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MATLAB是一种强大的数值计算和数据分析环境，特别适合于解决各种数学问题，包括微分方程的求解。本文档"A MATLAB Differentiation Matrix Suite"主要介绍了如何使用MATLAB软件套件来解决微分方程，特别是通过谱插值（即伪谱）方法。该套件包含了17个MATLAB函数，用于计算与Chebyshev、Hermite、Laguerre、Fourier以及sinc插值相关的任意阶导数。谱方法是数值分析中的一种高效技术，用于近似求解常微分方程和偏微分方程。它利用正交多项式在特定节点上的插值来表示解，然后通过计算这些节点上的导数来实现离散化。MATLAB中的这个套件提供了辅助功能，包括处理边界条件、使用barycentric公式进行插值以及求解正交多项式的根。在介绍中，作者提到了这个软件包可以应用于特殊函数、量子力学、非线性波和流体动力学稳定性等领域的问题。具体来说，它可以用于求解特征值问题、边界值问题和初值问题。特征值问题涉及到找到使得方程成立的参数值，边界值问题则涉及在特定边界条件下求解方程，而初值问题则需要找到满足特定初始条件的解。 MATLAB的矩阵运算能力使得这些复杂的计算变得相对简单，因为它的语法设计鼓励使用矩阵表达式，这在处理大量数据时非常有效。Fornberg（1996）和Canuto等人（1988）的工作强调了差异化矩阵在数值解法中的重要性，这种矩阵可以直接作用于插值多项式，从而高效地实现微分。 MATLAB套件中的函数分为几个类别：计算导数的函数、处理边界条件的函数、进行插值的函数和计算正交多项式根的函数。例如，Chebyshev多项式在解决周期性和对称性问题时特别有用，而Hermite多项式则适用于处理包含多个连续导数信息的问题。Laguerre多项式在处理半无限区间或指数衰减问题时表现良好，而Fourier多项式则常用于处理周期性问题。sinc函数则是对连续函数的一种插值方法，尤其适用于处理在整个实数轴上的问题。在实际应用中，用户可以结合这些工具解决各种实际问题，例如模拟物理系统的行为、分析工程结构的稳定性或研究金融模型的动力学。MATLAB的用户界面和脚本语言使得这些高级数值方法对非专家也相对友好，只需编写简单的代码就可以实现复杂的计算。 "A MATLAB Differentiation Matrix Suite"提供了一种强大且灵活的工具集，使得MATLAB用户能够利用谱方法有效地解决各种微分方程问题。通过熟练掌握这个套件，科研人员和工程师能够快速、准确地求解实际问题，而无需深入理解底层的数值算法。

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A MATLAB Differentiation Matrix Suite

J. A. C. WEIDEMAN

University of Stellenbosch

and

S. C. REDDY

Oregon State University

A software suite consisting of 17 MATLAB functions for solving differential equations by the

spectral collocation (i.e., pseudospectral) method is presented. It includes functions for

computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre,

Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary

conditions, performing interpolation using barycentric formulas, and computing roots of

orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue,

boundary value, and initial value problems arising in the fields of special functions, quantum

mechanics, nonlinear waves, and hydrodynamic stability.

Categories and Subject Descriptors: G.1.7 [Numerical Analysis]: Ordinary Differential

Equations—Boundary value problems; G.1.8 [Numerical Analysis]: Partial Differential

Equations—Spectral methods

General Terms: Algorithms

Additional Key Words and Phrases: MATLAB, spectral collocation methods, pseudospectral

methods, differentiation matrices

1. INTRODUCTION

This paper is about the confluence of two powerful ideas, both developed in

the last two or three decades. The first is the concept of a differentiation

matrix that has proven to be a very useful tool in the numerical solution of

differential equations [Canuto et al. 1988; Fornberg 1996]. The second is

the matrix-based approach to scientific computing that was introduced in

the MATLAB (Matrix Laboratory) software package [The MathWorks

Work by J.A.C. Weideman was supported in part by NSF grant DMS-9404599.

Authors’ addresses: J. A. C. Weideman, Department of Applied Mathematics, University of

Stellenbosch, Private Bag XI, Matieland, 7602, South Africa; email: weideman@na-net.ornl.gov;

S. C. Reddy, Department of Mathematics, Oregon State University, Corvallis, OR 97331;

email: reddy@math.orst.edu.

Permission to make digital/hard copy of part or all of this work for personal or classroom use

is granted without fee provided that the copies are not made or distributed for profit or

commercial advantage, the copyright notice, the title of the publication, and its date appear,

and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to

republish, to post on servers, or to redistribute to lists, requires prior specific permission

and/or a fee.

ACM Transactions on Mathematical Software, Vol. 26, No. 4, December 2000, Pages 465–519.

1998]. The basic unit in the MATLAB programming language is the matrix,

and this makes MATLAB the ideal tool for working with differentiation

matrices.

Differentiation matrices are derived from the spectral collocation (also

known as pseudospectral) method for solving differential equations of

boundary value type. This method is discussed in some detail below but for

more complete descriptions we refer to Canuto et al. [1988], Fornberg

[1996], Funaro [1992], and Gottlieb et al. [1984]. In the spectral collocation

method the unknown solution to the differential equation is expanded as a

global interpolant, such as a trigonometric or polynomial interpolant. In

other methods, such as finite elements or finite differences, the underlying

expansion involves local interpolants such as piecewise polynomials. In

practice this means that the accuracy of the spectral method is superior: for

problems with smooth solutions convergence rates of

O~e

2cN

! or O~e

are routinely achieved, where N is the number of degrees of freedom in the

expansion [Canuto et al. 1988; Stenger 1993; Tadmor 1986]. In contrast,

finite elements or finite differences yield convergence rates that are only

algebraic in

N, typically O~N

! or O~N

There is, however, a price to be paid for using a spectral method instead

of a finite element or a finite difference method: full matrices replace

sparse matrices; stability restrictions may become more severe; and com-

puter implementations, particularly for problems posed on irregular do-

mains, may not be straightforward. Nevertheless, provided the solution is

smooth the rapid convergence of the spectral method often compensates for

these shortcomings.

The Differentiation Matrix Suite introduced here consists of 17 MATLAB

functions, summarized in the Appendix, that enable the user to generate

spectral differentiation matrices based on Chebyshev, Fourier, Hermite,

and other interpolants. These functions enable one to solve, with just a few

lines of additional code, a variety of problems in scientific computation. We

picked the five important differential equations listed in Table I as exam-

ples. Each of these problems is solved in tutorial-style to illustrate the

usage of our codes.

To introduce the idea of a differentiation matrix we recall that the

spectral collocation method for solving differential equations is based on

weighted interpolants of the form [Canuto et al. 1988; Fornberg 1996;

Welfert 1997]

Table I. Examples

Example Domain Problem Type Solution Procedure Application

Error Function

@0, `!

Boundary value Chebyshev Probability

Mathieu periodic Eigenvalue Fourier Dynamical Systems

Schrödinger

@0, `!

Eigenvalue Laguerre Quantum Mechanics

Sine-Gordon

~2`, `!

Evolution Fourier, Hermite, sinc Nonlinear Waves

Orr-Sommerfeld

@21, 1#

Eigenvalue Chebyshev Fluid Mechanics

466 • J. A. C. Weideman and S. C. Reddy

ACM Transactions on Mathematical Software, Vol. 26, No. 4, December 2000.

f~x! ' p

N21

~x! 5

j51

~x!

~x!f

. (1)

Here $x

j51

is a set of distinct interpolation nodes;

~x! is a weight

function;

5 f~x

!; and the set of interpolating functions $

~x!%

j51

satis-

fies

! 5

(the Kronecker delta). This means that p

N21

~x! defined by

(1) is an interpolant of the function f~x! in the sense that

f~x

! 5 p

N21

!, k 5 1,...,N.

A list of commonly used nodes, weights, and interpolating functions are

tabulated in Section 3 below. These include the Chebyshev, Hermite, and

Laguerre expansions, in which case the interpolating functions

~x!% are

polynomials of degree N 2 1.

Two well-known nonpolynomial cases are

also included in our list, namely the trigonometric (Fourier) and sinc

(cardinal) interpolants.

Associated with an interpolant such as (1) is the concept of a collocation

derivative operator. This operator is generated by taking , derivatives of

(1) and evaluating the result at the nodes

$ x

~,!

! '

j51

~x!

x5x

, k 5 1,...,N.

The derivative operator may be represented by a matrix D

~,!

, the differen-

tiation matrix, with entries

k, j

~,!

~x!

x5x

. (2)

The numerical differentiation process may therefore be performed as the

matrix-vector product

~,!

5 D

~,!

f, (3)

where f (resp. f

~,!

) is the vector of function values (resp. approximate

derivative values) at the nodes $ x

When solving differential equations, the derivatives are approximated by

the discrete derivative operators (3). A linear two-point boundary value

problem may thus be converted to a linear system. A differential eigenvalue

problem may likewise be converted to a matrix eigenvalue problem. Solving

In the standard notation, one considers interpolating polynomials of degree N and sums, as

in (1), to have lower limit j 5 0 and upper limit N. Since MATLAB does not have a zero index

we begin sums with j 5 1, and consequently our notation will involve polynomials of degree

N 2 1.

A MATLAB Differentiation Matrix Suite • 467

ACM Transactions on Mathematical Software, Vol. 26, No. 4, December 2000.

linear systems and computing the eigenvalues of matrices are one-line

commands in MATLAB, involving the backslash operator and the function

eig respectively. Examples of the procedure are given in Sections 4 and 5.

After solving the matrix problem, approximations to the function values

at the nodes become available. It is often necessary to compute approxima-

tions at arbitrary points in the domain, however, so some form of interpo-

lation is required. MATLAB has a built-in function polyfit.m for polyno-

mial interpolation, but this function is intended for data fitting and may

not be the most efficient for the calculations we have in mind. For the same

reason the direct calculation of the interpolant (1) is not recommended. In

most cases the interpolant may be expressed in the so-called barycentric

form, which allows a more efficient implementation, and will therefore be

used here. For a discussion of barycentric interpolation formulas, and in

particular their superior stability properties, we refer to Henrici [1982,

Sect. 5.4] (polynomial) and to Henrici [1986, Sect. 13.6] (trigonometric).

The software suite introduced here consists of a set of MATLAB functions

that enable the user to compute differentiation matrices

~,!

, plus associ-

ated nodes $ x

%, for (a) all the important special cases (Chebyshev, Leg-

endre, Laguerre, Hermite, Fourier, sinc), and (b) an arbitrary number of

derivatives

, 5 1,...,M. Auxiliary codes include functions for comput-

ing the roots of some orthogonal polynomials (Legendre, Laguerre, Her-

mite), as well as barycentric interpolation formulas (Chebyshev, Fourier),

plus functions for implementing special boundary conditions. A summary of

all the functions in the suite is given in the Appendix.

This paper emphasizes the matrix-based implementation of the spectral

collocation method. It is well known, however, that certain methods—

Fourier, Chebyshev, sinc—can also be implemented by using the Fast

Fourier Transform (FFT). By applying the FFT technique the matrix-vector

product (3) can be computed in

O~N log N ! operations rather than the

O~N

! operations that the direct computation of such a product requires.

There are, however, situations where one might prefer the matrix approach

in spite of its inferior asymptotic operation count.

First, for small values of N the matrix approach is in fact faster than the

FFT implementation. Also, for the FFT to be optimally efficient

N has to be

a power of 2; otherwise the matrix approach may not be much slower in

practice even for large N. Second, the FFT approach places a limitation on

the type of algorithm that can be used for solving the linear system or

eigenvalue problem that arises after discretization of the differential equa-

tion. Only iterative algorithms based on matrix-vector products, such as

conjugate gradients or GMRES for linear systems and the Lanczos or

Arnoldi iterations for eigenvalues, can be used. These are not built-in codes

in MATLAB, so users will have to supply their own. (For these reasons the

solution procedures sketched in Section 5 apply only to the matrix ap-

proach.)

In spite of the advantages of the matrix-based approach in the MATLAB

setting, we have included three transform-based functions in our suite for

468 • J. A. C. Weideman and S. C. Reddy

ACM Transactions on Mathematical Software, Vol. 26, No. 4, December 2000.

educational purposes and for the sake of completeness. These functions

correspond to the Fourier, Chebyshev, and sinc methods, which are all

based on the FFT. (We point out that asymptotically fast algorithms for

polynomial interpolation and differentiation at arbitrary points have been

proposed in Dutt et al. [1996]. These algorithms are not based on the FFT,

and have not been included in our suite.)

A few comments about our MATLAB coding style are in order. It is well

known that efficient coding in MATLAB means vectorization, and the use

of built-in (compiled) functions wherever possible. Conditionals and loops,

particularly nested loops, are to be avoided. We have tried to adhere to this

guideline, even if it meant a more cryptic code. To compensate for this, we

made an effort to elucidate the logic of our codes in the text of this paper.

The execution times of MATLAB functions need not be proportional to

the number of floating-point operations performed. A code with an optimal

operation count may be slow in practice due to the reasons given above.

When faced with this situation, we chose the implementation which exe-

cutes quicker even if it meant a higher operation count.

Many of the functions in the suite will not run in versions of MATLAB

older than Version 5. This is primarily because of two reasons. First,

Version 5 is the first version of MATLAB that allows arrays with more than

two indices. Our functions typically generate arrays of dimension

N 3 N

3 M

where the third index is reserved for the order of the derivative.

Second, MATLAB 5 has a variable of type logical, not found in earlier

versions, which is utilized in several of the codes.

We are aware of two other general software packages for spectral

computations, both in Fortran. The first is Funaro [1993], and the second is

PseudoPack 2000 [Costa and Don 1999]. Funaro’s package computes first

and second derivatives and has support for general Jacobi polynomials,

many quadrature formulas, and routines for computing expansion coeffi-

cients. PseudoPack can compute up to fourth-order Fourier, Chebyshev,

and Legendre collocation derivatives. Additional features include routines

for filtering, coordinate mapping, and differentiation of functions of two

and three variables. The listings of various Fortran programs may be found

in Canuto et al. [1988] and Fornberg [1996]. Examples of spectral compu-

tations using MATLAB are also given in Trefethen [2000], where the reader

will find many applications that complement the ones listed in Table I.

Concerning higher derivatives, we remark that often the second- and

higher-derivative matrices are equal to the first-derivative matrix raised to

the appropriate power. But this is not always the case—for a sufficient

condition, involving the weight function

~x!, we refer to Welfert [1997].

Even when this is the case, it is not a good idea to compute higher-

derivative matrices by computing powers of the first-derivative matrix. The

computation of powers of a full matrix requires O~N

! operations, com-

pared to the

O~N

! for the recursive algorithm described in the next

section. Not only is this recursion faster, it also introduces less roundoff

error compared to the computation of matrix powers.

A MATLAB Differentiation Matrix Suite • 469

ACM Transactions on Mathematical Software, Vol. 26, No. 4, December 2000.

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