001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.math3.analysis.differentiation;
018
019import java.io.Serializable;
020
021import org.apache.commons.math3.analysis.UnivariateFunction;
022import org.apache.commons.math3.analysis.UnivariateMatrixFunction;
023import org.apache.commons.math3.analysis.UnivariateVectorFunction;
024import org.apache.commons.math3.exception.MathIllegalArgumentException;
025import org.apache.commons.math3.exception.NotPositiveException;
026import org.apache.commons.math3.exception.NumberIsTooLargeException;
027import org.apache.commons.math3.exception.NumberIsTooSmallException;
028import org.apache.commons.math3.util.FastMath;
029
030/** Univariate functions differentiator using finite differences.
031 * <p>
032 * This class creates some wrapper objects around regular
033 * {@link UnivariateFunction univariate functions} (or {@link
034 * UnivariateVectorFunction univariate vector functions} or {@link
035 * UnivariateMatrixFunction univariate matrix functions}). These
036 * wrapper objects compute derivatives in addition to function
037 * value.
038 * </p>
039 * <p>
040 * The wrapper objects work by calling the underlying function on
041 * a sampling grid around the current point and performing polynomial
042 * interpolation. A finite differences scheme with n points is
043 * theoretically able to compute derivatives up to order n-1, but
044 * it is generally better to have a slight margin. The step size must
045 * also be small enough in order for the polynomial approximation to
046 * be good in the current point neighborhood, but it should not be too
047 * small because numerical instability appears quickly (there are several
048 * differences of close points). Choosing the number of points and
049 * the step size is highly problem dependent.
050 * </p>
051 * <p>
052 * As an example of good and bad settings, lets consider the quintic
053 * polynomial function {@code f(x) = (x-1)*(x-0.5)*x*(x+0.5)*(x+1)}.
054 * Since it is a polynomial, finite differences with at least 6 points
055 * should theoretically recover the exact same polynomial and hence
056 * compute accurate derivatives for any order. However, due to numerical
057 * errors, we get the following results for a 7 points finite differences
058 * for abscissae in the [-10, 10] range:
059 * <ul>
060 *   <li>step size = 0.25, second order derivative error about 9.97e-10</li>
061 *   <li>step size = 0.25, fourth order derivative error about 5.43e-8</li>
062 *   <li>step size = 1.0e-6, second order derivative error about 148</li>
063 *   <li>step size = 1.0e-6, fourth order derivative error about 6.35e+14</li>
064 * </ul>
065 * This example shows that the small step size is really bad, even simply
066 * for second order derivative!
067 * </p>
068 * @version $Id: FiniteDifferencesDifferentiator.java 1420666 2012-12-12 13:33:20Z erans $
069 * @since 3.1
070 */
071public class FiniteDifferencesDifferentiator
072    implements UnivariateFunctionDifferentiator, UnivariateVectorFunctionDifferentiator,
073               UnivariateMatrixFunctionDifferentiator, Serializable {
074
075    /** Serializable UID. */
076    private static final long serialVersionUID = 20120917L;
077
078    /** Number of points to use. */
079    private final int nbPoints;
080
081    /** Step size. */
082    private final double stepSize;
083
084    /** Half sample span. */
085    private final double halfSampleSpan;
086
087    /** Lower bound for independent variable. */
088    private final double tMin;
089
090    /** Upper bound for independent variable. */
091    private final double tMax;
092
093    /**
094     * Build a differentiator with number of points and step size when independent variable is unbounded.
095     * <p>
096     * Beware that wrong settings for the finite differences differentiator
097     * can lead to highly unstable and inaccurate results, especially for
098     * high derivation orders. Using very small step sizes is often a
099     * <em>bad</em> idea.
100     * </p>
101     * @param nbPoints number of points to use
102     * @param stepSize step size (gap between each point)
103     * @exception NotPositiveException if {@code stepsize <= 0} (note that
104     * {@link NotPositiveException} extends {@link NumberIsTooSmallException})
105     * @exception NumberIsTooSmallException {@code nbPoint <= 1}
106     */
107    public FiniteDifferencesDifferentiator(final int nbPoints, final double stepSize)
108        throws NotPositiveException, NumberIsTooSmallException {
109        this(nbPoints, stepSize, Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY);
110    }
111
112    /**
113     * Build a differentiator with number of points and step size when independent variable is bounded.
114     * <p>
115     * When the independent variable is bounded (tLower &lt; t &lt; tUpper), the sampling
116     * points used for differentiation will be adapted to ensure the constraint holds
117     * even near the boundaries. This means the sample will not be centered anymore in
118     * these cases. At an extreme case, computing derivatives exactly at the lower bound
119     * will lead the sample to be entirely on the right side of the derivation point.
120     * </p>
121     * <p>
122     * Note that the boundaries are considered to be excluded for function evaluation.
123     * </p>
124     * <p>
125     * Beware that wrong settings for the finite differences differentiator
126     * can lead to highly unstable and inaccurate results, especially for
127     * high derivation orders. Using very small step sizes is often a
128     * <em>bad</em> idea.
129     * </p>
130     * @param nbPoints number of points to use
131     * @param stepSize step size (gap between each point)
132     * @param tLower lower bound for independent variable (may be {@code Double.NEGATIVE_INFINITY}
133     * if there are no lower bounds)
134     * @param tUpper upper bound for independent variable (may be {@code Double.POSITIVE_INFINITY}
135     * if there are no upper bounds)
136     * @exception NotPositiveException if {@code stepsize <= 0} (note that
137     * {@link NotPositiveException} extends {@link NumberIsTooSmallException})
138     * @exception NumberIsTooSmallException {@code nbPoint <= 1}
139     * @exception NumberIsTooLargeException {@code stepSize * (nbPoints - 1) >= tUpper - tLower}
140     */
141    public FiniteDifferencesDifferentiator(final int nbPoints, final double stepSize,
142                                           final double tLower, final double tUpper)
143            throws NotPositiveException, NumberIsTooSmallException, NumberIsTooLargeException {
144
145        if (nbPoints <= 1) {
146            throw new NumberIsTooSmallException(stepSize, 1, false);
147        }
148        this.nbPoints = nbPoints;
149
150        if (stepSize <= 0) {
151            throw new NotPositiveException(stepSize);
152        }
153        this.stepSize = stepSize;
154
155        halfSampleSpan = 0.5 * stepSize * (nbPoints - 1);
156        if (2 * halfSampleSpan >= tUpper - tLower) {
157            throw new NumberIsTooLargeException(2 * halfSampleSpan, tUpper - tLower, false);
158        }
159        final double safety = FastMath.ulp(halfSampleSpan);
160        this.tMin = tLower + halfSampleSpan + safety;
161        this.tMax = tUpper - halfSampleSpan - safety;
162
163    }
164
165    /**
166     * Get the number of points to use.
167     * @return number of points to use
168     */
169    public int getNbPoints() {
170        return nbPoints;
171    }
172
173    /**
174     * Get the step size.
175     * @return step size
176     */
177    public double getStepSize() {
178        return stepSize;
179    }
180
181    /**
182     * Evaluate derivatives from a sample.
183     * <p>
184     * Evaluation is done using divided differences.
185     * </p>
186     * @param t evaluation abscissa value and derivatives
187     * @param t0 first sample point abscissa
188     * @param y function values sample {@code y[i] = f(t[i]) = f(t0 + i * stepSize)}
189     * @return value and derivatives at {@code t}
190     * @exception NumberIsTooLargeException if the requested derivation order
191     * is larger or equal to the number of points
192     */
193    private DerivativeStructure evaluate(final DerivativeStructure t, final double t0,
194                                         final double[] y)
195        throws NumberIsTooLargeException {
196
197        // create divided differences diagonal arrays
198        final double[] top    = new double[nbPoints];
199        final double[] bottom = new double[nbPoints];
200
201        for (int i = 0; i < nbPoints; ++i) {
202
203            // update the bottom diagonal of the divided differences array
204            bottom[i] = y[i];
205            for (int j = 1; j <= i; ++j) {
206                bottom[i - j] = (bottom[i - j + 1] - bottom[i - j]) / (j * stepSize);
207            }
208
209            // update the top diagonal of the divided differences array
210            top[i] = bottom[0];
211
212        }
213
214        // evaluate interpolation polynomial (represented by top diagonal) at t
215        final int order            = t.getOrder();
216        final int parameters       = t.getFreeParameters();
217        final double[] derivatives = t.getAllDerivatives();
218        final double dt0           = t.getValue() - t0;
219        DerivativeStructure interpolation = new DerivativeStructure(parameters, order, 0.0);
220        DerivativeStructure monomial = null;
221        for (int i = 0; i < nbPoints; ++i) {
222            if (i == 0) {
223                // start with monomial(t) = 1
224                monomial = new DerivativeStructure(parameters, order, 1.0);
225            } else {
226                // monomial(t) = (t - t0) * (t - t1) * ... * (t - t(i-1))
227                derivatives[0] = dt0 - (i - 1) * stepSize;
228                final DerivativeStructure deltaX = new DerivativeStructure(parameters, order, derivatives);
229                monomial = monomial.multiply(deltaX);
230            }
231            interpolation = interpolation.add(monomial.multiply(top[i]));
232        }
233
234        return interpolation;
235
236    }
237
238    /** {@inheritDoc}
239     * <p>The returned object cannot compute derivatives to arbitrary orders. The
240     * value function will throw a {@link NumberIsTooLargeException} if the requested
241     * derivation order is larger or equal to the number of points.
242     * </p>
243     */
244    public UnivariateDifferentiableFunction differentiate(final UnivariateFunction function) {
245        return new UnivariateDifferentiableFunction() {
246
247            /** {@inheritDoc} */
248            public double value(final double x) throws MathIllegalArgumentException {
249                return function.value(x);
250            }
251
252            /** {@inheritDoc} */
253            public DerivativeStructure value(final DerivativeStructure t)
254                throws MathIllegalArgumentException {
255
256                // check we can achieve the requested derivation order with the sample
257                if (t.getOrder() >= nbPoints) {
258                    throw new NumberIsTooLargeException(t.getOrder(), nbPoints, false);
259                }
260
261                // compute sample position, trying to be centered if possible
262                final double t0 = FastMath.max(FastMath.min(t.getValue(), tMax), tMin) - halfSampleSpan;
263
264                // compute sample points
265                final double[] y = new double[nbPoints];
266                for (int i = 0; i < nbPoints; ++i) {
267                    y[i] = function.value(t0 + i * stepSize);
268                }
269
270                // evaluate derivatives
271                return evaluate(t, t0, y);
272
273            }
274
275        };
276    }
277
278    /** {@inheritDoc}
279     * <p>The returned object cannot compute derivatives to arbitrary orders. The
280     * value function will throw a {@link NumberIsTooLargeException} if the requested
281     * derivation order is larger or equal to the number of points.
282     * </p>
283     */
284    public UnivariateDifferentiableVectorFunction differentiate(final UnivariateVectorFunction function) {
285        return new UnivariateDifferentiableVectorFunction() {
286
287            /** {@inheritDoc} */
288            public double[]value(final double x) throws MathIllegalArgumentException {
289                return function.value(x);
290            }
291
292            /** {@inheritDoc} */
293            public DerivativeStructure[] value(final DerivativeStructure t)
294                throws MathIllegalArgumentException {
295
296                // check we can achieve the requested derivation order with the sample
297                if (t.getOrder() >= nbPoints) {
298                    throw new NumberIsTooLargeException(t.getOrder(), nbPoints, false);
299                }
300
301                // compute sample position, trying to be centered if possible
302                final double t0 = FastMath.max(FastMath.min(t.getValue(), tMax), tMin) - halfSampleSpan;
303
304                // compute sample points
305                double[][] y = null;
306                for (int i = 0; i < nbPoints; ++i) {
307                    final double[] v = function.value(t0 + i * stepSize);
308                    if (i == 0) {
309                        y = new double[v.length][nbPoints];
310                    }
311                    for (int j = 0; j < v.length; ++j) {
312                        y[j][i] = v[j];
313                    }
314                }
315
316                // evaluate derivatives
317                final DerivativeStructure[] value = new DerivativeStructure[y.length];
318                for (int j = 0; j < value.length; ++j) {
319                    value[j] = evaluate(t, t0, y[j]);
320                }
321
322                return value;
323
324            }
325
326        };
327    }
328
329    /** {@inheritDoc}
330     * <p>The returned object cannot compute derivatives to arbitrary orders. The
331     * value function will throw a {@link NumberIsTooLargeException} if the requested
332     * derivation order is larger or equal to the number of points.
333     * </p>
334     */
335    public UnivariateDifferentiableMatrixFunction differentiate(final UnivariateMatrixFunction function) {
336        return new UnivariateDifferentiableMatrixFunction() {
337
338            /** {@inheritDoc} */
339            public double[][]  value(final double x) throws MathIllegalArgumentException {
340                return function.value(x);
341            }
342
343            /** {@inheritDoc} */
344            public DerivativeStructure[][]  value(final DerivativeStructure t)
345                throws MathIllegalArgumentException {
346
347                // check we can achieve the requested derivation order with the sample
348                if (t.getOrder() >= nbPoints) {
349                    throw new NumberIsTooLargeException(t.getOrder(), nbPoints, false);
350                }
351
352                // compute sample position, trying to be centered if possible
353                final double t0 = FastMath.max(FastMath.min(t.getValue(), tMax), tMin) - halfSampleSpan;
354
355                // compute sample points
356                double[][][] y = null;
357                for (int i = 0; i < nbPoints; ++i) {
358                    final double[][] v = function.value(t0 + i * stepSize);
359                    if (i == 0) {
360                        y = new double[v.length][v[0].length][nbPoints];
361                    }
362                    for (int j = 0; j < v.length; ++j) {
363                        for (int k = 0; k < v[j].length; ++k) {
364                            y[j][k][i] = v[j][k];
365                        }
366                    }
367                }
368
369                // evaluate derivatives
370                final DerivativeStructure[][] value = new DerivativeStructure[y.length][y[0].length];
371                for (int j = 0; j < value.length; ++j) {
372                    for (int k = 0; k < y[j].length; ++k) {
373                        value[j][k] = evaluate(t, t0, y[j][k]);
374                    }
375                }
376
377                return value;
378
379            }
380
381        };
382    }
383
384}