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```SUBROUTINE DRTNI

PURPOSE
TO SOLVE GENERAL NONLINEAR EQUATIONS OF THE FORM F(X)=0
BY MEANS OF NEWTON-S ITERATION METHOD.

USAGE
CALL DRTNI (X,F,DERF,FCT,XST,EPS,IEND,IER)
PARAMETER FCT REQUIRES AN EXTERNAL STATEMENT.

DESCRIPTION OF PARAMETERS
X	  - DOUBLE PRECISION RESULTANT ROOT OF EQUATION F(X)=0.
F	  - DOUBLE PRECISION RESULTANT FUNCTION VALUE AT
ROOT X.
DERF   - DOUBLE PRECISION RESULTANT VALUE OF DERIVATIVE
AT ROOT X.
FCT	  - NAME OF THE EXTERNAL SUBROUTINE USED. IT COMPUTES
TO GIVEN ARGUMENT X FUNCTION VALUE F AND DERIVATIVE
DERF. ITS PARAMETER LIST MUST BE X,F,DERF, WHERE
ALL PARAMETERS ARE DOUBLE PRECISION.
XST	  - DOUBLE PRECISION INPUT VALUE WHICH SPECIFIES THE
INITIAL GUESS OF THE ROOT X.
EPS	  - SINGLE PRECISION 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 DERIVATIVE DERF WAS
EQUAL TO ZERO.

REMARKS
THE PROCEDURE IS BYPASSED AND GIVES THE ERROR MESSAGE IER=2
IF AT ANY ITERATION STEP DERIVATIVE OF F(X) IS EQUAL TO 0.
POSSIBLY THE PROCEDURE WOULD BE SUCCESSFUL IF IT IS STARTED
ONCE MORE WITH ANOTHER INITIAL GUESS XST.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
THE EXTERNAL SUBROUTINE FCT(X,F,DERF) MUST BE FURNISHED
BY THE USER.

METHOD
SOLUTION OF EQUATION F(X)=0 IS DONE BY MEANS OF NEWTON-S
ITERATION METHOD, WHICH STARTS AT THE INITIAL GUESS XST OF
A ROOT X. CONVERGENCE IS QUADRATIC IF THE DERIVATIVE OF
F(X) AT ROOT X IS NOT EQUAL TO ZERO. ONE ITERATION STEP
REQUIRES ONE EVALUATION OF F(X) AND ONE EVALUATION OF THE
DERIVATIVE OF F(X). FOR TEST ON SATISFACTORY ACCURACY SEE
FORMULAE (2) OF MATHEMATICAL DESCRIPTION.
FOR REFERENCE, SEE R. ZURMUEHL, PRAKTISCHE MATHEMATIK FUER
INGENIEURE UND PHYSIKER, SPRINGER, BERLIN/GOETTINGEN/
HEIDELBERG, 1963, PP.12-17.

```