Numerical integration trick
Suppose you want to evaluate the following integral: We’d like to do a change of variables to make the range of integration finite, and we’d like the transformed integral to be easy to evaluate numerically. The change of variables t = 1/x2 transforms the range of integration from (30, ∞) to (0, 1/900). I’ll explain where this choice came from and why it results in a nice function to integrate. The
Algorithm used for world record pi calculations
The following algorithm is based on work of Ramanujan and has been used in several world-record calculations of pi. Initialize a0 = 6 – 4 √2 and y0 = √2 – 1. Then compute and The terms an form a sequence of approximations to 1/π. The error in each approximation is given by This says that the number of correct digits roughly quadruples after each step. To see this, note that the number of correct d
The R language, for programmers
Outline Introduction Assignment and underscore Variable name gotchas Vectors Sequences Types Boolean operators Lists Matrices Missing values and NaNs Comments Functions Scope Misc. Other resources Introduction I have written software professionally in perhaps a dozen programming languages, and the hardest language for me to learn has been R. The language is actually fairly simple, but it is unconv
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