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Cancellation of the leading Simpson error coefficient

Example

For $f (x) = x^{6} - 5 x^{4}$ on $[0, 1]$ , the leading coefficient $- (1 / 180) \int_{0}^{1} f^{(4)}$ in Conditional asymptotic validity of the Simpson /15 indicator vanishes, but the one-level indicator and the refined Simpson error are nonzero. In fact, the indicator is $21 / 5$ times the true refined error in this exact example.

Facts & Assumptions

Given: The polynomial $f (x) = x^{6} - 5 x^{4}$ on $[0, 1]$ , the exact integral, and composite Simpson grids with $n = 2$ and $n = 4$ .

[F1]

Composite Simpson rule on an even uniform grid: on an even uniform grid, $Q_{S} = (h / 3) \sum_{j < n / 2} (f (x_{2 j}) + 4 f (x_{2 j + 1}) + f (x_{2 j + 2}))$ .

[F2]

Embedded Simpson refinement indicator: $η = ∣ Q_{n = 4} - Q_{n = 2} ∣ / 15$ for this parent and its two-half refinement.

[F3]

Conditional asymptotic validity of the Simpson /15 indicator: the leading coefficient is $- (1 / 180) \int_{0}^{1} f^{(4)}$ .

[F4]

For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term and Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : polynomial powers differentiate by the power rule, and derivatives distribute over finite sums and scalar multiples.

[F5]

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 are continuous; the power and derivative rules show the derivatives used here are also polynomial functions.

[F6]

Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable: continuous polynomials on $[0, 1]$ are Riemann integrable.

[F7]

The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : integrating a continuous derivative gives the endpoint difference of its antiderivative.

[F8]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{4}$ means the derivatives through order four are continuous on the closed interval.

The polynomial is $C^{4}$ on $[0, 1]$ .

Verification

1.1givenF3F4F5F6F7F8algebra

By [F4] and [F5], $f^{(4)} (x) = 360 x^{2} - 120$ and this derivative is continuous. An antiderivative is $120 x^{3} - 120 x$ , so [F6]–[F7] give $\int_{0}^{1} f^{(4)} (x) d x = 0$ . Hence $f \in C^{4} ([0, 1])$ by [F8], and the leading coefficient in [F3] is zero.

1.2givenF1F4F5F6F7algebra

By [F4]–[F5], $x^{7} / 7 - x^{5}$ has derivative $f$ and $f$ is continuous, so [F6]–[F7] give the exact integral $1 / 7 - 1 = - 6 / 7$ . Also $f (0) = 0$ , $f (1 / 2) = - 19 / 64$ , and $f (1) = - 4$ ; substitution in [F1] gives $Q_{n = 2} = - 83 / 96$ . The quarter-point values are $f (1 / 4) = - 79 / 4096$ and $f (3 / 4) = - 5751 / 4096$ , so [F1] also gives $Q_{n = 4} = \frac{1}{12} (4 (- \frac{79}{4096}) + 2 (- \frac{19}{64}) + 4 (- \frac{5751}{4096}) - 4) = - \frac{5267}{6144} .$ Thus the refined error is $- 6 / 7 - Q_{n = 4} = 5 / 43008$ , and the coarse error is $- 6 / 7 - Q_{n = 2} = 5 / 672$ .

2.1givenF2step 1.1step 1.2algebra∎

From [F2], $η = ∣ Q_{n = 4} - Q_{n = 2} ∣ / 15 = 1 / 2048$ . Hence $η / (5 / 43008) = 21 / 5$ . The nonzero refined error and zero leading coefficient show why the asymptotic /15 relation supplies no relative finite-level guarantee in the cancelled case.

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.7 (standard reference, not scraped)
L. Ridgway Scott, Numerical Analysis, 2nd ed., Chapter 13, §13.2 (standard reference, not scraped)