Consider two equations over two variables (in Reals) of the form:
$$f(x,y)= A_0[term 1]+A_1[term 2]+A_2[term 3]+A_3[term 4]+A_4$$
$$g(x,y)= B_0[term 1]+B_1[term 2]+B_2[term 3]+B_3[term 4]+B_4$$
where $A_0..A_4$ and $B_0..B_4$ are some real constants.
The maximum "degree of freedom" of each term in both equations is 2. The equations are not necessarily simple polynomials. It can also include trigonometric identities like sin(x), cos(y) etc. For eg (just a random example):
$$f(x,y):sin(x)∗cos(y)+sin(3x)−12*cos(5y)+.3=0$$
$$g(x,y):x−3y-.1=0$$
The terms and their "degree of freedom" would be as follows:
$$sin(x)∗cos(y), sin(3x), -12*cos(5y), .3: (2, 1, 1, 0)$$
$$x, 3y, .1: (1, 1, 0)$$
Query: Assuming this system of equations has a solution. Let x_0, y_0 be our initial guesses and the error or distance from 0 for each equation be E_0, E_1 respectively.
I tried Newton Raphson but it seems to work only when:
- our guess is very close to x, y.
- there is no guarantee that it will converge.
- there is no guaranteed rate of convergence.
Can some one help with the fastest known approach that is guaranteed to find a solution and has a guaranteed rate of convergence? By guaranteed rate of convergence we mean that there is a guarantee that for example error will half after some f() steps, where f() is some function defined over the two equations.
Given the arbitrariness of the terms you are adding up in the equations, it is pointless to even consider them... The structure does not rule out any functions $f(x,y), g(x,y)$.
In fact, as it is currently written, you are asking for a method that will surely work of any given system with two equations and two unknowns, and with a guaranteed rate of convergence. Let me save you some time: This does not exist.
If you are able to provide actual restrictions on the choice of $f$ and $g$, maybe something can be said.