bisection method to determine an approximation to the root that is accurate to at least within 10−4 for: in javascript

Here is an example of how you can implement the bisection method in JavaScript to determine an approximation to the root that is accurate to at least within 10^-4:

index.tsx
function bisectionMethod(a, b, tolerance) {
  // Check if the function has opposite signs at endpoints a and b
  if (Math.sign(func(a)) * Math.sign(func(b)) >= 0) {
    throw new Error("The function must have opposite signs at endpoints a and b.");
  }

  // Continue looping until the interval is smaller than the tolerance
  while (Math.abs(b - a) > tolerance) {
    // Find the midpoint of the interval
    let c = (a + b) / 2;

    // Check if the root is at midpoint or in the left half
    if (Math.sign(func(c)) * Math.sign(func(a)) < 0) {
      b = c;
    }
    // Otherwise, the root is in the right half
    else {
      a = c;
    }
  }

  // Return the approximate root
  return (a + b) / 2;
}

// Define your function here
function func(x) {
  return x * x - 4;
}

// Call the bisectionMethod function with your chosen parameters
let root = bisectionMethod(0, 2, 0.0001);
console.log("Approximate root:", root);
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In this example, the bisectionMethod function takes in three parameters: a, b, and tolerance. a and b define the interval within which the root is located, and tolerance specifies the required accuracy.

The method first checks if the function has opposite signs at the endpoints a and b. If not, it throws an error because the bisection method cannot guarantee convergence in such cases.

It then enters a loop where it calculates the midpoint c of the interval using the formula (a + b) / 2. It checks if the function has opposite signs at c and a. If it does, it updates b to c, otherwise it updates a to c. This effectively halves the interval at each iteration.

The loop continues until the interval's size is smaller than the specified tolerance. Finally, the method returns the average of a and b, which provides an approximation to the root that is accurate to at least within the given tolerance.

In the example above, the func function represents the equation or function for which you want to find the root. You can modify this function to match your specific problem.

Note: The bisection method assumes that the function is continuous in the given interval and the root exists within the interval. Make sure to choose appropriate values of a and b to encapsulate the root.