Introduction

Poncelet's porism or Poncelet's closure theorem is one of the most remarkable results in plane geometry. It is most easily described in terms of circles: suppose we have two circles \(C\) and \(D\), with \(D\) lying entirely inside \(C\). Pick a point \(p_0\) on \(C\), and find the tangent from \(p_0\) …

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In the previous post, we saw that a small change to the method of false position provided much faster convergence, while retaining its bracketing.

This was the Illinois method which is only one of a whole host of similar methods, some of which converge even faster.

And as a reminder, here's its definition, with a very …

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Such a long time since a last post! Well, that's academic life for you ...

If you look at pretty much any modern textbook on numerical methods, of which there are many, you'll find that the following methods will be given for the solution of a single non-linear equation \(f(x)=0\):

• direct iteration, also known as …

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This post illustrates the working of Ramanujan's generating functions for solving Euler's diophantine equation \(a^3+b^3=c^3+d^3\) as described by Andrews and Berndt in "Ramanujan's Lost Notebook, Part IV", pp 199 - 205 (Section 8.5). The text is available from Springer.

Ramanujan's result is that if

\[ …

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Steffensen's method is based on Newton's iteration for solving a non-linear equation \(f(x)=0\):

\[ x\leftarrow x-\frac{f(x)}{f'(x)} \]

Newton's method can fail to work in a number of ways, but when it does work it displays qudratic convergence; the number of correct signifcant figures roughly doubling at each step. …

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As is well known, tanh-sinh quadrature takes an integral

\[ \int_{-1}^1f(x)dx \]

and uses the substitution

\[ x = g(t) = \tanh\left(\frac{\pi}{2}\sinh t\right) \]

to transform the integral into

\[ \int_{-\infty}^{\infty}f(g(t))g'(t)dt. \]

The reason this works so well is that the derivative \(g'(t)\) dies away at a …

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An article by Bailey, Jeybalan and LI, "A comparison of three high-precision quadrature schemes", and available online here, compares Gauss-Legendre quadrature, tanh-sinh quadrature, and a rule where the nodes and weights are given by the error function and its integrand respectively.

However, Nick Trefethen …

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Note: This blog post is mainly computational, with a hint of proof-oriented mathematics here and there. For a more in-depth analysis, read the excellent article "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, pi, and the Ladies Diary" by Gert Akmkvist and Bruce Berndt, in The American …
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The rational numbers are well known to be countable, and one standard method of counting them is to put the positive rationals into an infinite matrix \(M=m_{ij}\), where \(m_{ij}=i/j\) so that you end up with something that looks like this:

\[ \left[\begin{array}{ccccc} …

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A few posts ago I showed how to do this in Python. Now it's Julia's turn. The data is the same: spread of influenza in a British boarding school with a population of 762. This was reported in the British Medical Journal on March 4, 1978, and you can read the original short article here.

As before we use the SIR …

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