— Function File: [X, FX, INFO] = fzero (FCN, APPROX, OPTIONS)

Given FCN, the name of a function of the form `F (X)', and an initial approximation APPROX, `fzero' solves the scalar nonlinear equation such that `F(X) == 0'. Depending on APPROX, `fzero' uses different algorithms to solve the problem: either the Brent's method or the Powell's method of `fsolve'. — Function File: [X, FX, INFO] = fzero (FCN, APPROX, OPTIONS,P1,P2,...)

Call FCN with FCN(X,P1,P2,...).

INPUT ARGUMENTS

APPROX can be a vector with two components,

               A = APPROX(1) and B = APPROX(2),

which localizes the zero of F, that is, it is assumed that X lies between A and B. If APPROX is a scalar, it is treated as an initial guess for X.

If APPROX is a vector of length 2 and F takes different signs at A and B, F(A)*F(B) < 0, then the Brent's zero finding algorithm [1] is used with error tolerance criterion

               reltol*|X|+abstol (see OPTIONS).

This algorithm combines superlinear convergence (for sufficiently regular functions) with the robustness of bisection.

Whether F has identical signs at A and B, or APPROX is a single scalar value, then `fzero' falls back to another method and `fsolve(FCN, X0)' is called, with the starting value X0 equal to (A+B)/2 or APPROX, respectively. Only absolute residual tolerance, abstol, is used then, due to the limitations of the `fsolve_options' function. See OPTIONS and `help fsolve' for details.

OPTIONS is a structure, with the following fields:

'abstol' - absolute (error for Brent's or residual for fsolve)

tolerance. Default = 1e-6.

'reltol' - relative error tolerance (only Brent's method). Default = 1e-6.

'prl' - print level, how much diagnostics to print. Default = 0, no

diagnostics output.

If OPTIONS argument is omitted, or a specific field is not present in the OPTIONS structure, default values will be used.

OUTPUT ARGUMENTS

The computed approximation to the zero of FCN is returned in X. FX is then equal

to FCN(X). If the iteration converged, INFO == 1. If Brent's method is used, and the function seems discontinuous, INFO is set to -5. If fsolve is used, INFO is determined by its convergence.

EXAMPLES

          fzero('sin',[-2 1]) will use Brent's method to find the solution to
          sin(x) = 0 in the interval [-2, 1]

          [x, fx, info] = fzero('sin',-2) will use fsolve to find a solution to
          sin(x)=0 near -2.

          options.abstol = 1e-2; fzero('sin',-2, options) will use fsolve to
          find a solution to sin(x)=0 near -2 with the absolute tolerance 1e-2.

REFERENCES

[1] Brent, R. P. "Algorithms for minimization without derivatives" (1971).