# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
General explicit difference formulas for numerical differentiation. (English) Zbl 1077.65021
The author uses numerical differentiation formulas in deriving numerical methods for solving ordinary and partial differential equations. The automatic differentiation and the regularization method on numerical differentiation are familiar techniques to obtain the derivatives of a function whose values are only obtained empirically at a set of points. First, explicit difference formulas of arbitrary order for approximating first and higher derivatives for unequally or equally spaced data, are proposed. Then, an $n$th generalized Vandermonde determinant is introduced and its basic properties are discussed. A linear algebraic system of Vandermonde type is solved analytically in terms of generalized Vandermonde determinants. A comparison with other implicit finite difference formulas based on interpolating polynomials shows that the new explicit formulas are very easy to implement for numerical approximation of arbitrary order to first and higher derivatives, and they need less computation time and storage and can be directly used for designing difference schemes for ordinary and partial differential equations. Numerical results suggest that the new explicit difference formulae are very efficient .

##### MSC:
 65D25 Numerical differentiation 15A06 Linear equations (linear algebra) 65L12 Finite difference methods for ODE (numerical methods) 15A15 Determinants, permanents, other special matrix functions
Full Text:
##### References:
 [1] Anderssen, R. S.; Bloomfield, P.: Numerical differentiation procedures for non-exact data. Numer. math. 22, 157-182 (1973/74) [2] Burden, R. L.; Faires, J. D.: Numerical analysis. (2001) · Zbl 0671.65001 [3] Chapra, S. C.; Canale, R. P.: Numerical methods for engineers. (1998) [4] Collatz, L.: The numerical treatment of differential equations. (1966) · Zbl 0173.17702 [5] Corliss, G.; Faure, C.; Griewank, A.; Hascoet, L.; Naumann, U.: Automatic differentiation of algorithms --- from simulation to optimization. (2002) · Zbl 0983.68001 [6] Cullum, J.: Numerical differentiation and regularization. SIAM J. Numer. anal. 8, No. 2, 254-265 (1971) · Zbl 0224.65005 [7] Dahlquist, G.; Bjorck, A.: Numerical methods. (1974) [8] Dokken, T.; Lyche, T.: A divided difference formula for the error in Hermite interpolation. Bit 19, 539-540 (1979) · Zbl 0428.41002 [9] Forst, W.: Interpolation und numerische differentiation. J. approx. Theory 39, No. 2, 118-131 (1983) · Zbl 0525.41008 [10] Gerald, C. F.; Wheatley, P. O.: Applied numerical analysis. (1989) · Zbl 0684.65002 [11] Grabar, L. P.: Numerical differentiation by means of Chebyshev polynomials orthonormalized on a system of equidistant points. Zh. vychisl, mat. I mat. Fiz. 7, No. 6, 1375-1379 (1967) · Zbl 0171.37301 [12] Griewank, A.: Evaluating derivatives, principles and techniques of algorithmic differentiation, number 19 in frontiers in applied mathematics. (2000) · Zbl 0958.65028 [13] Griewank, A.; Corliss, F.: Automatic differentiation of algorithms --- theory, implementation and application. (1991) · Zbl 0747.00030 [14] Hamming, R. W.: Numerical methods for scientists and engineers. (1962) · Zbl 0952.65500 [15] Hanke, M.; Scherzer, O.: Inverse problems light --- numerical differentiation. Amer. math. Monthly 6, 512-522 (2001) · Zbl 1002.65029 [16] Heath, M. T.: Science computing --- an introductory survey. (1997) [17] Khan, I. R.; Ohba, R.: Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series. J. comput. Appl. math. 107, 179-193 (1999) · Zbl 0939.65031 [18] Khan, I. R.; Ohba, R.: Digital differentiators based on Taylor series. IEICE trans. Fund. 82-A, No. 12, 2822-2824 (1999) [19] Khan, I. R.; Ohba, R.: New finite difference formulas for numerical differentiation. J. comput. Appl. math. 126, 269-276 (2001) · Zbl 0971.65014 [20] Khan, I. R.; Ohba, R.: Mathematical proof of explicit formulas for TAP-coefficients of Taylor series based FIR digital differentiators. IEICE trans. Fund. 84-A, No. 6, 1581-1584 (2001) [21] Khan, I. R.; Ohba, R.: Taylor series based finite difference approximations of higher-degree derivatives. J. comput. Appl. math. 154, 115-124 (2003) · Zbl 1018.65032 [22] Khan, I. R.; Ohba, R.; Hozumi, N.: Mathematical proof of closed form expressions for finite difference approximations based on Taylor series. J. comput. Appl. math. 150, 303-309 (2003) · Zbl 1017.65015 [23] King, T.; Murio, D.: Numerical differentiation by finite dimensional regularization. IMA J. Numer. anal. 6, 65-85 (1986) · Zbl 0584.65008 [24] Knowles, I.; Wallace, R.: A variational method for numerical differentiation. Numer. math. 70, 91-110 (1995) · Zbl 0818.65013 [25] Kreyzig, E.: Advanced engineering mathematics. (1994) [26] Krishnamurti, T. N.; Bounoua, L.: An introduction to numerical weather prediction technique techniques. (1995) [27] Kvasov, B. I.: Numerical differentiation and integration on the basis of interpolation parabolic splines. Chisl. metody mekh. Sploshn. sredy 14, No. 2, 68-80 (1983) [28] Mathews, J. H.; Fink, K. D.: Numerical methods using Matlab. (1999) [29] Murio, D. A.: The mollification method and the numerical solution of ill-posed problems. (1993) [30] Rail, L. B.: Automatic differentiation --- techniques and applications. (1981) [31] Ramm, A. G.: On numerical differentiation. Mathem. izvestija vuzov 11, 131-135 (1968) · Zbl 0187.10504 [32] Ramm, A. G.: Stable solutions of some ill-posed problems. Math. meth. Appl. sci. 3, 336-363 (1981) · Zbl 0469.65034 [33] Silvester, P.: Numerical formation of finite-difference operators. IEEE trans. Microwave theory technol. 18, No. 10, 740-743 (1970) [34] Tikhonov, A. N.; Arsenin, V. Y.: Solutions of ill-posed problems. (1977) · Zbl 0354.65028 [35] Vasin, V. V.: Regularization of the problem of numerical differentiation. Matem. zap. Ural’skii univ. 7, No. 2, 29-33 (1969) [36] Vershinin, V. V.; Pavlov, N. N.: Approximation of derivatives by smoothing splines. Vychisl. sistemy 98, 83-91 (1983) · Zbl 0566.41028 [37] Wahba, G.: Spline models for observational data. (1990) · Zbl 0813.62001