Newton-Raphson Method for systems of equations

The solution of a system of (non-linear) equations can be written in vector form as with . Let us call ( ) the estimated solution. We can expand each of these functions about the estimated solution, e.g.

whereby all partial derivatives are evaluated at the estimated solution. As in the case of one equation in one variable, the Newton-Raphson Method can be obtained by truncating the expansion after the linear terms. Let us use matrix notation:

> FF:=matrix(3,1,[F[1],`...`,F[n]]);

> XX:=matrix(3,1,[x[1],`...`,x[n]]);

> XE:=matrix(3,1,[xe[1],`...`,xe[n]]);

> JJ:=matrix(3,3,[[Diff(F[1],x[1]),`...`,Diff(F[1],x[n])],[`...`,`...`,`...`],[Diff(F[n],x[1]),`...`,Diff(F[n],x[n])]]);

This matrix is better known as the jacobian matrix. In vector notation, the Taylor series expansion up to the linear terms can be written as

> FF(XX)=FF(XE)+JJ*(XX-XE);

We can then evaluate this expression in the exact solution ,

> 0=FF(XE)+JJ*(XEX-XE);

The vector can then be isolated by multiplying the equation by the inverse of the jacobian matrix on the left, to find

> XEX=XE-`JJ^(-1)`*FF(XE);

>

This leads to the iteration process

This is the formula as given in the worksheet.

A practical example is given in the worksheet.

Exercise: write the theoretical derivation of the Newton-Raphson Method explicitly for a system of three equations in three unknowns.