View source for Differentiation, numerical

Finding the derivative of a function by numerical methods. Such differentiation is resorted to when the methods of [[Differential calculus|differential calculus]] are inapplicable (the function is obtained from tables) or involves considerable difficulties (the analytic expression of the function is complicated).

Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323201.png" /> be defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323202.png" /> and let the nodal points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323204.png" />, be given. The totality of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323206.png" />, is known as a table. The result of numerical differentiation of the table is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323207.png" /> which approximates, in some sense, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323208.png" />-th derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d0323209.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232010.png" /> on some sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232011.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232012.png" />. The use of numerical differentiation will be expedient if only an insignificant amount of computational effort is required to obtain the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232013.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232014.png" />. Linear methods of numerical differentiation are commonly employed; the result thus obtained is written in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232016.png" /> are functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232017.png" />. The most popular method for obtaining formulas (1) is as follows: One constructs the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232018.png" /></td> </tr></table>

interpolating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232019.png" />, and assumes that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232020.png" /></td> </tr></table>

The accuracy of the algorithms based on the interpolation formulas of Lagrange, Newton and others strongly depends on the selection of the manner of interpolation, and may sometimes be very low even if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232021.png" /> is sufficiently smooth and if the number of nodal points is large [[#References|[1]]]. Algorithms of numerical differentiation involving spline-interpolation [[#References|[2]]] are often free from this disadvantage. If all that is needed is the computation of the approximate values of the derivative at the nodal points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232022.png" /> only, formula (1) assumes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232024.png" /> is fully defined by specifying a coefficient matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232025.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232026.png" />. Formulas such as (2) are known as difference formulas for numerical differentiation. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232027.png" /> of such formulas are determined from the condition that the difference

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232028.png" /></td> </tr></table>

has highest order of smallness with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232029.png" />. As a rule, formulas (2) are very simple and easy to handle. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232030.png" />, they assume the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232031.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232032.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232033.png" /></td> </tr></table>

Numerical differentiation algorithms are often employed with tables in which the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032320/d03232034.png" /> are given (or obtained) inaccurately. In such cases there is need for a preliminary smoothing, since a direct application of the formula may result in large errors in the results [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.H. Ahlberg,   E.N. Nilson,   J.F. Walsh,   "Theory of splines and their applications" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Morozov,   "The differentiation problem and certain algorithms of approximation by experimental information" , ''Computing methods and programming'' , '''14''' , Moscow  (1970)  pp. 46–62  (In Russian)</TD></TR></table>



====Comments====
Explicit differentiation formulas are given in, e.g., [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Segun,   M. Abramowitz,   "Handbook of mathematical functions" , ''Appl. Math. Ser.'' , '''55''' , Nat. Bur. Standards  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.B. Hildebrand,   "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR></table>