An Extension of Chebfun to Two Dimensions

Abstract

Abstract. An object-oriented 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 the common zeros of two bivariate functions via Bézout resultants

“…These contouring algorithms can be very efficient at solving (1.1), and until this paper the roots(f,g) command in Chebfun2 exclusively employed such a contouring approach [43]. In the older version of Chebfun2 the zero level curves of f and g were computed separately using the Matlab command contourc, and then the intersections of these zero level curves were used as initial guesses for Newton's iteration.…”

“…Throughout, the zero contours of f and g in (1.1) are drawn as blue and red curves, respectively, computed by the command roots(f) in Chebfun2 [43]. The black dots are the solutions computed by our algorithm described in this paper as realized by the command roots(f,g) in Chebfun2.…”

“…Chebfun2 allows us to construct polynomials from interpolation data at a tensor Chebyshev grid [43], and for this example we take the interpolation data to be the Hadamard matrices H 32 and H 64 of size 32 × 32 and 64×64, i.e., we solve (1.1), where f (x i , x j ) = H 32 (i, j), g(x i , x j ) = H 64 (i, j), and x i are Chebyshev points. The Hadamard matrices contain ±1 entries and therefore, the constructed polynomials p = f and q = g (of degrees (m p , n p , m q , n q ) = (31, 31, 63, 63)) have many zero contours and common zeros.…”

“…In this typical situation we describe a robust, accurate, and fast numerical algorithm that can solve problems of much higher degree than in previous studies. Our algorithm has been implemented in the roots command in Chebfun2, an open source software system written in object-oriented Matlab that extends Chebfun [46] to functions of two variables defined on rectangles [43]. Chebfun2 approximates a bivariate function by a polynomial interpolant represented as a "chebfun2" object, which is accurate to relative machine precision [43].…”

“…Our algorithm has been implemented in the roots command in Chebfun2, an open source software system written in object-oriented Matlab that extends Chebfun [46] to functions of two variables defined on rectangles [43]. Chebfun2 approximates a bivariate function by a polynomial interpolant represented as a "chebfun2" object, which is accurate to relative machine precision [43]. Our first step is to use Chebfun2 to replace f (x, y) and g(x, y) in (1.1) by polynomial interpolants p(x, y) and q(x, y), respectively, before finding the common zeros of the corresponding polynomial system.…”