The NEWTON function solves a system of n non-linear equations in n dimensions using a globally-convergent Newton’s method.
NEWTON is based on the routine newt described in section 9.7 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.
Examples
Use NEWTON to solve an n-dimensional system of n non-linear equations. Systems of non-linear equations may have multiple solutions; starting the algorithms with different initial guesses enables detection of different solutions.
FUNCTION newtfunc, X
RETURN, [X[0] + X[1] - 3, X[0]^2 + X[1]^2 - 9]
END
PRO TEST_NEWTON
X = [1d, 5d]
result = NEWTON(X, 'newtfunc')
PRINT, 'For X=[1.0, 5.0], result = ', result
X = [1d, -1d]
result = NEWTON(X,'newtfunc')
PRINT, 'For X=[1.0, -1.0], result = ', result
END
TEST_NEWTON
IDL prints:
For X=[1.0, 5.0], result = -2.4871776e-006 3.0000025
For X=[1.0, -1.0], result = 3.0000000 -2.9985351e-008
Syntax
Result = NEWTON( X, Vecfunc [, CHECK=variable] [, /DOUBLE] [, ITMAX=value] [, STEPMAX=value] [, TOLF=value] [, TOLMIN=value] [, TOLX=value] )
Return Value
The result is an n-element vector containing the solution.
Arguments
X
An n-element vector containing an initial guess at the solution of the system.
Note: If NEWTON is complex then only the real part is used for the computation.
Vecfunc
A scalar string specifying the name of a user-supplied IDL function that defines the system of non-linear equations. This function must accept an n-element vector argument X and return an n-element vector result.
For example, suppose the non-linear system is defined by the following equations:
y0 = x0 + x1 - 3, y1 = x02 + x12 - 9
We write a function NEWTFUNC to express these relationships in the IDL language:
FUNCTION newtfunc, X
RETURN, [X[0] + X[1] -3.0, X[0]^2 + X[1]^2 - 9.0]
END
Keywords
CHECK
NEWTON calls an internal function named fmin() to determine whether the routine has converged to a local minimum rather than to a global minimum (see Numerical Recipes, section 9.7). Use the CHECK keyword to specify a named variable which will be set to 1 if the routine has converged to a local minimum or to 0 if it has not. If the routine does converge to a local minimum, try restarting from a different initial guess to obtain the global minimum.
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
ITMAX
The maximum allowed number of iterations. The default value is 200.
STEPMAX
The scaled maximum step length allowed in line search. The default value is 100.0.
TOLF
Set the convergence criterion on the function values. The default value is 1.0 x 10-4.
TOLMIN
Set the criterion for deciding whether spurious convergence to a minimum of the function fmin() has occurred. The default value is 1.0 x 10-6.
TOLX
Set the convergence criterion on X. The default value is 1.0 x 10-7.
Version History
See Also
BROYDEN, FX_ROOT, FZ_ROOTS