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17
Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
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Cited by 141 (31 self)
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In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
Fast algorithms for discrete polynomial transforms
 Math. Comput
, 1998
"... Abstract. Consider the Vandermondelike matrix P: = (Pk(cos jπ ..."
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Cited by 42 (6 self)
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Abstract. Consider the Vandermondelike matrix P: = (Pk(cos jπ
Fast and stable algorithms for discrete spherical Fourier transforms
, 1996
"... . In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ae R 3 , as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of ..."
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Cited by 20 (4 self)
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. In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ae R 3 , as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of discrete Legendre function transforms based on a modified and stabilized version of the DriscollHealy algorithm. 1991 Mathematics Subject Classification. Primary 65T99, 33C35, 33C25, 42C10 Key words and phrases. Discrete spherical Fourier transform, spherical harmonics, sampling theorem, discrete Legendre function transform, fast cosine transform, Chebyshev nodes, cascade summation 1 Introduction Fourier analysis on the sphere S ae R 3 has practical relevance in tomography, geophysics, seismology, meteorology and crystallography. It can be used in spectral methods for solving partial differential equations on the sphere (see [4], [16]). In [12], the authors utilize spherical Fourier...
A fast and wellconditioned spectral method
 SIAM Review
"... Abstract. A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes Ø(m2n) operations, where m is the number of Chebyshev points nee ..."
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Cited by 15 (3 self)
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Abstract. A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes Ø(m2n) operations, where m is the number of Chebyshev points needed to resolve the coefficients of the differential operator and n is the number of Chebyshev coefficients needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this implies stability in the standard 2norm. An adaptive QR factorization is developed to efficiently solve the resulting linear system and automatically choose the optimal number of Chebyshev coefficients needed to represent the solution. The resulting algorithm can efficiently and reliably solve for solutions that require as many as a million unknowns.
Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems
, 1998
"... . This paper is concerned with the solution of systems of linear equations T N xN = bN , where fT N gN2IN denotes a sequence of nonsingular nonsymmetricToeplitz matrices arising from a generating function of the Wiener class. We present a technique for the fast construction of optimal trigonometric ..."
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Cited by 15 (7 self)
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. This paper is concerned with the solution of systems of linear equations T N xN = bN , where fT N gN2IN denotes a sequence of nonsingular nonsymmetricToeplitz matrices arising from a generating function of the Wiener class. We present a technique for the fast construction of optimal trigonometric preconditioners M N = M N (T 0 N T N ) of the corresponding normal equation which can be extended to Toeplitz least squares problems in a straightforward way. Moreover, we prove that the spectrum of the preconditioned matrix M \Gamma1 N T 0 N T N is clustered at 1 such that the PCGmethod applied to the normal equation converges superlinearly. Numerical tests confirm the theoretical expectations. 1991 Mathematics Subject Classification. 65F10, 65F15, 65T10. Key words and phrases. Toeplitz matrix, Krylov space methods, CGmethod, preconditioners, normal equation, clusters of eigenvalues. 1 Introduction Consider the system of linear equations T N xN = bN ; (1.1) where T N 2 IR N;N...
Preconditioners for illconditioned Toeplitz matrices
, 1999
"... . This paper is concerned with the solution of systems of linear equations ANx = b, where fAN g N2N denotes a sequence of positive definite Hermitian illconditioned Toeplitz matrices arising from a (realvalued) nonnegative generating function f 2 C2ß with zeros. We construct positive definite ..."
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Cited by 8 (3 self)
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. This paper is concerned with the solution of systems of linear equations ANx = b, where fAN g N2N denotes a sequence of positive definite Hermitian illconditioned Toeplitz matrices arising from a (realvalued) nonnegative generating function f 2 C2ß with zeros. We construct positive definite Hermitian preconditioners MN such that the eigenvalues of M \Gamma1 N AN are clustered at 1 and the corresponding PCGmethod requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations. AMS subject classification: 65F10, 65F15, 65T10. Key words: Illconditioned Toeplitz matrices , CGmethod, clusters of eigenvalues, preconditioners. 1 Introduction. Systems of linear equations ANx = b with positive definite Hermitian T...
Preconditioners for illconditioned Toeplitz systems constructed from positive kernels
, 1999
"... . In this paper, we are interested in the iterative solution of ill{conditioned Toeplitz systems generated by continuous non{negative real{valued functions f with a nite number of zeros. We construct new w{circulant preconditioners without explicit knowledge of the generating function f by approxima ..."
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Cited by 8 (0 self)
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. In this paper, we are interested in the iterative solution of ill{conditioned Toeplitz systems generated by continuous non{negative real{valued functions f with a nite number of zeros. We construct new w{circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f KN with a suitable positive reproducing kernel KN . By the restriction to positive kernels we obtain positive denite preconditioners. Moreover, if f has only zeros of even order 2s, then we can prove that the property R t 2k KN (t) dt CN 2k (k = 0; : : : ; s) of the kernel is necessary and sucient to ensure the convergence of the PCG{method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were conrmed by numerical tests. 1991 Mathematics Subject Classication. 65F10, 65F15, 65T10. Key words and phrases. Ill{conditioned Toeplitz matrices , CG{method, preconditioners, reproducing kernels. 1 Intr...
Fast multidimensional scattered data approximation with Neumann boundary conditions
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A Fast Algorithm for Nonequispaced Fourier Transforms on the Rotation Group
"... In this paper we present algorithms to calculate fast Fourier transforms and its adjoint on the rotation group SO(3) for arbitrary sampling sets. It is based on the fast Fourier transform for nonequispaced nodes on the threedimensional torus. This algorithm evaluates the SO(3) Fourier transform of B ..."
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Cited by 3 (0 self)
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In this paper we present algorithms to calculate fast Fourier transforms and its adjoint on the rotation group SO(3) for arbitrary sampling sets. It is based on the fast Fourier transform for nonequispaced nodes on the threedimensional torus. This algorithm evaluates the SO(3) Fourier transform of Bbandlimited functions at M arbitrary input nodes in O(M + B 4) or even O(M + B 3 log 2 B) flops instead of O(MB 3). Numerical results will be presented establishing the algorithm’s numerical stability and time requirements.
A fast Fourier algorithm on the rotation group
, 2007
"... In this paper we present an algorithm for the fast Fourier transform on the rotation group SO(3) which is based on the fast Fourier transform for nonequispaced nodes on the threedimensional torus. This algorithm allows to evaluate the SO(3) Fourier transform of Bbandlimited functions at M arbitra ..."
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Cited by 3 (1 self)
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In this paper we present an algorithm for the fast Fourier transform on the rotation group SO(3) which is based on the fast Fourier transform for nonequispaced nodes on the threedimensional torus. This algorithm allows to evaluate the SO(3) Fourier transform of Bbandlimited functions at M arbitrary input nodes in O(M + B³ log² B) flops instead of O(MB 3). Some numerical results will be presented establishing the algorithm’s numerical stability and time requirements.