some simple numerical programs newton-raphson the most commonly used approximation algorithm is usually attributed to isaac newton it is typically called newton' methodbut is sometimes referred to as the newton-raphson method it can be used to find the real roots of many functionsbut we shall look at it only in the context of finding the real roots of polynomial with one variable the generalization to polynomials with multiple variables is straightforward both mathematically and algorithmically polynomial with one variable (by conventionwe will write the variable as xis either zero or the sum of finite number of nonzero termse each terme consists of constant (the coefficient of the term in this casemultiplied by the variable ( in this caseraised to nonnegative integer exponent ( in this casethe exponent on variable in term is called the degree of that term the degree of polynomial is the largest degree of any single term some examples are (degree ) (degree )and (degree in contrast / and are not polynomials if is polynomial and real numberwe will write (rto stand for the value of the polynomial when root of the polynomial is solution to the equation an such that ( sofor examplethe problem of finding an approximation to the square root of can be formulated as finding an such that newton proved theorem that implies that if valuecall it guessis an approximation to root of polynomialthen guess (guess)/ '(guess)where pis the first derivative of pis better approximation for any constant and any coefficient cthe first derivative of cx is cx for examplethe first derivative of is thereforewe know that we can improve on the current guesscall it yby choosing as our next guess ( )/ this is called successive approximation figure contains code illustrating how to use this idea to quickly find an approximation to the square root joseph raphson published similar method about the same time as newton the first derivative of function (xcan be thought of as expressing how the value of (xchanges with respect to changes in if you haven' previously encountered derivativesdon' worry you don' need to understand themor for that matter polynomialsto understand the implementation of newton' method