Ex21-01-21, R - rextester

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# Gráfico em 3 dimensões

# Curva
# Superfície z=f(x,y)

x <- seq(-2.5,2.5, by=0.05)
y <- seq(-2,2,by=0.05)
mf<-function(s,r){z=(r*s+s^(2)+cos(r*s)-2)^2+(s^2+r^2+s*r-4)^2# f(x,y); auxiliar para gráfico
z
}

require(grDevices) # for trans3d
z <- outer(x, y, mf)
z[is.na(z)] <- 4
op <- par(bg = "white")

persp(x, y, z, theta = 60, phi = 20, expand = 0.5)->res

# Método de Euler oara resolver a equação e'(t)=grad f(e(t)), e(0)=(0.5,1)

gradmf<-function(u){ # gradiente de f(x,y)
h=10^(-5)
s=u[1]
r=u[2]
dx=mf(s+h,r)-mf(s-h,r)
dy=mf(s,r+h)-mf(s,r-h)
c(dx,dy)/(2*h)
}

t0=0  # tempo inicial
tf=5  # t final 
e0=c(10,10) # condição inicial 
n=500000
h=(tf-t0)/n  # Tamanho do passo

tt=seq(t0,tf,by=h)

Y=matrix(0,2,length(tt))
Y[,1]=e0

for ( i in 1:(length(tt)-1)){
Y[,i+1]=Y[,i]-h*gradmf(Y[,i])
}

print("Aproximação para o ponto de máximo"); Y[,length(tt)]
print("Aproximação para o valor máximo"); mf(Y[1,length(tt)],Y[2,length(tt)])

lines(trans3d(Y[1,],Y[2,],mf(Y[1,],Y[2,]), res), col = "blue", lwd = 2) # Inclui curvas

plot(tt,mf(Y[1,],Y[2,]), col = "blue",'l')

contour(x, y, z,levels = c(0.5,1,3,5,10,20,30,50))
points(Y[1,],Y[2,], col = "blue", lwd = 2) # Inclui curvas

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Absolute running time: 8.18 sec, cpu time: 8.9 sec, memory peak: 120 Mb, absolute service time: 8,4 sec