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```SUBROUTINE RTWI

PURPOSE
TO SOLVE GENERAL NONLINEAR EQUATIONS OF THE FORM X=FCT(X)
BY MEANS OF WEGSTEIN-S ITERATION METHOD.

USAGE
CALL RTWI (X,VAL,FCT,XST,EPS,IEND,IER)
PARAMETER FCT REQUIRES AN EXTERNAL STATEMENT.

DESCRIPTION OF PARAMETERS
X	  - RESULTANT ROOT OF EQUATION X=FCT(X).
VAL	  - RESULTANT VALUE OF X-FCT(X) AT ROOT X.
FCT	  - NAME OF THE EXTERNAL FUNCTION SUBPROGRAM USED.
XST	  - INPUT VALUE WHICH SPECIFIES THE INITIAL GUESS OF
THE ROOT X.
EPS	  - INPUT VALUE WHICH SPECIFIES THE UPPER BOUND OF THE
ERROR OF RESULT X.
IEND   - MAXIMUM NUMBER OF ITERATION STEPS SPECIFIED.
IER	  - RESULTANT ERROR PARAMETER CODED AS FOLLOWS
IER=0 - NO ERROR,
IER=1 - NO CONVERGENCE AFTER IEND ITERATION STEPS,
IER=2 - AT ANY ITERATION STEP THE DENOMINATOR OF
ITERATION FORMULA WAS EQUAL TO ZERO.

REMARKS
THE PROCEDURE IS BYPASSED AND GIVES THE ERROR MESSAGE IER=2
IF AT ANY ITERATION STEP THE DENOMINATOR OF ITERATION
FORMULA WAS EQUAL TO ZERO. THAT MEANS THAT THERE IS AT
LEAST ONE POINT IN THE RANGE IN WHICH ITERATION MOVES WITH
DERIVATIVE OF FCT(X) EQUAL TO 1.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
THE EXTERNAL FUNCTION SUBPROGRAM FCT(X) MUST BE FURNISHED
BY THE USER.

METHOD
SOLUTION OF EQUATION X=FCT(X) IS DONE BY MEANS OF
WEGSTEIN-S ITERATION METHOD, WHICH STARTS AT THE INITIAL
GUESS XST OF A ROOT X. ONE ITERATION STEP REQUIRES ONE
EVALUATION OF FCT(X). FOR TEST ON SATISFACTORY ACCURACY SEE
FORMULAE (2) OF MATHEMATICAL DESCRIPTION.
FOR REFERENCE, SEE
(1) G. N. LANCE, NUMERICAL METHODS FOR HIGH SPEED COMPUTERS,
ILIFFE, LONDON, 1960, PP.134-138,
(2) J. WEGSTEIN, ALGORITHM 2, CACM, VOL.3, ISS.2 (1960),
PP.74,
(3) H.C. THACHER, ALGORITHM 15, CACM, VOL.3, ISS.8 (1960),
PP.475,
(4) J.G. HERRIOT, ALGORITHM 26, CACM, VOL.3, ISS.11 (1960),
PP.603.
```