Formulas of Newton-Dimensions

In numerical Analysis, the formulas of Newton-Dimensions, name of Isaac Newton and Roger Dimensions, is a whole of formulas for the numerical Calcul of an integral.

The function F is known at equidistant points $x_i$ , for I = 0,…, N . The formulas of degree N are as follows defined:

$\ int_a^b F (X) \, dx \ approx \ sum_ {i=0} ^n w_i \, F (x_i)$

where $x_i=ih+x_0$ , the $w_i$ are called the coefficients of squaring. As you can see it in the writing which follows, these weights derive from a Lagrangian base of polynomials. They depend only on the X _I and absolutely not of the function F . L ( X ) is the Lagrangian Interpolation for the points (( X ₀, F ( X ₀)). , ( X _n, F ( X _n)).

$\ int_a^b F (X) \, dx \ approx \ int_a^b L (X) \, dx = \ int_a^b \ sum_ {i=0} ^n F (x_i) \, l_i (X) \, dx$

= \ sum_ {i=0} ^n \ int_a^b F (x_i) l_i (X) dx= \ sum_ {i=0} ^n F (x_i) \ underbrace {\ int_a^b l_i (X) dx} _ {w_i}

A formula of Newton-Dimensions can be established with any degree. However, the not-stability and the not-convergence of the formulas of Newton-Dimensions constrained to use only degrees 1 or 2.

Demonstration

The polynomial of interpolation $L (X)$ of F is (See Lagrangian Interpolation):

L (X) =v_n (X) \ sum_ {i=0} ^n \ frac {y_i} {(x-x_i) v'_n (x_i)}

where

v_n (X) = \ prod^ {N} _ {j=0} (x-x_j)

. From where

w_i= \ int_a^b \ frac {v_n (X)}{(x-x_i) v'_n (x_i)}dx

One carries out the change of variable

y= \ frac {x-a} {H}

w_i= \ frac {Ba} {N} \ frac {(- 1) ^ {nor}} {I! (nor)!} \ int_0^n \ prod_ {k=0, K \ I} ^n (there - K) Dy

Application for N 1

$\begin{matrix}
w_0 &=& H \ frac {(- 1) ^ {1-0}} {0! (1-0)!} \ int_0^1 \ prod_ {k=0, K \ 0} ^1 (there - K) Dy \ \
&=& - H \ int_0^1 (y-1) Dy \ \
&=& - H \ left \ frac {(y-1) ^2} {2} \ right^1_0 \ \
&=& \ frac {H} {2}
\end{matrix}$
Idem for $w_1 = \ frac {H} {2}$

External bonds

Formulas of Newton-Dimensions on Math-Linux.com

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