[CAS problem] High-precision operations in numerical solution equations

[yangyongkang]

Hi everyone, I recently came across an x = tan (x) equation about x. Find x> 0, the solution over the interval [k * pi, (k + 1/2) * pi] (k is a positive integer). It is found that when k is taken large, the error occurs.

Code:

subst(tan(x)-x,x=fsolve(tan(x)=x,x=100000.5*pi))

Calculation output：142106699.971
Very large error。

mathematica supports high-precision operations

Code:

FindRoot[Tan[x] - x, {x, 100000.5*Pi}, WorkingPrecision -> 30]

Calculation output：{x -> 314160.836152123035796438894350}


I wrote it in C (dichotomy), compared it, and found that the error increases with increasing k.

Code:

#include<stdio.h>
#include<stdlib.h>
#include<math.h>
#define pi 3.14159265358
#define n 10000
double MidPoint(double (*function)(double),double x0,double x1,double error)
{
     double start=x0,end=x1;
    do{
            if((*function)((start+end)*0.5)<0)
            {
                   start=(start+end)*0.5;
            }else
            {
                    end=(start+end)*0.5;
            }
    }while(end-start>error);
    return (end+start)*0.5;
}
double equation(double x) {return tan(x)-x;}
int main()
{
    double next,last=0;
    char s[20];
    FILE *file=fopen("/Users/yangyongkang/Desktop/a.txt","w");
    for(int k=1;k<=n;k++)
    {
         next=MidPoint(equation,k*pi,(k+0.5)*pi,1e-11);
         sprintf(s,"%f\n",last*last*sin(next-last));
         fputs(s,file);
         last=next;
    }
    fclose(file);
}

Contrast with MMA, found this

Code:

Show[ListPlot @@@ {{#1^2*Sin[#2 - #1] & @@@ 
     Partition[
      ParallelTable[
       First@Values@
         FindRoot[Tan[x] - x, {x, n*3.14159265358}, 
          WorkingPrecision -> 20], {n, 1.5, 10000.5, 1}], 2, 1], 
    PlotStyle -> Red}, {ToExpression@
     StringSplit@Import["/Users/yangyongkang/Desktop/a.txt"]}}]

Red represents the MMA result, blue represents the C language calculation result, and the accuracy gap is widened.