Ritz Method implies Not Worse Approximation with Increased Number of Functions

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Theorem

Consider the Ritz method.

Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$.

Let $J \sqbrk {\eta_n} = \mu_n$.

Then:

$\mu_n \ge \mu_{n + 1}$

Proof

Denote $\eta_{n + 1} = \eta_n + \alpha_{n + 1} \phi_{n + 1}$.

For $\alpha_{n + 1} = 0$ we have:

$\eta_n = \eta_{n + 1}$

Suppose that $J \sqbrk {\eta_n}$ has been minimised with respect to $\boldsymbol \alpha$.

If:

$\exists \boldsymbol \alpha \in \R^n: \nexists \alpha_{n + 1} \ne 0: J \sqbrk {\eta_n} > J \sqbrk {\eta_{n + 1} }$

then $J \sqbrk {\eta_{n + 1} }$ is minimised for previously determined $\boldsymbol \alpha$ and $\alpha_{n + 1} = 0$:

$J \sqbrk {\eta_{n + 1} } = J \sqbrk {\eta_n}$

where $\eta_{n + 1} = \eta_n$.

Suppose:

$\exists \boldsymbol \alpha \in \R^n: \exists \alpha_{n + 1} \ne 0: J \sqbrk {\eta_n} > J \sqbrk {\eta_{n + 1} }$

Then $\eta_{n + 1} \ne \eta_n$, because their respective $\boldsymbol \alpha$ differ by at least one value: $\alpha_{n + 1}$.

Hence, for this supposition:

$J \sqbrk {\eta_{n + 1} } > J \sqbrk {\eta_n}$

Finally, both cases together imply that:

$J \sqbrk {\eta_{n + 1} } \ge J \sqbrk {\eta_n}$

or equivalently:

$\mu_n \ge \mu_{n + 1}$

$\blacksquare$

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 8.40 $: The Ritz Method and the Method of Finite Differences