How to find minimum range to put in a specific function

I pretty new to programming and just got enrolled in a college.
So far I've learned some basic loops, but the question is how do i determine the minimum range of a specific function?

Here's the case:
"Choose the first interval by finding a and b such that 𝑓(𝑎)∙𝑓(𝑏)<0 "

and the function that i have to use is: 8−4.5(𝑥−sin𝑥)=0

Now I was asked to input 2 numbers in that function and multiply it.

Here's example:

import java.util.Scanner;

public class Main {
public static void main(String args) {
Scanner sc = new Scanner(System.in);
double a = sc.nextDouble();
double b = sc.nextDouble();

System.out.println(function(a)*function(b);

public static double function(double x)
double resultF = 0;
resultF = 8 - (4.5 * (x - Math.sin(x)));
return resultF;



which I already have make a class for it.

Now the problem is how do I find the minimum value if I were to input 2 (function(a)*function(b)) to be as close as zero.

So it begins with making user input 2 random numbers. Then both of them will have to go through 8−4.5(𝑥−sin𝑥)=0 function.
After the numbers have been calculated then you take both of that "calculated numbers" and multiplied it together.
What I am trying to do here is trying to make a loop that would automatically shorter the interval of the "input" and making them as close to zero as possible. (Referring to the 8−4.5(𝑥−sin𝑥)=0 function)

So far what I can do is:

package com.company;

import java.util.Scanner;

public class Main {
public static void main(String args)
Scanner sc = new Scanner(System.in);
double a = sc.nextDouble();
double b = sc.nextDouble();
double resultA = function(a);
double resultB = function(b);
double newA = 0;
double newB = 0;

System.out.println(resultA * resultB);
System.out.println(resultA*resultB);
if (resultA*resultB >=0)
if(a<b)
while(function(a)*function(b)>=0)
newA = a++;
newB = b--;


else if(a > b)
while(function(a)*function(b)>=0)
newA = a--;
newB = b++;






I know that minimum interval is 3 and 2 but I don't know how to get my loop work.

Thank you for your replies !

Every Help is appreciated!
Thank you

1 Answer
1

To fully answer completely to your question if you want to get the shortest intervalle the intuitive methods is to reduce it by a step defined like this :

public static void main(String args)
Scanner sc = new Scanner(System.in);
double a = sc.nextDouble();
double b = sc.nextDouble();
double step = sc.nextDouble();
if (a > b)
double tmp = a;
a = b;
b = tmp;

double bResult = function(b);
while (function(a) * bResult < 0)
a += step;

a -= step;
double aResult = function(a);
while (aResult * function(b) < 0)
b -= step;

b += step;
System.out.println(a + " " + b);
System.out.println(function(a) + " " + function(b));
sc.close();



but if you want a precise result then the step is little then the computation can be very long ... A solution consists to use Dichotomy . The concept is simple : reduce the interval by divising le range by 2 until the range is shorter than your wanted interval like this :

public static void main(String args)
Scanner sc = new Scanner(System.in);
double a = sc.nextDouble();
double b = sc.nextDouble();
double step = sc.nextDouble();

if (a > b)
double tmp = a;
a = b;
b = tmp;

while (Math.abs(a - b) > step)
double mid = (a+b) / 2;
if (function(mid) == 0)
System.out.println(mid);
return;

if (function(a) * function(mid) < 0)
b = mid;
else
a = mid;


System.out.println(a + " " + b);
System.out.println(function(a) + " " + function(b));
sc.close();



Hoping that's will help you.

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