Opsætning (kan overspringes) 

>	with(LinearAlgebra):

 

>	with(plots):with(plottools):

 

>	viz:=(v,c)->arrow([0,0,0],v,0.05,0.1,0.1, color=c);

 

(1.1)

 

>

 

Fibonaccital 

>	A:=Matrix([[ 1 , 1 ],        [ 1 , 0 ]]);

 

(2.1)

 

>	seq(A^n,n=1..15);

 

(2.2)

 

>	seq(<1|0>.A^n.<1,0>,n=1..15);

 

(2.3)

 

>	seq(evalf((5+sqrt(5))/10*((1+sqrt(5))/2)^n,7),n=1..15);

 

(2.4)

 

>	(L,T):=Eigenvectors(A);

 

(2.5)

 

>	S:=T^(-1);

 

(2.6)

 

>	DD:=DiagonalMatrix(L);

 

(2.7)

 

>	S^(-1).DD.S;

 

(2.8)

 

>	simplify(%);

 

(2.9)

 

>	seq(simplify(<1|0>.S^(-1).DD^n.S.<1,0>),n=1..15);

 

(2.10)

 

>	simplify(<1|0>.S^(-1).DD^n.S.<1,0>);

 

(2.11)

 

>	simplify(<1|0>.S^(-1).DiagonalMatrix([L[1]^n,L[2]^n]).S.<1,0>);

 

(2.12)

 

>

 

 

	(2.13)

 

Befolkningsflytning 

Lad os kigge pÃ¥ afbildningen hørende til 

>	B:=<<85/100,15/100>|<10/100, 90/100>>;

 

>

 

(3.1)

 

>

 

,  =Andel byboere efter n Ã¥r .  

Flytningens forløb: 

>	<l[n+1], b[n+1]> = B.<l[n], b[n]>;

 

(3.2)

 

>	seq(evalf(B^n.<1,0>),n=0..20);

 

(3.3)

 

>	seq(evalf(B^n.<0,1>),n=0..20);

 

(3.4)

 

>	display([seq(display([viz(DiagonalMatrix([B^n,1]).<1,0,0>,red),viz(DiagonalMatrix([B^n,1]).<0,1,0>,green), viz(DiagonalMatrix([B^n,1]).<1/2,1/2,0>,yellow) ]),n=0..20)],insequence=true,scaling=constrained,axes=normal);

 

>

 

>	(L,T):=Eigenvectors(B);

 

(3.5)

 

>	s:=T^(-1);

 

(3.6)

 

>	simplify(s^(-1).DiagonalMatrix(L).s);

 

(3.7)

 

>	simplify(s^(-1).DiagonalMatrix(L)^n.s.<1,0>);

 

(3.8)

 

>	simplify(T.DiagonalMatrix([L[1]^n,L[2]^n]).T^(-1).<1,0>);

 

(3.9)

 

>	limit(%[1],n=infinity);

 

(3.10)

 

>

 

>

 

>

 

>

 

>