Abstract
In this paper, we provide further illustrations of prolate interpolation and pseudospectral differentiation based on the barycentric perspectives. The convergence rates of the barycentric prolate interpolation and pseudospectral differentiation are derived. Furthermore, we propose the new preconditioner, which leads to the wellconditioned prolate collocation scheme. Numerical examples are included to show the high accuracy of the new method. We apply this approach to solve the secondorder boundary value problem and Helmholtz problem.
Introduction
In many cases, we are confronted with wave phenomena, such as wave scattering, signal processing, and antenna theory, which are characterized by bandlimited functions (whose Fourier transforms are compactly supported) [3, 14]. It is well known that the natural tool for effectively representing bandlimited functions on an interval is prolate spheroidal wave functions (PSWFs) [7, 28, 32]. Hence, there has been a growing interest in developing prolate spheroidal wave functions, which also offer an alternative to Chebyshev and other orthogonal polynomials for pseudospectral/collocation and spectralelement algorithms [5].
It is well known that a simple way of approximating a function f(x) is to choose a sequence of points \(\{x_{j}\}_{j=0}^{n}\) and find the function P(x) from the values of f(x) at these interpolation nodes, i.e., set P(x_{j}) = f(x_{j}), 0 ≤ j ≤ n. The standard tool for interpolation and approximation algorithms was investigated in [15, 32]. We highlight that a very popular alternative nowadays is to use barycentric interpolation formula, and the favourable numerical aspects of this way are summarized by Berrut and Trefethen [1, 24]. So, the related issues are worthy of investigation.
The purpose of this paper is to have new insights into prolate interpolation and pseudospectral differentiation based on the ProlateGaussLobatto points. The inspiration behind the proposed numerical method is the remarkable advantage of barycentric interpolation formula [1]. The main contributions reside in the following aspects.

We give the barycentric prolate interpolation and differentiation formula, which enjoys a more stable approximability and efficiency than the formulas given early [32, 34].

We give the error analysis of the barycentric prolate interpolation and differentiation based on the error analysis of the standard prolate interpolation and differentiation [22].

We offer a preconditioning matrix that nearly inverts the secondorder prolate pseudospectral differentiation matrix, leading to a wellconditioned collocation approach for secondorder boundary value problems.
The structure of the paper is as follows. In Section 2, we review some results of PSWFs and prolate interpolation and pseudospectral differentiation, while Section 3 combines the barycentric form with the prolate interpolation, which yields the new barycentric prolate interpolation and pseudospectral differentiation scheme. Furthermore, the convergence analysis of barycentric prolate interpolation and pseudospectral differentiation is given. In Section 4, we introduce the preconditioning matrix that nearly inverts the secondorder barycentric prolate differentiation matrix. Sections 5 and 6 demonstrate the analysis via several numerical experiments and apply to the secondorder boundary value problem and Helmholtz problem.
Prolate interpolation and pseudospectral differentiation
Preliminaries
The PSWFs were introduced in the 1960s by Slepian et al. in a series of papers [20, 21]. Firstly, we briefly recall some preliminary properties of the PSWFs. All of these can be found in [2, 6, 13,14,15,16, 18, 20, 21, 30, 31, 34].
Prolate spheroidal wave functions of order zero are the eigenfunctions ψ_{j}(x) of the Helmholtz equation in prolate spheroidal coordinates:
for x ∈ (− 1,1) and c ≥ 0. A series of papers [18, 32, 34] have shown that the ψ_{j}(x) are also the eigenfunctions of the following integral eigenvalue problem:
Here, \(\{\chi _{j}:=\chi _{j}(c)\}_{j=0}^{\infty }\) and \(\{\lambda _{j}:=\lambda _{j}(c)\}_{j=0}^{\infty }\) are the associated eigenvalues corresponding to the differential operator and integral operator, and the constant c is known as the “bandwidth parameter.” The eigenvalues \(\{\lambda _{j}\}_{j=0}^{\infty }\) satisfy λ_{0} > λ_{1} > λ_{2} > … > 0, which decay exponentially to nearly 0. Specifically, based on the work of Wang and Zhang [31], we have:
where the notation \(A\simeq B\) means that for B≠ 0, the ratio \(\frac {A}{B}\rightarrow 1\) in the sense of some limiting process.
An important issue related to the PSWFs is the choice of bandlimit parameter c. For general functions, we do not have a simple optimal c. This is due to the fact that an arbitrary function has many different modes and each mode has a distinct optimal c [7]. Regardless of whether the function being represented is bandlimited or not, all the useful choices of c must satisfy [5]:
As recommended in [30, 32], in practice, a quite safe choice is \(c=\frac {N}{2}\). With this in mind, we choose \(c=\frac {N}{2}\). Guidelines on the suitable choice of c can be found in [3]. The practical rule for pairing up (c,N) has been given in [13, 32].
Denoting the zeros of (1 − x^{2})ψ_{n}(x) by the ProlateGaussLobatto points (PGL points). For computation, Boyd [6] described Newton’s iteration method with some care in selecting initial guesses. Generally, [12] gives the efficient algorithm for computing zeros of special functions, such as PSWFs. With the ProlateGaussLobatto points at our disposal, we will introduce the prolate interpolation and pseudospectal differentiation.
Prolate interpolation and pseudospectal differentiation
In this subsection, firstly, we review some facts about prolate interpolation and pseudospectal differentiation.
The key idea for interpolation is to search the prolate cardinal functions ℓ_{i}(x) := ℓ_{i}(x;c), which are designed to satisfy the interpolation property:
Then, the function f(x) is approximated by
The standard route to get the derivatives is by directly differentiating the prolate cardinal basis ℓ_{i}(x).
Generally, we define the prolate cardinal functions ℓ_{i}(x) as
where \(\{x_{k}\}_{k=0}^{N}\) are the Prolate points, which are zeros of s_{p}(x). It follows that the standard interpolation is [15, 32]:
The standard differentiation generated from the cardinal basis (2.7) can be computed by:
Taking derivative of \(s_{p}(x)=s_{p}^{\prime }(x_{j})(xx_{j})\ell _{j}(x)\) two times implies for
In the following, let us consider the prolate interpolation and pseudospectal differentiation scheme through the barycentric form.
Barycentric prolate interpolation and differentiation
In this section, we start with the barycentric interpolation [1, 9, 11], which are important pieces of the puzzle for our new approach, and then give the new insights into prolate interpolation, which are called the barycentric prolate interpolation. The differentiation matrices are derived through the barycentric interpolation formula. Then, the convergence analysis of barycentric prolate interpolation and differentiation is given.
Barycentric interpolation formula
Let \(\{x_{j}\}_{j=0}^{n}\) be a set of distinctive nodes in [− 1,1], which are arranged in ascending order, together with corresponding numbers f(x_{j}). We assume that the nodes are real and they are zeros of the function s(x), i.e., s(x_{j}) = 0 for 0 ≤ j ≤ N. Thus, the Lagrange interpolating basis is defined by:
Accordingly, the interpolation in Lagrange form for the function f(x) is
The barycentric formula is the alternative Lagrange form, and for computations, it is generally recommended that one should use barycentric interpolation formula [1, 11], which has stability or robustness property that proves advantageous in some application. The barycentric interpolation is defined as
where \(\{w_{j}\}_{j=0}^{N}\) are the barycentric weights. To this end, it suffices to note that the barycentric weights \(\{w_{j}\}_{j=0}^{N}\) can be written as different quantity. As with the polynomial interpolation [1, 24], s(x) can be written as:
such that barycentric weights become
For certain special sets of nodes \(\{x_{j}\}_{j=0}^{n}\), the explicit expressions of the barycentric weights w_{j} were available in [1, 26, 27]. For general point sets, the barycentric weights w_{j} can be evaluated by the fast multipole method [23]. These observations lead to an efficient method for computing prolate interpolants based on the ProlateGaussLobatto points through a new definition of nonzero barycentric weights.
Barycentric prolate interpolation and pseudospectral differentiation
Using the barycentric form, this subsection give a new definition of barycentric prolate weights, which leads to remarkably simple and efficient schemes for the construction of rational barycentric interpolation, which is denoted by barycentric prolate interpolation.
The fist question is how to choose barycentric weights. In the similar manner as deriving the barycentric formula (3.13) form Lagrange interpolation (3.11), since the ProlateGaussLobatto points are the roots of (1 − x^{2})ψ_{N}(x), it is straightforward to let s(x) := s_{p}(x) = (1 − x^{2})ψ_{N− 1}(x). Correspondingly, we define the prolate barycentric weights to be
According to the foregoing observations, it is desirable to define a new interpolation which is called barycentric prolate interpolation. Moreover, the interpolation property is stable with respect to the nonzero weights, as noticed in [33].
Definition 3.1
The barycentric prolate interpolation can be expressed as
where \(\{x_{j}\}_{j=0}^{n}\) are the ProlateGaussLobatto points and \(w_{j}=\frac {1}{2x_{j}\psi _{N1}(x_{j})+(1{x_{j}^{2}})\psi ^{\prime }_{N1}(x_{j})}\).
The error analysis will be derived in the next subsection. In fact, from the numerical evidences in Figs. 3 and 4, the barycentric prolate interpolation gives better approximations.
Remark 1
The barycentric prolate interpolation enjoys several advantages, which makes it very efficient in practice. (i) The barycentric prolate interpolation are scaleinvariant, thus avoiding any problems of underflow and overflow. (ii) Once w_{j} are computed, the interpolant at any points x will take only \(\mathcal {O}(N)\) floating point operations to compute.
Remark 2
The barycentric formula has natural advantages for applications to fast multipole method [1], which is an useful and efficient tool to improve the complexity of centain sums (3.15) from \(\mathcal {O}(N^{2})\) to \(\mathcal {O}(N)\). The idea of using FMM to accelerate the interpolation and pseudospectral differentiation can be traced back to [4, 8] and we see from [15] that the FMM was used to accelerate the standard prolate interpolation and differentiation. It is noteworthy to point out that the new scheme (3.15) can also be accelerated by the FMM through a very similar process in [17].
Remark 3
We have to calculate s_{p}(x) = (1 − x^{2})ψ_{N− 1}(x) in standard interpolation formula (2.8). Since \(\psi _{N}(x)={\sum }_{k}{\alpha ^{N}_{k}}\overline {P}_{k}(x)\), where the \(\overline {P}_{k}(x)\) is normalized Legendre polynomial and α_{k} is the eigenvector of a matrix, which is complex and timeconsuming. However, it is obvious that the factor s_{p}(x) has dropped out in the (3.15), and this feature has practical consequences.
Furthermore, defining the cardinal basis function of the barycentric prolate interpolation (3.15) as
It leads to the differentiation matrices
which have the explicit formulas [23]:
where \(\{x_{j}\}_{j=0}^{n}\) are the ProlateGaussLobatto points and \(w_{j}=\frac {1}{2x_{j}\psi _{N1}(x_{j})+(1{x_{j}^{2}})\psi ^{\prime }_{N1}(x_{j})}\).
Remark 4
It is obvious that the standard differentiation method (2.10) involves the firstorder differentiation value \(\ell ^{\prime }_{j}(x_{i})\) (2.9), which causes error propaganda for large number N. The barycentric prolate differentiation only involves the barycentric weights value. Hence, the barycentric prolate differentiation form is stable even for large N, which has been shown in Fig. 6.
Convergence properties of barycentric prolate interpolation and differentiation
Results can also be obtained for the convergence properties of barycentric prolate interpolation and differentiation.
Lemma 3.1
[22] Let f be the entire function, Γ_{R} be the boundary of the square [−R_{K},R_{K}] × [−i ⋅ R_{K},i ⋅ R_{K}], \(R_{K}>\frac {\pi }{2c}+\frac {8(c+1)}{c\cdot \lambda _{n}}\), ψ_{n}(R_{K})≠ 0, \(C_{1}=\max \limits _{z\in {\Gamma }_{R}}f(z)\). Suppose P_{n}(x) is the interpolant of f(x) at the ProlateGaussLobatto points (2.8), then it follows for χ_{n} > c^{2} and − 1 < x < 1 that
where \(\widetilde {C}\) is a constant.
Remark 5
We remark that the condition “Let f be the entire function” in Lemma 3.1 can be refined as “Let f be analytic in a region bounded by the square [−R_{k},R_{k}] × [−i ⋅ R_{k},i ⋅ R_{k}]” [18, 35].
Theorem 3.1
Let f be analytic in a region bounded by the square [−R_{k},R_{k}] × [−i ⋅ R_{k},i ⋅ R_{k}], Γ_{R} be the boundary of the square [−R_{K},R_{K}] × [−i ⋅ R_{K},i ⋅ R_{K}], \(R_{K}>\frac {\pi }{2c}+\frac {8(c+1)}{c\cdot \lambda _{n}}\), ψ_{n}(R_{K})≠ 0. Suppose P_{n}(x) and G_{n}(x) is the interpolant of f(x) at the ProlateGaussLobatto points by fomula (2.8) and (3.15), then it follows for χ_{n} > c^{2} and − 1 < x < 1 that
where \(\varepsilon _{n}:=\frac {2\cdot \lambda _{n}}{{R^{2}_{K}}1}\left (1+4\cdot c\cdot R_{K}\cdot e^{c\cdot R_{K}}\right )\) and \(\varepsilon _{n}^{\prime }=\left (2+\widetilde {C}\cdot \sqrt {2}\cdot n^{3}+\frac {1}{R_{K}}\right )\frac {2\cdot \lambda _{n}}{{R^{2}_{K}}1}\left (1+4\cdot c\cdot R_{K}\cdot e^{cR_{K}}\right )\).
Proof
Due to Lemma 3.1, when P_{n}[1](x) interpolates the constant function f(x) = 1, let P_{n}[1](x) = 1 + E_{n}(x), we provide the error that
It follows that
Then, we have:
and
Combining with (3.25), we obtain:
and
where ε_{n} and \(\varepsilon _{n}^{\prime }\) are defined in (3.24). The proof is completed. □
Remark 6
Theorem 3.1 shows a close connection between the barycentric prolate interpolation (3.15) and standard prolate interpolation (2.8). Roughly speaking, for λ_{n} satisfying (2.3), so f(x) − G_{n}(x) should decay exponentially with respect to n when c satisfies (2.4).
Remark 7
A function f may be less smooth than the case we have considered; numerical results illustrate that it might be also suited to this fast convergence. However, it appears open to know about exactly how the convergence rates of barycentric prolate interpolation depend on the degree of smoothness of f.
A wellconditioned prolatecollocation method
As everyone knows, the secondorder prolate differentiation matrix is apparently unstable even for slightly large N [19]. Fortunately, Wang et al. [32] offered a new basis leading to wellconditioned collocation linear systems. In this subsection, we give a different way to evaluate the Birkhoff interpolation basis, which generates the preconditioner P_{in}, such that the eigenvalues of \(P_{in}D_{in}^{(2)}\) are nearly concentrated around one.
Consider the secondorder BVPs with Dirichlet boundary conditions:
Following the work of Wang [32], the Birkhoff interpolation p(x) of f(x) can be uniquely determined by:
where \(\{B_{j}\}_{j=0}^{N}\) are the Birkhoff interpolation basis and satisfy:
Proposition 4.1
Let \(\{x_{j}\}_{j=0}^{n}\) be a set of ProlateGaussLobatto points. The Birkhoff interpolation basis \(\{B_{j}\}_{j=0}^{N}\) defined in (4.28)–(4.30) is given by:
where \(\{\widetilde {h}_{j}\}_{j=1}^{N1}\) are the prolate barycentric interpolation basis at \(\{x_{j}\}_{j=1}^{N1}\)
and \(\lambda _{j}=\{\frac {1}{\psi ^{\prime }_{N1}(x_{j})}\}_{j=1}^{N1}\). What’s more,
We omit the proof, since it is very similar to that in [29]. In order to avoid the instability and lowefficiency of the Lagrange interpolation, the barycentric form is used which is recommended by [1].
To construct the Birkhoff interpolation basis, we give the numerical scheme for integral (4.32) at x_{i}
Introducing the change of variable
allows us to rewrite the definite integrals (4.34) further as
Since the integrands in (4.36) can be computed exactly using an Gauss quadrature at Legendre points. Based on fast \(\mathcal {O}(N)\) operations for the computation of Gaussian quadrature due to Hale and Townsend [10], we get the fast scheme for the Birkhoff interpolation basis \(\{B_{j}\}_{j=0}^{N}\) and \(\{B^{(1)}_{j}\}_{j=0}^{N}\).
Let \(b^{(k)}_{ij}:=B^{(k)}_{j}(x_{i})\), and define the matrices
Due to (4.27), h_{k}(x) in (3.16) can be approximated by
According to the fact that h_{k}(x) satisfying h_{j}(x_{i}) = δ_{ij}, it follows that
where I_{M} is an M × M identity matrix, and the matrix \(\textbf {D}_{in}^{(2)}\) is the same as in (3.17). We depict in Fig. 1 the distribution of the largest and smallest eigenvalues of \(\textbf {B}_{in}\textbf {D}_{in}^{(2)}\) at the ProlateGaussLobatto points. This agrees with (4.39).
As we know, the usual collocation scheme is find f = (f(x_{1}),…,f(x_{N− 1})) by solving
where g = (g(x_{1}),…,g(x_{N− 1}))^{t}, Λ_{r} = diag(r(x_{1}),…,r(x_{N− 1})), Λ_{s} = diag(s(x_{1}),…,s(x_{N− 1})),
It is well known that the coefficient matrix of the usual collocation method has a high condition number. Below, let us consider the preconditioning method for solving BVP. On the one hand, due to \(B_{in}D_{in}^{(2)}=I_{N1}\), the matrix B_{in} can be used to precondition the illconditioned system by:
where
On the other hand, recall the formula (4.27): one can directly use {B_{k}} as basis. Then, the collocation scheme of BVP can be expressed as:
where \(\mathbf {u}=(f^{\prime \prime }_{N}(x_{1}),f^{\prime \prime }_{N}(x_{2}),\ldots ,f^{\prime \prime }_{N}(x_{N1}))^{T}\), and
We can obtain u by solving the system, and then recover f = (f_{N}(x_{1}),…,f_{N}(x_{N− 1}))^{T} from
where b_{j} = (B_{j}(x_{1}),B_{j}(x_{2}),…,B_{j}(x_{N− 1}))^{T} for j = 0,N.
Remark 8
Obviously, the new system (4.42) does not involve the direct multiplication of the preconditioner, and the roundoff errors in forming differentiation matrices can be alleviated.
Remark 9
The use of Birkhoff interpolation as basis functions for deriving precondition is mimic to the preconditioning technique in [32]. However, [32] search for the Birkhoff interpolation basis {B_{j}(x)} through expansion in a different finite dimensional space, and then solving the coefficients by the interpolation conditions. This process involves inverting a matrix of PSWF values, which is timeconsuming. My idea of constructing the basis {B_{j}(x)} in (4.31)–(4.32) is actually inspired by polynomialbased algorithms in [29] and the new insights reside in two aspects. First, in order to avoid the instability of the Lagrange interpolation, the barycentric form was used. Second, through changing the variable, the integrals in (4.32) were computed by the fast Gauss quadrature proposed by Hale and Townsend.
Numerical tests
In this section, we illustrate the numerical results in this paper. All the numerical results in this paper are carried out by using Matlab R2014a on a desktop (4.0 GB RAM, 2 Core2 (64 bit) processors at 3.17 GHz) with Windows 7 operating system.
Example 1
Figure 2 illustrates the convergence of the barycentric prolate interpolation formulas for the two analytic functions:
and
For each n, the error is defined by
which is measured at 1000 random points in [− 1,1]. As we can see, the convergence is exponential and is almost indistinguishable for different c. Moreover, it is shown that the optimal c depends on the function being approximated [7]. In the following, we will take c = n/2 for general functions, which is recommended in [30, 32].
Example 2
For the functions:
and
we focus on the comparison of the new barycentric prolate interpolation (c = n/2) (3.15) with the standard interpolation (2.8) in terms of the approximation error in \(L^{\infty }\) norm, which is measured at 1000 random points in [− 1,1]. Numerical results are shown in Figs. 3 and 4. It is seen that the errors for these approaches decrease very fast. Furthermore, the barycentric prolate interpolation has better stability than that of the Lagrange formulation for a large number of points.
Example 3
For the wave functions \(f(x)=\frac {\sin \limits (25x)}{2x^{2}}\) and \(f(x)=(\cos \limits (25x)+\sin \limits (x))/(1+4x^{2})\), we compare the barycentric prolate interpolation (c = n/2) with the barycentric interpolation in the polynomial case, whose nodes and barycentric weights are computed in the chebfun system by the command legpts [24]. Figure 5 illustrates the barycentric prolate interpolation yields spectrally accurate results using even fewer points than barycentric interpolation in the polynomial case.
Example 4
We compare the absolute errors of the derivatives for
at ProlateGaussLobatto points by the barycentric prolate differentiation (3.18)–(3.19) and standard method (2.9)–(2.10). Results of these calculations are shown in Fig. 6. As can be seen, since the standard method (2.10) involves the firstorder differentiation value, it causes error propaganda for a large number n. There is a good performance of prolate barycentric differentiation, which gives us the motivations for the application.
Application
Different from the usual collocation scheme using the standard Lagrange differentiation, barycentric prolate differentiation (3.18)–(3.19), combining with the usual spectral collocation method and GMRES, has been implemented and tested on the highly oscillatory problem and twodimensional Helmholtz problem. The comparison with CC pointsbased method is reminiscent when the solution is highly oscillatory.
Example 5
The second example is one where the solution is very oscillatory
The exact solution is
The behavior of the prolate barycentric differentiation matrix is demonstrated in Fig. 7. It is clear that this method is rapidly convergent and stable, which is better than the usual collocation method based on CC points.
Example 6
We extend the barycentric prolate pseudospectral method to 2D Helmholtz problem [25], which arises in the analysis of wave propagation:
where u = 0 on the boundary and k is a real parameter. For such a problem, we set up a grid based on ProlateGaussLobatto points independently in each direction called a tensor product grid. To solve such a problem for the particular choices k = 9, f(x,y) = exp(− 10[(y − 1)^{2} + (x − 1/2)^{2}]). The solution appears as a mesh plot in Fig. 8. Compared with the value u(0,0) is accurate to nine digits at Chebyshev grid [25] when N = 24, the new barycentric prolate differentiation scheme (3.19) achieves the accuracy to eleven digits at the same number of points. On the right side of Fig. 8, the absolute error at u(0,0) is illustrated when N = 4 : 2 : 38, which show the fast convergence rate at ProlateGaussLobatto points.
Example 7
We consider
with the exact solution \(u(x)=e^{(x^{2}1)/2}\). Below, Table 1 compares the condition number and errors of the spectral collocation (SC) scheme (4.40), direct preconditioned (MPC) scheme (4.41), and the new basis preconditioned collocation (BPC) scheme (4.42), respectively. We also show the iteration number for solving the systems by GMRES. Table 1 clearly indicates that the two preconditioned schemes are wellconditioned and the new basis preconditioned collocation (BPC) scheme has desired performance.
Conclusion
In this paper, we have developed a new scheme for the prolate interpolation and prolate spectral differentiation. The solver is based on the barycentric interpolation, which allows for stable approximation and the error analysis of barycentric prolate interpolation and differentiation are given. What’s more, the new preconditioning skill is proposed for the usual prolatecollocation scheme. The numerical examples demonstrate the performance of the proposed algorithms.
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Acknowledgments
I am deeply grateful to Prof. LiLian Wang, Prof. Shuhuang Xiang and Prof. Huiyuan Li for valuable comments on the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China with grant No. U1930402.
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Tian, Y. Barycentric prolate interpolation and pseudospectral differentiation. Numer Algor 88, 793–811 (2021). https://doi.org/10.1007/s11075020010577
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Keywords
 Barycentric interpolation
 Prolate spheroidal wave functions
 Pseudospectral differentiation
Mathematics Subject Classification (2010)
 65N35
 41A10
 42C05