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Use of Pythagorean Triples

Hugues’s paper. Look it up.

Posted by
Beetle B.
on
Tue 17 September 2019

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Gal’s Accurate Table Method

This method is useful when your intermediate precision is the same as the target precision.

Posted by
Beetle B.
on
Tue 17 September 2019

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Table Driven Methods

Given the input \(x\), the first step is to find a \(y\) such that \(f(x)\) can easily and accurately be calculated from \(f(y)\)....

Posted by
Beetle B.
on
Tue 17 September 2019

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Introduction to Table Based Methods

In general, when approximating a function, we’ll split the domain into multiple intervals and approximate each one with a polynomial....

Posted by
Beetle B.
on
Tue 17 September 2019

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Sequential Evaluation of Polynomials

Sequential Evaluation of Polynomials If you don’t have any parallelism available, Horner’s scheme is a good option. And if you have the...

Posted by
Beetle B.
on
Wed 12 June 2019

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Polynomials With Exact Representable Coefficients

An Iterative Method Compute the minimax approximation in a wider format. Then round the coefficient of the constant term. Then recompute...

Posted by
Beetle B.
on
Wed 24 April 2019

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Introduction to Polynomial Approximations In Finite Precision

We discussed calculating the minimax polynomial using Remez’s Algorithm, but we overlooked some subtleties. While the algorithm does...

Posted by
Beetle B.
on
Wed 24 April 2019

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Accurately Computing Supremum Norms

We never discussed how to calculate \(||f-p||_{\infty}\). Maple has a function to do this, but it can be inaccurate. Most people will...

Posted by
Beetle B.
on
Thu 14 March 2019

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Rational Approximations

Sometimes you need a fairly high degree polynomial to get reasonable accuracy, but can achieve a far greater accuracy with a much lower...

Posted by
Beetle B.
on
Wed 13 March 2019

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Remez’s Algorithm

Remez’s algorithm is one that converges to the minimax polynomial of a function. The author recommends using a polynomial approximation...

Posted by
Beetle B.
on
Tue 12 March 2019

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Miscellaneous (Chebyshev)

Chebyshev vs Minimax Note that the best minimax polynomial approximation need not be the Chebyshev polynomial. The latter is the best...

Posted by
Beetle B.
on
Tue 12 March 2019

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Least Maximum Polynomial Approximations

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...

Posted by
Beetle B.
on
Mon 11 March 2019

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Least Squares Polynomial Approximations

First, just a definition: A monic polynomial is one whose leading coefficient is 1. We want to find a polynomial of degree \(n\) that...

Posted by
Beetle B.
on
Mon 11 March 2019

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Introduction to The Classical Theory of Polynomial or Rational Approximations

We often will approximate functions as polynomial or rational functions. When doing this, we introduce two types of errors:...

Posted by
Beetle B.
on
Mon 04 March 2019

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Manipulating Double or Triple Word Numbers

The target format is the format of the result. The target precision is the precision of the target format. When computing polynomials,...

Posted by
Beetle B.
on
Thu 24 January 2019

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Computing the Error of a FP Addition or Multiplication

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...

Posted by
Beetle B.
on
Wed 23 January 2019

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Basic Notions of Floating Point Arithmetic

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...

Posted by
Beetle B.
on
Wed 23 January 2019