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AN EXTENSION OF CHEBFUN TO TWO DIMENSIONS
"... An objectoriented Matlab system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2 ” objects ..."
Abstract

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An objectoriented Matlab system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2 ” objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented.
Computing with functions in two dimensions
, 2014
"... New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs nearoptimal low rank approxim ..."
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Cited by 1 (0 self)
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New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs nearoptimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly efficient. Explicit convergence rates are shown for the singular values of differentiable and separately analytic functions, and examples are given to demonstrate some paradoxical features of low rank approximation theory. Analogues of QR, LU, and Cholesky factorizations are introduced for matrices that are continuous in one or both directions, deriving a continuous linear algebra. New notions of triangular structures are proposed and the convergence of the infinite series associated with these factorizations is proved under certain smoothness assumptions.