Newton–Raphson - Failure of the method to converge - example

The method does not converge , it means the approximation could not found

for certain equations like for ex 

y = 2 * x +2 * z +1 - b

here b is the value we need to find 

2 * x +2 * z +1 =  b

if we try the iteration method for this to find the value of x where the loss is zero

the cycle tends to repeat the same value for x alternatively .

finding dy/dx = 2 ,dy/dz =2 

x1 = x 0 - y(x0, z0)/ f '(x)

z1 = z 0 - y(x0, z0)/ f '(z) 

assume values for x0 = 0.5 , z0 = 0.4 , b = 24

y(0) = 2 *(0.5) +2 *(0.4) +1 - 24

 y0=-21.2

x1 = 0.5 - (-21.2 /2)

x1 = 11.1

z1 = 0.4 -  (-21.2 /2)

 z1 = 11

to check for convergence i wil substitute the value of x1 and z1 in the equation 

so that if it converges ,means  y =0 ,because iam expecting a value of 24 from the 

equation, lets see

y1 = 2(0.5) +2(11) +1 -24

y1= 21.2 

the loss here is too large not even close to zero.

ok we wil go for the next iteration and see

x2 = x1 - y1/f '(x1)

z2 = z1 - y1/f '(z1)

x2 = 11.1 - (21.2/2)

x2= 0.5

z2 = 11 - (21.2 /2)

z2 = 0.4

 

so whats this it looks like we have moved back to our initial guessed values for x ,z  

so the newton's method rotates and will not converge for this equation.

the reason why this is not used for machine learning convergence.