Differentiation matrices for univariate polynomials

Amiraslani, Amirhossein; Corless, Robert M.

doi:10.1007/s11075-019-00668-z

Cited by 11 publications

(30 citation statements)

References 24 publications

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“…Using the recurrence relations, which are satisfied by the Chebyshev polynomials of the first kind and their derivatives [3], [12] and equating the coefficients at the same polynomials in (4), we come [3] to the following dependence of the coefficients 𝑐 𝑘 on 𝑏 𝑘 :…”

Section: Let Us Differentiate the Function (3) The Derivative Is Expr...mentioning

confidence: 99%

Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order

Lovetskiy

Kulyabov

Hissein

2022

Discrete and Continuous Models

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The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, inverse with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.

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Section: Let Us Differentiate the Function (3) The Derivative Is Expr...mentioning

confidence: 99%

Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order

Lovetskiy

Kulyabov

Hissein

2022

Discrete and Continuous Models

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“…For instance, one can easily observe that the standard basis and Newton basis also satisfy (2.1) with α j = 1, β j = 0, γ j = 0 and α j = 1, β j = τ j , γ j = 0, respectively, where the τ j are the nodes when the function values are given (see e.g. [3] for more details).…”

Section: Direct Multiplication Of Polynomials In Degree-graded Basesmentioning

confidence: 99%

Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases

Karami¹,

Ahmadnasab²,

Hadizadeh³

et al. 2021

Preprint

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Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In this work, we present formulas and techniques for polynomial multiplications expressed in a variety of well-known polynomial bases without any change of basis. In particular, we take into consideration degree-graded polynomial bases including, but not limited to orthogonal polynomial bases and non-degree-graded polynomial bases including the Bernstein and Lagrange bases. All of the described polynomial multiplication formulas and techniques in this work, which are mostly presented in matrix-vector forms, preserve the basis in which the polynomials are given. Furthermore, using the results of direct multiplication of polynomials, we devise techniques for intra-basis polynomial division in the polynomial bases. A generalization of the well-known "long division" algorithm to any degree-graded polynomial basis is also given. The proposed framework deals with matrix-vector computations which often leads to well-structured matrices. Finally, an application of the presented techniques in constructing the Galerkin representation of polynomial multiplication operators is illustrated for discretization of a linear elliptic problem with stochastic coefficients.

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“…We note that it is a sparse matrix with only 3 nonzero elements in each column [29,Section 1.4.3]. So for each root α i , we get d i equations of the type (29). This gives us a linear system in the δp i 's.…”

Section: Computing Approximate Polynomialsmentioning

confidence: 99%

“…In order to find the derivatives in (29), we can use the differentiation matrix D B in the Bernstein basis which is introduced in [29]. We note that it is a sparse matrix with only 3 nonzero elements in each column [29,Section 1.4.3]. So for each root α i , we get d i equations of the type (29).…”

Section: Computing Approximate Polynomialsmentioning

confidence: 99%