Approximation of Functions

• To get the value of $sin(2.113)$ or $e^{-3.5}$.
• Does NOT look up in tables and interpolate!
• The computer approximates every function from some polynomial that is customized to give the values very accurately.
• We want the approximation to be efficient in that it obtains the values with the smallest error in the least number of arithmetic operations.
• Another way to approximate a function is with a series of sine and cosine terms, Fourier series (represents periodic functions).

• Chebyshev Polynomials and Chebyshev Series: Chebyshev polynomials are orthogonal polynomials that are the basis for fitting nonalgebraic functions with maximum efficiency.
• They can be used to modify a Taylor series so that there is greater efficiency.
• A series of such polynomials converges more rapidly than a Taylor series.
• Fourier Series: These are series of sine and cosine terms that can be used to approximate a function within a given interval very closely, even functions with discontinuities.
• Fourier series are important in many areas, particularly in getting an analytical solution to partial-differential equations.

• If we want to represent a known function as a polynomial, one way to do it is with a Taylor series.
• Given a function, $f(x)$, we write

  
  

  $$P_n(x)= a_0 + a_1(x - a) + a_2(x - a)^2 + a_3(x - a)^3 +\ldots + a_n(x - a)^n + \ldots$$
  
  

  
  
  

• Where
  
  $$a_i=\frac{f^{(i)}(a)}{i!}$$
  
  

• Then, rewriting this Taylor series expansion as

  
  

  $$P_n(x)= f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 +\ldots$$
  
  
  
  
  $$+ \frac{f^n(a)}{n!}(x-a)^n + \ldots$$
  
  
  
  
  
• Unless $f(x)$ is itself a polynomial, the series may have an infinite number of terms.
• Terminating the series incurs an error, truncation error.
• The error after the $(x - a)^n$ term,
  
  $$Error=\frac{(x-a)^{n+1}}{(n+1)!}f^{(n+1)}(\xi),where~ \xi~in~[a,x]$$
  
  

• A problem with using the Taylor series to get polynomial approximations to a transcendental function is that the error grows rapidly as $x$-values depart from $x = a$.
• For $f(x)=e^x$, the Taylor series is easy to write because the derivatives are so simple: $f^n{(a)}=e^a$ for all orders (n),
• For $a =0$, we have, (which is then called a Maclaurin series)
  
  $$e^x\approx 1+1(x-0)+1/2(x - 0)^2 + 1/6(x - 0)^3$$
  
  

• if we use only terms through $x^3$; the error term shows that the error of this will grow about proportional to $x^4$ as $x$-values depart from zero.
• There is a way to deal with this rapid growth of the errors,
• That is to write the polynomial approximation to $f(x)$ in terms of Chebyshev polynomials.