How to find roots in JavaScript

How to find roots in JavaScript

In mathematics, finding roots is a common problem. It can help us solve many practical problems, such as equation solving, image processing, etc. In computer science, the JavaScript language works well for mathematical problems, including root-finding problems. In this article, we will learn how to find roots in JavaScript.

1. What is root seeking

First of all, we need to clarify what root seeking is. In mathematics, the roots of an equation are the values ​​of the unknowns that make the equation true. For example, for a quadratic equation ax^2 bx c=0, the value of x is its root. In computer science, we often use numerical iterative methods to solve the roots of equations.

2. Numerical iteration method to solve roots

The numerical iteration method is a numerical analysis method that can be used to approximately solve mathematical problems. It gradually approximates the solution of a problem according to certain rules until it reaches a certain accuracy or a given termination condition.

In root-finding problems, the numerical iteration method is a widely used method. Its basic idea is to start from an initial value and gradually approach the target value according to an iterative formula until a certain accuracy is reached.

The steps of the numerical iteration method are as follows:

1. Determine the initial value x0.
2. Calculate the next approximation value xn 1 = f(xn) according to the iterative formula.
3. Determine whether the termination conditions are met. If it is not satisfied, continue calculating the next approximation value.
4. Repeat steps 2 and 3 until the termination condition is met.

In root-finding problems, the choice of iterative formula is very important. Different iteration formulas may lead to different convergence speeds and accuracy. Two commonly used iterative formulas are introduced below.

3. Root finding by bisection method

The bisection method is one of the simplest numerical iteration methods in root finding problems. Its basic idea is to continuously divide the interval to be determined into two, and then determine the next interval based on the values ​​of the function in the two sub-intervals. This process is repeated until the interval length is less than the given precision.

In JavaScript, the bisection root finding code is as follows:

function bisection(func, a, b, tol) {
    if (func(a) * func(b) >= 0) {
        throw "Error: f(a) and f(b) do not have opposite signs.";
    }
    let c = a;
    while ((b-a)/2 > tol) {
        c = (a+b)/2;
        if (func(c) === 0.0) {
            return c;
        } else if (func(c)*func(a) < 0) {
            b = c;
        } else {
            a = c;
        }
    }
    return c;
}

Parameter description:

• func: the function to be solved.
• a, b: Solution interval.
• tol: precision.

4. Newton’s method for finding roots

Newton’s method is a numerical iterative method for solving nonlinear equations. Its basic idea is to use local linear approximation of functions to perform iterative calculations. In each iteration, Newton's method will take the intersection of the tangent line at the current point and the x-axis as the next iteration point, and repeat this process until a certain accuracy is achieved.

In JavaScript, the code for finding the roots of Newton's method is as follows:

function newton(func, derivFunc, x0, tol) {
    let x1 = x0 - func(x0) / derivFunc(x0);
    while (Math.abs(x1 - x0) > tol) {
        x0 = x1;
        x1 = x0 - func(x0) / derivFunc(x0);
    }
    return x1;
}

Parameter description:

• func: the function to be solved.
• derivFunc: The derivative of the function.
• x0: Initial value.
• tol: precision.

5. Summary

This article introduces the basic methods of root finding in JavaScript, especially the bisection method and Newton's method in numerical iteration methods. In practical applications, appropriate methods can be selected according to specific problems to solve the roots of the equation.

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