nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

A.5 Newton-Raphson Method 135
system, ˙x = 0. The method uses a simple and a fast convergent algorithm provided
that the initial guess value is close to the root (Eldén et al. (2004)). When we walk
through the derivation and the applications it will be evident that the method is based
on differential calculus and the simple idea of linear approximation.
In order to derive it we start with the Taylor series expansion and since we applied the
method on a system of differential equations the derivation will be based here on.
Now let
˙x = f(x) (A.19)
And then let f(x) be a n-dimensional vector f = (f1, f2, ..., fn) T , x = (x1, x2, . . . , xn).
Then starting with the Taylor series expansion of the function fi(x) about the point x
fi(x + ∆x) = fi(x) +
n ∂fi
∆xj + O(∆x
∂xj
2 j) (A.20)
j=1
where the “big O” notation is an abbreviated way of writing the next terms in the series,
i.e. the quadratic and higher order terms, and denotes that they are small, since ∆x is
expected to be small. So by ignoring them we have the linearized version of equation
A.20, which we write using vector notation
f(x + ∆x) ≈ f(x) + J(f(x))∆x (A.21)
here J(f(x)) is the jacobian matrix. Since MATLAB is build to process calculations
numerically the elements in the Jacobian can not be derived analytically. Instead
we rewrite equation A.20 (again without the quadratic terms) to the form of a finite
difference which then can be used as an approximation of the elements in the Jacobian
∂fi
∂xj
≈ fi(x + hˆxj) − fi(x)
h
(A.22)
where ˆxj is a unit vector pointing in the direction of xj and h is the small increment of
x. Now, the purpose remains to find the roots of the function f(x) and the way of doing
it will be through iterative steps. Lets assume that x is the initial approximation of the
root x ∗ and hence solution to f(x) = 0. Then let x + ∆x be a better approximation.
To find the increment ∆x, and thereby the improved approximation for the root, we
set f(x + ∆x) = 0 in equation A.21
0 = f(x) + J(f(x))∆x (A.23)
Since ∆x = x k+1 − x k , where k is the number of iterations, we can rewrite equation
A.23 in the following way
x k+1 = x k − J(f(x)) −1 f(x) k
(A.24)
the procedure now repeats itself by setting x k+1 equal to x k until the increment |∆x| <
ε, where ε is a user defined error tolerance on the estimated value of x ∗ compared to
the analytical value. This is the essence in the Newton-Raphson method.