Fast structured Jacobi-Jacobi transforms

Jacobi polynomials are frequently used in scientific and engineering applications, and often times, one needs to use the so called Jacobi-Jacobi transforms which are transforms between two Jacobi expansions with different indices. In this paper, we develop a fast structured algorithm for Jacobi-Jacobi transforms. The algorithm is based on two main ingredients. (i) Derive explicit formulas for connection matrices of two Jacobi expansions with arbitrary indices. In particular, if the indices have integer differences, the connection matrices are relatively sparse or highly structured. The benefit of simultaneous promotion or demotion of the indices is shown. (ii) If the indices have noninteger differences, we explore analytically or numerically a low-rank property hidden in the connection matrices. Combining these two ingredients, we develop a fast structured Jacobi-Jacobi transform with nearly linear complexity, after an one-time precomputation with quadratic complexity, between coefficients of two Jacobi expansions with arbitrary indices. An important byproduct of the fast Jacobi-Jacobi transform is the fast Jacobi transform between the function values at a set of Chebyshev-Gauss-type points and coefficients of the Jacobi expansion with arbitrary indices. Ample numerical results are presented to illustrate the computational efficiency and accuracy of our algorithm.

[1]  Jianlin Xia,et al.  On the Complexity of Some Hierarchical Structured Matrix Algorithms , 2012, SIAM J. Matrix Anal. Appl..

[2]  Ben-Yu Guo,et al.  Jacobi Approximations in Certain Hilbert Spaces and Their Applications to Singular Differential Equations , 2000 .

[3]  Ryszard Szwarc.  CONNECTION COEFFICIENTS OF ORTHOGONAL POLYNOMIALS WITH APPLICATIONS TO CLASSICAL ORTHOGONAL POLYNOMIALS , 1993 .

[4]  V. Louis-Dorr,et al.  ECG Data Compression Using Jacobi Polynomials , 2007, Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[5]  Stanisław Lewanowicz,et al.  Quick construction of recurrence relations for the Jacobi coefficients , 1992 .

[6]  T. Koornwinder.  Two-Variable Analogues of the Classical Orthogonal Polynomials , 1975 .

[7]  Herbert E. Salzer,et al.  A recurrence scheme for converting from one orthogonal expansion into another , 1973, CACM.

[8]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[9]  Reiji Suda,et al.  A fast spherical harmonics transform algorithm , 2002, Math. Comput..

[10]  Ryszard Szwarc.  Connection Coefficients of Orthogonal Polynomials , 1992, Canadian Mathematical Bulletin.

[11]  Mark Tygert,et al.  Fast algorithms for spherical harmonic expansions, II , 2008, J. Comput. Phys..

[12]  A. Laforgia,et al.  On the asymptotic expansion of a ratio of gamma functions , 2012 .

[13]  Jie Shen,et al.  Highly Accurate Pseudospectral Approximations of the Prolate Spheroidal Wave Equation for Any Bandwidth Parameter and Zonal Wavenumber , 2017, J. Sci. Comput..

[14]  Jan S. Hesthaven,et al.  Computation of connection coefficients and measure modifications for orthogonal polynomials , 2012 .

[15]  Shivkumar Chandrasekaran,et al.  A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations , 2006, SIAM J. Matrix Anal. Appl..

[16]  Jenna Reis,et al.  The spectral connection matrix for classical real orthogonal polynomials , 2015 .

[17]  George Em Karniadakis,et al.  Supersensitivity due to uncertain boundary conditions , 2004 .

[18]  Richard Mikael Slevinsky,et al.  On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev--Jacobi transform , 2016, 1602.02618.

[19]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[20]  Bradley K. Alpert,et al.  A Fast Algorithm for the Evaluation of Legendre Expansions , 1991, SIAM J. Sci. Comput..

[21]  Jie Shen,et al.  Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle , 2010, Math. Comput..

[22]  Frank W. J. Olver,et al.  Airy and related functions , 2010, NIST Handbook of Mathematical Functions.

[23]  Jianlin Xia,et al.  Fast algorithms for hierarchically semiseparable matrices , 2010, Numer. Linear Algebra Appl..

[24]  John P. Boyd,et al.  Algorithm 840: computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions---prolate elements , 2005, TOMS.

[25]  Eduardo Godoy,et al.  Recurrence relations for connection coefficients between two families of orthogonal polynomials , 1995 .

[26]  N. Temme.  Special Functions: An Introduction to the Classical Functions of Mathematical Physics , 1996 .

[27]  Mark Tygert,et al.  Fast Algorithms for Spherical Harmonic Expansions , 2006, SIAM J. Sci. Comput..

[28]  Jianlin Xia,et al.  A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling , 2012, SIAM J. Matrix Anal. Appl..

[29]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[30]  Pascal Maroni,et al.  Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation , 2008, Numerical Algorithms.

[31]  T. Bella,et al.  The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials , 2015 .

[32]  Pascal Maroni,et al.  Connection coefficients for orthogonal polynomials: symbolic computations, verifications and demonstrations in the Mathematica language , 2012, Numerical Algorithms.

[33]  Dietrich Braess,et al.  On the efficient computation of high-dimensional integrals and the approximation by exponential sums , 2009 .

[34]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[35]  Jianlin Xia,et al.  On the Stability of Some Hierarchical Rank Structured Matrix Algorithms , 2016, SIAM J. Matrix Anal. Appl..

[36]  Nicholas Hale,et al.  A Fast, Simple, and Stable Chebyshev-Legendre Transform Using an Asymptotic Formula , 2014, SIAM J. Sci. Comput..

[37]  A. Erdélyi,et al.  The asymptotic expansion of a ratio of gamma functions. , 1951 .

[38]  Sheehan Olver,et al.  Fast polynomial transforms based on Toeplitz and Hankel matrices , 2016, Math. Comput..

[39]  P. Swarztrauber,et al.  Generalized Discrete Spherical Harmonic Transforms , 2000 .

[40]  T. Bella,et al.  The spectral connection matrix for classical orthogonal polynomials of a single parameter , 2014 .

[41]  Jianlin Xia,et al.  Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling , 2014, SIAM J. Matrix Anal. Appl..

[42]  John A. Board,et al.  Introduction to “A Fast Algorithm for Particle Simulations” , 1997 .

[43]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[44]  Nico M. Temme,et al.  Large degree asymptotics of generalized Bernoulli and Euler polynomials , 2009, 0909.3184.

[45]  Shivkumar Chandrasekaran,et al.  A divide-and-conquer algorithm for the eigendecomposition of symmetric block-diagonal plus semiseparable matrices , 2004, Numerische Mathematik.

[46]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[47]  C. L. Frenzen.  Error bounds for asymptotic expansions of the ratio of two gamma functions , 1987 .

[48]  Jens Keiner,et al.  Computing with Expansions in Gegenbauer Polynomials , 2009, SIAM J. Sci. Comput..

[49]  Jie Shen,et al.  E-cient Chebyshev-Legendre Galerkin Methods for Elliptic Problems , 1996 .

[50]  JENS KEINER,et al.  Gegenbauer polynomials and semiseparable matrices. , 2008 .

[51]  Jie Shen,et al.  Müntz-Galerkin Methods and Applications to Mixed Dirichlet-Neumann Boundary Value Problems , 2016, SIAM J. Sci. Comput..