5. Root Finding with Multiple Variables#

When we want to find a solution for the inverse kinematic we need to find a solution of a set of functions with multiple variables. We can use the Newton method to find the root of a function. In Exercise 2, we learned how to find the root of a function with one variable:

From previous Exercise: Newton Method with one variable#

To implement the Newton method, we used the first Taylor approximation (tangent) at the point \(x_n\):

\[f(x) = f(x_n) + f'(x_n)(x-x_n)\]

We set \(f(x_{n+1}) = 0\) and solved for \(x\):

\[0 = f(x_n) + f'(x_n)(x_{n+1}-x_n)\]
\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

The calculation rule for the next x which is closer to the root is thus iteratively called over and over again. The calculation rule according to the Newton method is therefore:

\[x_0 = start value\]
\[x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)}\]
\[x_{2} = x_{1} - \frac{f(x_1)}{f'(x_{1})}\]
\[\vdots\]
\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

We do this until we have reached the desired accuracy, i.e., the desired distance of \(f(x_{n+1})\) to 0.

Summary: The Newton method is an iterative method for finding the root of a function. The calculation rule for the next x which is closer to the root is thus iteratively called over and over again. When look at the calculation rule of the Newton method:

\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

We see that we need the function \(f(x)\) and its derivative \(f'(x)\).

Content of this Exercise#

  1. Defining the Root Finding Prblem with Multiple Variables

  2. Newton Method with multiple variables