*DECK SOS
SUBROUTINE SOS (FNC, NEQ, X, RTOLX, ATOLX, TOLF, IFLAG, RW, LRW,
+ IW, LIW)
C***BEGIN PROLOGUE SOS
C***PURPOSE Solve a square system of nonlinear equations.
C***LIBRARY SLATEC
C***CATEGORY F2A
C***TYPE SINGLE PRECISION (SOS-S, DSOS-D)
C***KEYWORDS BROWN'S METHOD, NEWTON'S METHOD, NONLINEAR EQUATIONS,
C ROOTS, SOLUTIONS
C***AUTHOR Watts, H. A., (SNLA)
C***DESCRIPTION
C
C SOS solves a system of NEQ simultaneous nonlinear equations in
C NEQ unknowns. That is, it solves the problem F(X)=0
C where X is a vector with components X(1),...,X(NEQ) and F
C is a vector of nonlinear functions. Each equation is of the form
C
C F (X(1),...,X(NEQ))=0 for K=1,...,NEQ.
C K
C
C The algorithm is based on an iterative method which is a
C variation of Newton's method using Gaussian elimination
C in a manner similar to the Gauss-Seidel process. Convergence
C is roughly quadratic. All partial derivatives required by
C the algorithm are approximated by first difference quotients.
C The convergence behavior of this code is affected by the
C ordering of the equations, and it is advantageous to place linear
C and mildly nonlinear equations first in the ordering.
C
C Actually, SOS is merely an interfacing routine for
C calling subroutine SOSEQS which embodies the solution
C algorithm. The purpose of this is to add greater
C flexibility and ease of use for the prospective user.
C
C SOSEQS calls the accompanying routine SOSSOL, which solves special
C triangular linear systems by back-substitution.
C
C The user must supply a function subprogram which evaluates the
C K-th equation only (K specified by SOSEQS) for each call
C to the subprogram.
C
C SOS represents an implementation of the mathematical algorithm
C described in the references below. It is a modification of the
C code SOSNLE written by H. A. Watts in 1973.
C
C **********************************************************************
C -Input-
C
C FNC -Name of the function program which evaluates the equations.
C This name must be in an EXTERNAL statement in the calling
C program. The user must supply FNC in the form FNC(X,K),
C where X is the solution vector (which must be dimensioned
C in FNC) and FNC returns the value of the K-th function.
C
C NEQ -Number of equations to be solved.
C
C X -Solution vector. Initial guesses must be supplied.
C
C RTOLX -Relative error tolerance used in the convergence criteria.
C Each solution component X(I) is checked by an accuracy test
C of the form ABS(X(I)-XOLD(I)) .LE. RTOLX*ABS(X(I))+ATOLX,
C where XOLD(I) represents the previous iteration value.
C RTOLX must be non-negative.
C
C ATOLX -Absolute error tolerance used in the convergence criteria.
C ATOLX must be non-negative. If the user suspects some
C solution component may be zero, he should set ATOLX to an
C appropriate (depends on the scale of the remaining variables)
C positive value for better efficiency.
C
C TOLF -Residual error tolerance used in the convergence criteria.
C Convergence will be indicated if all residuals (values of the
C functions or equations) are not bigger than TOLF in
C magnitude. Note that extreme care must be given in assigning
C an appropriate value for TOLF because this convergence test
C is dependent on the scaling of the equations. An
C inappropriate value can cause premature termination of the
C iteration process.
C
C IFLAG -Optional input indicator. You must set IFLAG=-1 if you
C want to use any of the optional input items listed below.
C Otherwise set it to zero.
C
C RW -A REAL work array which is split apart by SOS and used
C internally by SOSEQS.
C
C LRW -Dimension of the RW array. LRW must be at least
C 1 + 6*NEQ + NEQ*(NEQ+1)/2
C
C IW -An INTEGER work array which is split apart by SOS and used
C internally by SOSEQS.
C
C LIW -Dimension of the IW array. LIW must be at least 3 + NEQ.
C
C -Optional Input-
C
C IW(1) -Internal printing parameter. You must set IW(1)=-1 if
C you want the intermediate solution iterates to be printed.
C
C IW(2) -Iteration limit. The maximum number of allowable
C iterations can be specified, if desired. To override the
C default value of 50, set IW(2) to the number wanted.
C
C Remember, if you tell the code that you are using one of the
C options (by setting IFLAG=-1), you must supply values
C for both IW(1) and IW(2).
C
C **********************************************************************
C -Output-
C
C X -Solution vector.
C
C IFLAG -Status indicator
C
C *** Convergence to a Solution ***
C
C 1 Means satisfactory convergence to a solution was achieved.
C Each solution component X(I) satisfies the error tolerance
C test ABS(X(I)-XOLD(I)) .LE. RTOLX*ABS(X(I))+ATOLX.
C
C 2 Means procedure converged to a solution such that all
C residuals are at most TOLF in magnitude,
C ABS(FNC(X,I)) .LE. TOLF.
C
C 3 Means that conditions for both IFLAG=1 and IFLAG=2 hold.
C
C 4 Means possible numerical convergence. Behavior indicates
C limiting precision calculations as a result of user asking
C for too much accuracy or else convergence is very slow.
C Residual norms and solution increment norms have
C remained roughly constant over several consecutive
C iterations.
C
C *** Task Interrupted ***
C
C 5 Means the allowable number of iterations has been met
C without obtaining a solution to the specified accuracy.
C Very slow convergence may be indicated. Examine the
C approximate solution returned and see if the error
C tolerances seem appropriate.
C
C 6 Means the allowable number of iterations has been met and
C the iterative process does not appear to be converging.
C A local minimum may have been encountered or there may be
C limiting precision difficulties.
C
C 7 Means that the iterative scheme appears to be diverging.
C Residual norms and solution increment norms have
C increased over several consecutive iterations.
C
C *** Task Cannot Be Continued ***
C
C 8 Means that a Jacobian-related matrix was singular.
C
C 9 Means improper input parameters.
C
C *** IFLAG should be examined after each call to ***
C *** SOS with the appropriate action being taken. ***
C
C
C RW(1) -Contains a norm of the residual.
C
C IW(3) -Contains the number of iterations used by the process.
C
C **********************************************************************
C***REFERENCES K. M. Brown, Solution of simultaneous nonlinear
C equations, Algorithm 316, Communications of the
C A.C.M. 10, (1967), pp. 728-729.
C K. M. Brown, A quadratically convergent Newton-like
C method based upon Gaussian elimination, SIAM Journal
C on Numerical Analysis 6, (1969), pp. 560-569.
C***ROUTINES CALLED SOSEQS, XERMSG
C***REVISION HISTORY (YYMMDD)
C 801001 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 900510 Convert XERRWV calls to XERMSG calls, changed Prologue
C comments to agree with DSOS. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE SOS
DIMENSION X(*), RW(*), IW(*)
CHARACTER*8 XERN1
CHARACTER*16 XERN3, XERN4
EXTERNAL FNC
C***FIRST EXECUTABLE STATEMENT SOS
INPFLG = IFLAG
C
C CHECK FOR VALID INPUT
C
IF (NEQ .LE. 0) THEN
WRITE (XERN1, '(I8)') NEQ
CALL XERMSG ('SLATEC', 'SOS', 'THE NUMBER OF EQUATIONS ' //
* 'MUST BE A POSITIVE INTEGER. YOU HAVE CALLED THE ' //
* 'CODE WITH NEQ = ' // XERN1, 1, 1)
IFLAG = 9
ENDIF
C
IF (RTOLX .LT. 0.0D0 .OR. ATOLX .LT. 0.0D0) THEN
WRITE (XERN3, '(1PE15.6)') ATOLX
WRITE (XERN4, '(1PE15.6)') RTOLX
CALL XERMSG ('SLATEC', 'SOS', 'THE ERROR TOLERANCES FOR ' //
* 'THE SOLUTION ITERATES CANNOT BE NEGATIVE. YOU HAVE ' //
* 'CALLED THE CODE WITH RTOLX = ' // XERN3 //
* ' AND ATOLX = ' // XERN4,2, 1)
IFLAG = 9
ENDIF
C
IF (TOLF .LT. 0.0D0) THEN
WRITE (XERN3, '(1PE15.6)') TOLF
CALL XERMSG ('SLATEC', 'SOS', 'THE RESIDUAL ERROR ' //
* 'TOLERANCE MUST BE NON-NEGATIVE. YOU HAVE CALLED THE ' //
* 'CODE WITH TOLF = ' // XERN3, 3, 1)
IFLAG = 9
ENDIF
C
IPRINT = 0
MXIT = 50
IF (INPFLG .EQ. (-1)) THEN
IF (IW(1) .EQ. (-1)) IPRINT = -1
MXIT = IW(2)
IF (MXIT .LE. 0) THEN
WRITE (XERN1, '(I8)') MXIT
CALL XERMSG ('SLATEC', 'SOS', 'YOU HAVE TOLD THE CODE ' //
* 'TO USE OPTIONAL IN PUT ITEMS BY SETTING IFLAG=-1. ' //
* 'HOWEVER YOU HAVE CALLED THE CODE WITH THE MAXIMUM ' //
* 'ALLOWABLE NUMBER OF ITERATIONS SET TO IW(2) = ' //
* XERN1, 4, 1)
IFLAG = 9
ENDIF
ENDIF
C
NC = (NEQ*(NEQ+1))/2
IF (LRW .LT. 1 + 6*NEQ + NC) THEN
WRITE (XERN1, '(I8)') LRW
CALL XERMSG ('SLATEC', 'SOS', 'DIMENSION OF THE RW ARRAY ' //
* 'MUST BE AT LEAST 1 + 6*NEQ + NEQ*(NEQ+1)/2 . YOU HAVE ' //
* 'CALLED THE CODE WITH LRW = ' // XERN1, 5, 1)
IFLAG = 9
ENDIF
C
IF (LIW .LT. 3 + NEQ) THEN
WRITE (XERN1, '(I8)') LIW
CALL XERMSG ('SLATEC', 'SOS', 'DIMENSION OF THE IW ARRAY ' //
* 'MUST BE AT LEAST 3 + NEQ. YOU HAVE CALLED THE CODE ' //
* 'WITH LIW = ' // XERN1, 6, 1)
IFLAG = 9
ENDIF
C
IF (IFLAG .NE. 9) THEN
NCJS = 6
NSRRC = 4
NSRI = 5
C
K1 = NC + 2
K2 = K1 + NEQ
K3 = K2 + NEQ
K4 = K3 + NEQ
K5 = K4 + NEQ
K6 = K5 + NEQ
C
CALL SOSEQS(FNC, NEQ, X, RTOLX, ATOLX, TOLF, IFLAG, MXIT, NCJS,
1 NSRRC, NSRI, IPRINT, RW(1), RW(2), NC, RW(K1),
2 RW(K2), RW(K3), RW(K4), RW(K5), RW(K6), IW(4))
C
IW(3) = MXIT
ENDIF
RETURN
END