Deven Singh, Assignment 5, 09/20/2021
#OK To Post Homework

#Question 0
eq := {6*a(n - 1) + a(n + 3) + 5*a(n + 1) = 0};

subs(n = n - 3, eq);

# a(n) = 0*a(n-1)-5*a(n-2)+0*a(n-3)-6*a(n-4)

RecToSeq := proc(INI, REC, N) local i, k, L, newguy; if not (type(INI, list) and type(REC, list) and nops(INI) = nops(REC) and type(N, integer) and nops(INI) <= N) then print(`bad innput`); RETURN(FAIL); end if; k := nops(INI); L := INI; while nops(L) < N do newguy := add(REC[i]*L[-i], i = 1 .. k); L := [op(L), newguy]; end do; L; end proc;

A := RecToSeq([1, 2, 4, 11], [0, -5, 0, -6], 1000);

A[1000]:

evalf(%);

1.82*10^239

#Question 1

GrowthC := proc(INI, REC, K) local L, a, b; L := RecToSeq(INI, REC, K); a := L[-1]/L[-2]; b := L[-2]/L[-3]; if abs(a - b) < 1/10^(Digits + 3) then RETURN(evalf(a)); else print(` make `, K, `bigger `); RETURN(FAIL); end if; end proc;

GrowthC([0, 1, 1, 2, 4, 8, 16, 32, 64, 128], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 1000);
1.999


GrowthCe := proc(REC) local x, i; evalf([solve(1 - add(REC[i]/x^i, i = 1 .. nops(REC)))])[1]; end proc;

GrowthCe([1, 1, 1, 1, 1, 1, 1, 1, 1, 1]);

1.999


#Question 2
#Part i

LeslieMod := proc(SUR, FER) local i, L, A; if not (type(SUR, list) and type(FER, list) and nops(SUR) + 1 = nops(FER)) then print(`bad input`); RETURN(FAIL); end if; A := nops(SUR); L[0] := 1; for i to A do L[i] := L[i - 1]*SUR[i]; end do; [seq(FER[i + 1]*L[i], i = 0 .. A)]; end proc;

LeslieMod([0.99, 0.99], [0, 1/2, 1/4]):

GrowthCe(%);

.88

#Part ii

LeslieMat := proc(SUR, FER) local i, A; if not (type(SUR, list) and type(FER, list) and nops(SUR) + 1 = nops(FER)) then print(`bad input`); RETURN(FAIL); end if; A := nops(SUR); matrix([FER, seq([0 $ (i - 1), SUR[i], 0 $ (A + 1 - i)], i = 1 .. A)]); end proc;

with(LinearAlgebra):

LeslieMat([0.99, 0.99], [0, 1/2, 1/4]):

E := Matrix([[0, 1/2, 1/4], [0.99, 0, 0], [0, 0.99, 0]]);

evalf(Eigenvalues(E));
#The largest eigenvalue is .88


#Question 3

GrowthCe([0, 0, 0, 0.16, 0.41]);

.89

#Question 4
PlantGseq := proc(alpha, beta, gamma, sigma, INI, K) local a, b; if `not`(type(INI, list) and nops(INI) = 2 and type(K, integer) and nops(INI) <= K) then print(`bad input`); RETURN(FAIL); end if; a := alpha*sigma*gamma; b := beta*sigma^2*(1 - alpha)*gamma; RecToSeq(INI, [a, b], K); end proc;

PlantGseq(0.5, 0.25, 2.0, 0.8, [100, 80], 20);
PlantGseq(0.6, 0.3, 2.0, 0.8, [100, 96], 20);
#Both outputs match pg.18 Table 1.1

#Question 5
PlantGseq2 := proc(alpha, beta, gamma, sigma) local a, b; if `not`(type(alpha, float) and type(beta, float) and type(gamma, float) and type(sigma, float)) then print(`bad input`); RETURN(FAIL); end if; a := alpha*sigma*gamma; b := beta*sigma^2*(1 - alpha)*gamma; GrowthCe([a, b]); end proc;
PlantGseq2(0.5, 0.25, 2.0, 0.8);
.97
PlantGseq2(0.6, 0.3, 2.0, 0.8);
1.1