4.1.sagews

var('x')
var('y')

f1(x,y)=y+(1.5)*(cos(x-1))-1
f2(x,y)=(0.9)*(y^2)+(0.4)*(x^2)-1

df1_x(x,y)=diff(f1,x)
df1_x
df1_y(x,y)=diff(f1,y)

df1_y
df2_x(x,y)=diff(f2,x)
df2_x
df2_y(x,y)=diff(f2,y)
df2_y

def solve_s(s1,s2):
    a=s1[0]+s2[0]*(-(s1[1])/(s2[1]))
    b=s1[2]+s2[2]*(-(s1[1])/(s2[1]))
    g=b/a
    h=(s2[2]-s2[0]*g)/(s2[1])
    return [g,h]
eps=10^(-6)
k=0
x=1.7
y=-0.5
k=0
while (k<10):
    k+=1
    k
    a=df1_x(x,y)
    b=df1_y(x,y)
    e=-f1(x,y)
    c=df2_x(x,y)
    d=df2_y(x,y)
    f=f2(x,y)
    s1=[a,b,e]
    s2=[c,d,f]
    g,h=solve_s(s1,s2)[0],solve_s(s1,s2)[1]
    
    g
    h
    x1=x+g
    x1
    y1=y+h
    y1
    
    x=x1
    y=y1
x
y


plot(1-(1.5)*cos(x-1), (x,-2,2))

x y (x, y) |--> -1.50000000000000*sin(x - 1) (x, y) |--> 1.00000000000000 (x, y) |--> 0.800000000000000*x (x, y) |--> 1.80000000000000*y 1 1.42454482108129 1.72931217407840 3.12454482108129 1.22931217407840 2 0.568716728592658 1.28509758345359 3.69326154967395 2.51440975753199 3 1.84528113729822 1.03713724980056 5.53854268697217 3.55154700733255 4 -7.43450698983142 8.69157814421458 -1.89596430285925 12.2431251515471 5 -36.7400589573732 3.61274416748116 -38.6360232602325 15.8558693190283 6 -17.3672724898274 10.0055053981483 -56.0032957500599 25.8613747171766 7 -40.7542086066547 0.635466945943738 -96.7575043567145 26.4968416631203 8 -106.948734743279 -81.8293316252138 -203.706239099993 -55.3324899620934 9 -106.809883292866 -19.5458505287730 -310.516122392859 -74.8783404908664 10 -156.544119148952 -35.0614363494227 -467.060241541811 -109.939776840289 -467.060241541811 -109.939776840289