Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pseudo-spectral methods are a class of numerical methods used in applied mathematics and scientific computing for the solution of PDEs, such as the direct simulation of a particle with an arbitrary wavefunction interacting with an arbitrary potential. They are related to spectral methods and are used extensively in computational fluid dynamics and other areas, but are demonstrated below on an example from quantum physics. If the wavefunction is approximated by its value at n distinct points, each iteration requires 3 pointwise multiplications, one FFT, and one inverse FFT. The pointwise multiplications each require O(n) effort, and the FFT and inverse FFT each require O(nlgn) effort. The total computational effort is therefore determined largely by the FFT steps, so it is imperative to use an efficient (and accurate) implementation of the FFT. Fortunately, many are freely available.

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