Definition:Ritz Method

Definition
Let $ \mathcal M $ be a normed linear space.

Let $ J \left [ { y } \right ] $ be a functional defined on space $ \mathcal M $.

Let $ \{ { \phi_n } \} $ be an infinite sequence of mappings in $ \mathcal M $.

Let $ \mathcal M_n $ be an $ n $-dimensional linear subspace of $ \mathcal M $, spanned by first $ n $ mapping of $ \{ { \phi_n } \} $.

Let $ \eta_n = \boldsymbol \alpha \boldsymbol \phi $, where $ \boldsymbol \alpha $ is a real $ n $-dimensional vector.

Minimise $ J \left [ { \eta_n } \right ] $ $ \boldsymbol \alpha $.

Then $ J \left [ { \eta_n } \right ] $ is an approximate minimum of $ J \left [ { y } \right ] $, and is denoted by $ \mu_n $.

This method is known as Ritz method.