The Newton-Raphson second order method for a multivariables function can be written with the equation (I) form.
The solution is obtained with an efficient linear solving method like the partial pivoting gauss algorithm.
The second order Taylor's development gives the exact approximation of the function.
We have found with several examples that the above method converges in two iterations and the solution is obtained in the first iteration.
The conclusion is that this solving method corresponds to the exact one.
Another way is to obtain the solution with the inverting process as it is shown in equation (II) : this method can not always converge easily and it depends on the Hessian matrix conditioning because the inverse is a source of roundoff errors.
Indeed if the system is linear and it has a solution then you should receive a solution to the equation in one step, and be able to confirm it.
Thank you Mr Michael Patriksson for your comment. It is advised to use linear algebra algorithms to solve linear systems of equations. In this case, if the solution exists it is unique like you have cited. The Newton-Raphson algorithm uses the first and second derivatives of the multivariable function and can't be applied for a linear system of equations.
Hold on. That's not what I wrote.
Newton-Raphson is a method for a nonlinear equation in one (1) variable. Newton's method is designed for a nonlinear function in n variables and equations..
I will re-iterate: in an optimization problem of the form min -q^Tx + (1/2)x^TQx, the optimality condition is the linear system Qx = q. Newton's method applied to this system terminates in one step with the exact solution to the above equation; equivalently, if Q is positive definite then the linear system Qx=q identifies the unique solution, which then also is a minimizer of the function.
There is, of course, a very long array of approximate solution methods for linear equations. The above is special, since the system matrix is symmetric, and therefore is the optimality condition for the quadratic optimization described above.
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