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Gaussian and Simpson mass assembly give distinct exact and approximate checks

Example

Assume Countable Choice for the Lebesgue measure construction. For the degree-one monic Legendre basis $P_{0} = 1, P_{1} = x$ on $[- 1, 1]$ , the two-node Gaussian mass matrix is $diag (2, 2 / 3)$ . For the Chebyshev arcsine pairing, composite Simpson with two panels gives $M^{S} = (\begin{array}{cc} π & 0 \\ 0 & π / 3 \end{array}), G = (\begin{array}{cc} π & 0 \\ 0 & π / 2 \end{array}) .$ The Simpson error in the $(1, 1)$ entry is $π / 6$ and satisfies the itemwise error estimate.

Facts & Assumptions

Given: Countable Choice, Lebesgue measure on $[- 1, 1]$ , its two-node Gauss–Legendre rule, and the Chebyshev pairing sampled by two-panel composite Simpson.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed for the Lebesgue measure construction and Riemann–Lebesgue agreement used in the local two-node Gaussian calculation.

[F1]

Positive-measure Gaussian quadrature moment data, Monic orthogonal polynomials exist uniquely through the quadrature order, The m-node Gaussian quadrature rule for a positive measure, and Gaussian quadrature is exact through degree 2m−1 and has positive weights: valid order-three moment data for $m = 2$ give the unique monic orthogonal polynomial, its two root nodes, cardinal-integral weights, and exactness through degree three.

[F2]

Legendre polynomials have explicit norms and weighted derivative orthogonality: $\int_{- 1}^{1} P_{0}^{2} = 2$ , $\int_{- 1}^{1} P_{0} P_{1} = 0$ , and $\int_{- 1}^{1} P_{1}^{2} = 2 / 3$ .

[F3]

Assemble a weighted Galerkin mass matrix from Gaussian nodes and Gaussian quadrature exactly assembles the orthogonal Galerkin mass matrix: the mass entries are Gaussian sums and equal the corresponding exact polynomial integrals.

[F4]

Arcsine-weighted inner product on the Chebyshev interval: $G_{i j} = \int_{0}^{π} T_{i} (\cos θ) T_{j} (\cos θ) d θ$ for $i, j \in {0, 1}$ .

[F5]

Composite-Simpson assembly of the Chebyshev weighted mass matrix: for two panels, $M_{i j}^{S} = \frac{π}{6} (g_{i j} (0) + 4 g_{i j} (π / 2) + g_{i j} (π))$ .

[F6]

Composite Simpson error bound for Chebyshev weighted mass entries: $∣ G_{i j} - M_{i j}^{S} ∣ \leq π h^{4} (i + j)^{4} / 180$ for even panel count and $h = π / M$ .

[F7]

Chebyshev polynomials are orthogonal for the arcsine weight: the exact mass matrix for $T_{0}, T_{1}$ is $diag (π, π / 2)$ .

[F8]

Quarter-turn values and shifts by pi/2 and pi: $\cos 0 = 1$ , $\cos (π / 2) = 0$ , and $\cos π = - 1$ .

[F9]

$T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ : $T_{i} (\cos θ) = \cos (i θ)$ for $i = 0, 1$ .

[F10]

Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable, and A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included: under Countable Choice, restriction to the relative Borel sets of $[- 1, 1]$ is a positive measure, interval measure equals length, and positive-length intervals have positive measure. Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace and The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement identify relative neighborhoods with intersections of open real neighborhoods.

[F11]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, 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, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , and A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: continuous polynomial integrands on $[- 1, 1]$ are integrable, their ordinary integrals are evaluated by polynomial primitives, and under Countable Choice those values equal the Lebesgue integrals.

[F12]

The Lebesgue integral is linear on $L^{1} (μ)$ : finite linear combinations of integrable polynomial functions may be integrated termwise.

[F13]

Rational bounds for pi used in finite schedules: $π > 3$ .

[F14]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ supplies the positive root $\sqrt{3}$ ; Basic properties of the absolute value, A function differentiable at $c$ is continuous at $c$ , and 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 give continuity of $∣ x ∣^{k}$ and the polynomial integrands, as used with [F11]. Measure monotonicity in step 1.1 follows from the measure axioms in [F10]: if $A \subseteq B$ , then $B = A ⊔ (B ∖ A)$ and $μ (B) = μ (A) + μ (B ∖ A) \geq μ (A)$ .

Proof

1.1A1F1F10F11F14algebra

Under [A1], let $μ$ be Lebesgue measure restricted to the relative Borel sets of $[- 1, 1]$ . It is a positive measure with total mass $2$ by [F10]. Every relative open neighborhood of a point of $[- 1, 1]$ contains a positive-length interval, including a one-sided interval at either endpoint, so [F10] and [F14] give that neighborhood positive measure. Thus the relative support is all of $[- 1, 1]$ , containing at least three points. Every $∣ x ∣^{k}$ for $0 \leq k \leq 3$ is continuous and bounded on $[- 1, 1]$ by [F14], so [F11] gives its integrability. These are exactly the order-three, $m = 2$ moment-data conditions in [F1]. The primitives $x, x^{2} / 2, x^{3} / 3, x^{4} / 4$ give moments $μ_{0} = 2, μ_{1} = 0, μ_{2} = 2 / 3, μ_{3} = 0$ by [F11].

1.2F4F5F7F8F9algebra

By [F9], $g_{00} = 1$ , $g_{01} = \cos θ$ , and $g_{11} = \cos^{2} θ$ . Substitution of the quarter-turn values [F8] into [F5] gives $M_{00}^{S} = π$ , $M_{01}^{S} = 0$ , and $M_{11}^{S} = π / 3$ . The exact mass matrix is $G = diag (π, π / 2)$ by [F7].

2.1F1F12F14step 1.1algebra

The monic polynomial $p_{2} (x) = x^{2} - 1 / 3$ is orthogonal to $1$ and $x$ , since $μ_{2} - μ_{0} / 3 = 0$ and $μ_{3} - μ_{1} / 3 = 0$ by step 1.1 and [F12]. Finite linearity extends this to every polynomial of degree below two, so uniqueness in [F1] identifies $p_{2}$ with the Gaussian orthogonal polynomial. Its roots satisfy $x^{2} = 1 / 3$ , hence are $x_{1} = - 1 / \sqrt{3}$ and $x_{2} = 1 / \sqrt{3}$ by [F14]. Their cardinal polynomials are $ℓ_{1} (x) = (1 - \sqrt{3} x) / 2$ and $ℓ_{2} (x) = (1 + \sqrt{3} x) / 2$ , so [F12] and the first two moments give $w_{1} = w_{2} = 1$ . The rule Definition and exactness theorem in [F1] therefore give $Q_{2} (f) = f (- 1 / \sqrt{3}) + f (1 / \sqrt{3})$ , exact for degrees through three.

3.1A1F1F2F3step 2.1algebra

The Gaussian sum for the basis $1, x$ is ${(\begin{array}{cc} 1 & - 1 / \sqrt{3} \\ 1 & 1 / \sqrt{3} \end{array})}^{⁣ T} (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) (\begin{array}{cc} 1 & - 1 / \sqrt{3} \\ 1 & 1 / \sqrt{3} \end{array}) = (\begin{array}{cc} 2 & 0 \\ 0 & 2 / 3 \end{array}) .$ The entries agree with exact integrals by [F2], as also asserted by [F3].

4.1F6F13step 1.2algebra∎

The only nonzero entry error is $∣ G_{11} - M_{11}^{S} ∣ = π / 6$ . The Simpson theorem [F6] applies with $M = 2$ , $h = π / 2$ , and $i = j = 1$ , and gives its bound $π^{5} / 180$ . This bound is at least $π / 6$ , since [F13] gives $π^{4} > 81 > 30$ , equivalently $π^{5} / 180 > π / 6$ . Therefore the displayed finite calculation matches the general entrywise estimate.

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Sources

Walter Gautschi, The Interplay Between Classical Analysis and (Numerical) Linear Algebra, §2.1 (standard reference, not scraped)
Art B. Owen, Monte Carlo Theory, Methods and Examples, §7.2 (standard reference, not scraped)