Calculation method notes for personal use

This semester, we opened a course on computing methods and artificial intelligence algorithms , It's actually numerical analysis , So prepare to record the following learned contents .

• 1. Solution of nonlinear equations
  （1） Dichotomy , Newton method , Newton's iteration , Newton went down the mountain
  
  The first method is dichotomy , Dichotomy is easier to understand , That is to say, multiple half division operations are performed on an interval with different signs at one end of the function , Take the midpoint , Then the next interval is determined according to the function value of the midpoint , Continuously , Finally, the tolerance requirements are met , Or the maximum number of iterations .
  The characteristic of dichotomy is that it has low requirements for functions , You just need the function to be continuous , Unlike Newton's method mentioned later, it requires both the function and its first derivative . However, one of the obvious disadvantages of dichotomy is that its convergence speed is very slow , So it is generally used to find the initial value . The exact value can be solved by other methods .

#include<stdio.h> 
#include<math.h> 
int i=0; int n=0; 
double eps=1e-6;
double f(double x){
     return exp(x)-2; }
double root[50000]; 
double erfen(double down,double up)
{
    
double middle=(down+up)/2; 
if(i==100)
{
     
printf("%lf",middle); 
return middle;
  
   }
  if(f(down)*f(up)<0)
  {
     
  
  
  if(f(middle)*f(down)<0)
  {
     
  
  i++; 
  n++; 
  
  root[n]=middle;
  
  return erfen(down,middle);
  }
  
  else if(f(up)*f(middle)<0)
{
     
	i++;
	n++; 
	root[n]=middle;
return erfen(middle,up); 
}
  else if(f(middle)<1e-6) 
  {
    
   printf("%.18lf",middle); 
   return middle; 

  }
  }
}
int main(){
    
FILE*fp;
fp=fopen("out.txt","w");
int a,b; 
int p=0; 
int y;
scanf("%d%d",&a,&b); 
erfen(a,b);
int j=0;
for(j=0;j<i;j++){
     
fprintf(fp,"%.18lf ",root[j]);
p++;    
if(p==5)
{
    	
	p=0;
	fprintf(fp,"\n"); 
}
}
return 0;
 }

Dichotomy is certainly not enough to meet the demand, so we introduced the fixed-point method
The fixed-point method is like generating a sequence of numbers , The latter term of the sequence is always the function value of the former term as an independent variable
If this sequence converges , Then the function must have zero .
The way to do this is , Let the objective function be equal to 0 Add... To both sides of the equation x, So we have a fixed point equation , Then make x by a_n+1, Order on the other side f(a_n); So a sequence of numbers is generated , Then you can iterate continuously .
The following is the core idea

while(fabs(xk-xk+1)<e)
{
    
xk=xk+1;
xk+1=f(xk）;
}

The next chapter is Newton's method and Newton's downhill method .