1. Some exercises
Dr.Entesar Ganash

2. Exercise 1
Write a program to re-calculate the problem 1 but now using Newton’s method to solve numerically the last equation, starting with x=0 .5 and stopping either when the absolute value of the function is less than or after 20 repetitions.
Using IF statements & GO TO statement instead of Do While statement. Compare the results from both problems.
Solving
PROGRAM NEWTON
IMPLICIT NONE
Integer::k ! a counter
Real :: Eps ,x ! required accuracy (epsilon) & staring guess
k=0
Eps=1.0E-6
x=0.5
write(*,*)' ','x',' ', 'f(x)'
write(*,*)' ‘

3. continue Solving 1 10 If ((ABS(f(x))<=Eps) .OR. (k> 20)) then
write(*,*)x, f(x)
Else
x=x-f(x)/df(x)
k=k+1
Go to 10
End if
write(*,*)' '
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-0.2*sin(x)-0.5
End FUNCTION f
continue Solving 1

4. continue Solving 2 !--------------------------
!Anther Function Subprogram df
REAL FUNCTION df(x)
Implicit none
Real::df,x
df= *cos(x)
End FUNCTION df
END PROGRAM NEWTON
continue Solving 2

5. The result:

6. Exercise 2 Solving Write a program to calculate in radians
for ten terms, When
Then calculates the sine using the sine intrinsic function
Hint: n! = n*(n-1)*(n-2)*….*3*2*1.
Solving
ROGRAM SUMSIN
Implicit none
Integer:: I,K ! counters
Real :: theta ,factn,sum,PI ! angle, factorial value, summation & Constant
Real :: inf ! avariable symbol
PI=
THETA= !in degree
THETA=THETA*PI/ ! to convert to rad

7. continue Solving sum=0.0 ! intial value DO K=1,10 ! number of term
factn=1.0
DO I=2*K-1,1,-1
factn=factn*I
ENDDO
sum=sum+(-1)**(k-1)*theta**(2*k-1)/factn
print*,K,SUM
! FROM intrinsic function
inf=SIN(THETA)
print*,' '
PRINT*, 'SIN(THETA)from intrinsic function'
PRINT*, 'SIN(THETA)=',inf
END PROGRAM SUMSIN
continue Solving

8. The result: