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The singular value expansion of the Volterra integral equation associated to a numerical differentiation problem. (English) Zbl 1382.65063

Summary: We consider the Volterra integral equation of the first kind for the derivative of a given function with one-side boundary conditions. We give a method to obtain the singular value expansion for the corresponding integral kernel. This singular value expansion can be used to give algorithms for the solution of the numerical differentiation problem. A numerical experiment shows the results obtained by a simple version of such algorithms.

MSC:

65D25 Numerical differentiation
45D05 Volterra integral equations

Software:

nag; NAG
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Full Text: DOI

References:

[1] Abate, J.; Dubner, H., A new method for generating power series expansion of functions, SIAM J. Numer. Anal., 5, 102-112 (1968) · Zbl 0155.22101
[2] Brown, D. A.; Zingg, D. W., Efficient numerical differentiation of implicitly-defined curves for sparse systems, J. Comput. Appl. Math., 304, 138-159 (2016) · Zbl 1337.65020
[3] Bruno, O.; Hoch, D., Numerical differentiation of approximated functions with limited order-of-accuracy deterioration, SIAM J. Numer. Anal., 50, 1581-1603 (2012) · Zbl 1251.65022
[4] Conn, A. R.; Scheinberg, K.; Vicente, L. N., Introduction to Derivative-Free Optimization, MPS-SIAM Series on Optimization (2009), SIAM: SIAM Philadelphia · Zbl 1163.49001
[5] Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093
[6] Fernández, D. C.D. R.; Hicken, J. E.; Zingg, D. W., Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. & Fluids, 95, 171-196 (2014) · Zbl 1390.65064
[7] Fornberg, B., Numerical differentiation of analytic functions, ACM Trans. Math. Software, 7, 512-526 (1981) · Zbl 0465.65012
[8] Kress, R., Linear Integral Equations (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0671.45001
[9] Lu, S.; Pereverzev, S. V., Numerical differentiation from a viewpoint of regularization theory, Math. Comp., 75, 1853-1870 (2006) · Zbl 1115.65021
[10] Lyness, J. N.; Moler, C. B., Van der Monde systems and numerical differentiation, Numer. Math., 8, 458-464 (1966) · Zbl 0141.33404
[11] Lyness, J. N.; Moler, C. B., Numerical differentiation of analytic functions, SIAM J. Numer. Anal., 4, 2, 202-210 (1967) · Zbl 0155.48003
[12] Lyness, J. N.; Moler, C. B., Generalised Romberg methods for integrals of derivatives, Numer. Math., 14, 1-14 (1969) · Zbl 0175.16102
[13] Martins, J. R.R. A.; Sturdza, P.; Alonso, J. J., The complex-step derivative approximation, ACM Trans. Math. Software, 29, 3, 245-262 (2003) · Zbl 1072.65027
[14] NAG Fortran Library Manual (1997), NAG, Mark 18
[15] Nikolovski, F.; Stojkovska, I., Complex-step derivative approximation in noisy environment, J. Comput. Appl. Math., 327, 64-78 (2018) · Zbl 1372.65065
[16] Noschese, S.; Reichel, L., A modified truncated singular value decomposition method for discrete ill-posed problems, Numer. Linear Algebra Appl., 21, 813-822 (2014) · Zbl 1340.65070
[17] Shu, C., Differential Quadrature and its Application in Engineering (2000), Springer: Springer London · Zbl 0944.65107
[18] Squire, W.; Trap, G., Using complex variables to estimate derivatives of real function, SIAM Rev., 40, 1, 110-112 (1998) · Zbl 0913.65014
[19] Vladimirov, V. S., Equations of Mathematical Physics (1971), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0231.35002
[20] Wu, B.; Zhang, Q., Fast multiscale regularization methods for high-order numerical differentiation, IMA J. Numer. Anal., 36, 1432-1451 (2016) · Zbl 1433.65360
[21] Zhang, H.; Zhang, Q., Sparse discretization matrices for Volterra integral operators with applications to numerical differentiation, J. Integral Equations Appl., 23, 1, 137-156 (2011) · Zbl 1246.65255
[22] Zong, Z.; Zhang, Y., Advanced Differential Quadrature Methods (2009), CRC press: CRC press London · Zbl 1178.65125
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