Hugues’s paper. Look it up.
This method is useful when your intermediate precision is the same as the target precision.
Given the input \(x\), the first step is to find a \(y\) such that \(f(x)\) can easily and accurately be calculated from \(f(y)\)....
In general, when approximating a function, we’ll split the domain into multiple intervals and approximate each one with a polynomial....
Sequential Evaluation of Polynomials If you don’t have any parallelism available, Horner’s scheme is a good option. And if you have the...
An Iterative Method Compute the minimax approximation in a wider format. Then round the coefficient of the constant term. Then recompute...
We discussed calculating the minimax polynomial using Remez’s Algorithm, but we overlooked some subtleties. While the algorithm does...
We never discussed how to calculate \(||f-p||_{\infty}\). Maple has a function to do this, but it can be inaccurate. Most people will...
Sometimes you need a fairly high degree polynomial to get reasonable accuracy, but can achieve a far greater accuracy with a much lower...
Remez’s algorithm is one that converges to the minimax polynomial of a function. The author recommends using a polynomial approximation...
Chebyshev vs Minimax Note that the best minimax polynomial approximation need not be the Chebyshev polynomial. The latter is the best...
The supremum norm is given by \(||f-p||_{\infty}=\max_{a\le x\le b}|f(x)-p(x)|\). It is denoted by \(L^{\infty}\). Given a function...
First, just a definition: A monic polynomial is one whose leading coefficient is 1. We want to find a polynomial of degree \(n\) that...
We often will approximate functions as polynomial or rational functions. When doing this, we introduce two types of errors:...
The target format is the format of the result. The target precision is the precision of the target format. When computing polynomials,...
Let \(a,b\) be 2 floating point numbers. It can be shown that \((a+b)-\RN(a+b)\) is a floating point number. This may not be true for...
Basic Notions For a binary floating point system, if \(x\) is normal, then the leading bit is 1. Otherwise it is 0. If we have some...