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Solving Regula Falsi Method using Fortran
Posted Date: 12 Mar 2008 Resource Type: Articles/Knowledge Sharing Category: Computer & Technology
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Posted By: ashish singh Member Level: Diamond
Rating:  Points: 5
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3. REGULA FALSI METHOD
In the Bisection Method since the approximations are some times higher than and sometimes lower than the exact solution, the convergence of the method is very slow. To improve the rate of convergence, in Regula Falsi Method, the next approximation is takenas the point of intersection of the line (x1,y1)-(x2,y2) with X-axis,instead of the midpoint on the interval x1-x2.
Let the equation of line (x1,y1)-(x2,y2)
Be y=mx+c
Clearly,
y1=mx1+c
Y2=mx2+c
m=(y2-y1)/(x2-x1)
c=y1-mx1
c=y1-(y2-y1)/(x2-x1)*x1
c=(x2y1-x1y1-y2x1+x1y1)/x2-x1)
c=(x2y1-x1y2)/(x2-x1)
y=(y2-y1)/(x2-x1)*x+(x2y1-x1y2)/(x2-x1)
To find point of intersection of this line with X-axis, we can put y=0
(y2-y1)/(x2-x1)*x+(x2y1-x1y2)/(x2-x1)=0
(y2-y1)/(x2-x1)*x=-(x2y1-x1y2)/(x2-x1)
x=(x2y1-x1y2)/(y2-y1)
Thus starting with two values x1 and x2 such that F(x1) and F(x2) have opposite sign we get next approximation as
X=(x2y1-x1y2)/(y2-y1)
If F(x1) and F(x) have opposite sign then x2 is replaced by x otherwise x1 is replaced by x and this process is repeated until we get x such that F(x)~0.
Fortran Program for Regula Falsi Method :
C SOLUTION OF EQUATION BY USING REGULA-FALSI METHOD
FUNCTION F(X)
F=2*X*X-4*X+1
RETURN
END
2 WRITE(*,*)'ENTER TWO NUMBERS'
READ(*,*) X1,X2
Y1=F(X1)
Y2=F(X2)
IF(F(X1)*F(X2)) 3,4,2
4 IF(F(X1).EQ.0) THEN
WRITE(*,*)'SOLUTION=',X1
ELSE
WRITE(*,*)'SOLUTION=',X2
STOP
END IF
3 WRITE(*,*)'ENTER TOL'
READ(*,*) TOL
5 X=(X2*Y1-X1*Y2)/(Y1-Y2)
Y=F(X)
IF(ABS(Y).LE.TOL) THEN
WRITE(*,*)'SOLUTION=',X
STOP
ENDIF
IF(Y*Y1.GT.0) THEN
X1=X
Y1=Y
ELSE
X2=X
Y2=Y
END IF
GOTO 5
END
Output
REGULA FALSI METHOD
Microsoft (R) FORTRAN Optimizing Compiler Version 5.10
Copyright (c) Microsoft Corp 1982-1991. All rights reserved.
FALSI.FOR
D:\FORTRAN\BINB>FALSI
ENTER TWO NUMBERS
1
2
ENTER TOL
.005
SOLUTION= 1.705882
Stop - Program terminated.
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