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<document url="analysis.html">
  <properties>
    <title>The Commons Math User Guide - Numerical Analysis</title>
  </properties>
  <body>
    <section name="4 Numerical Analysis">
      <subsection name="4.1 Overview" href="overview">
        <p>
         The analysis package is the parent package for algorithms dealing with
         real-valued functions of one real variable. It contains dedicated sub-packages
         providing numerical root-finding, integration, interpolation and differentiation.
         It also contains a polynomials sub-package that considers polynomials with real
         coefficients as differentiable real functions.
        </p>
        <p>
         Functions interfaces are intended to be implemented by user code to represent
         their domain problems. The algorithms provided by the library will then operate
         on these function to find their roots, or integrate them, or ... Functions can
         be multivariate or univariate, real vectorial or matrix valued, and they can be
         differentiable or not.
        </p>
      </subsection>
      <subsection name="4.2 Error handling" href="errorhandling">
        <p>
          For user-defined functions, when the method encounters an error
          during evaluation, users must use their <em>own</em> unchecked exceptions.
          The following example shows the recommended way to do that, using root
          solving as the example (the same construct should be used for ODE
          integrators or for optimizations).
        </p>
        <source>private static class LocalException extends RuntimeException {

   // the x value that caused the problem
   private final double x;

   public LocalException(double x) {
     this.x = x;
   }

   public double getX() {
     return x;
   }

 }

 private static class MyFunction implements UnivariateFunction {
   public double value(double x) {
     double y = hugeFormula(x);
     if (somethingBadHappens) {
       throw new LocalException(x);
     }
     return y;
   }
 }

 public void compute() {
   try {
     solver.solve(maxEval, new MyFunction(a, b, c), min, max);
   } catch (LocalException le) {
     // retrieve the x value
   }
 }
 </source>
      <p>
        As shown in this example the exception is really something local to user code
        and there is a guarantee Apache Commons Math will not mess with it.
        The user is safe.
      </p>
      </subsection>
      <subsection name="4.3 Root-finding" href="rootfinding">
        <p>
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/UnivariateSolver.html">
          UnivariateSolver</a>, <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/UnivariateDifferentiableSolver.html">
          UnivariateDifferentiableSolver</a> and <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/PolynomialSolver.html">
          PolynomialSolver</a> provide means to find roots of
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a>,
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateDifferentiable.html">differentiable univariate real-valued functions</a>,
          and <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/polynomials/PolynomialFunction.html">polynomial functions</a> respectively.
          A root is the value where the function takes the value 0.  Commons-Math
          includes implementations of the several root-finding algorithms:
        </p>
          <table border="1" align="center">
            <tr BGCOLOR="#CCCCFF"><td colspan="5"><font size="+1">Root solvers</font></td></tr>
            <tr BGCOLOR="#EEEEFF"><font size="+1"><td>Name</td><td>Function type</td><td>Convergence</td><td>Needs initial bracketing</td><td>Bracket side selection</td></font></tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/BisectionSolver.html">Bisection</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>linear, guaranteed</td>
              <td>yes</td>
              <td>yes</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/BrentSolver.html">Brent-Dekker</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>super-linear, guaranteed</td>
              <td>yes</td>
              <td>no</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/BracketingNthOrderBrentSolver.html">bracketing n<sup>th</sup> order Brent</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>variable order, guaranteed</td>
              <td>yes</td>
              <td>yes</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/IllinoisSolver.html">Illinois Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>super-linear, guaranteed</td>
              <td>yes</td>
              <td>yes</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/LaguerreSolver.html">Laguerre's Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/polynomials/PolynomialFunction.html">polynomial functions</a></td>
              <td>cubic for simple root, linear for multiple root</td>
              <td>yes</td>
              <td>no</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/MullerSolver.html">Muller's Method</a> using bracketing to deal with real-valued functions</td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>quadratic close to roots</td>
              <td>yes</td>
              <td>no</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/MullerSolver2.html">Muller's Method</a> using modulus to deal with real-valued functions</td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>quadratic close to root</td>
              <td>yes</td>
              <td>no</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/NewtonRaphsonSolver.html">Newton-Raphson's Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateDifferentiableFunction.html">differentiable univariate real-valued functions</a></td>
              <td>quadratic, non-guaranteed</td>
              <td>no</td>
              <td>no</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/PegasusSolver.html">Pegasus Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>super-linear, guaranteed</td>
              <td>yes</td>
              <td>yes</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/RegulaFalsiSolver.html">Regula Falsi (false position) Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>linear, guaranteed</td>
              <td>yes</td>
              <td>yes</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/RiddersSolver.html">Ridder's Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>super-linear</td>
              <td>yes</td>
              <td>no</td>
            </tr>
            <tr>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/SecantSolver.html">Secant Method</a></td>
              <td><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">univariate real-valued functions</a></td>
              <td>super-linear, non-guaranteed</td>
              <td>yes</td>
              <td>no</td>
            </tr>
          </table>
        <p>
          Some algorithms require that the initial search interval brackets the root
          (i.e. the function values at interval end points have opposite signs).  Some
          algorithms preserve bracketing throughout computation and allow user to
          specify which side of the convergence interval to select as the root.  It is
          also possible to force a side selection after a root has been found even
          for algorithms that do not provide this feature by themselves.  This is
          useful for example in sequential search, for which a new search interval is
          started after a root has been found in order to find the next root.  In this
          case, user must select a side to ensure his loop is not stuck on one root
          and always return the same solution without making any progress.
        </p>
        <p>
          There are numerous non-obvious traps and pitfalls in root finding.
          First, the usual disclaimers due to the way 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 will not 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,  which may be 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
          ill-conditioned problems are met.  A way around this is to transform
          the problem in order to get a better conditioned function.  Proper
          selection of a root-finding algorithm and its configuration parameters
          requires knowledge of the analytical properties of the function under
          analysis and numerical analysis techniques.  Users are encouraged
          to consult a numerical analysis text (or a numerical analyst) when
          selecting and configuring a solver.
        </p>
        <p>
          In order to use the root-finding features, first a solver object must
          be created by calling its constructor, often providing relative and absolute
          accuracy. Using a solver object, roots of functions
          are easily found using the <code>solve</code> methods.  These methods takes
          a maximum iteration count <code>maxEval</code>, a function <code>f</code>,
          and either two domain values, <code>min</code> and <code>max</code>, or a
          <code>startValue</code> as parameters. If the maximal number of iterations
          count is exceeded, non-convergence is assumed and a <code>ConvergenceException</code>
          exception is thrown.  A suggested value is 100, which should be plenty, given 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
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/BracketedUnivariateSolver.html">Bracketed
          solvers</a> also take an <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/AllowedSolution.html">allowed solution</a>
          enum parameter to specify which side of the final convergence interval should be
          selected as the root. It can be <code>ANY_SIDE</code>, <code>LEFT_SIDE</code>, <code>RIGHT_SIDE</code>,
          <code>BELOW_SIDE</code> or <code>ABOVE_SIDE</code>. Left and right are used to specify the root along
          the function parameter axis while below and above refer to the function value axis. The solve methods
          compute a value <code>c</code> such that:
          <ul>
            <li><code>f(c) = 0.0</code> (see "function value accuracy")</li>
            <li><code>min &lt;= c &lt;= max</code> (except for the secant method, which may find a solution outside the interval)</li>
          </ul>
        </p>
        <p>
          Typical usage:
        </p>
        <source>UnivariateFunction function = // some user-defined function object
final double relativeAccuracy = 1.0e-12;
final double absoluteAccuracy = 1.0e-8;
final int    maxOrder         = 5;
UnivariateSolver solver   = new BracketingNthOrderBrentSolver(relativeAccuracy, absoluteAccuracy, maxOrder);
double c = solver.solve(100, function, 1.0, 5.0, AllowedSolution.LEFT_SIDE);</source>
        <p>
          Force bracketing, by refining a base solution found by a non-bracketing solver:
        </p>
        <source>UnivariateFunction function = // some user-defined function object
final double relativeAccuracy = 1.0e-12;
final double absoluteAccuracy = 1.0e-8;
UnivariateSolver nonBracketing = new BrentSolver(relativeAccuracy, absoluteAccuracy);
double baseRoot = nonBracketing.solve(100, function, 1.0, 5.0);
double c = UnivariateSolverUtils.forceSide(100, function,
                                           new PegasusSolver(relativeAccuracy, absoluteAccuracy),
                                           baseRoot, 1.0, 5.0, AllowedSolution.LEFT_SIDE);
</source>
        <p>
          The <code>BrentSolver</code> uses the Brent-Dekker algorithm which is
          fast and robust.  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.
        </p>
        <p>
          The <code>BracketingNthOrderBrentSolver</code> uses an extension of the
          Brent-Dekker algorithm which uses inverse n<sup>th</sup> order polynomial
          interpolation instead of inverse quadratic interpolation, and which allows
          selection of the side of the convergence interval for result bracketing.
          This is now the recommended algorithm for most users since it has the
          largest order, doesn't require derivatives, has guaranteed convergence
          and allows result bracket selection.
        </p>
        <p>
          The <code>SecantSolver</code> uses a straightforward secant
          algorithm which does not bracket the search and therefore does not
          guarantee convergence.  It may be faster than Brent on some well-behaved
          functions.
        </p>
        <p>
          The <code>RegulaFalsiSolver</code> is variation of secant preserving
          bracketing, but then it may be slow, as one end point of the search interval
          will become fixed after and only the other end point will converge to the root,
          hence resulting in a search interval size that does not decrease to zero.
        </p>
        <p>
          The <code>IllinoisSolver</code> and <code>PegasusSolver</code> are
          well-known variations of regula falsi that fix the problem of stuck
          end points by slightly weighting one endpoint to balance the interval
          at next iteration. Pegasus is often faster than Illinois. Pegasus may
          be the algorithm of choice for selecting a specific side of the convergence
          interval.
        </p>
        <p>
          The <code>BisectionSolver</code> 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
          adverse circumstances which might cause other algorithms to
          malfunction.  The drawback is of course that it is also guaranteed
          to be slow.
        </p>
        <p>
          The <code>UnivariateSolver</code> interface exposes many
          properties to control the convergence of a solver.  The accuracy properties
          are set at solver instance creation and cannot be changed afterwards,
          there are only getters to retriveve their values, no setters are available.
          <table>
            <tr><th>Property</th><th>Purpose</th></tr>
            <tr>
              <td>Absolute accuracy</td>
              <td>
                The Absolute Accuracy is (estimated) maximal difference between
                the computed root and the true root of the function.  This is
                what most people think of as "accuracy" intuitively.  The default
                value is chosen 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 default absolute
                accuracy may never be reached because the given relative
                accuracy is reached first.
              </td>
            </tr>
              <tr>
              <td>Relative accuracy</td>
              <td>
                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
                better suited for numerical calculations with computers, due to
                the way floating point numbers are represented.  The default
                value is chosen 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.
              </td>
            </tr>
            <tr>
              <td>Function value accuracy</td>
              <td>
                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 sped up by setting this value
                appropriately.
              </td>
            </tr>
          </table>
        </p>
      </subsection>
      <subsection name="4.4 Interpolation" href="interpolation">
        <p>
          A <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/UnivariateInterpolator.html">
          UnivariateInterpolator</a> is used to find a univariate real-valued
          function <code>f</code> which for a given set of ordered pairs
          (<code>x<sub>i</sub></code>,<code>y<sub>i</sub></code>) yields
          <code>f(x<sub>i</sub>)=y<sub>i</sub></code> to the best accuracy possible. The result
          is provided as an object implementing the <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">
          UnivariateFunction</a> interface. It can therefore be evaluated at any point,
          including point not belonging to the original set.
          Currently, only an interpolator for generating natural cubic splines and a polynomial
          interpolator are 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, as interpolation
          accuracy is in general determined by the algorithm.
        </p>
        <p>Typical usage:</p>
        <source>double x[] = { 0.0, 1.0, 2.0 };
double y[] = { 1.0, -1.0, 2.0};
UnivariateInterpolator interpolator = new SplineInterpolator();
UnivariateFunction function = interpolator.interpolate(x, y);
double interpolationX = 0.5;
double interpolatedY = function.value(x);
System.out println("f(" + interpolationX + ") = " + interpolatedY);</source>
        <p>
          A natural cubic spline is a function consisting of a polynomial of
          third degree for each subinterval determined by the x-coordinates of the
          interpolated points.  A function interpolating <code>N</code>
          value pairs consists of <code>N-1</code> polynomials. The function
          is continuous, smooth and can be differentiated twice.  The second
          derivative is continuous but not smooth.  The x values passed to the
          interpolator must be ordered in ascending order.  It is not valid to
          evaluate the function for values outside the range
          <code>x<sub>0</sub></code>..<code>x<sub>N</sub></code>.
        </p>
        <p>
          The polynomial function returned by the Neville's algorithm is a single
          polynomial guaranteed to pass exactly through the interpolation points.
          The degree of the polynomial is the number of points minus 1 (i.e. the
          interpolation polynomial for a three points set will be a quadratic
          polynomial). Despite the fact the interpolating polynomials is a perfect
          approximation of a function at interpolation points, it may be a loose
          approximation between the points. Due to <a
          href="http://en.wikipedia.org/wiki/Runge's_phenomenon">Runge's phenomenom</a>
          the error can get worse as the degree of the polynomial increases, so
          adding more points does not always lead to a better interpolation.
        </p>
        <p>
          Loess (or Lowess) interpolation is a robust interpolation useful for
          smoothing univariate scaterplots. It has been described by William
          Cleveland in his 1979 seminal paper <a
          href="http://www.math.tau.ac.il/~yekutiel/MA%20seminar/Cleveland%201979.pdf">Robust
          Locally Weighted Regression and Smoothing Scatterplots</a>. This kind of
          interpolation is computationally intensive but robust.
        </p>
        <p>
          Microsphere interpolation is a robust multidimensional interpolation algorithm.
          It has been described in William Dudziak's <a
          href="http://www.dudziak.com/microsphere.pdf">MS thesis</a>.
        </p>
        <p>
          <a href="http://en.wikipedia.org/wiki/Hermite_interpolation">Hermite interpolation</a>
          is an interpolation method that can use derivatives in addition to function values at sample points. The <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/HermiteInterpolator.html">HermiteInterpolator</a>
          class implements this method for vector-valued functions. The sampling points can have any spacing (there are
          no requirements for a regular grid) and some points may provide derivatives while others don't provide them
          (or provide derivatives to a smaller order). Points are added one at a time, as shown in the following example:
        </p>
        <source>HermiteInterpolator interpolator = new HermiteInterpolator;
// at x = 0, we provide both value and first derivative
interpolator.addSamplePoint(0.0, new double[] { 1.0 }, new double[] { 2.0 });
// at x = 1, we provide only function value
interpolator.addSamplePoint(1.0, new double[] { 4.0 });
// at x = 2, we provide both value and first derivative
interpolator.addSamplePoint(2.0, new double[] { 5.0 }, new double[] { 2.0 });
// should print "value at x = 0.5: 2.5625"
System.out.println("value at x = 0.5: " + interpolator.value(0.5)[0]);
// should print "derivative at x = 0.5: 3.5"
System.out.println("derivative at x = 0.5: " + interpolator.derivative(0.5)[0]);
// should print "interpolation polynomial: 1 + 2 x + 4 x^2 - 4 x^3 + x^4"
System.out.println("interpolation polynomial: " + interpolator.getPolynomials()[0]);</source>
        <p>
          A <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/BivariateGridInterpolator.html">
          BivariateGridInterpolator</a> is used to find a bivariate real-valued
          function <code>f</code> which for a given set of tuples
          (<code>x<sub>i</sub></code>,<code>y<sub>j</sub></code>,<code>f<sub>ij</sub></code>)
          yields <code>f(x<sub>i</sub>,y<sub>j</sub>)=f<sub>ij</sub></code> to the best accuracy
          possible. The result is provided as an object implementing the
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/BivariateFunction.html">
          BivariateFunction</a> interface. It can therefore be evaluated at any point,
          including a point not belonging to the original set.
          The arrays <code>x<sub>i</sub></code> and <code>y<sub>j</sub></code> must be
          sorted in increasing order in order to define a two-dimensional grid.
        </p>
        <p>
          In <a href="http://en.wikipedia.org/wiki/Bicubic_interpolation">bicubic interpolation</a>,
          the interpolation function is a 3rd-degree polynomial of two variables. The coefficients
          are computed from the function values sampled on a grid, as well as the values of the
          partial derivatives of the function at those grid points.
          From two-dimensional data sampled on a grid, the
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/BicubicSplineInterpolator.html">
          BicubicSplineInterpolator</a> computes a
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/BicubicSplineInterpolatingFunction.html">
          bicubic interpolating function</a>.
          Prior to computing an interpolating function, the
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/SmoothingPolynomialBicubicSplineInterpolator.html">
          SmoothingPolynomialBicubicSplineInterpolator</a> class performs smoothing of
          the data by computing the polynomial that best fits each of the one-dimensional
          curves along each of the coordinate axes.
        </p>
        <p>
          A <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/TrivariateGridInterpolator.html">
          TrivariateGridInterpolator</a> is used to find a trivariate real-valued
          function <code>f</code> which for a given set of tuples
          (<code>x<sub>i</sub></code>,<code>y<sub>j</sub></code>,<code>z<sub>k</sub></code>,
          <code>f<sub>ijk</sub></code>)
          yields <code>f(x<sub>i</sub>,y<sub>j</sub>,z<sub>k</sub>)=f<sub>ijk</sub></code>
          to the best accuracy possible. The result is provided as an object implementing the
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/TrivariateFunction.html">
          TrivariateFunction</a> interface. It can therefore be evaluated at any point,
          including a point not belonging to the original set.
          The arrays <code>x<sub>i</sub></code>, <code>y<sub>j</sub></code> and
          <code>z<sub>k</sub></code> must be sorted in increasing order in order to define
          a three-dimensional grid.
        </p>
        <p>
          In <a href="http://en.wikipedia.org/wiki/Tricubic_interpolation">tricubic interpolation</a>,
          the interpolation function is a 3rd-degree polynomial of three variables. The coefficients
          are computed from the function values sampled on a grid, as well as the values of the
          partial derivatives of the function at those grid points.
          From three-dimensional data sampled on a grid, the
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/TricubicSplineInterpolator.html">
          TricubicSplineInterpolator</a> computes a
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/interpolation/TricubicSplineInterpolatingFunction.html">
          tricubic interpolating function</a>.
        </p>
      </subsection>
      <subsection name="4.5 Integration" href="integration">
        <p>
          A <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/integration/UnivariateIntegrator.html">
          UnivariateIntegrator</a> provides the means to numerically integrate
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">
          univariate real-valued functions</a>.
          Commons-Math includes implementations of the following integration algorithms: <ul>
          <li><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/integration/RombergIntegrator.html">
          Romberg's method</a></li>
          <li><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/integration/SimpsonIntegrator.html">
          Simpson's method</a></li>
          <li><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/integration/TrapezoidIntegrator.html">
          trapezoid method</a></li>
          <li><a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/integration/LegendreGaussIntegrator.html">
          Legendre-Gauss method</a></li>
          </ul>
        </p>
      </subsection>
      <subsection name="4.6 Polynomials" href="polynomials">
        <p>
          The <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/polynomials/package-summary.html">
          org.apache.commons.math4.analysis.polynomials</a> package provides real
          coefficients polynomials.
        </p>
        <p>
          The <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/polynomials/PolynomialFunction.html">
          PolynomialFunction</a> class is the most general one, using traditional
          coefficients arrays. The
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/polynomials/PolynomialsUtils.html">
          PolynomialsUtils</a> utility class provides static factory methods to build
          Chebyshev, Hermite, Jacobi, Laguerre and Legendre polynomials. Coefficients are
          computed using exact fractions so these factory methods can build polynomials
          up to any degree.
        </p>
      </subsection>
      <subsection name="4.7 Differentiation" href="differentiation">
        <p>
          The <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/package-summary.html">
          org.apache.commons.math4.analysis.differentiation</a> package provides a general-purpose
          differentiation framework.
        </p>
        <p>
          The core class is <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/DerivativeStructure.html">
          DerivativeStructure</a> which holds the value and the differentials of a function. This class
          handles some arbitrary number of free parameters and arbitrary derivation order. It is used
          both as the input and the output type for the <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateDifferentiableFunction.html">
          UnivariateDifferentiableFunction</a> interface. Any differentiable function should implement this
          interface.
        </p>
        <p>
          The main idea behind the <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/DerivativeStructure.html">
          DerivativeStructure</a> class is that it can be used almost as a number (i.e. it can be added,
          multiplied, its square root can be extracted or its cosine computed... However, in addition to
          computed the value itself when doing these computations, the partial derivatives are also computed
          alongside. This is an extension of what is sometimes called Rall's numbers. This extension is
          described in Dan Kalman's paper <a
          href="http://www1.american.edu/cas/mathstat/People/kalman/pdffiles/mmgautodiff.pdf">Doubly Recursive
          Multivariate Automatic Differentiation</a>, Mathematics Magazine, vol. 75, no. 3, June 2002.
          Rall's numbers only hold the first derivative with respect to one free parameter whereas Dan Kalman's
          derivative structures hold all partial derivatives up to any specified order, with respect to any
          number of free parameters. Rall's numbers therefore can be seen as derivative structures for order
          one derivative and one free parameter, and primitive real numbers can be seen as derivative structures
          with zero order derivative and no free parameters.
        </p>
        <p>
          The workflow of computation of a derivatives of an expression <code>y=f(x)</code> is the following
          one. First we configure an input parameter <code>x</code> of type <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/DerivativeStructure.html">
          DerivativeStructure</a> so it will drive the function to compute all derivatives up to order 3 for
          example. Then we compute <code>y=f(x)</code> normally by passing this parameter to the f function.At
          the end, we extract from <code>y</code> the value and the derivatives we want. As we have specified
          3<sup>rd</sup> order when we built <code>x</code>, we can retrieve the derivatives up to 3<sup>rd</sup>
          order from <code>y</code>. The following example shows that (the 0 parameter in the DerivativeStructure
          constructor will be explained in the next paragraph):
       </p>
       <source>int params = 1;
int order = 3;
double xRealValue = 2.5;
DerivativeStructure x = new DerivativeStructure(params, order, 0, xRealValue);
DerivativeStructure y = f(x);
System.out.println("y    = " + y.getValue();
System.out.println("y'   = " + y.getPartialDerivative(1);
System.out.println("y''  = " + y.getPartialDerivative(2);
System.out.println("y''' = " + y.getPartialDerivative(3);</source>
       <p>
         In fact, there are no notions of <em>variables</em> in the framework, so neither <code>x</code>
         nor <code>y</code> are considered to be variables per se. They are both considered to be
         <em>functions</em> and to depend on implicit free parameters which are represented only by
         indices in the framework. The <code>x</code> instance above is there considered by the framework
         to be a function of free parameter <code>p0</code> at index 0, and as <code>y</code> is
         computed from <code>x</code> it is the result of a functions composition and is therefore also
         a function of this <code>p0</code> free parameter. The <code>p0</code> is not represented by itself,
         it is simply defined implicitely by the 0 index above. This index is the third argument in the
         constructor of the <code>x</code> instance. What this constructor means is that we built
         <code>x</code> as a function that depends on one free parameter only (first constructor argument
         set to 1), that can be differentiated up to order 3 (second constructor argument set to 3), and
         which correspond to an identity function with respect to implicit free parameter number 0 (third
         constructor argument set to 0), with current value equal to 2.5 (fourth constructor argument set
         to 2.5). This specific constructor defines identity functions, and identity functions are the trick
         we use to represent variables (there are of course other constructors, for example to build constants
         or functions from all their derivatives if they are known beforehand). From the user point of view,
         the <code>x</code> instance can be seen as the <code>x</code> variable, but it is really the identity
         function applied to free parameter number 0. As the identity function, it has the same value as its
         parameter, its first derivative is 1.0 with respect to this free parameter, and all its higher order
         derivatives are 0.0. This can be checked by calling the getValue() or getPartialDerivative() methods
         on <code>x</code>.
       </p>
       <p>
         When we compute <code>y</code> from this setting, what we really do is chain <code>f</code> after the
         identity function, so the net result is that the derivatives are computed with respect to the indexed
         free parameters (i.e. only free parameter number 0 here since there is only one free parameter) of the
         identity function x. Going one step further, if we compute <code>z = g(y)</code>, we will also compute
         <code>z</code> as a function of the initial free parameter. The very important consequence is that
         if we call <code>z.getPartialDerivative(1)</code>, we will not get the first derivative of <code>g</code>
         with respect to <code>y</code>, but with respect to the free parameter <code>p0</code>: the derivatives
         of g and f <em>will</em> be chained together automatically, without user intervention.
       </p>
       <p>
         This design choice is a very classical one in many algorithmic differentiation frameworks, either
         based on operator overloading (like the one we implemented here) or based on code generation. It implies
         the user has to <em>bootstrap</em> the system by providing initial derivatives, and this is essentially
         done by setting up identity function, i.e. functions that represent the variables themselves and have
         only unit first derivative.
       </p>
       <p>
         This design also allow a very interesting feature which can be explained with the following example.
         Suppose we have a two arguments function <code>f</code> and a one argument function <code>g</code>. If
         we compute <code>g(f(x, y))</code> with <code>x</code> and <code>y</code> be two variables, we
         want to be able to compute the partial derivatives <code>dg/dx</code>, <code>dg/dy</code>,
         <code>d2g/dx2</code> <code>d2g/dxdy</code> <code>d2g/dy2</code>. This does make sense since we combined
         the two functions, and it does make sense despite g is a one argument function only. In order to do
         this, we simply set up <code>x</code> as an identity function of an implicit free parameter
         <code>p0</code> and <code>y</code> as an identity function of a different implicit free parameter
         <code>p1</code> and compute everything directly. In order to be able to combine everything, however,
         both <code>x</code> and <code>y</code> must be built with the appropriate dimensions, so they will both
         be declared to handle two free parameters, but <code>x</code> will depend only on parameter 0 while
         <code>y</code> will depend on parameter 1. Here is how we do this (note that
         <code>getPartialDerivative</code> is a variable arguments method which take as arguments the derivation
         order with respect to all free parameters, i.e. the first argument is derivation order with respect to
         free parameter 0 and the second argument is derivation order with respect to free parameter 1):
       </p>
       <source>int params = 2;
int order = 2;
double xRealValue =  2.5;
double yRealValue = -1.3;
DerivativeStructure x = new DerivativeStructure(params, order, 0, xRealValue);
DerivativeStructure y = new DerivativeStructure(params, order, 1, yRealValue);
DerivativeStructure f = DerivativeStructure.hypot(x, y);
DerivativeStructure g = f.log();
System.out.println("g        = " + g.getValue();
System.out.println("dg/dx    = " + g.getPartialDerivative(1, 0);
System.out.println("dg/dy    = " + g.getPartialDerivative(0, 1);
System.out.println("d2g/dx2  = " + g.getPartialDerivative(2, 0);
System.out.println("d2g/dxdy = " + g.getPartialDerivative(1, 1);
System.out.println("d2g/dy2  = " + g.getPartialDerivative(0, 2);</source>
       <p>
          There are several ways a user can create an implementation of the <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateDifferentiableFunction.html">
          UnivariateDifferentiableFunction</a> interface. The first method is to simply write it directly using
          the appropriate methods from <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/DerivativeStructure.html">
          DerivativeStructure</a> to compute addition, subtraction, sine, cosine... This is often quite
          straigthforward and there is no need to remember the rules for differentiation: the user code only
          represent the function itself, the differentials will be computed automatically under the hood. The
          second method is to write a classical <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/UnivariateFunction.html">UnivariateFunction</a> and to
          pass it to an existing implementation of the <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateFunctionDifferentiator.html">
          UnivariateFunctionDifferentiator</a> interface to retrieve a differentiated version of the same function.
          The first method is more suited to small functions for which user already control all the underlying code.
          The second method is more suited to either large functions that would be cumbersome to write using the
          <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/DerivativeStructure.html">
          DerivativeStructure</a> API, or functions for which user does not have control to the full underlying code
          (for example functions that call external libraries).
        </p>
        <p>
          Apache Commons Math provides one implementation of the <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateFunctionDifferentiator.html">
          UnivariateFunctionDifferentiator</a> interface: <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/FiniteDifferencesDifferentiator.html">
          FiniteDifferencesDifferentiator</a>. This class creates a wrapper that will call the user-provided function
          on a grid sample and will use finite differences to compute the derivatives. It takes care of boundaries
          if the variable is not defined on the whole real line. It is possible to use more points than strictly
          required by the derivation order (for example one can specify an 8-points scheme to compute first
          derivative only). However, one must be aware that tuning the parameters for finite differences is
          highly problem-dependent. Choosing the wrong step size or the wrong number of sampling points can lead
          to huge errors. Finite differences are also not well suited to compute high order derivatives. Here is
          an example on how this implementation can be used:
        </p>
       <source>// function to be differentiated
UnivariateFunction basicF = new UnivariateFunction() {
                                public double value(double x) {
                                    return x * FastMath.sin(x);
                                }
                            });

// create a differentiator using 5 points and 0.01 step
FiniteDifferencesDifferentiator differentiator =
    new FiniteDifferencesDifferentiator(5, 0.01);

// create a new function that computes both the value and the derivatives
// using DerivativeStructure
UnivariateDifferentiableFunction completeF = differentiator.differentiate(basicF);

// now we can compute display the value and its derivatives
// here we decided to display up to second order derivatives,
// because we feed completeF with order 2 DerivativeStructure instances
for (double x = -10; x &lt; 10; x += 0.1) {
    DerivativeStructure xDS = new DerivativeStructure(1, 2, 0, x);
    DerivativeStructure yDS = f.value(xDS);
    System.out.format(Locale.US, "%7.3f %7.3f %7.3f%n",
                      yDS.getValue(),
                      yDS.getPartialDerivative(1),
                      yDS.getPartialDerivative(2));
}</source>
        <p>
          Note that using <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/FiniteDifferencesDifferentiator.html">
          FiniteDifferencesDifferentiator</a>a> in order to have a <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateDifferentiableFunction.html">
          UnivariateDifferentiableFunction</a> that can be provided to a <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/NewtonRaphsonSolver.html">Newton-Raphson's</a>
          solver is a very bad idea. The reason is that finite differences are not really accurate and needs lots
          of additional calls to the basic underlying function. If user initially have only the basic function
          available and needs to find its roots, it is <em>much</em> more accurate and <em>much</em> more
          efficient to use a solver that only requires the function values and not the derivatives. A good choice is
          to use <a href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/BracketingNthOrderBrentSolver.html">bracketing
          n<sup>th</sup> order Brent</a> method, which in fact converges faster than <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/solvers/NewtonRaphsonSolver.html">Newton-Raphson's</a> and
          can be configured to a highere order (typically 5) than Newton-Raphson which is an order 2 method.
        </p>
        <p>
          Another implementation of the <a
          href="../commons-math-docs/apidocs/org/apache/commons/math4/legacy/analysis/differentiation/UnivariateFunctionDifferentiator.html">
          UnivariateFunctionDifferentiator</a> interface is under development in the related project
          <a href="http://commons.apache.org/sandbox/nabla/">Apache Commons Nabla</a>. This implementation uses
          automatic code analysis and generation at binary level. However, at time of writing
          (end 2012), this project is not yet suitable for production use.
        </p>
      </subsection>
    </section>
  </body>
</document>