
f
,
and either two domain values, <code>min and max
, or a
<code>startValue as parameters. If the maximal number of iterations
count is exceeded, nonconvergence is assumed and a <code>ConvergenceException
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
illconditioned problems is to be solved, this number can be decreased in order
to avoid wasting time.
<a
href="../apidocs/org/apache/commons/math3/analysis/solvers/BracketedUnivariateSolver.html">Bracketed
solvers</a> also take an
enum parameter to specify which side of the final convergence interval should be
selected as the root. It can be <code>ANY_SIDE, LEFT_SIDE
, RIGHT_SIDE
,
<code>BELOW_SIDE or ABOVE_SIDE
. 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 such that:
<ul>
<li>f(c) = 0.0
(see "function value accuracy")
<li>min <= c <= max
(except for the secant method, which may find a solution outside the interval)
</ul>
</p>
<p>
Typical usage:
</p>
<source>UnivariateFunction function = // some user defined function object
final double relativeAccuracy = 1.0e12;
final double absoluteAccuracy = 1.0e8;
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 nonbracketing solver:
</p>
<source>UnivariateFunction function = // some user defined function object
final double relativeAccuracy = 1.0e12;
final double absoluteAccuracy = 1.0e8;
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 uses the BrentDekker 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 uses an extension of the
BrentDekker algorithm which uses inverse n<sup>th 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 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 wellbehaved
functions.
</p>
<p>
The <code>RegulaFalsiSolver 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 and PegasusSolver
are
wellknown 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 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 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>
<tr>
<td>Absolute accuracy
<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>
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>
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="../apidocs/org/apache/commons/math3/analysis/interpolation/UnivariateInterpolator.html">
UnivariateInterpolator</a> is used to find a univariate realvalued
function <code>f which for a given set of ordered pairs
(<code>x_{i},y_{i}
) yields
<code>f(x_{i})=y_{i} to the best accuracy possible. The result
is provided as an object implementing the <a
href="../apidocs/org/apache/commons/math3/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:
<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 xcoordinates of the
interpolated points. A function interpolating <code>N
value pairs consists of <code>N1 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_{0}..x_{N}
.
</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
is an interpolation method that can use derivatives in addition to function values at sample points. The <a
href="../apidocs/org/apache/commons/math3/analysis/interpolation/HermiteInterpolator.html">HermiteInterpolator</a>
class implements this method for vectorvalued 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="../apidocs/org/apache/commons/math3/analysis/interpolation/BivariateGridInterpolator.html">
BivariateGridInterpolator</a> is used to find a bivariate realvalued
function <code>f which for a given set of tuples
(<code>x_{i},y_{j}
,f_{ij}
)
yields <code>f(x_{i},y_{j})=f_{ij} to the best accuracy
possible. The result is provided as an object implementing the
<a href="../apidocs/org/apache/commons/math3/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_{i} and y_{j}
must be
sorted in increasing order in order to define a twodimensional grid.
</p>
<p>
In <a href="http://en.wikipedia.org/wiki/Bicubic_interpolation">bicubic interpolation,
the interpolation function is a 3rddegree 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 twodimensional data sampled on a grid, the
<a href="../apidocs/org/apache/commons/math3/analysis/interpolation/BicubicSplineInterpolator.html">
BicubicSplineInterpolator</a> computes a
<a href="../apidocs/org/apache/commons/math3/analysis/interpolation/BicubicSplineInterpolatingFunction.html">
bicubic interpolating function</a>.
Prior to computing an interpolating function, the
<a href="../apidocs/org/apache/commons/math3/analysis/interpolation/SmoothingPolynomialBicubicSplineInterpolator.html">
SmoothingPolynomialBicubicSplineInterpolator</a> class performs smoothing of
the data by computing the polynomial that best fits each of the onedimensional
curves along each of the coordinate axes.
</p>
<p>
A <a href="../apidocs/org/apache/commons/math3/analysis/interpolation/TrivariateGridInterpolator.html">
TrivariateGridInterpolator</a> is used to find a trivariate realvalued
function <code>f which for a given set of tuples
(<code>x_{i},y_{j}
,z_{k}
,
<code>f_{ijk})
yields <code>f(x_{i},y_{j},z_{k})=f_{ijk}
to the best accuracy possible. The result is provided as an object implementing the
<a href="../apidocs/org/apache/commons/math3/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_{i}, y_{j}
and
<code>z_{k} must be sorted in increasing order in order to define
a threedimensional grid.
</p>
<p>
In <a href="http://en.wikipedia.org/wiki/Tricubic_interpolation">tricubic interpolation,
the interpolation function is a 3rddegree 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 threedimensional data sampled on a grid, the
<a href="../apidocs/org/apache/commons/math3/analysis/interpolation/TricubicSplineInterpolator.html">
TricubicSplineInterpolator</a> computes a
<a href="../apidocs/org/apache/commons/math3/analysis/interpolation/TricubicSplineInterpolatingFunction.html">
tricubic interpolating function</a>.
</p>
</subsection>
<subsection name="4.5 Integration" href="integration">
<p>
A <a href="../apidocs/org/apache/commons/math3/analysis/integration/UnivariateIntegrator.html">
UnivariateIntegrator</a> provides the means to numerically integrate
<a href="../apidocs/org/apache/commons/math3/analysis/UnivariateFunction.html">
univariate realvalued functions</a>.
CommonsMath includes implementations of the following integration algorithms: <ul>
<li>
Romberg's method</a>
<li>
Simpson's method</a>
<li>
trapezoid method</a>
<li>
LegendreGauss method</a>
</ul>
</p>
</subsection>
<subsection name="4.6 Polynomials" href="polynomials">
<p>
The <a href="../apidocs/org/apache/commons/math3/analysis/polynomials/packagesummary.html">
org.apache.commons.math3.analysis.polynomials</a> package provides real
coefficients polynomials.
</p>
<p>
The <a href="../apidocs/org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html">
PolynomialFunction</a> class is the most general one, using traditional
coefficients arrays. The
<a href="../apidocs/org/apache/commons/math3/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="../apidocs/org/apache/commons/math3/analysis/differentiation/packagesummary.html">
org.apache.commons.math3.analysis.differentiation</a> package provides a generalpurpose
differentiation framework.
</p>
<p>
The core class is <a href="../apidocs/org/apache/commons/math3/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="../apidocs/org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableFunction.html">
UnivariateDifferentiableFunction</a> interface. Any differentiable function should implement this
interface.
</p>
<p>
The main idea behind the <a href="../apidocs/org/apache/commons/math3/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) is the following
one. First we configure an input parameter <code>x of type normally by passing this parameter to the f function.At
the end, we extract from <code>y the value and the derivatives we want. As we have specified
3<sup>rd order when we built x
, we can retrieve the derivatives up to 3^{rd}
order from <code>y. 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 in the framework, so neither x
nor <code>y are considered to be variables per se. They are both considered to be
<em>functions and to depend on implicit free parameters which are represented only by
indices in the framework. The <code>x instance above is there considered by the framework
to be a function of free parameter <code>p0 at index 0, and as y
is
computed from <code>x it is the result of a functions composition and is therefore also
a function of this <code>p0 free parameter. The p0
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 instance. What this constructor means is that we built
<code>x 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 instance can be seen as the x
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.
</p>
<p>
When we compute <code>y from this setting, what we really do is chain f
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), we will also compute
<code>z as a function of the initial free parameter. The very important consequence is that
if we call <code>z.getPartialDerivative(1), we will not get the first derivative of g
with respect to <code>y, but with respect to the free parameter p0
: the derivatives
of g and f <em>will 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 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 and a one argument function g
. If
we compute <code>g(f(x, y)) with x
and y
be two variables, we
want to be able to compute the partial derivatives <code>dg/dx, dg/dy
,
<code>d2g/dx2 d2g/dxdy
d2g/dy2
. 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 as an identity function of an implicit free parameter
<code>p0 and y
as an identity function of a different implicit free parameter
<code>p1 and compute everything directly. In order to be able to combine everything, however,
both <code>x and y
must be built with the appropriate dimensions, so they will both
be declared to handle two free parameters, but <code>x will depend only on parameter 0 while
<code>y will depend on parameter 1. Here is how we do this (note that
<code>getPartialDerivative 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="../apidocs/org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableFunction.html">
UnivariateDifferentiableFunction</a> interface. The first method is to simply write it directly using
the appropriate methods from <a href="../apidocs/org/apache/commons/math3/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="../apidocs/org/apache/commons/math3/analysis/UnivariateFunction.html">UnivariateFunction</a> and to
pass it to an existing implementation of the <a
href="../apidocs/org/apache/commons/math3/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="../apidocs/org/apache/commons/math3/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="../apidocs/org/apache/commons/math3/analysis/differentiation/UnivariateFunctionDifferentiator.html">
UnivariateFunctionDifferentiator</a> interface: in order to have a NewtonRaphson's
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 more accurate and much more
efficient to use a solver that only requires the function values and not the derivatives. A good choice is
to use <a href="../apidocs/org/apache/commons/math3/analysis/solvers/BracketingNthOrderBrentSolver.html">bracketing
n<sup>th order Brent method, which in fact converges faster than and
can be configured to a highere order (typically 5) than NewtonRaphson which is an order 2 method.
</p>
<p>
Another implementation of the <a
href="../apidocs/org/apache/commons/math3/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. 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>
Here is a short list of links related to this Java analysis.xml source code file:

Java example source code file (analysis.xml)
The analysis.xml Java example source code<?xml version="1.0"?> <! Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to You under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. > <?xmlstylesheet type="text/xsl" href="./xdoc.xsl"?> <document url="analysis.html"> <properties> <title>The Commons Math User Guide  Numerical Analysis </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 realvalued functions of one real variable. It contains dedicated subpackages providing numerical rootfinding, integration, interpolation and differentiation. It also contains a polynomials subpackage 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 userdefined functions, when the method encounters an error during evaluation, users must use their <em>own 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 Rootfinding" href="rootfinding"> <p> <a href="../apidocs/org/apache/commons/math3/analysis/solvers/UnivariateSolver.html"> UnivariateSolver</a>, UnivariateDifferentiableSolver</a> and PolynomialSolver</a> provide means to find roots of <a href="../apidocs/org/apache/commons/math3/analysis/UnivariateFunction.html">univariate realvalued functions, <a href="../apidocs/org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiable.html">differentiable univariate realvalued functions, and <a href="../apidocs/org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html">polynomial functions respectively. A root is the value where the function takes the value 0. CommonsMath includes implementations of the several rootfinding algorithms: </p> <table border="1" align="center"> <tr BGCOLOR="#CCCCFF">  Root solvers  
Name  Function type  Convergence  Needs initial bracketing  Bracket side selection  
Property  Purpose 

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