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12
Is Gauss Quadrature Better Than Clenshaw–Curtis?
, 2008
"... We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to exp ..."
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Cited by 40 (3 self)
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We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Sevenline MATLAB codes are presented that implement both methods, and experiments show that the supposed factorof2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O’Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z +1)/(z − 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw–Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [−1, 1].
Fast evaluation of quadrature formulae on the sphere
 Math. Comput
, 2006
"... Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 � S 2 �orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75–98, 2 ..."
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Cited by 12 (11 self)
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Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 � S 2 �orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75–98, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules. We give a formulation in matrixvector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical GaußLegendre and ClenshawCurtis quadrature rules as well as the HEALPix pixelation scheme and an equidistribution. 1.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
On the Fourier extension of nonperiodic functions
, 2009
"... We obtain exponentially accurate Fourier series for nonperiodic functions on the interval [−1,1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal p ..."
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Cited by 6 (2 self)
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We obtain exponentially accurate Fourier series for nonperiodic functions on the interval [−1,1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.
AN EXTENSION OF CHEBFUN TO TWO DIMENSIONS
"... Abstract. 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 ” ..."
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Cited by 1 (1 self)
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Abstract. 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. Key words. Matlab, Chebfun, Chebyshev polynomials, low rank approximation
Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave
 NUMERICAL ALGORITHMS
, 2010
"... We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the socalled Padua points on rectangles, and the corresponding versions for algebraic cubature. ..."
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Cited by 1 (0 self)
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We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the socalled Padua points on rectangles, and the corresponding versions for algebraic cubature.
WAGES AND INFORMALITY IN DEVELOPING COUNTRIES By
, 2012
"... It is often argued that informal labor markets in developing countries promote growth by reducing the impact of regulation. On the other hand informality may reduce the amount of social protection offered to workers. We extend the wageposting framework of Burdett and Mortensen (1998) to allow heter ..."
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It is often argued that informal labor markets in developing countries promote growth by reducing the impact of regulation. On the other hand informality may reduce the amount of social protection offered to workers. We extend the wageposting framework of Burdett and Mortensen (1998) to allow heterogeneous firms to decide whether to locate in the formal or the informal sector, as well as set wages. Workers engage in both off the job and on the job search. We estimate the model using Brazilian micro data and evaluate the labor market and welfare effects of policies towards informality.
World Bank
, 2012
"... Notes: Center discussion papers are preliminary materials circulated to stimulate discussion and critical comments. We thank Joe Altonji, Corina Mommaerts and seminar participants at Yale University for comments. Costas Meghir thanks the ESRC for funding under the Professorial Fellowship RES05127 ..."
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Notes: Center discussion papers are preliminary materials circulated to stimulate discussion and critical comments. We thank Joe Altonji, Corina Mommaerts and seminar participants at Yale University for comments. Costas Meghir thanks the ESRC for funding under the Professorial Fellowship RES051270204. Robin gratefully
A POLYNOMIAL INTERPOLATION PROCESS AT QUASICHEBYSHEV NODES WITH THE FFT
"... Abstract. Interpolation polynomial pn at the Chebyshev nodes cos πj/n (0 ≤ j ≤ n) for smooth functions is known to converge fast as n →∞. The sequence {pn} is constructed recursively and efficiently in O(n log2 n)flops for each pn by using the FFT, where n is increased geometrically, n = 2i (i =2, 3 ..."
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Abstract. Interpolation polynomial pn at the Chebyshev nodes cos πj/n (0 ≤ j ≤ n) for smooth functions is known to converge fast as n →∞. The sequence {pn} is constructed recursively and efficiently in O(n log2 n)flops for each pn by using the FFT, where n is increased geometrically, n = 2i (i =2, 3,...), until an estimated error is within a given tolerance of ε. This sequence {2j}, however, grows too fast to get pn of proper n, oftenamuch higher accuracy than ε being achieved. To cope with this problem we present quasiChebyshev nodes (QCN) at which {pn} can be constructed efficiently in the same order of flops as in the Chebyshev nodes by using the FFT, but with n increasing at a slower rate. We search for the optimum set in the QCN that minimizes the maximum error of {pn}. Numerical examples illustrate the error behavior of {pn} with the optimum nodes set obtained. 1.