I have a system of equations, and one of them is this : ##\int(1-U(y))Dy - H*\int(U(y)-U(y)^2)dy=0##(adsbygoogle = window.adsbygoogle || []).push({});

Can newtons method work if I approximate this integral to be ##\sum_y(1-U(y))-H\sum(U(y)-U(y)^2)=0##

y is a set integers in range ##[1,1000]##

I have newtons method working for this same system without this equation, but I am trying to solve for a new variable (a), so I added this equation and changed (a) from a constant to a new variable, but newtons method no longer converges.

Could this be because I am using rectangle left rule to approximate the integral, and maybe rectangle left has too much error for newtons method to work?

The other thing I can try is to take the derivative of both sides giving this equation ##1-U(y)-H(U(y)-U(y)^2=0##

The only problem with this is I would have to add this equation to every point on a grid, meaning it would be 1000 new equations for 1000 grid points instead of 1 equation. I would have to rewrite the whole algorithm, so I would rather use rectangle left rule if possible because it is much less coding.

Anyway, yeah, will newtons method work with rectangle left rule like this?

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# A Can Newton's method work with an approximated integral

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