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 [ f_1(x) = …

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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 of Oxford …

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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 Mathematical …

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The purpose of this post will be to see if we can implement the algorithm in Julia, and thus leverage Julia’s very fast execution time. We are working with polynomials defined on nilpotent variables, which means that the degree of any generator in a polynomial term will be 0 or 1. Assume that our generators are indexed …

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Introduction We know that there is no general sub-exponential algorithm for computing the permanent of a square matrix. But we may very reasonably ask – might there be a faster, possibly even polynomial-time algorithm, for some specific classes of matrices? For example, a sparse matrix will have most terms of the …

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Runge’s phenomenon says roughly that a polynomial through equally spaced points over an interval will wobble a lot near the ends. Runge demonstrated this by fitting polynomials through equally spaced point in the interval \([-1,1]\) on the function

\[ \frac{1}{1+25x^2} \]

and this function is now known as “Runge’s …

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