% -*- texinfo -*- % @deftypefn {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'. % % @deftypefnx {Function File} {} [X, FX, INFO] = fzero (FCN, APPROX, OPTIONS,P1,P2,...) % % Call FCN with FCN(X,P1,P2,...). % % @table @asis % @item INPUT ARGUMENTS % @end table % % @table @asis % @item APPROX can be a vector with two components, % @example % A = APPROX(1) and B = APPROX(2), % @end example % 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 % @example % reltol*|X|+abstol (see OPTIONS). % @end example % 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. % % @item OPTIONS is a structure, with the following fields: % % @table @asis % @item 'abstol' - absolute (error for Brent's or residual for fsolve) % tolerance. Default = 1e-6. % % @item 'reltol' - relative error tolerance (only Brent's method). Default = 1e-6. % % @item 'prl' - print level, how much diagnostics to print. Default = 0, no % diagnostics output. % @end table % % If OPTIONS argument is omitted, or a specific field is not present in the % OPTIONS structure, default values will be used. % @end table % % @table @asis % @item OUTPUT ARGUMENTS % @end table % % @table @asis % @item 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. % @end table % % @table @asis % @item EXAMPLES % @end table % % @example % fzero('sin',[-2 1]) will use Brent's method to find the solution to % sin(x) = 0 in the interval [-2, 1] % @end example % % @example % [x, fx, info] = fzero('sin',-2) will use fsolve to find a solution to % sin(x)=0 near -2. % @end example % % @example % 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. % @end example % % @table @asis % @item REFERENCES % [1] Brent, R. P. 'Algorithms for minimization without derivatives' (1971). % @end table % @end deftypefn % @seealso{fsolve}

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