## FX_ROOT

The FX_ROOT function computes real and complex roots of a univariate nonlinear function using an optimal Müller's method.

This routine is written in the IDL language. Its source code can be found in the file ``` fx_root.pro``` in the ``` lib``` subdirectory of the IDL distribution.

### Calling Sequence

Result = FX_ROOT( X, Func )

### Arguments

#### X

A 3-element real or complex initial guess vector. Real initial guesses may result in real or complex roots. Complex initial guesses will result in complex roots.

#### Func

A scalar string specifying the name of a user-supplied IDL function that defines the univariate nonlinear function. This function must accept the vector argument X.

For example, suppose we wish to find a root of the following function:

We write a function FUNC to express the function in the IDL language:

FUNCTION func, X

RETURN, EXP(SIN(X)^2 + COS(X)^2 - 1) - 1

END

### Keywords

#### DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

#### ITMAX

The maximum allowed number of iterations. The default is 100.

#### STOP

Use this keyword to specify the stopping criterion used to judge the accuracy of a computed root r( k ). Setting STOP = 0 (the default) checks whether the absolute value of the difference between two successively-computed roots, | r( k ) - r( k +1) | is less than the stopping tolerance TOL. Setting STOP = 1 checks whether the absolute value of the function FUNC at the current root, | FUNC(r( k )) |, is less than TOL.

#### TOL

Use this keyword to specify the stopping error tolerance. The default is 1.0  ¥  10 -4 .

### Example

To find the roots of the function FUNC defined above, first define a real 3-element initial guess vector:

x = [0.0, -!pi/2, !pi]

Compute a root of the function using double-precision arithmetic.

root = FX_ROOT(X, 'FUNC', /DOUBLE)

Check the accuracy of the computed root.

PRINT, EXP(SIN(ROOT)^2 + COS(ROOT)^2 - 1) - 1

IDL prints:

0.0000000

We can also define a complex 3-element initial guess vector:

X = [COMPLEX(-!PI/3, 0), COMPLEX(0, !PI), COMPLEX(0, -!PI/6)]

root = FX_ROOT(x, 'FUNC') ; Compute the root of the function.

Check the accuracy of the computed complex root.

PRINT, EXP(SIN(ROOT)^2 + COS(ROOT)^2 - 1) - 1

IDL prints:

( 0.00000, 0.00000)