Numerical Calculation
Part 1: Equations &Roots
(Newton- Raphson's Method)
Dr.Entesar Ganash
Solving equations
A numerical method is an approximate computer method to solve a mathematical
problem that often can not be solved analytically.
If we want to Solve the equation f(x)=0 that means we want to find its root x (or
roots) depends on the equation degree.
E.g.
Function: f(x) = x2 - 4
Roots: x = -2, x = 2
Because: f(-2) = (-2)2 - 4 = 4 - 4 = 0
f(2) = (2)2 - 4 = 4 - 4 = 0
For any f(x), there is no general technique to obtain roots analytically. Here we study
two numerical method for finding roots numerically.
Newton's Method
Bisection's Method
Newton's Method
A general idea:
 may be it is the easiest numerical method to apply for solving equations, i.e.
Newton’s method could be used to solve a general equation f(x)=0.
 it is an iterative method, i.e. it repeatedly attempts to improve a root estimation.
If is an approximation to root, the next approximation
is given as the
following:
Where
is the first derivative of
, until
has come close enough to
zero. It converges to a root only if the starting guess is ‘close enough’. since ‘close
enough’ depends on the nature of
A Structure Plan:
A structure plan to execute Newton's method:
1. Read the starting value (initial guess) of x and required accuracy (Epsilon)
2.
Define the function
3.
test whether
4.
If the condition in step 3 is right repeat until the condition in step 3 becomes
Wrong (that means Newton converged ) or up to k=20 (for example , and it is
necessary to limit k because the process may not converge) , find a new value of x
from Newton's method i.e. using
print
and
and its derivative
, end a program.
Example
Write a program using Newton’s method to solve this equation
starting with x= 2 and stopping either when the
absolute value of the function is less than
or after 20 repetitions.
Solving
PROGRAM NEWTON
IMPLICIT NONE
Integer::k
! a counter
Real :: Eps ,x ! required accuracy (epsilon) & staring guess
k=0
Eps=1.0E-6
x=2
write(*,*)'
','x','
', 'f(x)'
write(*,*)'----------------------------------'
Do WHILE ((ABS(f(x))>=Eps) .AND. (k< 20))
x=x-f(x)/df(x)
write(*,*)x, f(x)
k=k+1
END DO
IF (ABS(f(x))<=Eps)THEN
Write(*,*)'Newton converged'
ELSE
Write(*,*)'Newton diverged'
END IF
Contains
!-------------------------!Function Subprogram f is to solve f(x)=0
!-------------------------Real function f(x)
Implicit none
Real:: f,x
f=x**3+x-3
End function f
!-------------------------! Anther Function Subprogram df
!-------------------------Real function df(x)
Implicit none
Real::df,x
df=3*x**2+1
End function df
!-------------------------END PROGRAM NEWTON
continue
Solving
1
2
The result:
References
•Hahn, B.D., 1994, Fortran 90 For Scientists and Engineers, Elsevier.
•http://www.mathcs.emory.edu/~cheung/Courses/170/Syllabus/07/bisection.html
•Univ.,