Liste over kvadraturformler

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Denne artikel giver en liste over forskellige kvadraturformler til numerisk integration .

Notation

Generelt er den numeriske integrationsformel skrevet som følger:

\int \limits _{\Omega _{x}}f(x)dx=\sum _{i=1}^{m_{g}}({\widetilde {w_{i}}}f( x_{i})}=\sum _{i=1}^{m_{g}}{w_{i}\mathrm {det} (J(\xi _{i}))f(x(\xi _ {jeg})))}

$f(x)$ er en integrerbar funktion;
$w_{i}$ — integrationsvægte;
$\xi$ — koordinatsystem for masterelementet;
$J(\xi )={\frac {\partial (x_{1},\dots,x_{n})}{\partial (\xi _{1},\dots \xi _{n}) }}$ er Jacobi-matricen for overgangen til masterelementet.

På grund af additiviteten af integralet , vil simple områder ( trekant , firkant , tetraeder osv.) blive betragtet som integrationsområdet , med kompleks geometri kan området repræsenteres som en forening af simple og beregne integralet over dem eller brug en spline til at repræsentere tilknytningen til masterelementet. $\Omega$

I artiklen vil variabler blive brugt til at udpege naturlige koordinater og til at udpege koordinater for masterelementet - . $x,y,z$ $\xi ,\eta ,\zeta$

Endimensionelt integral

En-dimensionel integration er altid integration over et segment.

Integrationsområde: segment ; $[x_{0},x_{1}]$
Master element: cut ; $[-1,1]$
Hop til masterelement: ; $\xi (x)=2{\frac {x-x_{0}}{x_{1}-x_{0}}}-1$
Overgang fra masterelementet: ; ${\displaystyle x(\xi )={\frac {(x_{1}-x_{0})(\xi +1)}{2))+x_{0))$
Jacobian: . $\mathrm {det} (J(\xi ))={\frac {x_{1}-x_{0}}{2}}$

Nummer	Antal point	Integrationsrækkefølge	$\xi$	$w$	Derudover
en	en	en	$0$	$2$	Rektangel metode
2	2	en	$-en$	$en$	Trapezmetode
2	2	en	$en$	$en$	Trapezmetode
3	2	3	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$	Gauss metode -2
3	2	3	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$	Gauss metode -2
fire	3	3	$-en$	${\frac {1}{3))$	Simpson metode
			$0$	${\frac {4}{3}}$
			$en$	${\frac {1}{3))$
5	3	5	${\displaystyle -{\sqrt {\frac {3}{5))))$	${\frac {5}{9}}$	Gauss-3 metode
			$0$	${\frac {8}{9}}$
			${\displaystyle {\sqrt {\frac {3}{5))))$	${\frac {5}{9}}$
6	fire	7	$-{\sqrt {{\frac {3}{7}}-{\frac {2}{7}}{\sqrt {\frac {6}{5}}}}}$	${\frac {18+{\sqrt {30}}}{36}}$	Gauss-4 metode
			${\sqrt {{\frac {3}{7}}-{\frac {2}{7}}{\sqrt {\frac {6}{5}}}}}$	${\frac {18+{\sqrt {30}}}{36}}$
			$-{\sqrt {{\frac {3}{7}}+{\frac {2}{7}}{\sqrt {\frac {6}{5}}}}}$	${\frac {18-{\sqrt {30}}}{36}}$
			${\sqrt {{\frac {3}{7}}+{\frac {2}{7}}{\sqrt {\frac {6}{5}}}}}$	${\frac {18-{\sqrt {30}}}{36}}$
7	5	9	$0$	${\frac {128}{225}}$	Gauss-5 metode
			$-{\frac {1}{3}}{\sqrt {5-2{\sqrt {\frac {10}{7}}}}}$	${\frac {322+13{\sqrt {70}}}{900}}$
			${\frac {1}{3}}{\sqrt {5-2{\sqrt {\frac {10}{7}}}}}$	${\frac {322+13{\sqrt {70}}}{900}}$
			$-{\frac {1}{3}}{\sqrt {5+2{\sqrt {\frac {10}{7}}}}}$	${\frac {322-13{\sqrt {70}}}{900}}$
			${\frac {1}{3}}{\sqrt {5+2{\sqrt {\frac {10}{7}}}}}$	${\frac {322-13{\sqrt {70}}}{900}}$

Todimensionelt integral

Firkantet masterelement

Integrationsområde: rektangel $[x_{0},x_{1}]\ gange [y_{0},y_{1}]$
Hovedelement: kvadratisk $[-1,1]\ gange [-1,1]$
Skift til masterelement:

\xi (x,y)=2{\frac {x-x_{0}}{x_{1}-x_{0}}}-1

\eta (x,y)=2{\frac {y-y_{0}}{y_{1}-y_{0}}}-1

;

Overgang fra masterelementet:

x(\xi ,\eta )={\frac {(x_{1}-x_{0})(\xi +1)}{2}}+x_{0}

{\displaystyle y(\xi ,\eta )={\frac {(y_{1}-y_{0})(\eta +1)}{2))+y_{0))

;

Jacobian: . $\mathrm {det} (J(\xi ,\eta ))={\frac {(x_{1}-x_{0})(y_{1}-y_{0})}{4))$

Disse integrationsformler kan også bruges, når integrationsområdet er en konveks firkant, men så vil overgangsformlerne til masterelementet (og omvendt) ikke have så simpel en form. Du kan få et udtryk for overgangen ved hjælp af et interpolationspolynomium .
Mange af formlerne for kvadratintegration kan opnås som en kombination af formler for et segment: alle mulige par af endimensionelle punkter tages som integrationspunkter, og de tilsvarende produkter af integrationsvægte tages som vægte. Eksempler på sådanne metoder i tabellen nedenfor er rektangelmetoden, trapezmetoden og Gauss-2-metoden.

Nummer	Antal point	Integrationsrækkefølge	$\xi$	$\eta$	$w$	Derudover
en	en	en	$0$	$0$	$fire$	Rektangelmetode (gennemsnitsmetode)
2	fire	en	$-en$	$-en$	$en$	Trapezmetode
			$-en$	$en$	$en$
			$en$	$-en$	$en$
			$en$	$en$	$en$
3	fire	3	${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$	Gauss-2 metode
			${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$
fire	12	7	$-c$	$0$	${\displaystyle w_{c))$	$a={\sqrt {\frac {114-3{\sqrt {583}}}{287}}}$ $b={\sqrt {\frac {114+3{\sqrt {583}}}{287}}}$ ${\displaystyle c={\sqrt {\frac {6}{7))))$ $w_{a}={\frac {307}{810}}+{\frac {923}{270{\sqrt {583}}}}$ $w_{b}={\frac {307}{810}}-{\frac {923}{270{\sqrt {583}}}}$ $w_{c}={\frac {98}{405}}$ Antallet af noder er minimalt [1] .
			$c$	$0$	${\displaystyle w_{c))$
			$0$	$-c$	${\displaystyle w_{c))$
			$0$	$c$	${\displaystyle w_{c))$
			$-a$	$-en$	$w_a$
			$-en$	$-en$	$w_a$
			$-a$	$-en$	$w_a$
			$-en$	$-en$	$w_a$
			$-b$	$-b$	$w_{b}$
			$b$	$-b$	$w_{b}$
			$-b$	$b$	$w_{b}$
			$b$	$b$	$w_{b}$

Trekantet masterelement

Integrationsområde: trekant dannet af hjørner ; $(x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})$
Masterelement: en trekant dannet af toppunkter . $(0,0),(1,0),(0,1)$

For at gå til masterelementet bruges barycentriske koordinater (L-koordinater), betegnet med . ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3))$

{\displaystyle \lambda _{i}(x,y)=\alpha _{i1}x+\alpha _{i2}y+\alpha _{i3))

For at beregne koefficienterne for L-koordinater bruges matrixen : $D$

D={\begin{pmatrix}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\1&1&1\end{pmatrix))

Koefficientmatrixen er omvendt til : . $D$ ${\displaystyle \mathrm {A} =D^{-1))$

Hop til masterelement:

\xi (x,y)=\lambda _{1}(x,y)

\eta (x,y)=\lambda _{2}(x,y)

Overgang fra masterelementet:

{\begin{pmatrix}x\\y\\1\end{pmatrix}}=D{\begin{pmatrix}\xi \\\eta \\1-\xi -\eta \end{pmatrix} }

Jacobian: . $\mathrm {det} (J(\xi ,\eta ))=\mathrm {det} (D)$

Nummer	Antal point	Integrationsrækkefølge	$\xi$	$\eta$	$w$	Derudover
en	en	en	${\frac {1}{3}}$	${\frac {1}{3))$	${\frac {1}{2))$	Gennemsnitlig metode
2	3	2	${\frac {1}{2))$	$0$	${\frac {1}{6}}$	-
			$0$	${\frac {1}{2))$	${\frac {1}{6}}$
			${\frac {1}{2))$	${\frac {1}{2))$	${\frac {1}{6}}$
2	3	2	${\frac {1}{6}}$	${\frac {1}{6}}$	${\frac {1}{6}}$	Gauss-3 metode
			${\frac {2}{3}}$	${\frac {1}{6}}$	${\frac {1}{6}}$
			${\frac {1}{6}}$	${\frac {2}{3))$	${\frac {1}{6}}$
fire	fire	3	${\frac {1}{3}}$	${\frac {1}{3))$	$-{\frac {9}{32}}$	Gauss-4 metode
			${\frac {3}{5}}$	${\frac {1}{5}}$	${\frac {25}{96}}$
			${\frac {1}{5}}$	${\frac {3}{5))$	${\frac {25}{96}}$
			${\frac {1}{5}}$	${\frac {1}{5}}$	${\frac {25}{96}}$
5	7	3	${\frac {1}{3}}$	${\frac {1}{3))$	${\frac {9}{40}}$	Newton - Cotes metode _
			${\frac {1}{2))$	$0$	${\frac {1}{15}}$
			${\frac {1}{2))$	${\frac {1}{2))$	${\frac {1}{15}}$
			$0$	${\frac {1}{2))$	${\frac {1}{15}}$
			$en$	$0$	${\frac {1}{40}}$
			$0$	$en$	${\frac {1}{40}}$
			$0$	$0$	${\frac {1}{40}}$

Tredimensionelt integral

Kubisk hovedelement

Integrationsområde: parallelepipedum $[x_{0},x_{1}]\ gange [y_{0},y_{1}]\ gange [z_{0},z_{1}]$
Hovedelement: Terning $[-1,1]\ gange [-1,1]\ gange [-1,1]$
Skift til masterelement:

\xi (x,y,z)=2{\frac {x-x_{0}}{x_{1}-x_{0}}}-1

\eta (x,y,z)=2{\frac {y-y_{0}}{y_{1}-y_{0}}}-1

\zeta (x,y,z)=2{\frac {z-z_{0}}{z_{1}-z_{0}}}-1

Overgang fra masterelementet:

{\displaystyle x(\xi ,\eta ,\zeta )={\frac {(x_{1}-x_{0})(\xi +1)}{2))+x_{0))

{\displaystyle y(\xi ,\eta ,\zeta )={\frac {(y_{1}-y_{0})(\eta +1)}{2))+y_{0))

;

{\displaystyle z(\xi ,\eta ,\zeta )={\frac {(z_{1}-z_{0})(\zeta +1)}{2))+z_{0))

;

Jacobian: . $\mathrm {det} (J(\xi ,\eta ,\zeta ))={\frac {(x_{1}-x_{0})(y_{1}-y_{0})(z_ {1}-z_{0})}{8}}$

Såvel som for en firkant kan en terning bruges som et masterelement for en vilkårlig sekskant [ klargør ] , men så bliver overgangen og jakobiske formler mere komplicerede.
Ligeledes, ligesom en firkant, kan mange kubeintegrationsformler opnås fra segmentintegrationsformler, koordinaterne for knudepunkterne er alle mulige tripler af koordinater af den endimensionelle formel, og integrationsvægtene er produktet af de tilsvarende vægte af en-dimensionel formel.

Nummer	Antal point	Integrationsrækkefølge	$\xi$	$\eta$	$\zeta$	$w$	Derudover
en	en	en	$0$	$0$	$0$	$otte$	Rektangelmetode (gennemsnitsmetode)
2	otte	3	${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$	Gauss-2 metode
			${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle -{\frac {1}{\sqrt {3))))$	$en$
			${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	${\displaystyle {\frac {1}{\sqrt {3))))$	$en$
3	fjorten	5	${\displaystyle -{\sqrt {\frac {19}{30))))$	$0$	$0$	${\frac {320}{361}}$	Antallet af noder i klassen af formler med en tilnærmelsesvis rækkefølge på 5 og som ikke indeholder oprindelsen er minimal. [2]
			${\displaystyle {\sqrt {\frac {19}{30))))$	$0$	$0$	${\frac {320}{361}}$
			$0$	${\displaystyle -{\sqrt {\frac {19}{30))))$	$0$	${\frac {320}{361}}$
			$0$	${\displaystyle {\sqrt {\frac {19}{30))))$	$0$	${\frac {320}{361}}$
			$0$	$0$	${\displaystyle -{\sqrt {\frac {19}{30))))$	${\frac {320}{361}}$
			$0$	$0$	${\displaystyle {\sqrt {\frac {19}{30))))$	${\frac {320}{361}}$
			${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle -{\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$
			${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\displaystyle {\sqrt {\frac {19}{33))))$	${\frac {121}{361}}$

Da integrationsformlerne af høj orden indeholder mange punkter, præsenterer vi dem separat.

Ordre: 7, antal prikker: 34

Punktnummer	$\xi$	$\eta$	$\zeta$	$w$	Derudover
en	$-en$	$0$	$0$	$w_{1}$	${\displaystyle a={\sqrt {\frac {6}{7))))$ , , , , , _ $b={\sqrt {\frac {960-33{\sqrt {238}}}{2726}}}$ $c={\sqrt {\frac {960+33{\sqrt {238}}}{2726}}}$ $w_{1}={\frac {1078}{3645}}$ $w_{2}={\frac {343}{3645}}$ $w_{3}={\frac {43}{135}}+{\frac {829{\sqrt {238}}}{136323}}$ $w_{4}={\frac {43}{135}}-{\frac {829{\sqrt {238}}}{136323}}$
2	$-en$	$0$	$0$	$w_{1}$
3	$0$	$-en$	$0$	$w_{1}$
fire	$0$	$-en$	$0$	$w_{1}$
5	$0$	$0$	$-en$	$w_{1}$
6	$0$	$0$	$-en$	$w_{1}$
7	$-en$	$-en$	$0$	${\displaystyle w_{2))$
otte	$-en$	$-en$	$0$	${\displaystyle w_{2))$
9	$-en$	$-en$	$0$	${\displaystyle w_{2))$
ti	$-en$	$-en$	$0$	${\displaystyle w_{2))$
elleve	$-en$	$0$	$-en$	${\displaystyle w_{2))$
12	$-en$	$0$	$-en$	${\displaystyle w_{2))$
13	$-en$	$0$	$-en$	${\displaystyle w_{2))$
fjorten	$-en$	$0$	$-en$	${\displaystyle w_{2))$
femten	$0$	$-en$	$-en$	${\displaystyle w_{2))$
16	$0$	$-en$	$-en$	${\displaystyle w_{2))$
17	$0$	$-en$	$-en$	${\displaystyle w_{2))$
atten	$0$	$-en$	$-en$	${\displaystyle w_{2))$
19	$b$	$b$	$b$	${\displaystyle w_{3))$
tyve	$b$	$b$	$-b$	${\displaystyle w_{3))$
21	$b$	$-b$	$b$	${\displaystyle w_{3))$
22	$b$	$-b$	$-b$	${\displaystyle w_{3))$
23	$-b$	$b$	$b$	${\displaystyle w_{3))$
24	$-b$	$b$	$-b$	${\displaystyle w_{3))$
25	$-b$	$-b$	$b$	${\displaystyle w_{3))$
26	$-b$	$-b$	$-b$	${\displaystyle w_{3))$
27	$c$	$c$	$c$	$w_{4}$
28	$c$	$c$	$-c$	$w_{4}$
29	$c$	$-c$	$c$	$w_{4}$
tredive	$c$	$-c$	$-c$	$w_{4}$
31	$-c$	$c$	$c$	$w_{4}$
32	$-c$	$c$	$-c$	$w_{4}$
33	$-c$	$-c$	$c$	$w_{4}$
34	$-c$	$-c$	$-c$	$w_{4}$

Tetraedrisk masterelement

Integrationsområde: tetraeder dannet af hjørner . $(x_{1},y_{1},z_{1}),(x_{2},y_{2},z_{2}),(x_{3},y_{3},y_{ 3}),(x_{4},y_{4},z_{4})$
Hovedelement: tetraeder dannet af hjørner . $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$

På samme måde som trekanten bruges tetraederens L-koordinater til at gå til masterelementet, betegnet med : ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4))$

{\displaystyle \lambda _{i}(x,y,z)=\alpha _{i1}x+\alpha _{i2}y+\alpha _{i3}z+\alpha _{i4))

Koefficientmatricen er defineret som: , hvor ${\displaystyle \mathrm {A} =D^{-1))$

D={\begin{pmatrix}x_{1}&x_{2}&x_{3}&x_{4}\\y_{1}&y_{2}&y_{3}&y_{4}\\z_{1 }&z_{2}&z_{3}&z_{4}\\1&1&1&1\end{pmatrix}}

Skift til masterelement:

\xi (x,y,z)=\lambda _{1}(x,y,z)

\eta (x,y,z)=\lambda _{2}(x,y,z)

\zeta (x,y,z)=\lambda _{3}(x,y,z)

Overgang fra masterelementet:

{\begin{pmatrix}x\\y\\z\\1\end{pmatrix}}=D{\begin{pmatrix}\xi \\\eta \\\zeta \\1-\xi - \eta -\zeta \end{pmatrix}}

Jacobian: . $\mathrm {det} (J(\xi ,\eta ,\zeta ))=\mathrm {det} (D)$

Nummer	Antal point	Integrationsrækkefølge	$\xi$	$\eta$	$\zeta$	$w$	Derudover
en	en	en	${\frac {1}{4))$	${\frac {1}{4))$	${\frac {1}{4))$	${\frac {1}{6}}$	Gennemsnitlig metode
2	fire	2	${\frac {5+3{\sqrt {5}}}{20}}$	${\frac {5-{\sqrt {5}}}{20}}$	${\frac {5-{\sqrt {5}}}{20}}$	${\frac {1}{24}}$	Gauss-4 metode
			${\frac {5-{\sqrt {5}}}{20}}$	${\frac {5+3{\sqrt {5}}}{20}}$	${\frac {5-{\sqrt {5}}}{20}}$	${\frac {1}{24}}$
			${\frac {5-{\sqrt {5}}}{20}}$	${\frac {5-{\sqrt {5}}}{20}}$	${\frac {5+3{\sqrt {5}}}{20}}$	${\frac {1}{24}}$
			${\frac {5-{\sqrt {5}}}{20}}$	${\frac {5-{\sqrt {5}}}{20}}$	${\frac {5-{\sqrt {5}}}{20}}$	${\frac {1}{24}}$
3	5	3	${\frac {1}{4))$	${\frac {1}{4))$	${\frac {1}{4))$	$-{\frac {2}{15}}$
			${\frac {1}{2))$	${\frac {1}{6}}$	${\frac {1}{6}}$	${\frac {3}{40}}$
			${\frac {1}{6}}$	${\frac {1}{2))$	${\frac {1}{6}}$	${\frac {3}{40}}$
			${\frac {1}{6}}$	${\frac {1}{6}}$	${\frac {1}{2))$	${\frac {3}{40}}$
			${\frac {1}{6}}$	${\frac {1}{6}}$	${\frac {1}{6}}$	${\frac {3}{40}}$
fire	elleve	fire	${\frac {1}{4))$	${\frac {1}{4))$	${\frac {1}{4))$	$-{\frac {74}{5625}}$	Gauss-11 metode
			${\frac {11}{14}}$	${\frac {5}{70}}$	${\frac {5}{70}}$	${\frac {343}{45000}}$
			${\frac {5}{70}}$	${\frac {11}{14}}$	${\frac {5}{70}}$	${\frac {343}{45000}}$
			${\frac {5}{70}}$	${\frac {5}{70}}$	${\frac {11}{14}}$	${\frac {343}{45000}}$
			${\frac {5}{70}}$	${\frac {5}{70}}$	${\frac {5}{70}}$	${\frac {343}{45000}}$
			${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {56}{2250}}$
			${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {56}{2250}}$
			${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {56}{2250}}$
			${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {56}{2250}}$
			${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {56}{2250}}$
			${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {1+{\sqrt {5/14}}}{4}}$	${\frac {1-{\sqrt {5/14}}}{4}}$	${\frac {56}{2250}}$
5	fjorten	5	$a$	$a$	$a$	$A$	$A,\ B,\ C,\ a,\ b,\ c$ bestemmes ud fra følgende ligninger: ${\begin{cases}t={\frac {4}{27}}\left(71-4{\sqrt {79}}\sin \left({\frac {1}{3}}\ venstre[{\frac {\pi }{2}}+\arctan \left({\frac {27{\sqrt {10815}}}{67}}\right)\right]\right)\right)\\ a={\frac {1+\alpha }{4)),\ b={\frac {1+\beta }{4)),\ c={\frac {1+2\gamma }{4)) \\\gamma ={\sqrt {1/t)),\ C=t^{2}/7!,\ \alpha >0\\A\alpha ^{2}+B\beta ^{2}= (1-t/21)/5!\\A\alpha ^{3}+B\beta ^{3}=-2/6!\\A\alpha ^{4}+B\beta ^{4} =10/7!\\A\alpha ^{5}+B\beta ^{5}=-6/7!\\\end{cases}}$ $A\ca. 0,01878132095300264180$ $B\approx 0,01224884051939365826$ $C\approx 0,00709100346284691107$ $a\ca. 0,31088591926330060980$ $b\approx 0,09273525031089122640$ $c\approx 0,45449629587435035051$
			$1-3a$	$a$	$a$	$A$
			$a$	$1-3a$	$a$	$A$
			$a$	$a$	$1-3a$	$A$
			$b$	$b$	$b$	$B$
			$1-3b$	$b$	$b$	$B$
			$b$	$1-3b$	$b$	$B$
			$b$	$b$	$1-3b$	$B$
			${\frac {1}{2}}-c$	$c$	$c$	$C$
			$c$	${\frac {1}{2}}-c$	$c$	$C$
			$c$	$c$	${\frac {1}{2}}-c$	$C$
			${\frac {1}{2}}-c$	${\frac {1}{2}}-c$	$c$	$C$
			${\frac {1}{2}}-c$	$c$	${\frac {1}{2}}-c$	$C$
			$c$	${\frac {1}{2}}-c$	${\frac {1}{2}}-c$	$C$