The BROYDEN function solves a system of *n*nonlinear equations (where *n* ≥ 2) in *n* dimensions using a globally-convergent Broyden’s method.

## Examples

We can use BROYDEN to solve the non-linear system of equations defined by the BROYFUNC function above:

FUNCTION broyfunc, X

RETURN, [3.0 * X[0] - COS(X[1]*X[2]) - 0.5,$

X[0]^2 - 81.0*(X[1] + 0.1)^2 + SIN(X[2]) + 1.06,$

EXP(-X[0]*X[1]) + 20.0 * X[2] + (10.0*!PI - 3.0)/3.0]

END

;Provide an initial guess as the algorithm's starting point:

X = [-1.0, 1.0, 2.0]

;Compute the solution:

result = BROYDEN(X, 'BROYFUNC')

;Print the result:

PRINT, result

IDL prints:

0.500000 -1.10731e-07 -0.523599

The exact solution (to eight-decimal accuracy) is [0.5, 0.0, -0.52359877].

## Syntax

*Result* = BROYDEN( *X*, *Vecfunc* [, CHECK=*variable*] [, /DOUBLE] [, EPS=*value*] [, ITMAX=*value*] [, STEPMAX=*value*] [, TOLF=*value*] [, TOLMIN=*value*] [, TOLX=*value*] )

## Return Value

This function returns an *n*-element vector containing the solution.

## Arguments

### X

An *n*-element vector (where *n* ≥ 2) containing an initial guess at the solution of the system.

**Note: **If BROYDEN 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 a vector argument *X* and return a vector result.

For example, suppose we wish to solve the following system:

To represent this system, we define an IDL function named BROYFUNC:

`FUNCTION broyfunc, X`

RETURN, [3.0 * X[0] - COS(X[1]*X[2]) - 0.5,$

X[0]^2 - 81.0*(X[1] + 0.1)^2 + SIN(X[2]) + 1.06,$

EXP(-X[0]*X[1]) + 20.0 * X[2] + (10.0*!PI - 3.0)/3.0]

`END`

## Keywords

### CHECK

BROYDEN calls an internal function named fmin() to determine whether the routine has converged to a local rather than 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 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.

### EPS

Set this keyword to a number close to machine accuracy, used to remove noise from each iteration. The default is 10^{-7 }for single precision, and 10^{-14 }for double precision.

### ITMAX

Use this keyword to specify the maximum allowed number of iterations. The default is 200.

### STEPMAX

Use this keyword to specify the scaled maximum step length allowed in line searches. 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

Pre 4.0 |
Introduced |

## Resources and References

BROYDEN is based on the routine broydn 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.