Download FORTRAN PROGRAMS FOR SOLVIMG NUMERICAL PROBLEMS

FORTRAN PROGRAMS FOR SOLVIMG NUMERICAL PROBLEMS
Designed by
T K Rajan
Seln. Grade Lecturer in Mathematics
Govt Victoria College, Palakkad
BISECTION METHOD
c
40
50
PROGRAM bs
To find zero of an equation by bisection method
write(*,40)
format(1x,'To find a real root of an equation using Bisection',\)
write(*,50)
format(' method.')
write(*,*)
write(*,*)'Enter numbers between which the root is to be found:'
read(*,*)a,b
IF((f(a).GT.0).AND.(f(b).LT.0))THEN
w=a
a=b
b=w
ENDIF
20
30
10
write(*,*)'Input error value:'
read(*,*)e
c=(a+b)/2
IF(abs(f(c)).LT.e) GOTO 30
IF(f(c).LT.0)THEN
a=c
ELSE
b=c
ENDIF
GOTO 20
write(*,10)c
format(1x,'The approximate root of the function is:',F8.3)
STOP
END
function f(c)
f=c*c-5*c+6
return
END
GAUSS ELIMINATION
c
10
20
PROGRAM gelmn
Gauss elimination method to solve a linear system of eqns
dimension a(30,30),x(30)
write(*,10)
format(1x,'To solve a linear system of equations using Gauss ',\)
write(*,20)
format('Elimination method with pivoting(using subroutine).')
write(*,*)'Enter the number of variables:'
read(*,*)n
write(*,*)'Enter the coefficients in the equations:'
read(*,*)((a(i,j),j=1,n+1),i=1,n)
DO 30 k=1,n-1
100
90
call pivot(a,k,n)
DO 90 i=k+1,n
u=a(i,k)/a(k,k)
DO 100 j=k,n+1
a(i,j)=a(i,j)-u*a(k,j)
continue
continue
30
continue
IF(abs(a(n,n)).LE.(.00001))THEN
write(*,*)'Ill conditioned equations.'
STOP
ENDIF
70
60
110
80
x(n)=a(n,n+1)/a(n,n)
DO 60 i=n-1,1,-1
sum=0
DO 70 j=i+1,n
sum=sum+a(i,j)*x(j)
continue
x(i)=(a(i,n+1)-sum)/a(i,i)
continue
write(*,*)'Values of the variables are as follows:'
DO 80 i=1,n
write(*,110)x(i)
format(1x,F10.3)
continue
STOP
END
subroutine pivot(a,k,n)
dimension a(30,30)
real mx
integer p,q
mx=abs(a(k,k))
40
50
p=k
DO 40 m=k+1,n
IF(abs(a(m,k)).GT.mx)THEN
mx=abs(a(m,k))
p=m
ENDIF
continue
IF(mx.LE.(.00001))THEN
write(*,*)'Ill-conditioned equations.'
STOP
ENDIF
DO 50 q=k,n+1
temp=a(k,q)
a(k,q)=a(p,q)
a(p,q)=temp
continue
return
END
EULER’ METHOD
PROGRAM eu
c
Euler's method to solve a first order de
dimension x(50),y(50)
write(*,*)'To solve a differential equation using Euler method.'
write(*,*)'Enter initial values:'
read(*,*)a,c
write(*,*)'Enter value at which result is to be found:'
read(*,*)b
write(*,*)'Enter the number of subintervals required:'
read(*,*)n
x(1)=a
y(1)=c
h=(b-a)/n
write(*,*)'The function values at each step is given below.'
write(*,*)
write(*,*)'
x
Function value'
write(*,10)x(1),y(1)
10
format(1x,F10.4,5x,F10.4)
DO 20 i=1,n
x(i+1)=x(i)+h
y(i+1)=y(i)+h*f(x(i),y(i))
write(*,10)x(i+1),y(i+1)
20
continue
STOP
END
function f(x,y)
f=(y-x)/(y+x)
return
END
REGULA FALSI METHOD
PROGRAM regfal
Regula falsi Method to a find a root of an eqn
write(*,40)
40
format(1x,'To find the approximate root of a function using ')
write(*,50)
50
format(\,'Method of False Position.')
write(*,*)'Enter the numbers between which root is to be found:'
read(*,*)a,b
c
IF((f(a).GT.0).AND.(f(b).LT.0))THEN
w=a
a=b
b=w
ENDIF
20
10
30
write(*,*)'Input error value:'
read(*,*)e
c=(a*f(b)-b*f(a))/(f(b)-f(a))
IF(abs(f(c)).LT.e) GOTO 10
IF(f(c).LT.0)THEN
a=c
ELSE
b=c
ENDIF
GOTO 20
write(*,30)c
format(1x,'The approximate root of the function is:',F8.3)
STOP
END
function f(c)
f=c**3-9*c+1
return
END
GCD OF A SEQUENCE OF NUMBERS
PROGRAM ss
integer x(30)
write(*,*)'To find the gcd of a set of numbers.'
write(*,*)'Enter number of numbers:'
read(*,*)n
write(*,*)'Enter the numbers:'
read(*,*)(x(i),i=1,n)
j=x(1)
DO 10 i=2,n
k=gg(j,x(i))
j=k
10
continue
write(*,20)j
20
format(1x,' GCD= ',I4)
STOP
END
30
function gg(a,b)
integer a,b
IF(a.GT.b)THEN
j=a
a=b
b=j
ENDIF
i=mod(b,a)
IF(i.EQ.0)THEN
gg=a
return
ELSE
b=a
a=i
GOTO 30
ENDIF
END
FIBONACCI SEQUENCE
PROGRAM fbnc
c
Fobonacci sequence
integer f1,f2,f3
f1=1
f2=1
write(*,*)'To display fibonacci numbers.'
write(*,*)'Enter the number of terms to be displayed:'
read(*,*)n
write(*,*)'The fibonacci sequence is : '
IF(n.EQ.1)THEN
write(*,10)f1
10
format(1x,I6)
ELSE
write(*,10)f1
write(*,10)f2
DO 20 i=3,n
f3=f1+f2
write(*,10)f3
f1=f2
f2=f3
20
continue
ENDIF
STOP
END
GCD & LCM OF TWO NUMBERS
PROGRAM gcdlcm
integer a,b,l
write(*,*)'To find the gcd and lcm of two numbers.'
write(*,*)'Enter 2 numbers:'
read(*,*)a,b
m=a
n=b
IF(a.GT.b)THEN
j=a
a=b
b=j
ENDIF
10
i=mod(b,a)
IF(i.EQ.0)THEN
write(*,20)a
20
format(1x,'GCD= ',I4)
ELSE
b=a
a=i
GOTO 10
ENDIF
l=m*n/a
write(*,30)l
30
format(1x,'LCM= ',I4)
STOP
END
LAGRANGE’S METHOD OF INTERPOLATION
PROGRAM ll
c
Lagrange's Method of interpolation
dimension x(20),y(20)
write(*,50)
50
format(1x,'To find the function value for a particular value ',\)
write(*,60)
60
format('of x using Lagrange interpolation formula.')
write(*,*)
write(*,*)'Enter the number of known function values:'
read(*,*)n
write(*,*)'Enter value of x and corresponding function values:'
DO 10 i=1,n
read(*,*)x(i),y(i)
10
continue
write(*,*)'Enter x for which function value is to be found:'
read(*,*)a
s=0
DO 20 i=1,n
p=1
DO 30 j=1,n
IF(i.NE.j) p=p*(a-x(j))/(x(i)-x(j))
30
continue
s=s+p*y(i)
20
continue
write(*,*)
write(*,40)a,s
40
format(1x,'The function value at x =',F7.2,' is:',F9.3)
STOP
END
NEWTON’S METHOD OF INTERPOLATION
PROGRAM newt
c
Newton's Mthod of interpolation
dimension x(20),y(20,20)
write(*,100)
100
format(1x,'To find the value of a function corresponding to a',\)
write(*,110)
110
format(' value of x using Newtons divided difference method.')
write(*,*)
write(*,*)'Enter the number of known function values:'
read(*,*)n
write(*,*)'Enter the values of x and the corresponding function
*values:'
DO 10 i=1,n
read(*,*)x(i),y(i,1)
10
continue
write(*,*)'Enter the value of x for which function value is to be
*found:'
read(*,*)a
k=0
DO 20 j=2,n
k=k+1
DO 30 i=1,n-k
y(i,j)=(y(i+1,j-1)-y(i,j-1))/(x(i+k)-x(i))
30
continue
20
continue
90
80
70
50
40
60
write(*,*)'The divided difference table is : '
write(*,*)
DO 70 i=1,n
write(*,90)x(i)
DO 80 j=1,n-i+1
write(*,90)y(i,j)
format(1x,F10.3\)
continue
write(*,*)
write(*,*)
continue
s=y(1,1)
DO 40 j=2,n
p=1
DO 50 i=1,j-1
p=p*(a-x(i))
continue
s=s+p*y(1,j)
continue
write(*,60)s
format(1x,'The corresponding function value is:',F10.3)
STOP
END
NEWTON RAPHSON METHOD
PROGRAM nr
c
Newton Raphson Method to fin a root of an eqn
integer count
count=0
write(*,40)
40
format(1x,'To find a root of an equation using Newton Raphson',\)
write(*,50)
50
format(' Method.')
write(*,*)'Enter initial root and error value:'
read(*,*)x,e
10
IF(g(x).EQ.0)THEN
write(*,*)'Incorrect initial root.'
STOP
ENDIF
y=x-(f(x)/g(x))
IF(abs(f(y)).LT.e) GOTO 20
count=count+1
IF(count.GT.500)THEN
write(*,*)'An error has occured.'
STOP
ENDIF
x=y
GOTO 10
20
write(*,30)x
30
format(1x,'Approximate root= ',F10.3)
STOP
END
function f(x)
f=x*x-5*x+6
return
END
function g(x)
g=2*x-5
return
END
NUMERICAL DIFFERENTAION
PROGRAM ndff
c
Numerical Differentiation
dimension x(20),y(20,20)
write(*,70)
70
format(1x,'To find the value of derivative of a function at a',\)
write(*,80)
80
format(' given value using Numerical Differentiation.')
write(*,*)
write(*,*)'Enter the number of known function values:'
read(*,*)n
write(*,*)'Enter values of x and corresponding function values:'
read(*,*)(x(i),y(i,1),i=1,n)
write(*,*)'Enter the value at which derivative is to be found:'
read(*,*)a
DO 10 j=2,n
DO 10 i=1,n-j+1
y(i,j)=(y(i+1,j-1)-y(i,j-1))/(x(i+j-1)-x(i))
10
continue
v=y(1,2)
DO 20 i=3,n
s=0
DO 30 j=1,i-1
p=1
DO 40 k=1,i-1
IF(k.NE.j) p=p*(a-x(k))
40
continue
s=s+p
30
continue
v=v+s*y(1,i)
20
continue
write(*,50)a,v
50
format(1x,'The value of the derivative at x= ',F9.4,' is: ',F9.4)
STOP
END
PRODUCT OF TWO MATRICES
PROGRAM matsub
c
Product of two matrices
dimension a(10,10),b(10,10),c(10,10)
integer p,q
write(*,*)'To find the product of two matrices.'
write(*,*)'Enter the order of the first matrix:'
read(*,*)m,n
write(*,*)'Enter the order of the second matrix:'
read(*,*)p,q
IF(n.NE.p)THEN
write(*,*)'The matrices are not conformable for multiplication.'
STOP
ENDIF
call inpt(a,m,n)
call inpt(b,p,q)
call pro(a,b,c,m,n,q)
call outpt(c,m,q)
STOP
END
subroutine inpt(a,m,n)
dimension a(10,10)
write(*,10)m,n
10
format(1x,'Enter the matrix of order',I2,' *'I2,' :')
read(*,*)((a(i,j),j=1,n),i=1,m)
END
subroutine pro(a,b,c,m,n,q)
dimension a(10,10),b(10,10),c(10,10)
integer q
DO 20 i=1,m
DO 30 j=1,q
c(i,j)=0
DO 40 k=1,n
c(i,j)=c(i,j)+a(i,k)*b(k,j)
40
continue
30
continue
20
continue
END
subroutine outpt(c,m,q)
dimension c(10,10)
integer q
write(*,*)'The product matrix is: '
DO 50 i=1,m
DO 60 j=1,q
write(*,70)c(i,j)
70
format(3x,F6.2,4x\)
60
continue
write(*,*)
50
continue
END
RUNGE-KUTTA METHOD
PROGRAM rr
c
Rung- Kutta Method to solve a d.e.
integer count
real k1,k2
count=0
write(*,10)
10
format(1x,'To solve a differential equation using second ',\)
write(*,20)
20
format('order Runge Kutta method.')
write(*,*)'Enter the initial values:'
read(*,*)x1,y1
write(*,*)'Enter value at which function value is to be found:'
read(*,*)a
write(*,*)'Enter the number of subintervals:'
read(*,*)n
h=(a-x1)/n
write(*,*)'Values of x and corresponding function values are: '
write(*,*)
write(*,30)
30
format(8x,' x ',5x,' f(x) ')
write(*,40)x1,y1
40
format(1x,F10.4,F10.4)
50
k1=h*f(x1,y1)
k2=h*f(x1+h,y1+k1)
y2=y1+(k1+k2)/2
x2=x1+h
write(*,40)x2,y2
count=count+1
IF(count.LT.n) THEN
x1=x2
y1=y2
GOTO 50
ENDIF
write(*,*)
write(*,60)x2,y2
60
format(1x,'The value of the function at x=',F10.4,' is:',F10.4)
STOP
END
function f(x,y)
f=x+y
return
END
RUNGE-KUTTA METHOD OF ORDER 4
PROGRAM rrkk
c
Runge-Kutta method of order 4
integer count
count=0
write(*,10)
10
format(1x,'To solve a differential equation using fourth ',\)
write(*,20)
20
format('order Runge Kutta method.')
write(*,*)
write(*,*)'Enter the initial values:'
read(*,*)x1,y1
write(*,*)'Enter value at which function value is to be found:'
read(*,*)a
write(*,*)'Enter the number of subintervals:'
read(*,*)n
h=(a-x1)/n
write(*,*)'The function values are as follows: '
write(*,*)
write(*,30)
30
format(8x,' x ',5x,' f(x) ')
write(*,40)x1,y1
40
format(1x,F10.4,F10.4)
50
s1=f(x1,y1)
s2=f(x1+h/2,y1+s1*h/2)
s3=f(x1+h/2,y1+s2*h/2)
s4=f(x1+h,y1+s3*h)
y2=y1+(s1+2*s2+2*s3+s4)*h/6
x2=x1+h
write(*,40)x2,y2
count=count+1
IF(count.LT.n)THEN
x1=x2
y1=y2
GOTO 50
ENDIF
write(*,*)
write(*,60)x2,y2
60
format(1x,'The value at ',F10.4,' is :',F10.4)
STOP
END
function f(x,y)
f=x+y
return
END
SIMPSON’S RULE OF INTEGRATION
PROGRAM zz
c
Simpson's rule of integration
integer count
count=1
write(*,30)
30
format(1x,'To find the value of integral of a function using',\)
write(*,40)
40
format(' Simpsons rule. ')
write(*,*)
write(*,*)'Enter the limits of integration:'
read(*,*)a,b
write(*,*)'Enter the number of subintervals(an even no.) :'
read(*,*)n
h=(b-a)/n
x=a
y=f(x)
sum=y
10
IF(count.LT.n)THEN
x=x+h
y=f(x)
count=count+1
IF(mod(count,2).EQ.0)THEN
sum=sum+4*y
ELSE
sum=sum+2*y
ENDIF
GOTO 10
ENDIF
x=x+h
y=f(x)
sum=sum+y
sum=h*sum/3
write(*,20)sum
20
format(1x,'The value of the integral is: ',F10.4)
STOP
END
function f(x)
f=1/(1+x*x)
return
END
SIMPSON’S METHOD OD INTEGRATION (ARRAY GIVEN)
PROGRAM zz
Simpson's method of integration
dimension x(20),y(20)
write(*,*)'To find the value of integral of a function using
*Simpsons rule. '
write(*,*)
write(*,*)'Enter the limits of integration:'
read(*,*)a,b
write(*,*)'Enter the number of subintervals(an even no.) :'
read(*,*)n
h=(b-a)/n
x(1)=a
y(1)=f(x(1))
DO 10 i=2,n+1
x(i)=x(i-1)+h
y(i)=f(x(i))
10
continue
s=y(1)+y(n+1)
DO 20 i=2,n,2
s=s+4*y(i)
20
continue
DO 30 i=3,n-1,2
s=s+2*y(i)
30
continue
s=h*s/3
write(*,40)s
40
format(1x,'The value of the integral is: ',F10.4)
STOP
END
c
function f(x)
f=sin(x)
return
END
TRAPEZOIDAL RULE OF INTEGRATION
PROGRAM trp
c
Trapezoidal rule of integration
integer count
count=1
write(*,30)
30 format(1x,'To find the value of the integral of a function ',\)
write(*,40)
40 format('using trapezoidal rule.')
write(*,*)
write(*,*)'Enter the limits of integration: '
read(*,*)a,b
IF(b.LT.a)THEN
c=a
a=b
b=c
ENDIF
write(*,*)'Enter the number of subintervals(an even no.) :'
read(*,*)n
h=(b-a)/n
x=a
y=f(x)
sum=y
10 IF(count.LT.n)THEN
x=x+h
y=f(x)
sum=sum+2*y
count=count+1
GOTO 10
ENDIF
x=x+h
y=f(x)
sum=sum+y
sum=h*sum/2
write(*,*)
write(*,20)sum
20 format(1x,'The value of the integral is :',F10.4)
STOP
END
function f(x)
f=1/(1+x*x)
return
END
TRAPEZOIDAL RULE OF INTEGRATION (ARRAY )
PROGRAM ttrr
c
Trapezoidal rule of integration
dimension x(20),y(20)
write(*,40)
40 format(1x,'To find the integral of a given function using ',\)
write(*,50)
50 format('trapezoidal rule.')
write(*,*)'Enter the limits of integration:'
read(*,*)a,b
write(*,*)'Enter the number of subintervals:'
read(*,*)n
IF(a.GT.b)THEN
c=a
a=b
b=c
ENDIF
h=(b-a)/n
x(1)=a
y(1)=f(x(1))
write(*,*)x(1),y(1)
DO 10 i=2,n+1
x(i)=x(i-1)+h
y(i)=f(x(i))
write(*,*)x(i),y(i)
10 continue
s=y(1)+y(n+1)
DO 20 i=2,n
s=s+2*y(i)
20 continue
v=h*s/2
write(*,30)v
30 format(1x,'The value of the integral is: ',F10.4)
STOP
END
function f(x)
f=sin(x)
return
END
TRIANGULAR FACTORISATION TO SOLVE A LINEAR SYSTEM
PROGRAM tri
c
Triangular factorisation to solve a linear system of eqns
real l(20,20)
dimension a(20,20),b(20),u(20,20),x(20),y(20)
write(*,130)
130 format(1x,'To solve a system of equations using Method of ')
write(*,140)
140 format(\,'Triangularisation.')
write(*,*)'Enter the number of equations:'
read(*,*)n
write(*,*)'Enter the coefficients of each of the equations:'
read(*,*)((a(i,j),j=1,n),b(i),i=1,n)
DO 10 i=1,n
DO 20 j=i+1,n
l(i,j)=0
20
continue
DO 30 j=1,i-1
u(i,j)=0
30
continue
u(1,i)=a(1,i)
l(i,i)=1
IF(i.GT.1) l(i,1)=a(i,1)/u(1,1)
10 continue
DO 40 i=2,n
DO 50 j=2,i-1
s=0
DO 60 k=1,j-1
s=s+l(i,k)*u(k,j)
60
continue
l(i,j)=(a(i,j)-s)/u(j,j)
50
continue
DO 70 j=i,n
s=0
DO 80 k=1,i-1
s=s+l(i,k)*u(k,j)
80
continue
u(i,j)=a(i,j)-s
70
continue
40 continue
write(*,*)((l(i,j),i=1,n),j=1,n)
write(*,*)((u(i,j),i=1,n),j=1,n)
y(1)=b(1)
DO 90 i=2,n
sum=0
DO 100 j=1,n-1
sum=sum+l(i,j)*y(j)
100
continue
y(i)=b(i)-sum
90 continue
write(*,*)(y(i),i=1,n)
x(n)=y(n)/u(n,n)
DO 110 i=n-1,1,-1
sum=0
DO 120 j=i+1,n
sum=sum+u(i,j)*x(j)
120
continue
x(i)=(y(i)-sum)/u(i,i)
110 continue
write(*,*)'The solution system is: '
DO 150 i=1,n
write(*,160)x(i)
160
format(1x,F6.3)
150 continue
STOP
END