Test funktioner til optimering

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I anvendt matematik er testfunktioner kendt som kunstige landskaber nyttige til at evaluere ydeevnen af optimeringsalgoritmer, såsom:

konvergenshastighed.
Nøjagtighed.
robusthed .
Overordnet ydelse.

Denne artikel introducerer nogle testfunktioner for at give dig en idé om de forskellige situationer, du skal stå over for, når du overvinder sådanne problemer.

Artiklen præsenterer den generelle formel for ligningen, stedet for den objektive funktion, grænserne for variablerne og koordinaterne for det globale minimum.

Test funktioner for et enkelt optimeringsmål

Navn	Formel	Globalt minimum	Søgemetode
Rastrigin funktion	$f(\mathbf {x} )=En+\sum _{i=1}^{n}\venstre[x_{i}^{2}-A\cos(2\pi x_{i})\ ret]$ ${\text{hvor: }}A=10$	$f(0,\dots ,0)=0$	$-5.12\leq x_{i}\leq 5.12$
Ackley funktion	$f(x,y)=-20\exp \left[-0.2{\sqrt {0.5\left(x^{2}+y^{2}\right)))\right]$ $-\exp \left[0.5\left(\cos(2\pi x)+\cos(2\pi y)\right)\right]+e+20$	$f(0,0)=0$	$-5\leq x,y\leq 5$
Kuglefunktion	$f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}$	$f(x_{1},\dots ,x_{n})=f(0,\dots ,0)=0$	$-\infty \leq x_{i}\leq \infty$ , $1\leq i\leq n$
Rosenbrock funktion	$f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2} \right)^{2}+\left(x_{i}-1\right)^{2}\right]$	${\text{Min))={\begin{cases}n=2&\rightarrow \quad f(1,1)=0,\\n=3&\rightarrow \quad f(1,1,1) =0,\\n>3&\højrepil \quad f(\underbrace {1,\dots ,1} _{n{\text{ gange}}})=0\\\end{cases}}$	$-\infty \leq x_{i}\leq \infty$ , $1\leq i\leq n$
Beals funktion	$f(x,y)=\left(1,5-x+xy\right)^{2}+\left(2,25-x+xy^{2}\right)^{2}$ $+\left(2.625-x+xy^{3}\right)^{2}$	$f(3,0.5)=0$	$-4.5\leq x,y\leq 4.5$
Goldstein-Price funktion	$f(x,y)=\left[1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{ 2}\right)\right]$ $\left[30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right]$	$f(0,-1)=3$	$-2\leq x,y\leq 2$
Stand funktion	${\displaystyle f(x,y)=\left(x+2y-7\right)^{2}+\left(2x+y-5\right)^{2))$	$f(1,3)=0$	$-10\leq x,y\leq 10$
Bukin funktion N 6	$f(x,y)=100{\sqrt {\left\|y-0.01x^{2}\right\|}}+0.01\left\|x+10\right\|.\quad$	$f(-10,1)=0$	$-15\leq x\leq -5$ , $-3\leq y\leq 3$
Matthias funktion	$f(x,y)=0,26\left(x^{2}+y^{2}\right)-0,48xy$	$f(0,0)=0$	$-10\leq x,y\leq 10$
Afgiftsfunktion N 13	$f(x,y)=\sin ^{2}3\pi x+\left(x-1\right)^{2}\left(1+\sin ^{2}3\pi y\right )$ $+\left(y-1\right)^{2}\left(1+\sin ^{2}2\pi y\right)$	$f(1,1)=0$	$-10\leq x,y\leq 10$
Himmelblau funktion	$f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad$	${\text{Min))={\begin{cases}f\left(3.0,2.0\right)&=0.0\\f\left(-2.805118,3.131312\right)&=0.0\\f\ left(-3.779310,-3.283186\right)&=0.0\\f\left(3.584428,-1.848126\right)&=0.0\\\end{cases}}$	$-5\leq x,y\leq 5$
Funktion af den trepuklede kamel	$f(x,y)=2x^{2}-1,05x^{4}+{\frac {x^{6}}{6}}+xy+y^{2}$	$f(0,0)=0$	$-5\leq x,y\leq 5$
Isom funktion	$f(x,y)=-\cos \left(x\right)\cos \left(y\right)\exp \left(-\left(\left(x-\pi \right)^{ 2}+\venstre(y-\pi \right)^{2}\right)\right)$	$f(\pi ,\pi )=-1$	$-100\leq x,y\leq 100$
"Cross on tray" funktion (Cross-in-bakke funktion)	$f(x,y)=-0,0001\left[\left\|\sin x\sin y\exp \left(\left\|100-{\frac {\sqrt {x^{2}+y^{ 2}}}{\pi }}\right\|\right)\right\|+1\right]^{0.1}$	${\text{Min))={\begin{cases}f\left(1.34941,-1.34941\right)&=-2.06261\\f\left(1.34941,1.34941\right)&=-2.06261\\ f\left(-1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,-1.34941\right)&=-2.06261\\\end{sager}}$	$-10\leq x,y\leq 10$
Æggestandsfunktion (æggeholder funktion)	$f(x,y)=-\left(y+47\right)\sin {\sqrt {\left\|{\frac {x}{2))+\left(y+47\right)\ højre\|}}-x\sin {\sqrt {\left\|x-\left(y+47\right)\right\|}}$	$f(512,404.2319)=-959.6407$	$-512\leq x,y\leq 512$
Bordholder funktion	$f(x,y)=-\venstre\|\sin x\cos y\exp \left(\left\|1-{\frac {\sqrt {x^{2}+y^{2))} {\pi ))\right\|\right)\right\|$	${\text{Min))={\begin{cases}f\left(8.05502,9.66459\right)&=-19.2085\\f\left(-8.05502,9.66459\right)&=-19.2085\\ f\left(8.05502,-9.66459\right)&=-19.2085\\f\left(-8.05502,-9.66459\right)&=-19.2085\end{cases}}$	$-10\leq x,y\leq 10$
McCormick funktion	$f(x,y)=\sin \left(x+y\right)+\left(xy\right)^{2}-1,5x+2,5y+1$	$f(-0.54719,-1.54719)=-1.9133$	$-1.5\leq x\leq 4$ , $-3\leq y\leq 4$
Shaffer funktion N2	$f(x,y)=0,5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0,5}{\left[1+0,001\ venstre(x^{2}+y^{2}\right)\right]^{2}}}$	$f(0,0)=0$	$-100\leq x,y\leq 100$
Shaffer funktion N4	${\displaystyle f(x,y)=0,5+{\frac {\cos ^{2}\left[\sin \left(\left\|x^{2}-y^{2}\right\|\right) \right]-0,5}{\left[1+0,001\left(x^{2}+y^{2}\right)\right]^{2))))$	$f(0.1.25313)=0.292579$	$-100\leq x,y\leq 100$
Stybinsky-Tang funktion	$f({\boldsymbol {x)))={\frac {\sum _{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{ i}}{2}}$	$-39.16617n<f(\underbrace {-2.903534,\ldots ,-2.903534} _{n{\text{ gange}}})<-39.16616n$	$-5\leq x_{i}\leq 5$ .. _ $1\leq i\leq n$

Test funktioner til betinget optimering

Navn	Formel	Globalt minimum	Søgemetode
rosenbrock-funktion, begrænset til kubisk og direkte [1]	${\displaystyle f(x,y)=(1-x)^{2}+100(yx^{2})^{2))$ , udsat for: $(x-1)^{3}-y+1<0{\text{ og }}x+y-2<0$	$f(1.0,1.0)=0$	$-1.5\leq x\leq 1.5$ , $-0.5\leq y\leq 2.5$
Rosenbrocks funktion begrænset af en disk [2]	${\displaystyle f(x,y)=(1-x)^{2}+100(yx^{2})^{2))$ , udsat for: $x^{2}+y^{2}<2$	$f(1.0,1.0)=0$	$-1.5\leq x\leq 1.5$ , $-1.5\leq y\leq 1.5$
Bounded Mishra-Bird funktion [3] [4]	$f(x,y)=\sin(y)e^{\left[(1-\cos x)^{2}\right]}+\cos(x)e^{\left[(1 -\sin y)^{2}\right]}+(xy)^{2}$ , udsat for: $(x+5)^{2}+(y+5)^{2}<25$	$f(-3.1302468,-1.5821422)=-106.7645367$	$-10\leq x\leq 0$ , $-6.5\leq y\leq 0$
Ændret Townsend-funktion [5]	$f(x,y)=-[\cos((x-0.1)y)]^{2}-x\sin(3x+y)$ , udsat for: hvor: t = Atan2(x,y) $x^{2}+y^{2}<\left[2\cos t-{\frac {1}{2}}\cos 2t-{\frac {1}{4}}\cos 3t -{\frac {1}{8}}\cos 4t\right]^{2}+[2\sin t]^{2}$	$f(2.0052938,1.1944509)=-2.0239884$	$-2.25\leq x\leq 2.5$ , $-2,5\leq y\leq 1,75$
Simonescu funktion [6]	$f(x,y)=0.1xy$ , udsat for: $x^{2}+y^{2}\leq \left[r_{T}+r_{S}\cos \left(n\arctan {\frac {x}{y}}\right)\ højre]^{2}$ ${\text{hvor: }}r_{T}=1,r_{S}=0.2{\text{ og }}n=8$	$f(\pm 0,85586214,\mp 0,85586214)=-0,072625$	$-1.25\leq x,y\leq 1.25$

Test funktioner til multiobjektiv optimering

Titel/billede	Formel	Minimum	Søgeområde
Bean og Korn funktion	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=4x^{2}+4y^{2}\\f_{2}\ left(x,y\right)&=\left(x-5\right)^{2}+\left(y-5\right)^{2}\\\end{cases}}$	${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&=\left(x-5\right)^{2}+y^{2 }\leq 25\\g_{2}\left(x,y\right)&=\left(x-8\right)^{2}+\left(y+3\right)^{2}\geq 7.7\\\end{cases}}$	$0\leq x\leq 5$ , $0\leq y\leq 3$
Chakong og Haimes funktion	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=2+\left(x-2\right)^{2}+\left (y-1\right)^{2}\\f_{2}\left(x,y\right)&=9x-\left(y-1\right)^{2}\\\end{cases} }$	${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&=x^{2}+y^{2}\leq 225\\g_{ 2}\left(x,y\right)&=x-3y+10\leq 0\\\end{cases}}$	$-20\leq x,y\leq 20$
Fonseca og Fleming funktion	${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=1-\exp \left(-\sum _{i= 1}^{n}\left(x_{i}-{\frac {1}{\sqrt {n))}\right)^{2}\right)\\f_{2}\left({\boldsymbol {x}}\right)&=1-\exp \left(-\sum _{i=1}^{n}\left(x_{i}+{\frac {1}{\sqrt {n}} }\right)^{2}\right)\\\end{cases}}$		$-4\leq x_{i}\leq 4$ , $1\leq i\leq n$
testfunktion 4	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=x^{2}-y\\f_{2}\left(x, y\right)&=-0.5xy-1\\\end{cases}}$	${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&=6.5-{\frac {x}{6}}-y\geq 0\ \g_{2}\left(x,y\right)&=7.5-0.5xy\geq 0\\g_{3}\left(x,y\right)&=30-5x-y\geq 0\\ \end{sager}}$	$-7\leq x,y\leq 4$
Kursiv funktion	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{2}\left [-10\exp \left(-0.2{\sqrt {x_{i}^{2}+x_{i+1}^{2}}}\right)\right]\\&\\f_{2} \left({\boldsymbol {x}}\right)&=\sum _{i=1}^{3}\left[\left\|x_{i}\right\|^{0.8}+5\sin \left (x_{i}^{3}\right)\right]\\\end{cases}}$		$-5\leq x_{i}\leq 5$ , . $1\leq i\leq 3$
Schaffer funktion N. 1	${\text{Minimer}}={\begin{cases}f_{1}\left(x\right)&=x^{2}\\f_{2}\left(x\right)&= \left(x-2\right)^{2}\\\end{cases}}$		$-A\leq x\leq A$ . Formværdier , der er blevet brugt med succes. Højere værdier øger problemets sværhedsgrad. $EN$ $ti$ $10^{5}$ $EN$
Schaffer funktion N.2	${\text{Minimer}}={\begin{cases}f_{1}\left(x\right)&={\begin{cases}-x,&{\text{if }}x\leq 1\\x-2,&{\tekst{if }}1<x\leq 3\\4-x,&{\text{if }}3<x\leq 4\\x-4,&{\ tekst{if }}x>4\\\end{cases}}\\f_{2}\left(x\right)&=\left(x-5\right)^{2}\\\end{cases }}$		$-5\leq x\leq 10$ .
Poloni2 objektiv funktion	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=\left[1+\left(A_{1}-B_{1}\ venstre(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\f_ {2}\left(x,y\right)&=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}$ ${\text{hvor}}={\begin{cases}A_{1}&=0.5\sin \left(1\right)-2\cos \left(1\right)+\sin \left( 2\right)-1.5\cos \left(2\right)\\A_{2}&=1.5\sin \left(1\right)-\cos \left(1\right)+2\sin \left( 2\right)-0.5\cos \left(2\right)\\B_{1}\left(x,y\right)&=0.5\sin \left(x\right)-2\cos \left(x \right)+\sin \left(y\right)-1.5\cos \left(y\right)\\B_{2}\left(x,y\right)&=1.5\sin \left(x\right) )-\cos \left(x\right)+2\sin \left(y\right)-0,5\cos \left(y\right)\end{cases}}$		$-\pi \leq x,y\leq \pi$
Zister-Dieb-Teri funktion N. 1	${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({ \boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\venstre ({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i= 2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right )&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right))}\\ \end{sager}}$		$0\leq x_{i}\leq 1$ , . $1\leq i\leq 30$
Zister-Dieb-Teri funktion N. 2	${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({ \boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\venstre ({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i= 2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right )&=1-\venstre({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)))\right) ^{2}\\\end{cases}}$		$0\leq x_{i}\leq 1$ , . $1\leq i\leq 30$
Zister-Dieb-Terin funktion N. 3	${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({ \boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\venstre ({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i= 2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right )&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)))}-\ venstre({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)))\right)\sin \left(10 \pi f_{1}\left({\boldsymbol {x}}\right)\right)\end{cases}}$		$0\leq x_{i}\leq 1$ , . $1\leq i\leq 30$
Zister-Dieb-TeriN funktion. fire	${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({ \boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\venstre ({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=91+\sum _{i=2}^{10}\left(x_ {i}^{2}-10\cos \left(4\pi x_{i}\right)\right)\\h\left(f_{1}\left({\boldsymbol {x}}\right) ,g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\ venstre({\boldsymbol {x}}\right)}}}\end{cases}}$		$0\leq x_{1}\leq 1$ ... _ $-5\leq x_{i}\leq 5$ $2\leq i\leq 10$
Zister-Dieb-Teri funktion N. 6	${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=1-\exp \left(-4x_{1}\right )\sin ^{6}\left(6\pi x_{1}\right)\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x }}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left( {\boldsymbol {x}}\right)&=1+9\venstre[{\frac {\sum _{i=2}^{10}x_{i}}{9}}\right]^{0.25} \\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-\venstre({\ frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{sager }}$		$0\leq x_{i}\leq 1$ , . $1\leq i\leq 10$
Winnet funktion	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=0.5\left(x^{2}+y^{2}\right) +\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)&={\frac {\left(3x-2y+4\ højre)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y \right)&={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\ højre)\right)\\\end{sager}}$		$-3\leq x,y\leq 3$ .
Funktion af Osyzki og Kundu	$F_{1}(x)=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}$ $-\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right) ^{2}$ ${\text{Minimer}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=F_{1}(x)\\f_{2}\ venstre({\boldsymbol {x}}\right)&=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}$	${\text{st}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)&=x_{1}+x_{2}-2\geq 0 \\g_{2}\left({\boldsymbol {x}}\right)&=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x} }\right)&=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)&=2-x_{1}+3x_{ 2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)&=4-\left(x_{3}-3\right)^{2}-x_{4}\ geq 0\\g_{6}\left({\boldsymbol {x}}\right)&=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\ slut{cases}}$	$0\leq x_{1},x_{2},x_{6}\leq 10$ , , . $1\leq x_{3},x_{5}\leq 5$ $0\leq x_{4}\leq 6$
CTP1-funktion (2 variable)	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&= \left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}$	${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)} {0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases} }$	$0\leq x,y\leq 1$ .
Constr-Ex problem	${\text{Minimer}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&= {\frac {1+y}{x}}\\\end{cases}}$	${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&=y+9x\geq 6\\g_{1}\left(x,y \right)&=-y+9x\geq 1\\\end{cases}}$	$0.1\leq x\leq 1$ , $0\leq y\leq 5$

Se også

Litteratur

Panteleev A.V., Metlitskaya D.V., E.A. Aleshina metoder til global optimering. Metaheuristiske strategier og algoritmer // M.: Vuzovskaya kniga. 2013. 244 s. ISBN 978-5-9502-0743-3
Sergienko AB Test funktioner til global optimering.

Noter

↑ Simionescu, PA (29. september – 2. oktober 2002). Nye koncepter i grafisk visualisering af målfunktioner (PDF) . ASME 2002 Internationale Design Engineering Tekniske Konferencer og Computere og Information i Engineering Conference. Montreal, Canada. pp. 891-897. Arkiveret (PDF) fra originalen 2017-01-08 . Hentet 7. januar 2017 . Forældet parameter brugt |deadlink=( hjælp )
↑ Løs et begrænset ikke-lineært problem - MATLAB & Simulink . www.mathworks.com . Hentet 29. august 2017. Arkiveret fra originalen 29. august 2017. (ubestemt)
↑ Fugleproblem (begrænset) | Phoenix Integration (utilgængeligt link) . wayback.archive.org . Hentet 29. august 2017. Arkiveret fra originalen 29. december 2016. (ubestemt)
↑ Mishra, Sudhanshu. Nogle nye testfunktioner til global optimering og ydeevne af frastødende partikelsværmmetode (engelsk) // MPRA Paper : journal. - 2006. Arkiveret 4. november 2018.
↑ Townsend, Alex Begrænset optimering i Chebfun . chebfun.org (januar 2014). Hentet 29. august 2017. Arkiveret fra originalen 29. august 2017. (ubestemt)
↑ Simionescu , PA Computerstøttede grafer og simuleringsværktøjer til AutoCAD-brugere . — 1. — Boca Raton, FL: CRC Press , 2014. — ISBN 978-1-4822-5290-3 .