A Fully-Nested Interpolatory Quadrature Based on Fejér’s Second Rule

Abstract

Our goal is to alleviate the constraint of the classical 1D interpolatory nested quadratures than one should go from a set of n to a set of (2$n+1$) points (for Fejér second rule [5]) or (2$n-1$) points (for Clenshaw-Curtis rule [1]) to benefit from the nesting property. In this work a sequence of recursively included quadrature sets for all odd number of quadrature points is proposed to define interpolatory rules. These sets are confounded with the one of Fejér’s second rule when the cardinal is a power of two minus one and different if not. The weights of the corresponding interpolatory rule are studied. This rule is efficient for calculating integrals of very regular functions with control of accuracy via application of successive formulas of increasing order.

Appendices

Appendices

1.1 Properties of the Non-ordered Rules

In the definition of the proposed rule, in Sect. 1.4, the way to successively add the points of $\mathcal{F}_{2M+1}$ that do not belong to $\mathcal{F}_{M}$, to define the intermediate sets of points $\mathcal{S}_{M+2}$ to $\mathcal{S}_{2M-1}$ has been explicitely prescribed: the opposite largest and lowest missing values are successively added. This procedure (qualified here as “ordered” as in Sect. 1.3) may seem arbitrary and the reader may wonder whether there are other ways of defining a fully nested rule with positive weights. The possible other sets (qualified as “non-ordered” as in Sect. 1.3.) of 5 points (resp. 9 11 and 13 points) are considered in this appendix. In order to avoid confusions between formulae and numerical values for ordered and non-ordered rules, an underline is added to the notation of the sets, the weights and the delta functions related to non-ordered rules.

Fig. 3.

Convergence of Gauss quadrature (denoted GAU), Fejer quadrature (denoted FEJ) and proposed method (denoted PM) for the six test functions proposed by L.N. Trefethen [6]

Fig. 4.

Convergence of Gauss quadrature (denoted GAU), Fejer quadrature (denoted FEJ) and proposed method (denoted PM) for $f(x)=exp(-x^2)$. Error plotted as a function of the total number of f evaluations needed to calculate all quadrature with odd number of points up to the current.

The first proposed rule distinct from the ones of Fejér is the five-point rule. Instead of $\pm \cos (\pi /8)$, $\pm \cos (3\pi /8)$ could be added to $\mathcal{S}_3$ to build $\underline{\mathcal{S}_5}$ while preserving the property of successive inclusion of the sets. The corresponding weights are

$$\begin{aligned} (\underline{w}^{\pi /4}_5,\underline{w}^{3\pi /8}_5,\underline{w}^{\pi /2}_5)&= (\frac{10+2\sqrt{2}}{15},\frac{-4-4 \sqrt{2}}{15},\frac{18+4 \sqrt{2}}{15}) \\ (\underline{w}^{3\pi /4}_5,\underline{w}^{5\pi /8}_5)&=(\underline{w}^{\pi /4}_5,\underline{w}^{3\pi /8}_5) \end{aligned}$$

A weight is negative; that is not satisfactory for the reason discussed in Sect. 1.1 and in the conclusion. The definition of sequentially included 9-point, 11-point and 13-point sets between $\mathcal{F}_7$ and $\mathcal{F}_{15}$ is discussed. The numerical values of the weights have been calculated using LAPACK. For systems of corresponding sizes with known solution, an accuracy of eight to nine digits is obtained. The weights of the non-ordered rules are printed below with four digits for the sake of readability.

The abscissae of $\mathcal{F}_{15}$ that can be added to $\mathcal{F}_7$ without having a negative weight in the 9-point rule are $\pm \cos (\pi /16)$ (proposed standard choice) and $\pm \cos (5\pi /16)$. The two choices lead to positive weights for the corresponding 9-point rule. Unfortunately, no 11-point rule with positive weights can be built by adding $\pm \cos (\pi /16)$, $\pm \cos (3\pi /16)$ or $\pm \cos (7\pi /16)$ to the non-ordered set $\underline{\mathcal{S}}_9$:

$$\begin{aligned} \text {If}~&~\underline{\mathcal{S}}_{11} =(\pm \cos (\pi /16), \pm \cos (\pi /8), \pm \cos (\pi /4),\pm \cos (5\pi /16), \pm \cos (3\pi /8),\cos (\pi /2)) \\&(\underline{w}^{\pi /16}_{11},\underline{w}^{\pi /8}_{11},\underline{w}^{\pi /4}_{11}, \underline{w}^{5\pi /16}_{11}, \underline{w}^{3\pi /8}_{11},\underline{w}^{\pi /2}_{11}) \simeq \\&~~~~~~~~~~~~~~~~~~~~~ (0.0281,0.1292,0.2953,-0.0216,0.3754,0.3869) \\ \text {If}~&~\underline{\mathcal{S}}_{11} =(\pm \cos (\pi /8), \pm \cos (3\pi /16),\pm \cos (\pi /4),\pm \cos (5\pi /16),\pm \cos (3\pi /8),\cos (\pi /2)) \\&(\underline{w}^{\pi /8}_{11},\underline{w}^{3\pi /16}_{11}, \underline{w}^{\pi /4}_{11}, \underline{w}^{5\pi /16}_{11},\underline{w}^{3\pi /8}_{11},\underline{w}^{\pi /2}_{11}) \simeq \\&~~~~~~~~~~~~~~~~~~~~~ (0.2247,-0.1613,0.4801,-0.1713,0.4495,0.3564) \\ \text {If}~&~\underline{\mathcal{S}}_{11}=(\pm \cos (\pi /8), \pm \cos (\pi /4),\pm \cos (5\pi /16), \pm \cos (3\pi /8),\pm \cos (7\pi /16), \cos (\pi /2)) \\&(\underline{w}^{\pi /8}_{11},\underline{w}^{\pi /4}_{11},\underline{w}^{5\pi /16}_{11}, \underline{w}^{3\pi /8}_{11},\underline{w}^{7\pi /16}_{11},\underline{w}^{\pi /2}_{11}) \simeq \\&~~~~~~~~~~~~~~~~~~~~~ (0.1941,0.0339,0.7011,-0.8951,1.7167,-1.5017) \end{aligned}$$

A sequence of sets corresponding to positive weights is then searched based on $\mathcal{S}_9$. Adding $\pm \cos (3 \pi /16)$ (as in the 11-point ordered rule) or $\pm \cos (7 \pi /16)$ leads to a set of positive weights. If the second choice is retained, neither $\pm \cos (3 \pi /16)$ nor $\pm \cos (5 \pi /16)$ can be added to the non-ordered 11-point set while still obtaining positive weights.

If the ordered set $\mathcal{S}_{11}$ is considered, then the last four abscissae $\pm \cos (5 \pi /16)$, $\pm \cos (7 \pi /16)$ can only be added to the set in this order to build the 13- and then 15-point rule (if not, the 13-point rule has a negative weight).

Hence, for small numbers of points, it is impossible (set of 5 points) or only partly possible (set of 9, 11, 13 points) to define fully nested successive sets of quadrature points between two standard sets of Fejér quadrature that lead to interpolatory quadratures with only positive weights.

1.2 Calculation of $\varphi ^j_{{\tiny {M}}}$

Let us first define $\overline{\varPhi }^j_s(\theta )$ (s being an even integer) as

$$\begin{aligned} \overline{\varPhi }^1_s(\theta ) = \frac{\sin ((s-1)\theta ) + \sin ((s+1)\theta )}{4},~~~~~~ \overline{\varPhi }^{j+1}_s(\theta ) =\overline{\varPhi }^j_s(\theta )~\frac{\cos (2\theta )-1}{2}. \end{aligned}$$

(32)

Using the trigonometric identity for $\sin (a)\cos (b)$ the $\overline{\varPhi }^j_s$ can be expanded in a series of sine functions with odd-integer-times-$\theta$ arguments. This series is “symmetric” about $\sin (s \theta )$. Let $\overline{\varphi }^j_s$ denote the integral of $\overline{\varPhi }^j_s$ over $[0,\pi ]$. Another consequence of $\sin (a)\cos (b)$ identity is that $\overline{\varPhi }^{j+1}_s(\theta )$, $\overline{\varPhi }^{j}_{s-2}(\theta )$, $\overline{\varPhi }^{j}_{s}(\theta )$, and $\overline{\varPhi }^{j}_{s+2}(\theta )$ satisfy a simple recurrence relation:

$$ \overline{\varPhi }^{j+1}_s(\theta ) = \frac{1}{4} \overline{\varPhi }^j_s(\theta ) ( 2\cos (2\theta )-2 ) =\frac{1}{4} ( 2 \overline{\varPhi }^j_s(\theta ) \cos (2\theta ) -2\overline{\varPhi }^j_s(\theta ) ) $$

When applying the trigonometric identity $\sin (a)\cos (b)$ with $B=2\theta$ for all $\sin ( k \theta )$ terms of $\overline{\varPhi }^j_s(\theta )$, the sum of two series with the same coefficients $\overline{\varPhi }^j_s$ but shifted factors $\sin ((k-2)\theta )$ and $\sin ((k+2)\theta )$ is obtained so that

$$ \overline{\varPhi }^{j+1}_s(\theta ) = \frac{1}{4} ( \overline{\varPhi }^j_{s-2}(\theta ) -2\overline{\varPhi }^j_s(\theta )+\overline{\varPhi }^j_{s+2}(\theta )), $$

and

$$\begin{aligned} \overline{\varphi }^{j+1}_s = \frac{1}{4} ( \overline{\varphi }^j_{s-2} -2\overline{\varphi }^j_s+\overline{\varphi }^j_{s+2}). \end{aligned}$$

(33)

From the first values of $\overline{\varphi }^j_s$, the general following form is assumed for these integrals:

$$\begin{aligned} \overline{\varphi }^j_s = \frac{a_j ~~s}{(s^2-1)(s^2-3^3)...(s^2-(2j-1)^2)} \end{aligned}$$

(34)

Plugging this expression in the right-hand side of Eq. (33), a similar expression for $\overline{\varphi }^j_{s+1}$ with $a_{j+1}= (4j^2 - 2j) a_j$ is obtained. The latter and the expression for $\overline{\varphi }^1_s$ validate Eq. (34). Finally, $\overline{\varphi }^j_s$ takes the following form:

$$ \overline{\varphi }^j_s = \frac{(2j-2)!~s}{\prod _{l=1}^{l=j}(s^2-(2l-1)^2)}. $$

The convention for the index s of $\overline{\varPhi }$ and $\overline{\varphi }$ leads to symmetric expressions about s and makes the derivation of $\overline{\varphi }^j_s$ not do difficult. On the contrary, for the derivation of $\varDelta _{M+2m}$, $w_{M+2m}$ expressions in Sect. 4, its is more practical to introduce

$$ \varPhi ^{j}_{{\tiny {M}}}(\theta ) = \overline{\varPhi }^{j}_{{{\tiny {M}}}+1}(\theta ) ~~~~ \varphi ^{j}_{{\tiny {M}}}= \overline{\varphi }^{j}_{{{\tiny {M}}}+1} $$

$$\begin{aligned} \varphi ^{j}_{{\tiny {M}}}= \frac{(2j-2)!~(M+1)}{\prod _{l=1}^{l=j}((M+1)^2-(2l-1)^2)} \end{aligned}$$

(35)

1.3 Computational Weight Calculations

Among the four methods presented in Sect. 3, those of Subsects. 3.1, 3.2 and 3.3 were used for weight calculation and the values were checked with respect to the theoretical values obtained by Fejér (Eq. (7)) whenever $\mathcal{S}_n$ and $\mathcal{F}_n$ are the same.The resolution of a linear system for the weights (Eq. (2)) and the polynomial method (Eq. (13)) appeared to be more prone to numerical errors than the trigonometric polynomial method (Eq. (15)).

The focus was then put on the last method. Eq. (15) was used to calculate the successive trigonometric polynomials of the points already involved in the quadrature starting from Eq. (23) for M=3.

$$\begin{aligned} \varDelta ^{\pi /4}_3(\theta )&=0.5 \sin (\pi /4)*(\sin (\pi /4) \sin (\theta ) + \sin (3\pi /4) \sin (3 \theta )) = \frac{\sqrt{2}}{4}(\sin (\theta )+\sin (3\theta )) \\ \varDelta ^{\pi /2}_3(\theta )&=0.5 \sin (\pi /2)*(\sin (\pi /2) \sin (\theta ) + \sin (3\pi /2) \sin (3 \theta )) = \frac{1}{2}(\sin (\theta )-\sin (3\theta )) \end{aligned}$$

The trigonometric polynomial of a new abscissa in [0, 1], $\cos (\xi )$, is calculated for the one corresponding to the null abscissa times $(\cos (2 \theta ) + 1 )$ plus a proper renormalisation of the coefficients:

$$\begin{aligned} \varDelta ^{\xi }_{n+2}(\theta ) \sim (\cos (2 \theta ) + 1)\varDelta ^{\pi /2}_n(\theta ) ~~~ \varDelta ^{\xi }_{n+2}(\xi ) = 1/2~\sin (\xi ) \end{aligned}$$

This non-intuitive normalisation is simply the counterpart of:

$$\begin{aligned} EL^{\xi }_{n+2}(x) \sim x^2~EL^{\pi /2}_n(x) ~~~ EL^{\xi }_{n+2}(\cos (\xi )) = 1/2 \end{aligned}$$

For this method, the $\varDelta _n^\alpha$ functions and the weights have been calculated for increasing n. The numerical error was checked (a) for the standard Fejér rule for $n=7,15,31...$ based on Eqs. (7) and (23); (b) with respect to analytical expressions obtained in Sect. 4. The difference between theoretical coefficients of Fejér up to the 31-point rule and the computed values appeared to be less than $8.~10^{-14}$ and the residual of the first four equations of the linear system for the weights (exact quadrature of $0,x^2, x^4, x^6$) appeared to be less than $5.~10^{-13}$. The author can provide the calculated weights for odd n from 3 to 31 (not printed here for the sake of conciseness) and/or the program used to calculate them.

1.4 Lebesgue Constants of Sets $\mathcal{S}_n$

As an interpolatory rule calculates the exact sum over $[-1,+1]$ of the Lagragian polynomial that interpolates the considered function in its set, it is interesting to check the Lebesgue constants of the sets $\mathcal{S}_n$ proposed for quadrature. Their values are plotted in Fig. 5. They unfortunately strongly grow between two successive $2^Q-1$ integers. The lowest values for large sets are $\Lambda _{2^Q-1}$ that is found numerically to be equal to the cardinal of the set, $2^Q-1$, and $\Lambda _{2^Q+1}$ that is found numerically to be close to $ln ( 2^Q+1)$.

Fig. 5.

Lebesgue constants for the sets $\mathcal{S}_n$

The corresponding Gauss-Lobatto sets $\mathcal{L}_n = \{-1,1\} \cup \mathcal{S}_{n-2}$ were considered. They do not exhibit significantly better or worse Lebesgue constants.