9.7 Techniques for numerical  9.6 Solution of simultaneous  9.6.1 Gauss-Seidel iteration

9.6.2 The Newton-Raphson method

The iterative method described earlier, Newton's method, can be extended to sets of simultaneous equations. In a single dimension, the slope was used as a first approximation to find the change needed to move to the root of the equation. With multiple dimensions, the analog to the slope is the multidimensional gradient, and the Newton-Raphson method consists of using the gradient to obtain successively better approximations.

The Taylor expansion of

$$f_i(x_1,x_2,x_3,\dots) = 0$$ (9.55)

 

can be written to first order using vector notation as

$$f_i({\bf x}) = f_i({\bf x}_0) = \Delta f_i\cdot({\bf x}-{\bf x}_0).$$ (9.56)

 

An approximation to the equations to be solved is then, in matrix notation,

$${\bf f} = -{\bf F}{\bf\delta}$$ (9.57)

 

where ${\bf f}$ is the matrix of functions of the form (9.6.2) to be solved, evaluated at an estimated location for the root ${\bf x}_0$, ${\bf F}$ is the matrix of derivatives of the functions ${\bf f}$with respect to the variables ${\bf x}$ for those values ${\bf x}_0$, and ${\bf\delta}$ is the matrix of differences $({\bf x}-{\bf x}_0)$ between the values that solve the equation and the current values.

The values of ${\bf\delta}$ are estimates of the needed correction to an estimate of the solution, so the form

$${\bf\delta} = -{\bf F}^{-1}{\bf f}$$ (9.58)

 

will provide successively better estimates of the solution if applied iteratively. This method is readily applied to non-linear equations, and can use finite-difference estimates of the derivatives to evaluate the gradients.

   
 9.7 Techniques for numerical  9.6 Solution of simultaneous  9.6.1 Gauss-Seidel iteration