ZEROFCN Function

Finds the real zeros of a real function using Müller’s method.

Usage

result = ZEROFCN(f)

Input Parameters

f—Scalar string specifying a user-supplied function for which the zeros are to be found. The f function accepts one scalar parameter from which the function is evaluated and returns a scalar of the same type.

Returned Value

result—An array containing the zeros x of the function.

Input Keywords

Double—If present and nonzero, double precision is used.

N_Roots—Number of roots for ZEROFCN to find. Default: N_Roots = 1

XGuess—Array with N_Roots components containing the initial guesses for the zeros. Default: XGuess = 0

Err_Abs—First stopping criterion. A zero, x_i, is accepted if | f (x_i) | < Err_Abs. Default: Err_Abs = SQRT(e),where e is the machine precision

Err_Rel—Second stopping criterion. A zero, x_i, is accepted if the relative change of two successive approximations to x_i is less than Err_Rel. Default: Err_Rel = SQRT(e),where e is the machine precision

Eta—Spread criteria for multiple zeros. If the zero, x_i, has been computed and | x_i – x_j | < Eps, where x_j is a previously computed zero, then the computation is restarted with a guess equal to x_i + Eta. Default: Eta = 0.01

Eps—See Eta. Default: Eps = SQRT(e), where e is the machine precision

Itmax—Maximum number of iterations per zero. Default: Itmax = 100

Info—Array of length N_Roots containing convergence information. The value Info (j – 1) is the number of iterations used in finding the j^th zero when convergence is achieved. If convergence is not obtained in Itmax iterations, Info (j – 1) is greater than Itmax.

Discussion

Function ZEROFCN computes n real zeros of a real function f. Given a user-supplied function f (x) and an n-vector of initial guesses x₀, x₁, ..., x_n–1, the function uses Müller’s method to locate n real zeros of f. The function has two convergence criteria. The first criterion requires that | f (x_i^(m)) | be less than Err_Abs. The second criterion requires that the relative change of any two successive approximations to an x_i be less than Err_Rel. Here, x_i^(m) is the m-th approximation to x_i. Let Err_Abs be denoted by e₁, and Err_Rel be denoted by e₂. The criteria can be stated mathematically as follows:

ZEROFCN has two convergence criteria; “convergence” is the satisfaction of either criterion.

Criterion 1:

 

Criterion 2:

 

“Convergence” is the satisfaction of either criterion.

Example

This example finds a real zero of the third-degree polynomial:

f(x) = x³ – 3x² + 3x – 1

The results are shown in ZEROFCN Function.

.RUN

; Define function f.

FUNCTION f, x

   RETURN, x^3 - 3 * x^2 + 3 * x - 1

END

; Use hardware characters for the plot.

!P.Font = 0

; Compute the real zero(s).

zero = ZEROFCN('f')

x = 2 * FINDGEN(100)/99

; Plot results.

PLOT, x, f(x)

OPLOT, [zero], [f(zero)], Psym = 6

XYOUTS, .5, .5, 'Computed zero is at x = ' + $

   STRING(zero(0)), Charsize = 1.5

ZEROFCN Function

Warning Errors

MATH_NO_CONVERGE_MAX_ITER—Function failed to converge within Itmax iterations for at least one of the N_Roots roots.