# The EquationSolver Class

The EquationSolver class is the abstract base class for all root finder classes. Each root finding algorithm is implemented by a different class, derived from EquationSolver.

EquationSolver inherits from the
IterativeAlgorithm class.
The AbsoluteTolerance
and RelativeTolerance properties set the desired
precision as specified by the ConvergenceCriterion property. The default
value for both tolerances is SqrtEpsilon (roughly 10^{-8}).
MaxIterations sets the maximum number of
iterations. The default value for this property depends on the algorithm used. IterationsNeeded returns the actual number of
iterations performed after the algorithm has completed.

The function to solve is passed as a Func<T, TResult> delegate to a constructor, or it can be set later using the TargetFunction property.

The Solve method does the actual work of solving the equation. When called without parameters, it returns the approximation to the zero of the target function. When called with a single Double argument, it attempts to find a point where the target function equals the specified value.

## Verifying the result

The Solve method always returns the best estimate for the root of the target function. Successive calls to the Result property will also return this value, until the next call to Solve. An optional parameter lets you specify the right-hand side of the equation TargetFunction(x) = rightHandSide that is to be solved. The default is zero.

If the ThrowExceptionOnFailure property is set to true, an exception is thrown if the algorithm has failed to converge to a solution within the desired accuracy. If false, the Solve method returns the best approximation to the zero, regardless of whether it is within the requested tolerance.

The Status property indicates how the algorithm terminated. Its possible values and their meaning are listed below.

Value | Description |
---|---|

NoResult | The algorithm has not been executed. |

Normal | The algorithm ended normally. The desired accuracy has been achieved. |

IterationLimitExceeded | The number of iterations needed to achieve the desired accuracy is greater than MaxIterations. |

RoundOffError | Round-off prevented the algorithm from achieving the desired accuracy. |

BadFunction | Bad behavior of the target function prevented the algorithm from achieving the desired accuracy. |

Divergent | Doesn't apply. |

Equation solvers fall in two basic categories: those that use root bracketing, and those that use the derivative of the target function. Root bracketing solvers require an interval that is known to contain a zero of the target function. The interval is reduced in successive iterations until it is within the desired tolerance. Other algorithms require only a single starting value, but require the derivative of the function.

### References

Chabert, J.-L. (Ed.). *A History of Algorithms: From the Pebble to the Microchip.* New York:
Springer-Verlag, 1999.