Discrete least squares approximation by trigonometric polynomials

Authors:
L. Reichel, G. S. Ammar and W. B. Gragg

Journal:
Math. Comp. **57** (1991), 273-289

MSC:
Primary 65D15

DOI:
https://doi.org/10.1090/S0025-5718-1991-1079030-8

MathSciNet review:
1079030

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Abstract: We present an efficient and reliable algorithm for discrete least squares approximation of a real-valued function given at arbitrary distinct nodes in by trigonometric polynomials. The algorithm is based on a scheme for the solution of an inverse eigenproblem for unitary Hessenberg matrices, and requires only arithmetic operations as compared with operations needed for algorithms that ignore the structure of the problem. Moreover, the proposed algorithm produces consistently accurate results that are often better than those obtained by general QR decomposition methods for the least squares problem. Our algorithm can also be used for discrete least squares approximation on the unit circle by algebraic polynomials.

**[1]**G. S. Ammar, W. B. Gragg, and L. Reichel,*Constructing a unitary matrix from spectral data*, Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (G. H. Golub and P. Van Dooren, eds.), Springer-Verlag, New York, 1990, pp. 385-396.**[2]**J.-P. Berrut,*Baryzentrische Formeln zur trigonometrischen Interpolation*. I, II, J. Appl. Math. Phys. (ZAMP)**35**(1984), 91-105, 193-205. MR**753088 (85m:65007)****[3]**Å. Björck and V. Pereyra,*Solution of Vandermonde systems of equations*, Math. Comp.**24**(1970), 893-903. MR**0290541 (44:7721)****[4]**C. J. Demeure,*Fast QR factorization of Vandermonde matrices*, Linear Algebra Appl.**122-124**(1989), 165-194. MR**1019987 (91a:65068)****[5]**J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart,*LINPACK users' guide*, SIAM, Philadelphia, PA, 1979.**[6]**G. E. Forsythe,*Generation and use of orthogonal polynomials for data-fitting with a digital computer*, J. Soc. Indust. Appl. Math.**5**(1957), 74-88. MR**0092208 (19:1079e)****[7]**W. Gautschi,*On generating orthogonal polynomials*, SIAM J. Sci. Statist. Comput.**3**(1982), 289-317. MR**667829 (84e:65022)****[8]**-,*Orthogonal polynomials--constructive theory and applications*, J. Comput. Appl. Math.**12 & 13**(1985), 61-76. MR**793944 (87a:65045)****[9]**G. H. Golub and C. F. Van Loan,*Matrix computations*, 2nd ed., Johns Hopkins Univ. Press, 1989. MR**1002570 (90d:65055)****[10]**W. B. Gragg,*The QR algorithm for unitary Hessenberg matrices*, J. Comput. Appl. Math.**16**(1986), 1-8.**[11]**W. B. Gragg and W. J. Harrod,*The numerically stable reconstruction of Jacobi matrices from spectral data*, Numer. Math.**44**(1984), 317-335. MR**757489 (85i:65052)****[12]**U. Grenander and G. Szegö,*Toeplitz forms and their applications*, Chelsea, New York, 1984. MR**890515 (88b:42031)****[13]**P. Henrici,*Applied and computational complex analysis*, vol. 3, Wiley, New York, 1986. MR**822470 (87h:30002)****[14]**A. C. R. Newbery,*Trigonometric interpolation and curve-fitting*, Math. Comp.**24**(1970), 869-876. MR**0279966 (43:5687)****[15]**L. Reichel,*Fast QR decomposition of Vandermonde-like matrices and polynomial least squares approximation*, Numerical Analysis Report 89-5, Department of Mathematics, M.I.T., 1989.

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DOI:
https://doi.org/10.1090/S0025-5718-1991-1079030-8

Article copyright:
© Copyright 1991
American Mathematical Society