二等分法程序

给定浮点数x以及两个数字’a’和’b’的函数f(x)，以使f(a)* f(b)<0且f(x)在[a，b]中是连续的。在此，f(x)表示代数或先验方程。在区间[a，b]中找到函数的根(或找到x的值，使得f(x)为0)。
例子：

Input: A function of x, for example x3 - x2 + 2.     
       And two values: a = -200 and b = 300 such that
       f(a)*f(b) < 0, i.e., f(a) and f(b) have
       opposite signs.
Output: The value of root is : -1.0025
        OR any other value with allowed
        deviation from root.

什么是代数和先验函数?
代数函数可以用多项式形式表示，如f(x)= a ₁ x ³ + a ₂ x ² +….. + e其中aa ₁ ，a ₂ ，…是常数，x是变量。
超越函数是非代数函数，例如f(x)= sin(x)* x – 3或f(x)= e ^x + x ²或f(x)= ln(x)+ x…。
什么是对分法?
该方法也称为间隔减半法，二分查找法或二分法。此方法用于查找给定间隔中方程的根，该间隔为f(x)= 0的’x’值。
该方法基于中间值定理，该中间值定理指出如果f(x)是连续函数，并且存在两个实数a和b，使得f(a)* f(b)0和f(b)<0) ，那么可以确保它们之间至少有一个根。
假设：

1. f(x)是区间[a，b]中的连续函数
2. f(a)* f(b)<0

脚步：

1. 求出中间点c =(a + b)/ 2。
2. 如果f(c)== 0，则c是解的根。
3. 否则f(c)！= 0
   1. 如果值f(a)* f(c)<0，则根位于a和c之间。因此，我们再次求出a和c
   2. 否则，如果f(b)* f(c)<0，则根位于b和c之间。因此，我们重复b和c。
   3. 其他给定函数不遵循假设之一。

由于root可能是浮点数，因此我们在a和b之间的差小于值?时重复上述步骤。 (非常小的值)。

以下是上述步骤的实现。

C++

// C++ program for implementation of Bisection Method for
// solving equations
#include
using namespace std;
#define EPSILON 0.01
 
// An example function whose solution is determined using
// Bisection Method. The function is x^3 - x^2  + 2
double func(double x)
{
    return x*x*x - x*x + 2;
}
 
// Prints root of func(x) with error of EPSILON
void bisection(double a, double b)
{
    if (func(a) * func(b) >= 0)
    {
        cout << "You have not assumed right a and b\n";
        return;
    }
 
    double c = a;
    while ((b-a) >= EPSILON)
    {
        // Find middle point
        c = (a+b)/2;
 
        // Check if middle point is root
        if (func(c) == 0.0)
            break;
 
        // Decide the side to repeat the steps
        else if (func(c)*func(a) < 0)
            b = c;
        else
            a = c;
    }
    cout << "The value of root is : " << c;
}
 
// Driver program to test above function
int main()
{
    // Initial values assumed
    double a =-200, b = 300;
    bisection(a, b);
    return 0;
}

Java

// Java program for implementation of Bisection Method
// for solving equations
class GFG{
    static final float EPSILON = (float)0.01;
 
    // An example function whose solution is determined using
    // Bisection Method. The function is x^3 - x^2  + 2
    static double func(double x)
    {
        return x*x*x - x*x + 2;
    }
 
    // Prints root of func(x) with error of EPSILON
    static void bisection(double a, double b)
    {
        if (func(a) * func(b) >= 0)
        {
            System.out.println("You have not assumed"
                        + " right a and b");
            return;
        }
 
        double c = a;
        while ((b-a) >= EPSILON)
        {
            // Find middle point
            c = (a+b)/2;
 
            // Check if middle point is root
            if (func(c) == 0.0)
                break;
 
            // Decide the side to repeat the steps
            else if (func(c)*func(a) < 0)
                b = c;
            else
                a = c;
        }
                //prints value of c upto 4 decimal places
        System.out.printf("The value of root is : %.4f"
                        ,c);
    }
 
    // Driver program to test above function
    public static void main(String[] args)
    {
        // Initial values assumed
        double a =-200, b = 300;
        bisection(a, b);
    }
    // This code is contributed by Nirmal Patel
}

Python3

# Python program for implementation
# of Bisection Method for
# solving equations
 
  
# An example function whose
# solution is determined using
# Bisection Method.
# The function is x^3 - x^2  + 2
def func(x):
    return x*x*x - x*x + 2
  
# Prints root of func(x)
# with error of EPSILON
def bisection(a,b):
 
    if (func(a) * func(b) >= 0):
        print("You have not assumed right a and b\n")
        return
  
    c = a
    while ((b-a) >= 0.01):
 
        # Find middle point
        c = (a+b)/2
  
        # Check if middle point is root
        if (func(c) == 0.0):
            break
  
        # Decide the side to repeat the steps
        if (func(c)*func(a) < 0):
            b = c
        else:
            a = c
             
    print("The value of root is : ","%.4f"%c)
     
# Driver code
# Initial values assumed
a =-200
b = 300
bisection(a, b)
 
# This code is contributed
# by Anant Agarwal.

C#

// C# program for implementation
// of Bisection Method for
// solving equations
using System;
 
class GFG
{
static float EPSILON = (float)0.01;
 
// An example function whose
// solution is determined using
// Bisection Method. The function
// is x^3 - x^2 + 2
static double func(double x)
{
    return x * x * x -
           x * x + 2;
}
 
// Prints root of func(x)
// with error of EPSILON
static void bisection(double a,
                      double b)
{
    if (func(a) * func(b) >= 0)
    {
        Console.WriteLine("You have not assumed" +
                                " right a and b");
        return;
    }
 
    double c = a;
    while ((b - a) >= EPSILON)
    {
        // Find middle point
        c = (a + b) / 2;
 
        // Check if middle
        // point is root
        if (func(c) == 0.0)
            break;
 
        // Decide the side
        // to repeat the steps
        else if (func(c) * func(a) < 0)
            b = c;
        else
            a = c;
    }
     
    // prints value of c
    // upto 4 decimal places
    Console.WriteLine("The value of " +
                      "root is : "+ c);
}
 
// Driver Code
static public void Main ()
{
    // Initial values assumed
    double a = -200, b = 300;
    bisection(a, b);
}
}
 
// This code is contributed by ajit

PHP

= 0)
    {
        echo "You have not assumed " .
                 "right a and b","\n";
        return;
    }
 
    $c = $a;
    while (($b - $a) >= $EPSILON)
    {
        // Find middle point
        $c = ($a + $b) / 2;
 
        // Check if middle
        // point is root
        if (func($c) == 0.0)
            break;
 
        // Decide the side to
        // repeat the steps
        else if (func($c) * func($a) < 0)
            $b = $c;
        else
            $a = $c;
    }
    echo "The value of root is : " , $c;
}
 
// Driver Code
 
// Initial values assumed
$a =-200;
$b = 300;
bisection($a, $b);
 
// This code is contributed by ajit
?>

Javascript

 
// JavaScript program for implementation
// of Bisection Method for
// solving equations
 
let EPSILON = 0.01;
 
    // An example function whose solution is determined using
    // Bisection Method. The function is x^3 - x^2  + 2
    function func(x)
    {
        return x*x*x - x*x + 2;
    }
   
    // Prints root of func(x) with error of EPSILON
    function bisection(a, b)
    {
        if (func(a) * func(b) >= 0)
        {
            document.write("You have not assumed"
                        + " right a and b");
            return;
        }
   
        let c = a;
        while ((b-a) >= EPSILON)
        {
            // Find middle point
            c = (a+b)/2;
   
            // Check if middle point is root
            if (func(c) == 0.0)
                break;
   
            // Decide the side to repeat the steps
            else if (func(c)*func(a) < 0)
                b = c;
            else
                a = c;
        }
                //prints value of c upto 4 decimal places
        document.write("The value of " +
                      "root is : "+ c.toFixed(4));
    }
 
 
// Driver program
 
        // Initial values assumed
        let a =-200, b = 300;
        bisection(a, b);
         
        // This code is contributed by susmitakundugoaldanga.

输出：

The value of root is : -1.0025