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==Technical discussion==

In a more abstract way, the pseudo-spectral method deals with the multiplication of two functions <math>V(x)</math> and <math>f(x)</math> as part of a partial differential equation. To simplify the notation, the time-dependence is dropped. Conceptually, it consists of three steps:

# <math>f(x), \tilde{f}(x) = V(x)f(x)</math> are expanded in a finite set of basis functions (this is the [[spectral method]]).
# For a given set of basis functions, a quadrature is sought that converts scalar products of these basis functions into a weighted sum over grid points.
# The product is calculated by multiplying <math>V,f</math> at each grid point.

===Expansion in a basis===

The functions <math>f, \tilde f</math> can be expanded in a finite basis <math>\{\phi_n\}_{n = 0,\ldots,N}</math> as

:<math>f(x) = \sum_{n=0}^N c_n \phi_n(x)</math>
:<math>\tilde f(x) = \sum_{n=0}^N \tilde c_n \phi_n(x)</math>

For simplicity, let the basis be orthogonal and normalized, <math>\langle \phi_n, \phi_m \rangle = \delta_{nm}</math> using the [[inner product]] <math>\langle f, g \rangle = \int_a^b f(x) \overline{g(x)} dx</math> with appropriate boundaries <math>a,b</math>. The coefficients are then obtained by

:<math>c_n = \langle f, \phi_n \rangle</math>
:<math>\tilde c_n = \langle \tilde f, \phi_n \rangle</math>

A bit of calculus yields then

:<math>\tilde c_n = \sum_{m=0}^N V_{n-m} c_m</math>

with <math>V_{n-m} = \langle V\phi_m, \phi_n \rangle</math>. This forms the basis of the spectral method. To distinguish the basis of the <math>\phi_n</math> from the quadrature basis, the expansion is sometimes called Finite Basis Representation (FBR).

===Quadrature===

For a given basis <math>\{\phi_n\}</math> and number of <math>N+1</math> basis functions, one can try to find a quadrature, i.e., a set of <math>N+1</math> points and weights such that

:<math>\langle \phi_n, \phi_m \rangle = \sum_{i=0}^N w_i \phi_n(x_i) \overline{\phi_m(x_i)} \qquad\qquad n,m = 0,\ldots,N</math>

Special examples are the [[Gaussian quadrature]] for polynomials and the [[Discrete Fourier Transform]] for plane waves. It should be stressed that the grid points and weights, <math>x_i,w_i</math> are a function of the basis ''and'' the number <math>N</math>.

The quadrature allows an alternative numerical representation of the function <math>f(x), \tilde f(x)</math> through their value at the grid points. This representation is sometimes denoted Discrete Variable Representation (DVR), and is completely equivalent to the expansion in the basis.

:<math>f(x_i) = \sum_{n=0}^N c_n \phi_n(x_i)</math>
:<math>c_n = \langle f, \phi_n \rangle = \sum_{i=0}^{N} w_i f(x_i) \overline{\phi_n(x_i)}</math>

===Multiplication===

The multiplication with the function <math>V(x)</math> is then done at each grid point,

:<math>\tilde f(x_i) = V(x_i) f(x_i).</math>

This generally introduces an additional approximation. To see this, we can calculate one of the coefficients <math>\tilde c_n</math>:

:<math>\tilde c_n = \langle \tilde f, \phi_n \rangle = \sum_i w_i \tilde f(x_i) \overline{\phi_n(x_i)} = \sum_i w_i V(x_i) f(x_i) \overline{\phi_n(x_i)}</math>

However, using the spectral method, the same coefficient would be <math>\tilde c_n = \langle Vf, \phi_n \rangle</math>. The pseudo-spectral method thus introduces the additional approximation

:<math>\langle Vf, \phi_n \rangle \approx \sum_i w_i V(x_i) f(x_i) \overline{\phi_n(x_i)}.</math>

If the product <math>Vf</math> can be represented with the given finite set of basis functions, the above equation is exact due to the chosen quadrature.