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Newtonin summat – Wikipedia

Newtonin summat

Wikipedia

Newtonin summia voidaan käyttää polynomiyhtälön juurten potenssien summien laskemiseen.

Olkoon P(x) n-asteen polynomi

$\displaystyle P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Olkoon yhtälön $P (x) = 0$ juuret $x_1,x_2,\ldots,x_n$ . Määritellään seuraavat summat:

$\displaystyle S_1 = x_1 + x_2 + \cdots + x_n$

$\displaystyle S_2 = x_1^2 + x_2^2 + \cdots + x_n^2$

$\vdots$

$\displaystyle S_k = x_1^k + x_2^k + \cdots + x_n^k$

$\vdots$

Newtonin summien mukaan,

$\displaystyle a_nS_1 + a_{n-1} = 0$

$\displaystyle a_nS_2 + a_{n-1}S_1 + 2a_{n-2}=0$

$\displaystyle a_nS_3 + a_{n-1}S_2 + a_{n-2}S_1 + 3a_{n-3}=0$

$\vdots$

[muokkaa] Aiheesta muualla

AOPS-wikin artikkeli

Luokka: Alkeisalgebra

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