C Jacobian is calculated at the starting point by either the
C user-supplied subroutine or a forward-difference approximation,
C but it is not recalculated until the rank-1 method fails to
C produce satisfactory progress.
C
C Timing. The time required by DNSQ 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 DNSQ 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 DNSQ will be strongly influenced by the time
C spent in FCN and JAC.
C
C Storage. DNSQ 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 *Long Description:
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 DNSQ example.
C C
C INTEGER J,IOPT,N,MAXFEV,ML,MU,MODE,NPRINT,INFO,NFEV,LDFJAC,LR,
C * NWRITE
C DOUBLE PRECISION XTOL,EPSFCN,FACTOR,FNORM
C DOUBLE PRECISION X(9),FVEC(9),DIAG(9),FJAC(9,9),R(45),QTF(9),
C * WA1(9),WA2(9),WA3(9),WA4(9)
C DOUBLE PRECISION DENORM,D1MACH
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
C LDFJAC = 9
C LR = 45
C C
C C SET XTOL 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 XTOL = SQRT(D1MACH(4))
C C
C MAXFEV = 2000
C ML = 1
C MU = 1
C EPSFCN = 0.E0
C MODE = 2
C DO 20 J = 1, 9
C DIAG(J) = 1.E0
C 20 CONTINUE
C FACTOR = 1.E2
C NPRINT = 0
C C
C CALL DNSQ(FCN,JAC,IOPT,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,ML,MU,
C * EPSFCN,DIAG,MODE,FACTOR,NPRINT,INFO,NFEV,NJEV,
C * R,LR,QTF,WA1,WA2,WA3,WA4)
C FNORM = DENORM(N,FVEC)
C WRITE (NWRITE,1000) FNORM,NFEV,INFO,(X(J),J=1,N)
C STOP
C 1000 FORMAT (5X,' FINAL L2 NORM OF THE RESIDUALS',E15.7 //
C * 5X,' NUMBER OF FUNCTION EVALUATIONS',I10 //
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 DOUBLE PRECISION X(N),FVEC(N)
C INTEGER K
C DOUBLE PRECISION ONE,TEMP,TEMP1,TEMP2,THREE,TWO,ZERO
C DATA ZERO,ONE,TWO,THREE /0.E0,1.E0,2.E0,3.E0/
C C
C IF (IFLAG .NE. 0) GO TO 5
C C
C C INSERT PRINT STATEMENTS HERE WHEN NPRINT IS POSITIVE.
C C
C RETURN
C 5 CONTINUE
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 Number of function evaluations 14
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 D1MACH, D1MPYQ, D1UPDT, DDOGLG, DENORM, DFDJC1,
C DQFORM, DQRFAC, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800301 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
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 DNSQ