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## de.jstacs.algorithms.optimization
Class Optimizer

    java.lang.Object
      de.jstacs.algorithms.optimization.Optimizer

    public class Optimizer

extends Object

This class can be used for optimization purposes. The function will be minimized. To maximize a function use the classes `NegativeFunction` and `NegativeDifferentiableFunction` as input. All methods for multidimensional optimization need an implementation of `DifferentiableFunction`, that will be minimized. The optimizing methods try to do as less as possible function evaluation, which is important for functions that are costly to evaluate. So if you have a costly function and you will fasten the computation, try to do as much effort as you can to make the methods `evaluateFunction()` and `evaluateGradientOfFunction()` in your implementation as fast as possible.
You are enabled to choose different termination modes.

Author:
  Jens Keilwagen
See Also:
  `OneDimensionalFunction`, `DifferentiableFunction`, `NegativeOneDimensionalFunction`, `NegativeDifferentiableFunction`, `NumericalDifferentiableFunction`

Field Summary
`static byte` | `CONJUGATE_GRADIENTS_FR` This constant can be used to specify that conjugate gradients (by Fletcher and Reeves) should be used in the `optimize`-method.
`static byte` | `CONJUGATE_GRADIENTS_PRP` This constant can be used to specify that conjugate gradients (by Polak and Ribière should be used in the `optimize`-method.
`static byte` | `QUASI_NEWTON_BFGS` This constant can be used to specify that the quasi-Newton method of Broyden-Fletcher-Goldfarb-Shanno should be used in the `optimize`-method.
`static byte` | `QUASI_NEWTON_DFP` This constant can be used to specify that the quasi-Newton method of Davidon-Fletcher-Powell should be used in the `optimize` -method.
`static byte` | `STEEPEST_DESCENT` This constant can be used to specify that the steepest descent should be used in the `optimize`-method.

Constructor Summary
`Optimizer()`

Method Summary
`static double[]` | `brentsMethod(OneDimensionalFunction f, double a, double x, double b, double tol)` Approximates a minimum (not necessary the global) in the interval `[lower,upper]`.
`static double[]` | `brentsMethod(OneDimensionalFunction f, double a, double x, double fx, double b, double tol)` Approximates a minimum (not necessary the global) in the interval `[lower,upper]`.
`static int` | `conjugateGradientsFR(DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The conjugate gradient algorithm by Fletcher and Reeves.
`static int` | `conjugateGradientsPR(DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The conjugate gradient algorithm by Polak and Ribière.
`static int` | `conjugateGradientsPRP(DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The conjugate gradient algorithm by Polak and Ribière called "Polak-Ribière-Positive".
`static double[]` | `findBracket(OneDimensionalFunction f, double lower, double startDistance)` This method returns a bracket containing a minimum.
`static double[]` | `findBracket(OneDimensionalFunction f, double lower, double fLower, double startDistance)` This method returns a bracket containing a minimum.
`static double[]` | `goldenRatio(OneDimensionalFunction f, double lower, double upper, double eps)` Approximates a minimum (not necessary the global) in the interval `[lower,upper]`.
`static double[]` | `goldenRatio(OneDimensionalFunction f, double lower, double p1, double fP1, double upper, double eps)` Approximates a minimum (not necessary the global) in the interval `[lower,upper]`.
`static int` | `limitedMemoryBFGS(DifferentiableFunction f, double[] currentValues, byte m, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The Broyden-Fletcher-Goldfarb-Shanno version of limited memory quasi-Newton methods.
`static int` | `optimize(byte algorithm, DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out)` This method enables you to use all different implemented optimization algorithms by only one method.
`static int` | `optimize(byte algorithm, DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` This method enables you to use all different implemented optimization algorithms by only one method.
`static double[]` | `parabolicInterpolation(OneDimensionalFunction f, double lower, double p1, double upper, double eps)` Approximates a minimum (not necessary the global) in the interval `[lower,upper]`.
`static double[]` | `parabolicInterpolation(OneDimensionalFunction f, double lower, double fLower, double p1, double fP1, double upper, double fUpper, double eps)` Approximates a minimum (not necessary the global) in the interval `[lower,upper]`.
`static int` | `quasiNewtonBFGS(DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The Broyden-Fletcher-Goldfarb-Shanno version of the quasi-Newton method.
`static int` | `quasiNewtonDFP(DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The Davidon-Fletcher-Powell version of the quasi-Newton method.
`static int` | `steepestDescent(DifferentiableFunction f, double[] currentValues, TerminationCondition terminationMode, double linEps, StartDistanceForecaster startDistance, OutputStream out, Time t)` The steepest descent.

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Field Detail

### STEEPEST_DESCENT

    public static final byte STEEPEST_DESCENT

  This constant can be used to specify that the steepest descent should be used in the `optimize`-method.

  See Also:
    Constant Field Values

### CONJUGATE_GRADIENTS_FR

    public static final byte CONJUGATE_GRADIENTS_FR

  This constant can be used to specify that conjugate gradients (by Fletcher and Reeves) should be used in the `optimize`-method.

  See Also:
    Constant Field Values

### CONJUGATE_GRADIENTS_PRP

    public static final byte CONJUGATE_GRADIENTS_PRP

  This constant can be used to specify that conjugate gradients (by Polak and Ribière should be used in the `optimize`-method.

  See Also:
    Constant Field Values

### QUASI_NEWTON_DFP

    public static final byte QUASI_NEWTON_DFP

  This constant can be used to specify that the quasi-Newton method of Davidon-Fletcher-Powell should be used in the `optimize` -method.

  See Also:
    Constant Field Values

### QUASI_NEWTON_BFGS

    public static final byte QUASI_NEWTON_BFGS

  This constant can be used to specify that the quasi-Newton method of Broyden-Fletcher-Goldfarb-Shanno should be used in the `optimize`-method.

  See Also:
    Constant Field Values

Constructor Detail

### Optimizer

    public Optimizer()

Method Detail

### findBracket

    public static double[] findBracket(OneDimensionalFunction f,
                                       double lower,
                                       double fLower,
                                       double startDistance)
                                throws EvaluationException

  This method returns a bracket containing a minimum. This method can be used to find an initial bracket for a minimization method.

  Parameters:
    `f` - the function to be minimized
    `lower` - the initial value of `x`, lower bound
    `fLower` - the value `f(x)`
    `startDistance` - the initial distance for bracketing the minimum
  Returns:
    an array containing `x_lower`, `f(x_lower)` , `x_middle`, `f(x_middle)`, `x_upper` and `f(x_upper)` at positions 0 to 5
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
  See Also:
    `brentsMethod(OneDimensionalFunction f, double a, double x, double fx, double b, double tol )`, `goldenRatio(OneDimensionalFunction f, double lower, double p1, double fP1, double upper, double eps )`

### findBracket

    public static double[] findBracket(OneDimensionalFunction f,
                                       double lower,
                                       double startDistance)
                                throws EvaluationException

  This method returns a bracket containing a minimum.

  Parameters:
    `f` - the function to be minimized
    `lower` - the initial value of `x`, lower bound
    `startDistance` - the initial distance for bracketing the minimum
  Returns:
    an array containing `x_lower`, `f(x_lower)` , `x_inside`, `f(x_inside)`, `x_upper` and `f(x_upper)` at positions 0 to 5
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
  See Also:
    `findBracket(OneDimensionalFunction, double, double, double)`

### brentsMethod

    public static double[] brentsMethod(OneDimensionalFunction f,
                                        double a,
                                        double x,
                                        double fx,
                                        double b,
                                        double tol)
                                 throws EvaluationException

  Approximates a minimum (not necessary the global) in the interval `[lower,upper]`. The method is using the "Brent's method".
  This method is a modified copy of a method downloaded from the internet.

  Parameters:
    `f` - the function to be minimized
    `a` - the initial lower boundary
    `x` - an abscissa inside the interval `[a,b]`
    `fx` - the ordinate corresponding to `x`
    `b` - the initial upper bound
    `tol` - the termination parameter (interval size)
  Returns:
    an array containing the minimum `x` (position 0) and the value of the minimum `f(x)` (position 1)
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function

### brentsMethod

    public static double[] brentsMethod(OneDimensionalFunction f,
                                        double a,
                                        double x,
                                        double b,
                                        double tol)
                                 throws EvaluationException

  Approximates a minimum (not necessary the global) in the interval `[lower,upper]`. The method is using the "Brent's method".
  This method is an modified copy of a method downloaded from the internet.

  Parameters:
    `f` - the function to be minimized
    `a` - the initial lower boundary
    `x` - an abscissa inside the interval `[a,b]`
    `b` - the initial upper bound
    `tol` - the termination parameter (interval size)
  Returns:
    an array containing the minimum `x` (position 0) and the value of the minimum `f(x)` (position 1)
  Throws:
    `EvaluationException` - if there was a mistake in evaluating the function
  See Also:
    `brentsMethod(OneDimensionalFunction, double, double, double, double, double)`

### goldenRatio

    public static double[] goldenRatio(OneDimensionalFunction f,
                                       double lower,
                                       double p1,
                                       double fP1,
                                       double upper,
                                       double eps)
                                throws EvaluationException

  Approximates a minimum (not necessary the global) in the interval `[lower,upper]`. The method is using the "golden section".

  Parameters:
    `f` - the function to be minimized
    `lower` - the lower interval boundary
    `p1` - an abscissa in the interval `[lower,upper]`
    `fP1` - the ordinate corresponding to `p1`
    `upper` - the upper interval boundary
    `eps` - the minimal interval size
  Returns:
    an array containing the minimum `x` (position 0) and the value of the minimum `f(x)` (position 1)
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function

### goldenRatio

    public static double[] goldenRatio(OneDimensionalFunction f,
                                       double lower,
                                       double upper,
                                       double eps)
                                throws EvaluationException

  Approximates a minimum (not necessary the global) in the interval `[lower,upper]`. The method is using the "golden section".

  Parameters:
    `f` - the function to be minimized
    `lower` - the lower interval boundary
    `upper` - the upper interval boundary
    `eps` - the minimal interval size
  Returns:
    an array containing the minimum `x` (position 0) and the value of the minimum `f(x)` (position 1)
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
  See Also:
    `goldenRatio(OneDimensionalFunction, double, double, double, double, double)`

### parabolicInterpolation

    public static double[] parabolicInterpolation(OneDimensionalFunction f,
                                                  double lower,
                                                  double fLower,
                                                  double p1,
                                                  double fP1,
                                                  double upper,
                                                  double fUpper,
                                                  double eps)
                                           throws Exception

  Approximates a minimum (not necessary the global) in the interval `[lower,upper]`. The method is using the "parabolic interpolation".
  This method should work well for convex functions.

  Parameters:
    `f` - the function to be minimized
    `lower` - the lower interval boundary
    `fLower` - the ordinate corresponding to lower
    `p1` - an abscissa in the interval `[lower,upper]`
    `fP1` - the ordinate corresponding to `p1`
    `upper` - the upper interval boundary
    `fUpper` - the ordinate corresponding to upper
    `eps` - the minimal interval size
  Returns:
    an array containing the minimum `x` (position 0) and the value of the minimum `f(x)` (position 1)
  Throws:
    `Exception` - if there was something wrong during the evaluation of the function

### parabolicInterpolation

    public static double[] parabolicInterpolation(OneDimensionalFunction f,
                                                  double lower,
                                                  double p1,
                                                  double upper,
                                                  double eps)
                                           throws Exception

  Approximates a minimum (not necessary the global) in the interval `[lower,upper]`. The method is using the "parabolic interpolation".

  Parameters:
    `f` - the function to be minimized
    `lower` - the lower interval boundary
    `p1` - an abscissa in the interval `[lower,upper]`
    `upper` - the upper interval boundary
    `eps` - the minimal interval size
  Returns:
    an array containing the minimum `x` (position 0) and the value of minimum `f(x)` (position 1)
  Throws:
    `Exception` - if there was something wrong during the evaluation of the function
  See Also:
    `parabolicInterpolation(OneDimensionalFunction, double, double, double, double, double, double, double)`

### steepestDescent

    public static int steepestDescent(DifferentiableFunction f,
                                      double[] currentValues,
                                      TerminationCondition terminationMode,
                                      double linEps,
                                      StartDistanceForecaster startDistance,
                                      OutputStream out,
                                      Time t)
                               throws DimensionException,
                                      TerminationException,
                                      EvaluationException,
                                      IOException

  The steepest descent.

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO

### conjugateGradientsFR

    public static int conjugateGradientsFR(DifferentiableFunction f,
                                           double[] currentValues,
                                           TerminationCondition terminationMode,
                                           double linEps,
                                           StartDistanceForecaster startDistance,
                                           OutputStream out,
                                           Time t)
                                    throws DimensionException,
                                           TerminationException,
                                           IOException,
                                           EvaluationException

  The conjugate gradient algorithm by Fletcher and Reeves.

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO

### conjugateGradientsPR

    public static int conjugateGradientsPR(DifferentiableFunction f,
                                           double[] currentValues,
                                           TerminationCondition terminationMode,
                                           double linEps,
                                           StartDistanceForecaster startDistance,
                                           OutputStream out,
                                           Time t)
                                    throws DimensionException,
                                           TerminationException,
                                           IOException,
                                           EvaluationException

  The conjugate gradient algorithm by Polak and Ribière.

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO
  See Also:
    `conjugateGradientsPRP(DifferentiableFunction, double[], TerminationCondition, double, StartDistanceForecaster, OutputStream, boolean, Time)`

### conjugateGradientsPRP

    public static int conjugateGradientsPRP(DifferentiableFunction f,
                                            double[] currentValues,
                                            TerminationCondition terminationMode,
                                            double linEps,
                                            StartDistanceForecaster startDistance,
                                            OutputStream out,
                                            Time t)
                                     throws DimensionException,
                                            TerminationException,
                                            IOException,
                                            EvaluationException

  The conjugate gradient algorithm by Polak and Ribière called "Polak-Ribière-Positive".

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written.
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong with the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO
  See Also:
    `conjugateGradientsPRP(DifferentiableFunction, double[], TerminationCondition, double, StartDistanceForecaster, OutputStream, boolean, Time)`

### quasiNewtonDFP

    public static int quasiNewtonDFP(DifferentiableFunction f,
                                     double[] currentValues,
                                     TerminationCondition terminationMode,
                                     double linEps,
                                     StartDistanceForecaster startDistance,
                                     OutputStream out,
                                     Time t)
                              throws DimensionException,
                                     TerminationException,
                                     IOException,
                                     EvaluationException

  The Davidon-Fletcher-Powell version of the quasi-Newton method.

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written.
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO

### quasiNewtonBFGS

    public static int quasiNewtonBFGS(DifferentiableFunction f,
                                      double[] currentValues,
                                      TerminationCondition terminationMode,
                                      double linEps,
                                      StartDistanceForecaster startDistance,
                                      OutputStream out,
                                      Time t)
                               throws DimensionException,
                                      TerminationException,
                                      IOException,
                                      EvaluationException

  The Broyden-Fletcher-Goldfarb-Shanno version of the quasi-Newton method.

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written.
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO

### limitedMemoryBFGS

    public static int limitedMemoryBFGS(DifferentiableFunction f,
                                        double[] currentValues,
                                        byte m,
                                        TerminationCondition terminationMode,
                                        double linEps,
                                        StartDistanceForecaster startDistance,
                                        OutputStream out,
                                        Time t)
                                 throws DimensionException,
                                        TerminationException,
                                        IOException,
                                        EvaluationException

  The Broyden-Fletcher-Goldfarb-Shanno version of limited memory quasi-Newton methods.

  Parameters:
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `m` - the current number of vectors used to approximate the Hessian, typically between 3 and 10
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written.
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO

### optimize

    public static int optimize(byte algorithm,
                               DifferentiableFunction f,
                               double[] currentValues,
                               TerminationCondition terminationMode,
                               double linEps,
                               StartDistanceForecaster startDistance,
                               OutputStream out)
                        throws DimensionException,
                               TerminationException,
                               IOException,
                               EvaluationException

  This method enables you to use all different implemented optimization algorithms by only one method. You just have to change the parameter `algorithm`.

  Parameters:
    `algorithm` - the algorithm that should be used, either you use 2 < x < 11 for `limitedMemoryBFGS(DifferentiableFunction, double[], byte, TerminationCondition, double, StartDistanceForecaster, OutputStream, Time)` with `m=x` or the constants of the class: `STEEPEST_DESCENT`, `CONJUGATE_GRADIENTS_FR`, `CONJUGATE_GRADIENTS_PRP`, `QUASI_NEWTON_DFP`, `QUASI_NEWTON_BFGS`
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written.
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO
  See Also:
    `optimize(byte, DifferentiableFunction, double[], TerminationCondition, double, StartDistanceForecaster, OutputStream, Time)`

### optimize

    public static int optimize(byte algorithm,
                               DifferentiableFunction f,
                               double[] currentValues,
                               TerminationCondition terminationMode,
                               double linEps,
                               StartDistanceForecaster startDistance,
                               OutputStream out,
                               Time t)
                        throws DimensionException,
                               TerminationException,
                               IOException,
                               EvaluationException

  This method enables you to use all different implemented optimization algorithms by only one method. You just have to change the parameter `algorithm`.

  Parameters:
    `algorithm` - the algorithm that should be used, either you use 2 < x < 11 for `limitedMemoryBFGS(DifferentiableFunction, double[], byte, TerminationCondition, double, StartDistanceForecaster, OutputStream, Time)` with `m=x` or the constants of the class: `STEEPEST_DESCENT`, `CONJUGATE_GRADIENTS_FR`, `CONJUGATE_GRADIENTS_PRP`, `QUASI_NEWTON_DFP`, `QUASI_NEWTON_BFGS`
    `f` - the function to be minimized
    `currentValues` - at the begin the start vector and at the end the minimum vector
    `terminationMode` - the choice how to terminate the algorithm
    `linEps` - the bound for stopping the linear search
    `startDistance` - the initial step for the linear search
    `out` - the `OutputStream` for writing some information about the iterations, if `null` nothing will be written.
    `t` - a `Time` instance used to measure the elapsed time
  Returns:
    the number of iterations
  Throws:
    `EvaluationException` - if there was something wrong during the evaluation of the function
    `TerminationException` - if the termination condition is unknown
    `DimensionException` - if the dimension of the scope of the function and the array of arguments do not match
    `IOException` - if there was something wrong with the IO

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