In what situation would a taylor series for a polynomial be necessary?

Question

I'm having a hard time understanding why it would be useful to use the Taylor series for a function in order to gain an approximation of a function, instead of just using the function itself when programming. If I can tell my computer to compute e^(.1) and it will give me an exact value, why would I take an approximation instead?

Answer 1

Taylor series are generally not used to approximate functions. Usually, some form of minimax polynomial is used.

Taylor series converge slowly (it takes many terms to get the accuracy desired) and are inefficient (they are more accurate near the point around which they are centered and less accurate away from it). The largest use of Taylor series is likely in mathematics classes and papers, where they are useful for examining the properties of functions and for learning about calculus.

To approximate functions, minimax polynomials are often used. A minimax polynomial has the minimum possible maximum error for a particular situation (interval over which a function is to be approximated, degree available for the polynomial). There is usually no analytical solution to finding a minimax polynomial. They are found numerically, using the Remez algorithm. Minimax polynomials can be tailored to suit particular needs, such as minimizing relative error or absolute error, approximating a function over a particular interval, and so on. Minimax polynomials need fewer terms than Taylor series to get acceptable results, and they “spread” the error over the interval instead of being better in the center and worse at the ends.

When you call the exp function to compute e^x, you are likely using a minimax polynomial, because somebody has done the work for you and constructed a library routine that evaluates the polynomial. For the most part, the only arithmetic computer processors can do is addition, subtraction, multiplication, and division. So other functions have to be constructed from those operations. The first three give you polynomials, and polynomials are sufficient to approximate many functions, such as sine, cosine, logarithm, and exponentiation (with some additional operations of moving things into and out of the exponent field of floating-point values). Division adds rational functions, which is useful for functions like arctangent.

Answer 2

For two reasons. First and foremost - most processors do not have hardware implementations of complex operations like exponentials, logarithms, etc... In such cases the programming language may provide a library function for computing those - in other words, someone used a taylor series or other approximation for you.

Second, you may have a function that not even the language supports.

I recently wanted to use lookup tables with interpolation to get an angle and then compute the sin() and cos() of that angle. Trouble is that it's a DSP with no floating point and no trigonometric functions so those two functions are really slow (software implementation). Instead I put sin(x) in the table instead of x and then used the taylor series for y=sqrt(1-x*x) to compute the cos(x) from that. This taylor series is accurate over the range I needed with only 5 terms (denominators are all powers of two!) and can be implemented in fixed point using plain C and generates code that is faster than any other approach I could think of.