IDL Analyst Reference Guide: Nonlinear Equations

IMSL_ZEROFCN

The IMSL_ZEROFCN function finds the real zeros of a real function using Müller's method.

Syntax

Result = IMSL_ZEROFCN(f [, /DOUBLE] [, ERR_ABS=value] [, ERR_REL=value] [, ETA=value] [, EPS=value] [, INFO=array] [, ITMAX=value] [, N_ROOTS=value] [, XGUESS=array] )

Return Value

An array containing the zeros x of the function.

Arguments

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.

Keywords

DOUBLE

If present and nonzero, double precision is used.

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.

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.

ITMAX

Maximum number of iterations per zero. Default: ITMAX = 100.

N_ROOTS

Number of roots for IMSL_ZEROFCN to find. Default: N_ROOTS = 1.

XGUESS

Array with N_ROOTS components containing the initial guesses for the zeros. Default: XGUESS = 0

Discussion

The IMSL_ZEROFCN function 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.

IMSL_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 Figure 10-1.

.RUN  
; Define function f.  
FUNCTION f, x  
   return, x^3 - 3 * x^2 + 3 * x - 1  
END  
  
zero = IMSL_ZEROFCN('f')  
; Compute the real zero(s).  
x = 2 * FINDGEN(100)/99  
PLOT, x, f(x)  
; Plot results.  
OPLOT, [zero], [f(zero)], Psym = 6  
XYOUTS, .5, .5, 'Computed zero is at x = ' + $  
   STRING(zero(0)), Charsize = 1.5  

Figure 10-1: IMSL_ZEROFCN Function

Figure 10-1: IMSL_ZEROFCN Function

Errors

Warning Errors

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

Version History

  IDL Online Help (March 06, 2007)