Mathematics
Taylor Polynomials
Taylor polynomials are approximations of functions using a finite number of terms from their Taylor series. They are used to estimate the value of a function at a particular point or to approximate the behavior of a function near a certain point. The accuracy of the approximation depends on the number of terms used in the polynomial.
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7 Key excerpts on "Taylor Polynomials"
CHAPTER 8 Representation of Functions by Infinite Taylor Series INTRODUCTION The polynomials are a large and varied class of functions formed by the simplest arithmetic operations and presenting no obstacle to the performance of differentiation and integration. Although a natural wish that every func-tion might be a polynomial goes unrealized, we will see in this chapter that most ordinary functions can be arbitrarily well approximated by polynomials of successively higher degrees, and this in a systematic way. There results a uniform representation of functions as polynomial-like expressions of infinitely many terms, somewhat analogous to the decimal system by which numbers are expressed as infinite “sums” of powers of ten with coefficients. The benefits for both understanding and calculation are very great. In the first article this representation by the Taylor series, as it is called, is derived from polynomial approximations, and evidence of its value is provided by an uncritical look at the series associated with the exponential function and the sine and cosine. The second article begins with a theorem by which the Taylor series can be justified, and goes on to establish the validity of several important series by means of the theorem and certain special arguments. 8.1 POLYNOMIAL APPROXIMATION AND THE TAYLOR SERIES 8.1.1 The first three approximating polynomials. A. Constant and linear approximation. I will begin with a review of the approxima-tions considered in §§ 3.7.11 and 3.8.6–8, with somewhat different emphasis. 1 1 It is advisable to read or re-read those sections. 389 390 REPRESENTATION BY INFINITE SERIES § 8.1.1 Suppose that the value of a continuous function f ( x ) is known at x = a , but unknown elsewhere. Because of continuity, it is reasonable to estimate its value at points x near a by f ( x ) ≈ f ( a ) .
- Joe D. Hoffman, Joe D. Hoffman, Steven Frankel(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
192 Chapter 4 Consequently, any continuous function can be approximated to any accuracy by a polynomial of high enough degree. In practice, low-degree polynomials are employed, so care must be taken to achieve the desired accuracy. Polynomials satisfy a uniqueness theorem: A polynomial of degree n passing exactly through n + I discrete points is unique The polynomial through a specific set of points may take many different forms, but all forms are equivalent. Any form can be manipulated into any other form by simple algebraic rearrangement. The Taylor series is a polynomial of infinite order. Thus, (4.4) It is, of course, impossible to evaluate an infinite number of terms. The Taylor polynomial of degree n is defined by f(x) = Pn(x) + R n + I (x) where the Taylor polynomial Pn(x), and the remainder term R n + 1 (x) are given by Pn(x) = f(xo) + f'(xo)(x -xo) + ... + ~ fn)(xo)(x -xot n. Rn+l(x) = (n ~ I)! fn+I)«()(x -XO)+1 Xo S ~ S x (4.5) (4.6) (4.7) The Taylor polynomial is a truncated Taylor series, with an explicit remainder, or error, term. The Taylor polynomial cannot be used as an approximating function for discrete data because the derivatives required in the coefficients cannot be determined. It does have great significance, however, for polynomial approximation, because it has an explicit error term. When a polynomial of degree n, Pn(x), is fit exactly to a set of n + I discrete data points, (xo,fo), (XI ,fj), ... , (xnJn), as illustrated in Figure 4.5, the polynomial has no error at the data points themselves. However, at the locations between the data points, there is an error which is defined by Error(x) = Pn(x) -f(x) It can be shown that the error term, Error(x), has the form I Error(x) = )' (x -xo)(x -XI)· .. (x -xn)fn+l)(~) (n + I . (4.8) (4.9) where Xo S ~ S x n . This form of the error term is used extensively in the error analysis of procedures based on approximating polynomials.
eBook - PDF- Tom M. Apostol(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Thus only odd powers of x appear in the Taylor Polynomials generated by the sine function at 0. The Taylor polynomial of degree 2n + 1 has the form T 2n+1 (sin x) = x − x 3 3! + x 5 5! − x 7 7! + · · · + (−1) n x 2n+1 (2n + 1)! . example 3. Arguing as in Example 2, we find that the Taylor Polynomials generated by the cosine function at 0 contain only even powers of x. The polynomial of degree 2n is given by T 2n (cos x) = 1 − x 2 2! + x 4 4! − x 6 6! + · · · + (−1) n x 2n (2n)! . Note that each Taylor polynomial T 2n (cos x) is the derivative of the Taylor polynomial T 2n+1 (sin x). This is due to the fact that the cosine itself is the derivative of the sine. In the next section we learn that certain relations which hold between functions are transmitted to their Taylor Polynomials. 7.3 Calculus of Taylor Polynomials If a function f has derivatives of order n at a point a, we can always form its Taylor polynomial T n f by the formula T n f (x) = n ∑ k=0 f (k) (a) k! (x − a) k . Sometimes the calculation of the derivatives f (k) (a) may become lengthy, so it is desirable to have alternate methods for determining Taylor Polynomials. The next theorem describes prop- erties of the Taylor operator that often enable us to obtain new Taylor Polynomials from given ones. In this theorem it is understood that all Taylor Polynomials are generated at a common point a. 276 Polynomial approximations to functions theorem 7.2. The Taylor operator T n has the following properties: (a) Linearity property. If c 1 and c 2 are constants, then T n (c 1 f + c 2 g) = c 1 T n (f ) + c 2 T n (g). (b) Differentiation property. The derivative of a Taylor polynomial of f is a Taylor polynomial of f ′ ; in fact, we have (T n f ) ′ = T n−1 (f ′ ). (c) Integration property. An indefinite integral of a Taylor polynomial of f is a Taylor polyno- mial of an indefinite integral of f . More precisely, if g(x) = ∫ x a f (t) dt, then we have T n+1 g(x) = ∫ x a T n f (t) dt.
eBook - PDF- P Adams, K Smith;R V??born??;;(Authors)
- 2004(Publication Date)
- WSPC(Publisher)
Chapter 6 Polynomials Polynomial functions have always been important, if for nothing else than because, in the past, they were the only functions which could be readily evaluated. In this chapter we define polynomials as algebraic entities rather than functions, establish the long divi- sion algorithm in an abstract setting, we also look briefly at zeros of polynomials and prove the Taylor Theorem for polynomials in a generality which cannot be obtained by using methods of calculus. 6.1 Polynomial functions If M is a ring and ao, al, a2, . . . , an E M then a function of the form is called a polynomial, or sometimes more explicitly, a polynomial with coefficients in M . Obviously, one can add any number of zero coefficients, or rewrite Equation (6.1) in ascending order of powers of z without changing the polynomial. The domain of definition of the polynomial is naturally M , but the definition of A(x) makes sense for any x in a ring which contains M . This natural extension of the domain of definition is often understood without explicitly saying so. If A and B are two polynomials then the polynomials A + B, -A and AB are defined in the obvious way as A + B : x H A(x) + B(x) -A : x -A(x) AB : z H A(x)B(x) The coefficients of A+B are obvious; they are the sums of the corresponding coefficients of A and B. The zero polynomial function is the zero function, 167 168 Introduction to Mathematics with Maple that is 10 : x H 0. Similarly, the coefficients of -A have opposite signs to the coefficients of A. The coefficients of AB are obtained by multiplying through, collecting terms with the same power of x and sorting them in descending (or ascending) powers of x. If and P = AB, then There is a clear pattern to the formulae + an-ibm (6.2). In order to subsume them in a compact formula we set a k = 0 for k > n and bk = 0 for k > rn. Then we can rewrite Equations (6.2) as k pk = ak-jbj j=O for' k = 1,2,.
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
C h a p te r 24 Introduction to Taylor Polynomials, Taylor Series, and Power Series We now come to the important topics of power series and Taylor Polynomials and series. In this chapter, we’ll see a general overview of these topics. The following two chapters will deal with problem-solving techniques in the context of the material in this chapter. Here’s what we’ll look at first: • approximations, Taylor Polynomials, and a Taylor approximation theo-rem; • how good our approximations are, and the full Taylor Theorem; • the definition of power series; • the definition of Taylor series and Maclaurin series; and • convergence issues involving Taylor series. 24.1 Approximations and Taylor Polynomials Here’s a nice fact: for any real number x , we have e x ∼ = 1 + x + x 2 2 + x 3 6 . Also, the closer x is to 0, the better the approximation. Let’s play around with this for a little bit. Start off with x = 0. Actually, both sides are then equal to 1, so the approximation is perfect! What about when x isn’t 0? Let’s try x = -1 / 10. The above equation says that e -1 / 10 ∼ = 1 -1 10 + 1 / 100 2 -1 / 1000 6 , which simplifies to e -1 / 10 ∼ = 5429 6000 . My calculator says that e -1 / 10 is equal to 0 . 9048374180 (to ten decimal places), while 5429 / 6000 is equal to 0 . 9048333333 (also to ten decimal places). 520 • Taylor Polynomials, Taylor Series, and Power Series These numbers are pretty close to each other! In fact, the difference is only about 0 . 0000040847. How on earth did I come up with the polynomial 1 + x + x 2 / 2 + x 3 / 6? It’s clearly not just any old polynomial; it’s specially related to e x . Rather than concentrate on e x itself, let’s get a little more general and consider other functions. Also, there’s nothing special about the degree 3 of our polynomial: we could have used any degree.
- Vladimir Dobrushkin(Author)
- 2017(Publication Date)
- Chapman and Hall/CRC(Publisher)
3.3 The Polynomial ApproximationOne of the alternative approaches to the discrete numerical methods discussed in §3.2 is the Taylor series method. If the slope functionin the initial value problemf ( x , y )(3.3.1)y= f′( x , y ), y(=x 0)y 0,is sufficiently differentiable or is itself given by a power series, then the numerical integration by Taylor’s expansion is possible. In what follows, it is assumed thatpossesses continuous derivatives with respect to both x and y of all orders required to justify the analytical operations to be performed.f ( x , y )There are two ways in which a Taylor series can be used to construct an approximation to the solution of the initial value problem. One can either utilize the differential equation to generate Taylor Polynomials that approximate the solution, or one can use a Taylor polynomial as a part of a numeric integration scheme similar to Euler’s methods. We are going to illustrate both techniques.It is known from calculus that a smooth function g (x ) can be approximated by its Taylor series polynomial of the order n :(3.3.2)p n( x )= g(+x 0)g′(x 0)( x -+ ⋯ +x 0)g( n )(x 0)n !,( x -nx 0)which is valid in some neighborhood of the point x = x 0 . How good this approximation is depends on the existence of the derivatives of the function g (x ) and their values at x = x 0 , as the following statement shows (consult an advanced calculus course).Theorem 3.1: [Lagrange] Let g (x ) - p (x ) measure the accuracy of the polynomial approximation (3.3.2 of the function g (x ) that possesses (n + 1) continuous derivatives on an interval containing x 0 and x , theng( x )-p n( x )=g( n + 1 )( ξ )( n + 1 ) !,( x -x 0)n + 1where ξ , although unknown, is guaranteed to be between x 0 and x .Therefore, one might be tempted to find the solution to the initial value problem (3.3.1
eBook - PDF- Mark J. Christensen(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
However, with the inter-polating polynomial you must decide what points the polynomial shall actually pass through. With the Taylor polynomial you need only specify a single point. On the other hand in order to calculate the Taylor polynomial of degree Ν we must, evidently, calculate all the derivatives up to and including the Nth. This is not a serious disadvantage since, in order to calculate the error estimate for the interpolating polynomial of the same degree we will also need the formula for the derivative of order N+1. Thus, in many ways, the amount of information needed to use either estimate is approximately the same. Bearing in mind these considerations and looking to our example we now see how to write the formula for the general Taylor polynomial for a function F about the fixed point XO: PN(X) = F(XO)+F'(XO)*(X-XO)+(F»»(X0)/2!)*(X-X0)t2 + (F* · · (X0)/3i) *(X-XO) >h3 + (F' · · · (X0)/41 ) *(X-XO) f 4 + ... + (F* ·... (XO)/N!)*(X-XO)fN where by F'* '*'(XO) we mean the Nth derivative of F at XO. As was indicated before, the multiplicative factor l/Il is necessary to compensate for the fact that the term with the Ith power of X-XO will be differ-entiated I times, giving rise to multiplication by I, I-l,...,2,l. Thus the coefficients of the Taylor polynomial will be calculated in an entirely different way from the interpolating polynomial. For example, suppose that we wished to calculate the Taylor polynomial of degree Ν for the SIN func-tion. In order to do so we must first compute the first Ν derivatives of the SIN. Fortunately this is easy for the SIN function, the derivatives are; SECTION 8.2 Taylor Polynomials 4000 T=(X-X0)f2 4010 P=l 4020 D=l 4030 FOR J=l TO 4 197 SIN'(X) = COS(X) , SIN''(X) = -SIN(X) , SIN*''(X) = -COS(X) 5ΙΝ·'·'(Χ) = SIN(X) and so on. In fact we see a pattern. The derivatives from order 5 to order 8 will be identical to those of order 1 to 4.
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