Wednesday, October 2, 2019

Newtons Method :: Newton-Raphson Method

Finding roots of a function is often a task which faces mathematicians. For simple functions, such as linear ones, the task is simple. When functions become more complex, such as with cubic and quadratic functions, mathematicians call upon more convoluted methods of finding roots. For many functions, there exist formulas which allow us to find roots. The most common such formula is, perhaps, the quadratic formula. When functions reach a degree of five and higher, a convenient, root-finding formula ceases to exist. Newton’s method is a tool used to find the roots of nearly any equation. Unlike the cubic and quadratic equations, Newton’s method – more accurately, the Newton-Raphson Method – can help to find roots of nearly any type of function, including all polynomial functions.   Ã‚  Ã‚  Ã‚  Ã‚  Newton’s method use derivative calculus to find the roots of a function or relation by first taking an approximation and then improving the accuracy of that approximation until the root is found. The idea behind the method is as follows. Given a point, P(Xn,Yn), on a curve, a line tangent to the curve at P crosses the X axis at a point whose X coordinate is closer to the root than Xn. This X coordinate, we will call Xn+1. Repeating this process using Xn+1 in place of Xn will return a new Xn+1 which will be closer to the root. Eventually, our Xn will equal our Xn+1. When this is the case, we have found a root of the equation. This method may be unnecessarily complex when we are solving a quadratic or cubic equation. However, the Newton-Raphson Method compensates for its complexity in its breadth. The following examples show the versatility of the Newton Raphson Method.   Ã‚  Ã‚  Ã‚  Ã‚  Example 1 is a simple quadratic function. The most practical approach to finding the roots of this equation would be to use the quadratic equation or to factor the polynomial. However, the Nowton-Raphson method still works and allows us to find the roots of the equation. The initial number, Xn, 3, is a relatively poor approximation. The choice of 3 illustrates that the initial guess can be any number. However, as the initial approximation worsens, the calculation becomes more laborious.   Ã‚  Ã‚  Ã‚  Ã‚  Example 2 demonstrates one of the advantages to Newton’s method. Function 2 is a Quintic function. Mathematician, Niels Henrik Abels proved that there exists no convenient equation, such as the cubic equation, which can help us find the function’s roots.

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