1. 1.

   Most of the linear approximation formulas involved in the numerical integration processes of differential equations can be put into the form

   Report issue for preceding element

For specific formulas one can consult various books on numerical analysis and, in particular, the well-known book by L. Collatz [1].

Formula (1), also called a numerical derivation formula, allows us to approximately calculate the value of the derivative of a certain order (of order$m+r$) of a function, on a given point (the point$x_{0}$), by a given linear combination of the values, in finite number, of the function and of some of its successive derivatives, on given points. The$a_{j},a_{i,j}$are given constants, the distinct points$x_{0},x_{1},x_{2},\ldots,x_{s}$of the real axis belong to the bounded and closed interval$[a,b]$on which the function$f$is defined. Finally the$s,r_{1},r_{2},\ldots,r_{s}$are natural numbers and$r,m$non-negative integers also given.

The difference$R=R(f)$, between the first and second members of formula (1), is the remainder of this numerical derivation formula. In a previous work [3] we made a fairly detailed discussion of this remainder and later we completed these results [6].
2. Note that the remainder$R(f)$is a linear functional defined on a certain set$S$of functions$f$defined on the interval$[a,b]$. The study of the rest, which obviously depends on the function$f$, therefore of the structure of the whole$S$, leads to various delimitations of the error committed by the approximation (1) of$f^{(m+r)}\left(x_{0}\right)$.

Let us assume that, in general,$R(f)$be a linear functional defined on a linear set$S$of real and continuous functions, defined on an interval$I$of the real axis. The theory of higher-order convex functions or the theory of various generalizations of these functions can be used to specify the structure of the functional$R(f)$when it has a degree of accuracy or something analogous.

Suppose, in particular, that$S$contains all polynomials. Then the degree of exactness, if it exists, is an integer$n\geqq-1$well determined by the property that
$R(f)$is zero on any polynomial of degree$n$and is different from zero on at least one polynomial of degree$n+1^{*}$). We then have the following theorem:

When$R(f)$is nonzero on any function$f\in S$, convex of order$n$, We have

Or$K\neq 0$is independent of the function$f$and the knots$\xi_{1},\xi_{2},\ldots,\xi_{n+2}$generally depend on the function$f$, are distinct and even within the interval$I$if$n\geqq 0$.

In this case we say that the linear functional$R(f)$is of the simple form. Moreover, for simplicity, in a certain sense, the condition that$R(f)$be different from zero for$f\in S$convex of a certain order, is at the same time necessary and sufficient. Various criteria allow us to decide whether a linear functional$R(f)$is of the simple form or not. Of the simple form are the remainders in many linear approximation formulas of analysis. For example, in the Taylor formula, in the more general Lagrange-Hermite interpolation formula, in many classical mechanical quadrature formulas and in most linear formulas used in the numerical integration of differential equations. As for the factor$K$, it is equal to$R\left(x^{n+1}\right)$.

I introduced (first under another name) the notion of linear functional of the simple form in another work [2].

The function$f$is said to be convex of order$n$on$I$if all its differences divided$\left[x_{1},x_{2},\ldots,x_{n+2};f\right]$, of order$n+1$on knots$x_{1},x_{2},\ldots,x_{n+2}\in I$distinct, are positive.

For the definitions and properties of divided differences on distinct or non-distinct nodes, of higher-order convex functions, for the notion of simplicity of a linear functional and for various other properties used in this work, one can consult my previous works. For example, my dissertation of "Mathematica" [4].

Finally, let us note that, in the case of the simplicity of$R(f)$, formula (2) must be considered in close connection with various theorems and formulas of the mean of divided differences. The right-hand side of formula (2) can also be represented in various forms. For example, if$n\geqq 0$and if the function$f$has a derivative$(n+1)$th on the inside of$I$, We have

$\xi$being a point of the interior of$I$.
3. The linear functional$R(f)$may not be of the simple form, but if it

has a determined degree of accuracy,$n$, under fairly general assumptions (see [4]), we have a formula of the form

Or$A,B$are independent of the function$f$And$\xi_{v},v=1,2,\ldots,n+2,\xi_{v}^{\prime},v=1,2,\ldots$,$n+2$are two groups of$n+2$distinct points of$I$, generally depending on the function$f$. We have$A+B=R\left(x^{n+1}\right)\neq 0$and the simplicity comes down to the fact that we can choose one of the coefficients$A,B$equal to zero. If the function$f$has a derivative$(n+1)$th on the inside of$I(n\geqq 0)$, from (4) we deduce the formula

$\xi,\xi^{\prime}$being two points of the interior of$I$.
Note that this time instead of a single group of points$\xi_{\nu}$respectively of a single point$\xi$we encounter two groups of points$\xi_{v},\xi_{v}^{\prime}$, respectively two points$\xi$,$\xi^{\prime}$, generally without connection between them, of the interval$I$of function definition$f$.

The preceding considerations apply to a linear functional of the form

where the nodes$z_{i}$are distinct and the coefficients$c_{i,j}$are independent of the function$f$. When$R(f)$is not zero identically, so if the$c_{i,j}$are not all zero, the linear functional has a degree of accuracy$n$well determined.

In particular, the remainder of the approximation formula (1) is of the form (6).
Note also that by passing to a primitive of the function$f$, the study of the remainder of a quadrature formula allowing the approximate calculation of an integral of the form$\int_{a}^{b}f(x)\mathrm{d}x$, returns to the study of a functional of the form (6) (see [4]).
4. We have sought to remedy the defect highlighted by the italicized lines of the previous no.

When$R(f)$is not of the simple form we can seek to reestablish simplicity by defining a degree of simplicity not in relation to the successive powers of$x$, but with respect to a suitably defined Tschebycheff system on$I$. We have applied such a method in a previous work [5]. Assuming that$f$has a sufficient number of derivatives, we can then, in certain cases, obtain$R(f)$as a linear combination of the values ​​of some of the derivatives of$f$on a single point$\xi$(generally depending on the function$f$). This linear combination can have coefficients also depending on$\xi$, as shown by the example of our cited work [5].
5. In the particular case of the linear functional (6) we can obtain such a formula also in the following way. We can find a function$g$, defined on$I$, depending only on the coefficients$a_{i,j}$and knots$z_{i}$(hands not of the function$f$and functional$R$), so that we have

The total number of nodes of the divided difference of the second member is$q+1==k_{1}+k_{2}+\ldots+k_{p}$. We can always find for$g$a polynomial, even of degree$q$. It is determined as a Lagrange-Hermite interpolation polynomial on the$q+1$nodes considered.

So if we assume that$q>0$and that$f$either$q$-times derivable (as well as$g$) on the inside of$I$, we get

$\xi$being an interior point of$I$depending, in general, on the function$f$This result is obtained by taking into account the classical Cauchy formula

$\xi$being inside the smallest interval containing the nodes$x_{1},x_{2},\ldots,x_{q+1}$.
6. The function$g$not being determined in a unique way, there may be several representations of the previous form.

Example. Consider the linear functional

Or$0<a<b$And$f$is continuous on$[a,b]$and derivable on the open interval$]a,b[$.
This linear functional is of the form

by choosing, or$g=\frac{1}{x}$, or else$g=\frac{a+bx}{ab}$.
Formula (8) gives us

We therefore obtain, either

either

In both formulas the number$\xi$is obviously not the same. To justify this statement it is enough to take$f=x^{2}$.