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Collocation quadrature data are well defined

Statement

For the distinct nodes and cardinal polynomials in Collocation tableau from distinct quadrature nodes, every entry of $Q$ , every component of $b$ , and the Lebesgue constant $Λ_{M}$ is a finite real number.

Facts & Assumptions

Given: An integer $M \geq 1$ and distinct nodes $0 \leq c_{1} < \dots < c_{M} \leq 1$ as in Collocation tableau from distinct quadrature nodes.

[F1]

Cardinal-polynomial interpolation and Cardinal interpolation functions are polynomials of the stated degrees: each cardinal function $ℓ_{j}$ is a polynomial of degree at most $M - 1$ .

[F2]

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: polynomial functions, absolute values of continuous functions, and finite sums of continuous functions are continuous.

[F3]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous real function on a nondegenerate compact interval is Riemann integrable. For a degenerate interval, The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ stipulates that the integral is zero.

[F4]

Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length and Lower bound, bounded below, bounded set: $[0, 1]$ is closed and bounded. A subset of $R$ is compact if and only if it is closed and bounded makes it compact, so Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value gives a finite maximum of the continuous cardinal sum.

Proof

1.1F1F2F3construct

By [F1], each $ℓ_{j}$ is a polynomial, hence continuous by [F2]. For every $i, j$ , its restriction to $[0, c_{i}]$ is Riemann integrable by [F3] when $c_{i} > 0$ , and its integral is $0$ when $c_{i} = 0$ . Its integral on $[0, 1]$ is also finite by [F3]. Thus all defining entries $Q_{i j}$ and $b_{j}$ are finite.

2.1F2F4algebra∎

The function $s \mapsto \sum_{j = 1}^{M} ∣ ℓ_{j} (s) ∣$ is continuous by [F2]; [F4] supplies its finite maximum on $[0, 1]$ . This is exactly $Λ_{M}$ from the definition. The matrices $Q$ and the vector $b$ have finitely many finite entries, so their displayed finite sums and row norms are finite as well.

Depends on

Used by

Cited to discharge well-definedness by Collocation tableau from distinct quadrature nodes.

Dependency tree · two levels

63 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner, Solving Ordinary Differential Equations I, 2nd ed., Chapter II §II.7 (standard reference, not scraped)