CMOLER EXERCISESsource: Experiments with MATLAB - CMoler -
book
1. INTRO and ITERATIONS
notes: exp(1)=e=2.718281828459046, log(exp(1))=1, log10(10)=1,
Loga(x)=logb(x)/loga(b)lim x(x^n/n^x) lim(ln(x^n)/ln(n^x)) 1/ln(x)
0. pi is not exactly , pi=355/113=3.14159265358979
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1. INTRO and ITERATIONS
format longq=(1+sqrt(5))/2 = 1.6180339887498951.1 43^2 = 1849
(-3)^4 = 81 sin(1) = 0.841470984807897 4^(3^2) = 262144 (-3)^(0.25)
= 0.930604859102100 + 0.930604859102099i sin(pi/180) =
0.017452406437284 (4^3)^2 = 4096 sin(pi/3) = 0.866025403784439
(32)^0.25 = 2.378414230005442 (-2)^(-4/3) = -0.198425131496025 +
0.343682409249651i arcsin(1)/pi = ??? Undefined function or method
'arcsin' for input arguments of type 'double'.1.2.a F =
62C=5/9*(F-32)C = 16.6667
1.2.bC = 100F=C*9/5+32F = 212
1.3 pg7 from barn-megaparsec to teaspoon units conversion: 1
megaparsec = 0.61742486 teaspoons.
1.4 phi=(1+sqrt(5))/2 =
1.618033988749895x=-1:.025:4;plot(x,x,'-',x,sqrt(1+x),'-',phi,phi,'o')
Crossing on x=pi . Improving graph presentation:
set(axes_handle,XGrid,on); set(axes_handle,YGrid,on); Grid, box,
axes, set
1.5 example FOR:format longfor k=1:31 x=sqrt(1+x) end
1.6.- example WHILE:format longx=3while x ~= sqrt(1+x)x =
sqrt(1+x)end
things that can be improved: sqrt(1+x) is evaluated twice each
loop, and comparing R to Z is ok as long as working floating point
as it is the case, otherwise equality may never be reached
regardless of how many loops iterated.
example improved WHILE:x = 3y = Inf;while abs(x-y) > eps(x)y
= x;x = sqrt(1+x)end1.7.- PLOT
examplephi=(1+sqrt(5))/2x=-1:.025:4;plot(x,x,'-',x,sqrt(1+x),'-',phi,phi.'o')
1.8 sq(x)=x+1 solution, phi, approximated with Taylor seriesn=10
% n=10 p/q error < 1.47e-04, n=100 error abs(xn+1-xn)/abs(x1-x0)
(f'(x0))^n
x=sqrt(1+x) x0=1.618033988770158f(x)=sqrt(1+x))
f'(x)=-0.5*(1+x)^(-0.5)(f'(x0))^10 =
7.940057378387408e-6(f'(x0))^32 = 4.780085345013851e-17
MSWindows calculator (ver 6.1 build 7600): x=sqrt(1+x)
x0=1.618033988770158, abs(x0-sqrt(1+x0)) =
1.4001493531043385682988887303586e-11
1.36%LOTKA Lotka-Volterra predator-prey model.function yp =
lotka(t,y)global ALPHA BETAyp = [y(1) - ALPHA*y(1)*y(2); -y(2) +
BETA*y(1)*y(2)];% ALPHA = BETA = 0.1 race mode
%file lotka2.mglobal ALPHA BETAALPHA = 0.01BETA = 0.03[t,y] =
ode23(@lotka,[0,10],[1; 1]);plot(t,y);
% ALPHA = 1, BETA = 0.03 persecution mode
% ALPHA = BETA = 0.4 , combat modeglobal ALPHA BETAALPHA =
0.4BETA = 0.4[t,y] = ode23(@lotka,[0,10],[1; 1]);plot(t,y);
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