The numerical analysis package provides basic blocks for common tasks in numerical computation. Currently, only real valued, univariate (depending on one variable) functions are handled.

Available blocks:

• A framework for solving non-linear equations (root finding)
• Generating functions by interpolation.

Possible future additions may include numerical differentation, numerical integration and finding minimal or maximal values of a function. Functionality dealing with multivariate functions, complex valued functions or special type functions will probably go into other packages.

A org.apache.commons.math.analysis.UnivariateRealSolver provides the means to find roots of univariate, real valued, functions. A root is the value where the function value vanishes. Commons-Math supports various implementations of UnivariateRealSolver to solve functions with differing characteristics. The current interface allows for computing exactly one root. If the given interval contains more than one root, an indication may be given or not. The current implementations all wont notify the user and return simply the first root found.

There are numerous non-obvious traps and pitfalls in root finding. Firstly, the usual disclaimers due to the nature how real world computers calculate values apply. If the computation of the function provides numerical instabilities, for example due to bit cancellation, the root finding algorithms may behave badly and fail to converge or even return bogus values. There wouldn't necessarily be an indication that the computed root is way off the true value. Secondly, The root finding problem itself may be inherently ill conditioned. There is a "domain of indeterminacy", the interval for which the function has near zero absolute values around the true root, may be very large. Even worse, small problems like roundoff error may cause the function value to "numerically oscillate" between negative and positive values. This may again result in roots way off the true value, without indication. There is not much a generic algorithm can do if such ill conditioned problems are met. A way around this is to transform the problem in order to get a better conditioned function.

The package provides several implementations off root finding algorithms, each with advantages and drawbacks. The may algorithms differ in

• Number of iterations for computing a specific root of a specific function.
• Number of function evaluations per iteration. An algorithm needing less iterations may still need multiple function evaluations, which may be more costly (for example, involve a numerical integration).
• Whether the interval must bracket a root or not (function values have different signs at interval endpoints).
• Behaviour in case of problems (indicate the error, return bogus values...).

In order to use the root-finding features, first a solver object must be created. It is encouraged that all solver object creation occurs via the org.apache.commons.math.analysis.UnivariateRealSolverFactory class. UnivariateRealSolverFactory is a simple factory used to create all of the solver objects supported by Commons-Math. The typical usage of UnivariateRealSolverFactory to create a solver object would be:

The solvers that can be instantiated via the UnivariateRealSolverFactory are detailed below:

Solver	Factory Method	Notes on Use
Bisection	newBisectionSolver	Root must be bracketted. Linear, guaranteed convergence
Brent	newBrentSolver	Root must be bracketted. Super-linear, guaranteed convergence
Secant	newSecantSolver	Root must be bracketted. Super-linear, non-guaranteed convergence

Using a solver object, roots of functions are easily found using the solve methods. For a function f, and two domain values, min and max, solve computes the value c such that:

• f(c) = 0.0 (see "function value accuracy")
• min <= c <= max

Typical usage:

The BrentSolver uses the Brent-Dekker algorithm which is fast and robust. This algorithm is recommended for most users. If there are multiple roots in the interval, or there is a large domain of indeterminacy, the algorithm will converge to a random root in the interval without indication that there are problems. Interestingly, the examined text book implementations all disagree in details of the convergence criteria. Also each implementation had problems for one of the test cases, so the expressions had to be fudged further. Don't expect to get exactly the same root values as for other implementations of this algorithm.

The SecantSolver uses a variant of the well known secant algorithm. It may be a bit faster than the Brent solver for a class of well behaved functions.

The BisectionSolver is included for completeness and for establishing a fall back in cases of emergency. The algorithm is simple, most likely bug free and guaranteed to converge even in very advert circumstances which might cause other algorithms to malfunction. The drawback is of course that it is also guaranteed to be slow.

Along with the solve methods, the UnivariateRealSolver interface provides many properties to control the convergence of a solver. For the most part, these properties should not have to change from their default values to produce quality results. In the circumstances where changing these property values is needed, it is easily done through getter and setter methods on UnivariateRealSolver:

Property	Methods	Purpose
Absolute accuracy	getAbsoluteAccuracy resetAbsoluteAccuracy setAbsoluteAccuracy	The Absolute Accuracy is maximal difference between the computed root and the true root of the function. This is what most people think of as "accuracy" intuitively. The initial value is choosen as a sane value for most real world problems, for roots in the range from -100 to +100. For accurate computation of roots near zero, in the range form -0.0001 to +0.0001, the value may be decreased. For computing roots much larger in absolute value than 100, the absolute accuracy may never be reached because the given relative accuracy is reached first.
Relative accuracy	getRelativeAccuracy resetRelativeAccuracy setRelativeAccuracy	The Relative Accuracy is the maximal difference between the computed root and the true root, divided by the maximum of the absolute values of the numbers. This accuracy measurement is more suited for numerical calculations with computers, due to the way floating point numbers are represented. The default value is choosen so that algorithms will get a result even for roots with large absolute values, even while it may be impossible to reach the given absolute accuracy.
Function value accuracy	getFunctionValueAccuracy resetFunctionValueAccuracy setFunctionValueAccuracy	This value is used by some algorithms in order to prevent numerical instabilities. If the function is evaluated to an absolute value smaller than the Function Value Accuracy, the algorithms assume they hit a root and return the value immediately. The default value is a "very small value". If the goal is to get a near zero function value rather than an accurate root, computation may be speed up by setting this value appropriately.
Maximum iteration count	getMaximumIterationCount resetMaximumIterationCount setMaximumIterationCount	This is the maximal number of iterations the algorith will try. If this number is exceeded, non-convergence is assumed and an exception is thrown. The default value is 100, which should be plenty giving that a bisection algorithm can't get any more accurate after 52 iterations because of the number of mantissa bits in a double precision floating point number. If a number of ill conditioned problems is to be solved, this number can be decreased in order to avoid wasting time.

A org.apache.commons.math.analysis.UnivariateRealInterpolator is used to find a univariate, real valued function f which for a given set of pairs (xi,yi) yields f(xi)=yi to the best accuracy possible. Currently, only an interpolator for generating natural cubic splines is available. There is no interpolator factory, mainly because the interpolation algorithm is more determined by the kind of the interpolated function rather than the set of points to interpolate. There aren't currently any accuracy controls either.

Typical usage:

A natural spline is a function consisting of a polynominal of third degree for each interval. A function interpolating N value pairs consists of N-1 polynominals. The function is continuous, smooth and can be derived two times. The second derivative is continuous but not smooth. The curvature at the first and the last point is zero (that's the "natural" part, coming from the flexible rulers used in construction drawing). The x values passed to the interpolator must be ordered in ascendig order (there is no such restriction for the y values, of course). It is not valid to evaluate the function for values outside the range x0..xN. Currently, the original array for the x values is referenced by the generated function, which is probably a bad idea.