Re: Solving two-dimensional systems

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, May 26, 2015 at 11:16:12AM -0400, Charles Greathouse wrote:
> In computing https://oeis.org/A258042 I found myself solving this
> two-dimensional system:
> 
> solve(a=1.7, 1.8, my(x=solve(y=1.8, 2,
> y*(a+y)*log(a+y)-(a+y^2)*log(a+y^2))); log(a+x^2)/log(a+x)^2-1)
> 
> Is there any better way to do this than nested solve() calls?

First, you can simplify to

solve(a=1.7,1.8,my(x=solve(y=1.8,2,
  y*(a+y)-(a+y^2)*log(a+y)));log(a+x^2)-log(a+x)^2)

Secondly, you could use a two dimensional Newton iteration:
Set:
F(a,x)=(x*(a+x)-(a+x^2)*log(a+x),log(a+x^2)-log(a+x)^2)

Compute the differential dF so that

F(a+h1,x+h2)=F(a,x)+dF(a,x)*[h1,h2]~+O(||(h1,h2)||^2)

Solve 
0=F(a,x)+dF(a,x)*[h1,h2]~
for h1, h2 and Iterate.

If that does not converge, replace the system by an equivalent one so that it
does.

Cheers,
Bill.