Generation of Pseudospectral Differentiation Matrices I

Welfert, Bruno D.

doi:10.1137/s0036142993295545

Cited by 97 publications

(74 citation statements)

References 6 publications

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“…The computation of spectral collocation differentiation matrices for derivatives of arbitrary order has been considered by Huang and Sloan [1994] (constant weights) and Welfert [1997] (arbitrary ␣͑x͒). The algorithm implemented in poldif.m and chebdif.m follows these references closely.…”

Section: An Algorithm For Polynomial Differentiationmentioning

confidence: 99%

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A MATLAB differentiation matrix suite

Weideman

Reddy

2000

ACM Trans. Math. Softw.

755

564

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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.

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Section: An Algorithm For Polynomial Differentiationmentioning

confidence: 99%

“…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] …”

Section: Introductionmentioning

confidence: 99%

A MATLAB differentiation matrix suite

Weideman

Reddy

2000

ACM Trans. Math. Softw.

755

564

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“…. , n. However, more accurate and stable computational methods to compute these entries can be found in Baltensperger and Trummer (2003); Weideman and Reddy (2000); Welfert (1997).…”

Section: Differentiation Matrixmentioning

confidence: 99%

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

Foroozandeh

Shamsi

Pinho

2017

International Journal of Control

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This paper presents a new approach for the efficient and accurate solution of Singular Optimal Control Problems (SOCP). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of the solution. As the first step of this method, the SOCP is converted into a binary optimal control problem. Then, by utilizing the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the original optimal control and to compute the approximating solution of it (getting both the optimal state and control). The main advantages of the present method are that: (i) without a priori information, the structure of optimal control is detected; (ii) it produces good results even using a small number of collocation points; and (iii) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.

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“…See Section 5 for a discussion of how to accurately evaluate the expressions in (3.3) and (3.15). Welfert (1997) derives an expression for the entries of a (p + 1)th-order square differentiation matrix in terms of its pth-order counterpart, and based on this gives a recursive algorithm to compute higher-order square differentiation matrices in O(n 2 ) operations. Following the same approach we derive a similar recursive method for high-order rectangular differentiation matrices.…”

Section: Rectangular Differentiation Matrixmentioning

confidence: 99%