Abstract
Contiguous submatrices of the Fourier matrix are known to be ill-conditioned. In a recent paper in SIAM review A. Barnett has provided new bounds on the rate of ill-conditioning (Barnett in SIAM Rev 64:105–131, 2022). In this paper we focus on the corresponding singular value decomposition. The singular vectors can be computed from the so-called periodic discrete prolate spheroidal sequences, named in analogy to spheroidal wave functions which are associated with the continuous Fourier transform. Their numerical computation is hampered by the clustering of singular values. We collect and expand known results on the stable numerical computation of the singular value decomposition of Fourier submatrices. The prolate sequences are eigenvectors of a tridiagonal matrix whose spectrum is free of clusters and this enables their computation. We collect these observations in a simple and convenient algorithm. The corresponding singular values can be accurately computed as well, except when they are small. Even then, small singular values can be computed in high-precision arithmetic with modest computational effort, even for large and extremely ill-conditioned submatrices. We illustrate the computations and point out a few applications in which Fourier submatrices arise.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
Notes
In fact, his construction produces a commuting tridiagonal matrix for both Gramians \(C^{*}C\) and \(CC^{*}\).
The name ‘twiddle factor’ originated with [4] and is prevalent in the literature on FFT. It is used here as a synonym for a complex root of unity. In general, powers of \(\omega \) are also often called twiddle factors.
In particular, their arguments can be used to derive that \(A^{*}A\) has separated eigenvalues if and only if \(q\le p\) and \(p+q \le N\). As a consequence, repeated singular values in the (compact) SVD of A only occur whenever \(p+q>N\). This does not affect our computations, as the eigenvalues of the tridiagonal matrices in this paper are always unique even if the corresponding singular values of A are not.
We have to fix branch cuts in case p or q is even. The (k, k) entry of \(D_p^{-\frac{(q-1)}{2}}\) here is \(\omega ^{(k-1)(q-1)/2}\) and it is best implemented that way explicitly. Numerically raising the diagonal matrix \(D_p\) to the power \(-\frac{(q-1)}{2}\) may yield a different outcome, due to choosing a different square root. A similar comment applies to powers of \(D_q\).
For each n the Dirichlet kernel \(\mathcal {D}_n\) is a radial kernel given (up to a factor) by \(\mathcal {D}_n(r)=\frac{\sin {((2n+1)r/2)}}{\sin (r/2)}\).
A complete and user-friendly implementation of the algorithm is available at the time of writing in a Julia package called ‘FourierSubmatrices.jl’, available from https://github.com/daanhb/FourierSubmatrices.jl. The experiments of this paper have been computed in Matlab, except when high precision was required.
Using the Julia package ‘FourierSubmatrices.jl‘, the command cond(DFTBlock{BigFloat}(512,256,256)) takes 0.05 seconds when using 512 bit precision on a contemporary laptop.
References
Barnett, A.H.: How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix? SIAM Rev. 64, 105–131 (2022)
Coppé, V., Huybrechs, D., Matthysen, R., Webb, M.: The AZ algorithm for least squares problems with a known incomplete generalized inverse. SIAM J. Mat. Anal. Appl. 46, 1237–1259 (2020)
Edelman, A., McCorquodale, P., Toledo, S.: The future Fast Fourier transform? SIAM J. Sci. Comput. 20(3), 1094–1114 (1999)
Gentleman, W.M., Sande, G.: Fast Fourier transforms: for fun and profit. In: Proceedings of the November 7–10, 1966, Fall Joint Computer Conference, AFIPS ’66 (Fall), pp. 563–578. Association for Computing Machinery, New York (1966). https://doi.org/10.1145/1464291.1464352
Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Grünbaum, F.A.: Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions. SIAM J. Alg. Discrete Methods 2(2), 136–141 (1981)
Jain, A., Ranganath, S.: Extrapolation algorithms for discrete signals with application in spectral estimation. IEEE Trans. Acoust. Speech Signal Process. 29(4), 830–845 (1981)
Matthysen, R., Huybrechs, D.: Fast algorithms for the computation of Fourier extensions of arbitrary length. SIAM J. Sci. Comput. 38(2), A899–A922 (2016)
Matthysen, R., Huybrechs, D.: Function approximation on arbitrary domains using Fourier frames. SIAM J. Numer. Anal. 56, 1360–1385 (2018)
Osipov, A., Rokhlin, V., Xiao, H.: Prolate Spheroidal Wave Functions of Order Zero. Springer, New York (2013)
Ruiz-Antolín, D., Townsend, A.: A nonuniform fast Fourier transform based on low rank approximation. SIAM J. Sci. Comput. 40(1), A529–A547 (2018)
Slepian, D.: Prolate spheroidal wave functions, Fourier analysis, and uncertainty V: the discrete case. Bell Syst. Tech. J. 57, 1371–1430 (1978)
Slepian, D., Pollak, H.: Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I. Bell Syst. Tech. J. 40(1), 43–63 (1961)
Van Dooren, P., Laudadio, T., Mastronardi, N.: Computing the eigenvectors of nonsymmetric tridiagonal matrices. Comput. Math. Math. Phys. 61, 733–749 (2021)
Xu, W.Y., Chamzas, C.: On the periodic discrete prolate spheroidal sequences. SIAM J. Appl. Math. 44(6), 1210–1217 (1984)
Yang, H.: A unified framework for oscillatory integral transforms: when to use nufft or butterfly factorization? J. Comput. Phys. 388, 103–122 (2019)
Zhu, Z., Karnik, S., Davenport, M.A., Romberg, J., Wakin, M.B.: The eigenvalue distribution of discrete periodic time-frequency limiting operators. IEEE Signal Process. Lett. 25(1), 95–99 (2018). https://doi.org/10.1109/LSP.2017.2751578
Acknowledgements
The authors are grateful to Alex Barnett, Nick Trefethen and Thijs Steel for their suggestions and discussions on the topic, to Emiel Dhondt for his contributions in the implementation, and to the anonymous reviewers for their constructive remarks.
Funding
This work was supported in part by the FWO-Flanders (Project G088622N).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dirckx, S., Huybrechs, D. & Ongenae, R. On the Computation of the SVD of Fourier Submatrices. J Sci Comput 95, 68 (2023). https://doi.org/10.1007/s10915-023-02171-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-023-02171-z