The number (0.25 Pi) is a third order root of f[x] below.

f[x_] := Sin[x] - Cos[x] - Sqrt[2] (x - 0.25 Pi);
Chop@Normal@Series[f[x], {x, 0.25 Pi, 4}]

-0.235702 (-0.785398 + x)^3

Newton's method converges slowly to a multiple root such as the third order root above, but if f[x] is sufficiently well behaved, g[x_]:=If[ f[x]==0, 0, f[x]/f'[x] ] only has simple roots. I can't understand why, but very few books on numerical methods mention that. Anyway we can implement Newton's method to find a root of g[x] above and have quadratic convergence on the root. That can be done as follows:



0.25 Pi - %

{-4.2146, -3.08437, -1.50089, -0.211427, -0.000629266, -7.64783*10^-10}

How do we extend the code above to find a root of a system of nonlinear equations in two or more dimensions? Example given:


find the root of f[x,y] at { x-> -0.514933, y-> 0.514933 } . I know how to find that root with the built-in FindRoot. I want to know how to extend the method above.

  • 1
    $\begingroup$ It is certainly unfortunate that Schröder's method is not more well-known; nevertheless, this is more a math question than a Mathematica question, no? $\endgroup$ – J. M. will be back soon Jan 15 '17 at 17:38
  • $\begingroup$ I think I will put a similar question on the stack-exchange math site. $\endgroup$ – Ted Ersek Jan 15 '17 at 17:51
  • $\begingroup$ Really not a Mathematica question. There is a substantial literature on root deflation, though. Macaulay matrices get used in many such, at least in the case of polynomial systems. I should add that in general this is more complicated than the univariate case, because the "multiplicity structure" can be more complicated: in the univariate case multiplicity n implies n-1 derivatives vanish at the root and these provide a basis for what is called the "dual space. Dual to what? I won't get into that. But the idea is that in the multivariate setting things need not be so straightforward. $\endgroup$ – Daniel Lichtblau Jan 15 '17 at 20:22
  • $\begingroup$ As for f[x]/f'[x] having simple roots, just factor out the root, that is, write f as f[x] = (x-r)^n*g[x] where g does not vanish at the root r. Then the quotient is (x-r) * g/(n*g+(x-r)*g'` and the second factor is simply 1/n at x=r, hence this quotient vanishes to first order. $\endgroup$ – Daniel Lichtblau Jan 15 '17 at 20:29

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