IDL Analyst Reference Guide: Nonlinear Equations

IMSL_ZEROSYS

The IMSL_ZEROSYS function solves a system of n nonlinear equations, f_i (x) = 0, using a modified Powell hybrid algorithm.

Syntax

Result = IMSL_ZEROSYS(f, n [, /DOUBLE] [, ERR_REL=value] [, FNORM=value] [, JACOBIAN=string] [, ITMAX=value] [, XGUESS=array])

Return Value

An array containing a solution of the system of equations.

Arguments

f

Scalar string specifying a user-supplied function to evaluate the system of equations to be solved. The f function accepts one parameter containing the point at which the functions are to be evaluated and returns the computed function values at the given point.

n

Number of equations to be solved and the number of unknowns.

Keywords

DOUBLE

If present and nonzero, double precision is used.

ERR_REL

Stopping criterion. The root is accepted if the relative error between two successive approximations to this root is less than ERR_REL. Default: ERR_REL = SQRT(e), where e is the machine precision.

FNORM

Scalar with the value f ²₀ + ... + f ²_n–1 at the point x.

JACOBIAN

Scalar string specifying a user-supplied function to evaluate the x n Jacobian. The function accepts as parameter the point at which the Jacobian is to be evaluated and returns a two-dimensional matrix defined by result (i, j) = ¶f_i/¶x_j.

ITMAX

Maximum allowable number of iterations. Default: ITMAX = 200.

XGUESS

Array with N components containing the initial estimate of the root. Default: XGUESS = 0.

Discussion

The IMSL_ZEROSYS function is based on the MINPACK subroutine HYBRDJ, which uses a modification of the hybrid algorithm due to M.J.D. Powell. This algorithm is a variation of Newton's Method, which takes precautions to avoid undesirable large steps or increasing residuals. For further discussion, see Moré et al. (1980).

Example

The following 2 x 2 system of nonlinear equations is solved:

f(x) = x₀ + x₁ – 3

f(x) = x₀² + x₁² – 9

.RUN  
; Define the system through the function f.  
FUNCTION f, x  
   RETURN, [x(0)+x(1)-3, x(0)^2+x(1)^2-9]  
END  
  
PM, IMSL_ZEROSYS('f', 2), $  
Title = 'Solution of the system:', FORMAT = '(f10.5)'  
; Compute the solution and output the results.  
Solution of the system:  
      0.00000  
      3.00000