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

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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

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

f(x) = x³ – 3x² + 3x – 1

The results are shown in in the figure that follows.

.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

Errors

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Warning Errors

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

Syntax

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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

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An array containing the zeros x of the function.

Arguments

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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

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DOUBLE (optional)

If present and nonzero, then double precision is used.

ERR_ABS (optional)

First stopping criterion. A zero, x_i, is accepted if | f (x_i) | < ERR_ABS. Default: SQRT(ε), where ε is the machine precision .

ERR_REL (optional)

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

ETA (optional)

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: 0.01

EPS (optional)

See ETA. Default: SQRT(ε), where ε is the machine precision.

INFO (optional)

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: 100.

N_ROOTS (optional)

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

XGUESS (optional)

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

Version History

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6.4	Introduced