MR a1 Loesung Fossy

...und wie geht's weiter?

zurück zur Aufgabenstellung

Gegeben sind 7 Werte - die ersten 7 Werte. Gesucht ist eine Regel für die weiteren Werte. Nichts liegt näher, als das über ein Polynom zu lösen. $p(i)$ wobei $i$ die gewünschte Zeile ist. Wir kennen $p(0)$ ... $p(6)$.

$p(0)=1$
$p(1)=11$
$p(2)=21$
$p(3)=1211$
$p(4)=111221$
$p(5)=312211$
$p(6)=13112221$

Also gesucht ist ein Polynom, das genau das oben stehende erfüllt - sonst nix. Die Suche ist einfach, wenn man andere Polynome addiert. Ich nenne sie $q_{j}(i)$ - dieses Polynom (ich brauche 7 verschiedene solche) hat an der Stelle $j$ den Wert 1 - an den anderen (ganzzahligen) Stellen hat es den Wert 0:

$q_{j}(x)={\prod _{k\in \{0..6\}-j}{(x-k)} \over \prod _{k\in \{0..6\}-j}{(j-k)}}$

Das gesuchte $p(x)$ ist dann blos die Summe der geiegneten q's mal dem gewünschten Wert an der jeweiligen Stelle:

$p(x)=1\cdot q_{0}(x)+11\cdot q_{1}(x)+21\cdot q_{2}(x)+...+13112221\cdot q_{6}(x)$

$q_{0}(x)={{(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)} \over {(0-1)(0-2)(0-3)(0-4)(0-5)(0-6)}}={{x^{6}-21x^{5}+175x^{4}-735x^{3}+1624x^{2}-1764x+720} \over 720}$

$q_{1}(x)={{(x-0)(x-2)(x-3)(x-4)(x-5)(x-6)} \over {(1-0)(1-2)(1-3)(1-4)(1-5)(1-6)}}={{x^{6}-20x^{5}+155x^{4}-580x^{3}+1044x^{2}-720x} \over -120}$

$q_{2}(x)={{(x-0)(x-1)(x-3)(x-4)(x-5)(x-6)} \over {(2-0)(2-1)(2-3)(2-4)(2-5)(2-6)}}={{x^{6}-19x^{5}+137x^{4}-461x^{3}+702x^{2}-360x} \over 48}$

$q_{3}(x)={{(x-0)(x-1)(x-2)(x-4)(x-5)(x-6)} \over {(3-0)(3-1)(3-2)(3-4)(3-5)(3-6)}}={{x^{6}-18x^{5}+121x^{4}-372x^{3}+508x^{2}-240x} \over -36}$

$q_{4}(x)={{(x-0)(x-1)(x-2)(x-3)(x-5)(x-6)} \over {(4-0)(4-1)(4-2)(4-3)(4-5)(4-6)}}={{x^{6}-17x^{5}+107x^{4}-307x^{3}+396x^{2}-180x} \over 48}$

$q_{5}(x)={{(x-0)(x-1)(x-2)(x-3)(x-4)(x-6)} \over {(5-0)(5-1)(5-2)(5-3)(5-4)(5-6)}}={{x^{6}-16x^{5}+95x^{4}-260x^{3}+324x^{2}-144x} \over -120}$

$q_{6}(x)={{(x-0)(x-1)(x-2)(x-3)(x-4)(x-5)} \over {(6-0)(6-1)(6-2)(6-3)(6-4)(6-5)}}={{x^{6}-15x^{5}+85x^{4}-225x^{3}+274x^{2}-120x} \over 720}$

So jetzt weden nur mehr die Polynome $q_{j}(x)$ addiert. Das KGV der Nenner ist 720 (das führt dann zu den Faktoren 1,-6,15,-20,15,-6 und 1). Ich addiere die Faktoren vor den Potenzen von x. Das Ergebnis ist folgendes Polynom - die Koefizienten sind in der Tabelle dahinter...

$p(x)={{a_{6}x^{6}+a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}} \over 720}$

$a_{6}=1\cdot 1+11\cdot (-6)+21\cdot 15+1211\cdot (-20)+111221\cdot 15+312211\cdot (-6)+13112221\cdot 1=12.883.300$
$a_{5}=1\cdot (-21)+11\cdot (-6)\cdot (-20)+21\cdot 15\cdot (-19)+1211\cdot (-20)\cdot (-18)+111221\cdot 15\cdot (-17)+312211\cdot (-6)\cdot (-16)+13112221\cdot (-15)=-194.641.140$
$a_{4}=1\cdot 175+11\cdot (-6)\cdot 155+21\cdot 15\cdot 137+1211\cdot (-20)\cdot 121+111221\cdot 15\cdot 107+312211\cdot (-6)\cdot 95+13112221\cdot 85=1.112.190.700$
$a_{3}=1\cdot (-735)+11\cdot (-6)\cdot (-580)+21\cdot 15\cdot (-461)+1211\cdot (-20)\cdot (-372)+111221\cdot 15\cdot (-307)+312211\cdot (-6)\cdot (-260)+13112221\cdot (-225)=-2.966.471.100$
$a_{2}=1\cdot 1624+11\cdot (-6)\cdot 1044+21\cdot 15\cdot 702+1211\cdot (-20)\cdot 508+111221\cdot 15\cdot 396+312211\cdot (-6)\cdot 324+13112221\cdot 274=3.634.313.200$
$a_{1}=1\cdot (-1764)+11\cdot (-6)\cdot (-720)+21\cdot 15\cdot (-360)+1211\cdot (-20)\cdot (-240)+111221\cdot 15\cdot (-180)+312211\cdot (-6)\cdot (-144)+13112221\cdot (-120)=-1.598.267.760$
$a_{0}=720$

$a_{6}=$	$1\cdot 1$	$+11\cdot (-6)$	$+21\cdot 15$	$+1211\cdot (-20)$	$+111221\cdot 15$	$+312211\cdot (-6)$	$+13112221\cdot 1$	=12.883.300
$a_{5}=$	$1\cdot (-21)$	$+11\cdot (-6)\cdot (-20)$	$+21\cdot 15\cdot (-19)$	$+1211\cdot (-20)\cdot (-18)$	$+111221\cdot 15\cdot (-17)$	$+312211\cdot (-6)\cdot (-16)$	$+13112221\cdot (-15)$	=-194.641.140
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{4}=}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1\cdot 175}	$+11\cdot (-6)\cdot 155$	$+21\cdot 15\cdot 137$	$+1211\cdot (-20)\cdot 121$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 111221 \cdot 15 \cdot 107}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 312211 \cdot (-6) \cdot 95}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 13112221 \cdot 85}	=1.112.190.700
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_3=}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \cdot (-735)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 11 \cdot (-6) \cdot (-580)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 21 \cdot 15 \cdot (-461)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 1211 \cdot (-20) \cdot (-372)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 111221 \cdot 15 \cdot (-307)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 312211 \cdot (-6) \cdot (-260)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 13112221 \cdot (-225)}	=-2.966.471.100
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2=}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \cdot 1624}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 11 \cdot (-6) \cdot 1044}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 21 \cdot 15 \cdot 702}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 1211 \cdot (-20) \cdot 508}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 111221 \cdot 15 \cdot 396}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 312211 \cdot (-6) \cdot 324}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 13112221 \cdot 274}	=3.634.313.200
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1=}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \cdot (-1764)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 11 \cdot (-6) \cdot (-720)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 21 \cdot 15 \cdot (-360)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 1211 \cdot (-20) \cdot (-240)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 111221 \cdot 15 \cdot (-180)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 312211 \cdot (-6) \cdot (-144)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 13112221 \cdot (-120)}	=-1.598.267.760
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0=}								=720

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)={{12.883.300 x^6 -194.641.140 x^5 + 1.112.190.700 x^4 -2.966.471.100 x^3 + 3.634.313.200 x^2 -1.598.267.760 x + 720}\over 720}}