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Posted to commits@commons.apache.org by tn...@apache.org on 2015/02/16 23:40:19 UTC
[49/82] [partial] [math] Update for next development iteration:
commons-math4
http://git-wip-us.apache.org/repos/asf/commons-math/blob/a7b4803f/src/main/java/org/apache/commons/math3/analysis/differentiation/DSCompiler.java
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diff --git a/src/main/java/org/apache/commons/math3/analysis/differentiation/DSCompiler.java b/src/main/java/org/apache/commons/math3/analysis/differentiation/DSCompiler.java
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-/*
- * 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/LICENSE-2.0
- *
- * 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.
- */
-package org.apache.commons.math3.analysis.differentiation;
-
-import java.util.ArrayList;
-import java.util.Arrays;
-import java.util.List;
-import java.util.concurrent.atomic.AtomicReference;
-
-import org.apache.commons.math3.exception.DimensionMismatchException;
-import org.apache.commons.math3.exception.MathArithmeticException;
-import org.apache.commons.math3.exception.MathInternalError;
-import org.apache.commons.math3.exception.NotPositiveException;
-import org.apache.commons.math3.exception.NumberIsTooLargeException;
-import org.apache.commons.math3.util.CombinatoricsUtils;
-import org.apache.commons.math3.util.FastMath;
-import org.apache.commons.math3.util.MathArrays;
-
-/** Class holding "compiled" computation rules for derivative structures.
- * <p>This class implements the computation rules 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. However, in order to avoid performances bottlenecks, the recursive
- * rules are "compiled" once in an unfold form. This class does this recursion unrolling
- * and stores the computation rules as simple loops with pre-computed indirection arrays.</p>
- * <p>
- * This class maps all derivative computation into single dimension arrays that hold the
- * value and partial derivatives. The class does not hold these arrays, which remains under
- * the responsibility of the caller. For each combination of number of free parameters and
- * derivation order, only one compiler is necessary, and this compiler will be used to
- * perform computations on all arrays provided to it, which can represent hundreds or
- * thousands of different parameters kept together with all theur partial derivatives.
- * </p>
- * <p>
- * The arrays on which compilers operate contain only the partial derivatives together
- * with the 0<sup>th</sup> derivative, i.e. the value. The partial derivatives are stored in
- * a compiler-specific order, which can be retrieved using methods {@link
- * #getPartialDerivativeIndex(int...) getPartialDerivativeIndex} and {@link
- * #getPartialDerivativeOrders(int)}. The value is guaranteed to be stored as the first element
- * (i.e. the {@link #getPartialDerivativeIndex(int...) getPartialDerivativeIndex} method returns
- * 0 when called with 0 for all derivation orders and {@link #getPartialDerivativeOrders(int)
- * getPartialDerivativeOrders} returns an array filled with 0 when called with 0 as the index).
- * </p>
- * <p>
- * Note that the ordering changes with number of parameters and derivation order. For example
- * given 2 parameters x and y, df/dy is stored at index 2 when derivation order is set to 1 (in
- * this case the array has three elements: f, df/dx and df/dy). If derivation order is set to
- * 2, then df/dy will be stored at index 3 (in this case the array has six elements: f, df/dx,
- * df/dxdx, df/dy, df/dxdy and df/dydy).
- * </p>
- * <p>
- * Given this structure, users can perform some simple operations like adding, subtracting
- * or multiplying constants and negating the elements by themselves, knowing if they want to
- * mutate their array or create a new array. These simple operations are not provided by
- * the compiler. The compiler provides only the more complex operations between several arrays.
- * </p>
- * <p>This class is mainly used as the engine for scalar variable {@link DerivativeStructure}.
- * It can also be used directly to hold several variables in arrays for more complex data
- * structures. User can for example store a vector of n variables depending on three x, y
- * and z free parameters in one array as follows:
- * <pre>
- * // parameter 0 is x, parameter 1 is y, parameter 2 is z
- * int parameters = 3;
- * DSCompiler compiler = DSCompiler.getCompiler(parameters, order);
- * int size = compiler.getSize();
- *
- * // pack all elements in a single array
- * double[] array = new double[n * size];
- * for (int i = 0; i < n; ++i) {
- *
- * // we know value is guaranteed to be the first element
- * array[i * size] = v[i];
- *
- * // we don't know where first derivatives are stored, so we ask the compiler
- * array[i * size + compiler.getPartialDerivativeIndex(1, 0, 0) = dvOnDx[i][0];
- * array[i * size + compiler.getPartialDerivativeIndex(0, 1, 0) = dvOnDy[i][0];
- * array[i * size + compiler.getPartialDerivativeIndex(0, 0, 1) = dvOnDz[i][0];
- *
- * // we let all higher order derivatives set to 0
- *
- * }
- * </pre>
- * Then in another function, user can perform some operations on all elements stored
- * in the single array, such as a simple product of all variables:
- * <pre>
- * // compute the product of all elements
- * double[] product = new double[size];
- * prod[0] = 1.0;
- * for (int i = 0; i < n; ++i) {
- * double[] tmp = product.clone();
- * compiler.multiply(tmp, 0, array, i * size, product, 0);
- * }
- *
- * // value
- * double p = product[0];
- *
- * // first derivatives
- * double dPdX = product[compiler.getPartialDerivativeIndex(1, 0, 0)];
- * double dPdY = product[compiler.getPartialDerivativeIndex(0, 1, 0)];
- * double dPdZ = product[compiler.getPartialDerivativeIndex(0, 0, 1)];
- *
- * // cross derivatives (assuming order was at least 2)
- * double dPdXdX = product[compiler.getPartialDerivativeIndex(2, 0, 0)];
- * double dPdXdY = product[compiler.getPartialDerivativeIndex(1, 1, 0)];
- * double dPdXdZ = product[compiler.getPartialDerivativeIndex(1, 0, 1)];
- * double dPdYdY = product[compiler.getPartialDerivativeIndex(0, 2, 0)];
- * double dPdYdZ = product[compiler.getPartialDerivativeIndex(0, 1, 1)];
- * double dPdZdZ = product[compiler.getPartialDerivativeIndex(0, 0, 2)];
- * </p>
- * @see DerivativeStructure
- * @since 3.1
- */
-public class DSCompiler {
-
- /** Array of all compilers created so far. */
- private static AtomicReference<DSCompiler[][]> compilers =
- new AtomicReference<DSCompiler[][]>(null);
-
- /** Number of free parameters. */
- private final int parameters;
-
- /** Derivation order. */
- private final int order;
-
- /** Number of partial derivatives (including the single 0 order derivative element). */
- private final int[][] sizes;
-
- /** Indirection array for partial derivatives. */
- private final int[][] derivativesIndirection;
-
- /** Indirection array of the lower derivative elements. */
- private final int[] lowerIndirection;
-
- /** Indirection arrays for multiplication. */
- private final int[][][] multIndirection;
-
- /** Indirection arrays for function composition. */
- private final int[][][] compIndirection;
-
- /** Private constructor, reserved for the factory method {@link #getCompiler(int, int)}.
- * @param parameters number of free parameters
- * @param order derivation order
- * @param valueCompiler compiler for the value part
- * @param derivativeCompiler compiler for the derivative part
- * @throws NumberIsTooLargeException if order is too large
- */
- private DSCompiler(final int parameters, final int order,
- final DSCompiler valueCompiler, final DSCompiler derivativeCompiler)
- throws NumberIsTooLargeException {
-
- this.parameters = parameters;
- this.order = order;
- this.sizes = compileSizes(parameters, order, valueCompiler);
- this.derivativesIndirection =
- compileDerivativesIndirection(parameters, order,
- valueCompiler, derivativeCompiler);
- this.lowerIndirection =
- compileLowerIndirection(parameters, order,
- valueCompiler, derivativeCompiler);
- this.multIndirection =
- compileMultiplicationIndirection(parameters, order,
- valueCompiler, derivativeCompiler, lowerIndirection);
- this.compIndirection =
- compileCompositionIndirection(parameters, order,
- valueCompiler, derivativeCompiler,
- sizes, derivativesIndirection);
-
- }
-
- /** Get the compiler for number of free parameters and order.
- * @param parameters number of free parameters
- * @param order derivation order
- * @return cached rules set
- * @throws NumberIsTooLargeException if order is too large
- */
- public static DSCompiler getCompiler(int parameters, int order)
- throws NumberIsTooLargeException {
-
- // get the cached compilers
- final DSCompiler[][] cache = compilers.get();
- if (cache != null && cache.length > parameters &&
- cache[parameters].length > order && cache[parameters][order] != null) {
- // the compiler has already been created
- return cache[parameters][order];
- }
-
- // we need to create more compilers
- final int maxParameters = FastMath.max(parameters, cache == null ? 0 : cache.length);
- final int maxOrder = FastMath.max(order, cache == null ? 0 : cache[0].length);
- final DSCompiler[][] newCache = new DSCompiler[maxParameters + 1][maxOrder + 1];
-
- if (cache != null) {
- // preserve the already created compilers
- for (int i = 0; i < cache.length; ++i) {
- System.arraycopy(cache[i], 0, newCache[i], 0, cache[i].length);
- }
- }
-
- // create the array in increasing diagonal order
- for (int diag = 0; diag <= parameters + order; ++diag) {
- for (int o = FastMath.max(0, diag - parameters); o <= FastMath.min(order, diag); ++o) {
- final int p = diag - o;
- if (newCache[p][o] == null) {
- final DSCompiler valueCompiler = (p == 0) ? null : newCache[p - 1][o];
- final DSCompiler derivativeCompiler = (o == 0) ? null : newCache[p][o - 1];
- newCache[p][o] = new DSCompiler(p, o, valueCompiler, derivativeCompiler);
- }
- }
- }
-
- // atomically reset the cached compilers array
- compilers.compareAndSet(cache, newCache);
-
- return newCache[parameters][order];
-
- }
-
- /** Compile the sizes array.
- * @param parameters number of free parameters
- * @param order derivation order
- * @param valueCompiler compiler for the value part
- * @return sizes array
- */
- private static int[][] compileSizes(final int parameters, final int order,
- final DSCompiler valueCompiler) {
-
- final int[][] sizes = new int[parameters + 1][order + 1];
- if (parameters == 0) {
- Arrays.fill(sizes[0], 1);
- } else {
- System.arraycopy(valueCompiler.sizes, 0, sizes, 0, parameters);
- sizes[parameters][0] = 1;
- for (int i = 0; i < order; ++i) {
- sizes[parameters][i + 1] = sizes[parameters][i] + sizes[parameters - 1][i + 1];
- }
- }
-
- return sizes;
-
- }
-
- /** Compile the derivatives indirection array.
- * @param parameters number of free parameters
- * @param order derivation order
- * @param valueCompiler compiler for the value part
- * @param derivativeCompiler compiler for the derivative part
- * @return derivatives indirection array
- */
- private static int[][] compileDerivativesIndirection(final int parameters, final int order,
- final DSCompiler valueCompiler,
- final DSCompiler derivativeCompiler) {
-
- if (parameters == 0 || order == 0) {
- return new int[1][parameters];
- }
-
- final int vSize = valueCompiler.derivativesIndirection.length;
- final int dSize = derivativeCompiler.derivativesIndirection.length;
- final int[][] derivativesIndirection = new int[vSize + dSize][parameters];
-
- // set up the indices for the value part
- for (int i = 0; i < vSize; ++i) {
- // copy the first indices, the last one remaining set to 0
- System.arraycopy(valueCompiler.derivativesIndirection[i], 0,
- derivativesIndirection[i], 0,
- parameters - 1);
- }
-
- // set up the indices for the derivative part
- for (int i = 0; i < dSize; ++i) {
-
- // copy the indices
- System.arraycopy(derivativeCompiler.derivativesIndirection[i], 0,
- derivativesIndirection[vSize + i], 0,
- parameters);
-
- // increment the derivation order for the last parameter
- derivativesIndirection[vSize + i][parameters - 1]++;
-
- }
-
- return derivativesIndirection;
-
- }
-
- /** Compile the lower derivatives indirection array.
- * <p>
- * This indirection array contains the indices of all elements
- * except derivatives for last derivation order.
- * </p>
- * @param parameters number of free parameters
- * @param order derivation order
- * @param valueCompiler compiler for the value part
- * @param derivativeCompiler compiler for the derivative part
- * @return lower derivatives indirection array
- */
- private static int[] compileLowerIndirection(final int parameters, final int order,
- final DSCompiler valueCompiler,
- final DSCompiler derivativeCompiler) {
-
- if (parameters == 0 || order <= 1) {
- return new int[] { 0 };
- }
-
- // this is an implementation of definition 6 in Dan Kalman's paper.
- final int vSize = valueCompiler.lowerIndirection.length;
- final int dSize = derivativeCompiler.lowerIndirection.length;
- final int[] lowerIndirection = new int[vSize + dSize];
- System.arraycopy(valueCompiler.lowerIndirection, 0, lowerIndirection, 0, vSize);
- for (int i = 0; i < dSize; ++i) {
- lowerIndirection[vSize + i] = valueCompiler.getSize() + derivativeCompiler.lowerIndirection[i];
- }
-
- return lowerIndirection;
-
- }
-
- /** Compile the multiplication indirection array.
- * <p>
- * This indirection array contains the indices of all pairs of elements
- * involved when computing a multiplication. This allows a straightforward
- * loop-based multiplication (see {@link #multiply(double[], int, double[], int, double[], int)}).
- * </p>
- * @param parameters number of free parameters
- * @param order derivation order
- * @param valueCompiler compiler for the value part
- * @param derivativeCompiler compiler for the derivative part
- * @param lowerIndirection lower derivatives indirection array
- * @return multiplication indirection array
- */
- private static int[][][] compileMultiplicationIndirection(final int parameters, final int order,
- final DSCompiler valueCompiler,
- final DSCompiler derivativeCompiler,
- final int[] lowerIndirection) {
-
- if ((parameters == 0) || (order == 0)) {
- return new int[][][] { { { 1, 0, 0 } } };
- }
-
- // this is an implementation of definition 3 in Dan Kalman's paper.
- final int vSize = valueCompiler.multIndirection.length;
- final int dSize = derivativeCompiler.multIndirection.length;
- final int[][][] multIndirection = new int[vSize + dSize][][];
-
- System.arraycopy(valueCompiler.multIndirection, 0, multIndirection, 0, vSize);
-
- for (int i = 0; i < dSize; ++i) {
- final int[][] dRow = derivativeCompiler.multIndirection[i];
- List<int[]> row = new ArrayList<int[]>(dRow.length * 2);
- for (int j = 0; j < dRow.length; ++j) {
- row.add(new int[] { dRow[j][0], lowerIndirection[dRow[j][1]], vSize + dRow[j][2] });
- row.add(new int[] { dRow[j][0], vSize + dRow[j][1], lowerIndirection[dRow[j][2]] });
- }
-
- // combine terms with similar derivation orders
- final List<int[]> combined = new ArrayList<int[]>(row.size());
- for (int j = 0; j < row.size(); ++j) {
- final int[] termJ = row.get(j);
- if (termJ[0] > 0) {
- for (int k = j + 1; k < row.size(); ++k) {
- final int[] termK = row.get(k);
- if (termJ[1] == termK[1] && termJ[2] == termK[2]) {
- // combine termJ and termK
- termJ[0] += termK[0];
- // make sure we will skip termK later on in the outer loop
- termK[0] = 0;
- }
- }
- combined.add(termJ);
- }
- }
-
- multIndirection[vSize + i] = combined.toArray(new int[combined.size()][]);
-
- }
-
- return multIndirection;
-
- }
-
- /** Compile the function composition indirection array.
- * <p>
- * This indirection array contains the indices of all sets of elements
- * involved when computing a composition. This allows a straightforward
- * loop-based composition (see {@link #compose(double[], int, double[], double[], int)}).
- * </p>
- * @param parameters number of free parameters
- * @param order derivation order
- * @param valueCompiler compiler for the value part
- * @param derivativeCompiler compiler for the derivative part
- * @param sizes sizes array
- * @param derivativesIndirection derivatives indirection array
- * @return multiplication indirection array
- * @throws NumberIsTooLargeException if order is too large
- */
- private static int[][][] compileCompositionIndirection(final int parameters, final int order,
- final DSCompiler valueCompiler,
- final DSCompiler derivativeCompiler,
- final int[][] sizes,
- final int[][] derivativesIndirection)
- throws NumberIsTooLargeException {
-
- if ((parameters == 0) || (order == 0)) {
- return new int[][][] { { { 1, 0 } } };
- }
-
- final int vSize = valueCompiler.compIndirection.length;
- final int dSize = derivativeCompiler.compIndirection.length;
- final int[][][] compIndirection = new int[vSize + dSize][][];
-
- // the composition rules from the value part can be reused as is
- System.arraycopy(valueCompiler.compIndirection, 0, compIndirection, 0, vSize);
-
- // the composition rules for the derivative part are deduced by
- // differentiation the rules from the underlying compiler once
- // with respect to the parameter this compiler handles and the
- // underlying one did not handle
- for (int i = 0; i < dSize; ++i) {
- List<int[]> row = new ArrayList<int[]>();
- for (int[] term : derivativeCompiler.compIndirection[i]) {
-
- // handle term p * f_k(g(x)) * g_l1(x) * g_l2(x) * ... * g_lp(x)
-
- // derive the first factor in the term: f_k with respect to new parameter
- int[] derivedTermF = new int[term.length + 1];
- derivedTermF[0] = term[0]; // p
- derivedTermF[1] = term[1] + 1; // f_(k+1)
- int[] orders = new int[parameters];
- orders[parameters - 1] = 1;
- derivedTermF[term.length] = getPartialDerivativeIndex(parameters, order, sizes, orders); // g_1
- for (int j = 2; j < term.length; ++j) {
- // convert the indices as the mapping for the current order
- // is different from the mapping with one less order
- derivedTermF[j] = convertIndex(term[j], parameters,
- derivativeCompiler.derivativesIndirection,
- parameters, order, sizes);
- }
- Arrays.sort(derivedTermF, 2, derivedTermF.length);
- row.add(derivedTermF);
-
- // derive the various g_l
- for (int l = 2; l < term.length; ++l) {
- int[] derivedTermG = new int[term.length];
- derivedTermG[0] = term[0];
- derivedTermG[1] = term[1];
- for (int j = 2; j < term.length; ++j) {
- // convert the indices as the mapping for the current order
- // is different from the mapping with one less order
- derivedTermG[j] = convertIndex(term[j], parameters,
- derivativeCompiler.derivativesIndirection,
- parameters, order, sizes);
- if (j == l) {
- // derive this term
- System.arraycopy(derivativesIndirection[derivedTermG[j]], 0, orders, 0, parameters);
- orders[parameters - 1]++;
- derivedTermG[j] = getPartialDerivativeIndex(parameters, order, sizes, orders);
- }
- }
- Arrays.sort(derivedTermG, 2, derivedTermG.length);
- row.add(derivedTermG);
- }
-
- }
-
- // combine terms with similar derivation orders
- final List<int[]> combined = new ArrayList<int[]>(row.size());
- for (int j = 0; j < row.size(); ++j) {
- final int[] termJ = row.get(j);
- if (termJ[0] > 0) {
- for (int k = j + 1; k < row.size(); ++k) {
- final int[] termK = row.get(k);
- boolean equals = termJ.length == termK.length;
- for (int l = 1; equals && l < termJ.length; ++l) {
- equals &= termJ[l] == termK[l];
- }
- if (equals) {
- // combine termJ and termK
- termJ[0] += termK[0];
- // make sure we will skip termK later on in the outer loop
- termK[0] = 0;
- }
- }
- combined.add(termJ);
- }
- }
-
- compIndirection[vSize + i] = combined.toArray(new int[combined.size()][]);
-
- }
-
- return compIndirection;
-
- }
-
- /** Get the index of a partial derivative in the array.
- * <p>
- * If all orders are set to 0, then the 0<sup>th</sup> order derivative
- * is returned, which is the value of the function.
- * </p>
- * <p>The indices of derivatives are between 0 and {@link #getSize() getSize()} - 1.
- * Their specific order is fixed for a given compiler, but otherwise not
- * publicly specified. There are however some simple cases which have guaranteed
- * indices:
- * </p>
- * <ul>
- * <li>the index of 0<sup>th</sup> order derivative is always 0</li>
- * <li>if there is only 1 {@link #getFreeParameters() free parameter}, then the
- * derivatives are sorted in increasing derivation order (i.e. f at index 0, df/dp
- * at index 1, d<sup>2</sup>f/dp<sup>2</sup> at index 2 ...
- * d<sup>k</sup>f/dp<sup>k</sup> at index k),</li>
- * <li>if the {@link #getOrder() derivation order} is 1, then the derivatives
- * are sorted in increasing free parameter order (i.e. f at index 0, df/dx<sub>1</sub>
- * at index 1, df/dx<sub>2</sub> at index 2 ... df/dx<sub>k</sub> at index k),</li>
- * <li>all other cases are not publicly specified</li>
- * </ul>
- * <p>
- * This method is the inverse of method {@link #getPartialDerivativeOrders(int)}
- * </p>
- * @param orders derivation orders with respect to each parameter
- * @return index of the partial derivative
- * @exception DimensionMismatchException if the numbers of parameters does not
- * match the instance
- * @exception NumberIsTooLargeException if sum of derivation orders is larger
- * than the instance limits
- * @see #getPartialDerivativeOrders(int)
- */
- public int getPartialDerivativeIndex(final int ... orders)
- throws DimensionMismatchException, NumberIsTooLargeException {
-
- // safety check
- if (orders.length != getFreeParameters()) {
- throw new DimensionMismatchException(orders.length, getFreeParameters());
- }
-
- return getPartialDerivativeIndex(parameters, order, sizes, orders);
-
- }
-
- /** Get the index of a partial derivative in an array.
- * @param parameters number of free parameters
- * @param order derivation order
- * @param sizes sizes array
- * @param orders derivation orders with respect to each parameter
- * (the lenght of this array must match the number of parameters)
- * @return index of the partial derivative
- * @exception NumberIsTooLargeException if sum of derivation orders is larger
- * than the instance limits
- */
- private static int getPartialDerivativeIndex(final int parameters, final int order,
- final int[][] sizes, final int ... orders)
- throws NumberIsTooLargeException {
-
- // the value is obtained by diving into the recursive Dan Kalman's structure
- // this is theorem 2 of his paper, with recursion replaced by iteration
- int index = 0;
- int m = order;
- int ordersSum = 0;
- for (int i = parameters - 1; i >= 0; --i) {
-
- // derivative order for current free parameter
- int derivativeOrder = orders[i];
-
- // safety check
- ordersSum += derivativeOrder;
- if (ordersSum > order) {
- throw new NumberIsTooLargeException(ordersSum, order, true);
- }
-
- while (derivativeOrder-- > 0) {
- // as long as we differentiate according to current free parameter,
- // we have to skip the value part and dive into the derivative part
- // so we add the size of the value part to the base index
- index += sizes[i][m--];
- }
-
- }
-
- return index;
-
- }
-
- /** Convert an index from one (parameters, order) structure to another.
- * @param index index of a partial derivative in source derivative structure
- * @param srcP number of free parameters in source derivative structure
- * @param srcDerivativesIndirection derivatives indirection array for the source
- * derivative structure
- * @param destP number of free parameters in destination derivative structure
- * @param destO derivation order in destination derivative structure
- * @param destSizes sizes array for the destination derivative structure
- * @return index of the partial derivative with the <em>same</em> characteristics
- * in destination derivative structure
- * @throws NumberIsTooLargeException if order is too large
- */
- private static int convertIndex(final int index,
- final int srcP, final int[][] srcDerivativesIndirection,
- final int destP, final int destO, final int[][] destSizes)
- throws NumberIsTooLargeException {
- int[] orders = new int[destP];
- System.arraycopy(srcDerivativesIndirection[index], 0, orders, 0, FastMath.min(srcP, destP));
- return getPartialDerivativeIndex(destP, destO, destSizes, orders);
- }
-
- /** Get the derivation orders for a specific index in the array.
- * <p>
- * This method is the inverse of {@link #getPartialDerivativeIndex(int...)}.
- * </p>
- * @param index of the partial derivative
- * @return orders derivation orders with respect to each parameter
- * @see #getPartialDerivativeIndex(int...)
- */
- public int[] getPartialDerivativeOrders(final int index) {
- return derivativesIndirection[index];
- }
-
- /** Get the number of free parameters.
- * @return number of free parameters
- */
- public int getFreeParameters() {
- return parameters;
- }
-
- /** Get the derivation order.
- * @return derivation order
- */
- public int getOrder() {
- return order;
- }
-
- /** Get the array size required for holding partial derivatives data.
- * <p>
- * This number includes the single 0 order derivative element, which is
- * guaranteed to be stored in the first element of the array.
- * </p>
- * @return array size required for holding partial derivatives data
- */
- public int getSize() {
- return sizes[parameters][order];
- }
-
- /** Compute linear combination.
- * The derivative structure built will be a1 * ds1 + a2 * ds2
- * @param a1 first scale factor
- * @param c1 first base (unscaled) component
- * @param offset1 offset of first operand in its array
- * @param a2 second scale factor
- * @param c2 second base (unscaled) component
- * @param offset2 offset of second operand in its array
- * @param result array where result must be stored (it may be
- * one of the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void linearCombination(final double a1, final double[] c1, final int offset1,
- final double a2, final double[] c2, final int offset2,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < getSize(); ++i) {
- result[resultOffset + i] =
- MathArrays.linearCombination(a1, c1[offset1 + i], a2, c2[offset2 + i]);
- }
- }
-
- /** Compute linear combination.
- * The derivative structure built will be a1 * ds1 + a2 * ds2 + a3 * ds3 + a4 * ds4
- * @param a1 first scale factor
- * @param c1 first base (unscaled) component
- * @param offset1 offset of first operand in its array
- * @param a2 second scale factor
- * @param c2 second base (unscaled) component
- * @param offset2 offset of second operand in its array
- * @param a3 third scale factor
- * @param c3 third base (unscaled) component
- * @param offset3 offset of third operand in its array
- * @param result array where result must be stored (it may be
- * one of the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void linearCombination(final double a1, final double[] c1, final int offset1,
- final double a2, final double[] c2, final int offset2,
- final double a3, final double[] c3, final int offset3,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < getSize(); ++i) {
- result[resultOffset + i] =
- MathArrays.linearCombination(a1, c1[offset1 + i],
- a2, c2[offset2 + i],
- a3, c3[offset3 + i]);
- }
- }
-
- /** Compute linear combination.
- * The derivative structure built will be a1 * ds1 + a2 * ds2 + a3 * ds3 + a4 * ds4
- * @param a1 first scale factor
- * @param c1 first base (unscaled) component
- * @param offset1 offset of first operand in its array
- * @param a2 second scale factor
- * @param c2 second base (unscaled) component
- * @param offset2 offset of second operand in its array
- * @param a3 third scale factor
- * @param c3 third base (unscaled) component
- * @param offset3 offset of third operand in its array
- * @param a4 fourth scale factor
- * @param c4 fourth base (unscaled) component
- * @param offset4 offset of fourth operand in its array
- * @param result array where result must be stored (it may be
- * one of the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void linearCombination(final double a1, final double[] c1, final int offset1,
- final double a2, final double[] c2, final int offset2,
- final double a3, final double[] c3, final int offset3,
- final double a4, final double[] c4, final int offset4,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < getSize(); ++i) {
- result[resultOffset + i] =
- MathArrays.linearCombination(a1, c1[offset1 + i],
- a2, c2[offset2 + i],
- a3, c3[offset3 + i],
- a4, c4[offset4 + i]);
- }
- }
-
- /** Perform addition of two derivative structures.
- * @param lhs array holding left hand side of addition
- * @param lhsOffset offset of the left hand side in its array
- * @param rhs array right hand side of addition
- * @param rhsOffset offset of the right hand side in its array
- * @param result array where result must be stored (it may be
- * one of the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void add(final double[] lhs, final int lhsOffset,
- final double[] rhs, final int rhsOffset,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < getSize(); ++i) {
- result[resultOffset + i] = lhs[lhsOffset + i] + rhs[rhsOffset + i];
- }
- }
- /** Perform subtraction of two derivative structures.
- * @param lhs array holding left hand side of subtraction
- * @param lhsOffset offset of the left hand side in its array
- * @param rhs array right hand side of subtraction
- * @param rhsOffset offset of the right hand side in its array
- * @param result array where result must be stored (it may be
- * one of the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void subtract(final double[] lhs, final int lhsOffset,
- final double[] rhs, final int rhsOffset,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < getSize(); ++i) {
- result[resultOffset + i] = lhs[lhsOffset + i] - rhs[rhsOffset + i];
- }
- }
-
- /** Perform multiplication of two derivative structures.
- * @param lhs array holding left hand side of multiplication
- * @param lhsOffset offset of the left hand side in its array
- * @param rhs array right hand side of multiplication
- * @param rhsOffset offset of the right hand side in its array
- * @param result array where result must be stored (for
- * multiplication the result array <em>cannot</em> be one of
- * the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void multiply(final double[] lhs, final int lhsOffset,
- final double[] rhs, final int rhsOffset,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < multIndirection.length; ++i) {
- final int[][] mappingI = multIndirection[i];
- double r = 0;
- for (int j = 0; j < mappingI.length; ++j) {
- r += mappingI[j][0] *
- lhs[lhsOffset + mappingI[j][1]] *
- rhs[rhsOffset + mappingI[j][2]];
- }
- result[resultOffset + i] = r;
- }
- }
-
- /** Perform division of two derivative structures.
- * @param lhs array holding left hand side of division
- * @param lhsOffset offset of the left hand side in its array
- * @param rhs array right hand side of division
- * @param rhsOffset offset of the right hand side in its array
- * @param result array where result must be stored (for
- * division the result array <em>cannot</em> be one of
- * the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void divide(final double[] lhs, final int lhsOffset,
- final double[] rhs, final int rhsOffset,
- final double[] result, final int resultOffset) {
- final double[] reciprocal = new double[getSize()];
- pow(rhs, lhsOffset, -1, reciprocal, 0);
- multiply(lhs, lhsOffset, reciprocal, 0, result, resultOffset);
- }
-
- /** Perform remainder of two derivative structures.
- * @param lhs array holding left hand side of remainder
- * @param lhsOffset offset of the left hand side in its array
- * @param rhs array right hand side of remainder
- * @param rhsOffset offset of the right hand side in its array
- * @param result array where result must be stored (it may be
- * one of the input arrays)
- * @param resultOffset offset of the result in its array
- */
- public void remainder(final double[] lhs, final int lhsOffset,
- final double[] rhs, final int rhsOffset,
- final double[] result, final int resultOffset) {
-
- // compute k such that lhs % rhs = lhs - k rhs
- final double rem = FastMath.IEEEremainder(lhs[lhsOffset], rhs[rhsOffset]);
- final double k = FastMath.rint((lhs[lhsOffset] - rem) / rhs[rhsOffset]);
-
- // set up value
- result[resultOffset] = rem;
-
- // set up partial derivatives
- for (int i = 1; i < getSize(); ++i) {
- result[resultOffset + i] = lhs[lhsOffset + i] - k * rhs[rhsOffset + i];
- }
-
- }
-
- /** Compute power of a double to a derivative structure.
- * @param a number to exponentiate
- * @param operand array holding the power
- * @param operandOffset offset of the power in its array
- * @param result array where result must be stored (for
- * power the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- * @since 3.3
- */
- public void pow(final double a,
- final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- // [a^x, ln(a) a^x, ln(a)^2 a^x,, ln(a)^3 a^x, ... ]
- final double[] function = new double[1 + order];
- if (a == 0) {
- if (operand[operandOffset] == 0) {
- function[0] = 1;
- double infinity = Double.POSITIVE_INFINITY;
- for (int i = 1; i < function.length; ++i) {
- infinity = -infinity;
- function[i] = infinity;
- }
- } else if (operand[operandOffset] < 0) {
- Arrays.fill(function, Double.NaN);
- }
- } else {
- function[0] = FastMath.pow(a, operand[operandOffset]);
- final double lnA = FastMath.log(a);
- for (int i = 1; i < function.length; ++i) {
- function[i] = lnA * function[i - 1];
- }
- }
-
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute power of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param p power to apply
- * @param result array where result must be stored (for
- * power the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void pow(final double[] operand, final int operandOffset, final double p,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- // [x^p, px^(p-1), p(p-1)x^(p-2), ... ]
- double[] function = new double[1 + order];
- double xk = FastMath.pow(operand[operandOffset], p - order);
- for (int i = order; i > 0; --i) {
- function[i] = xk;
- xk *= operand[operandOffset];
- }
- function[0] = xk;
- double coefficient = p;
- for (int i = 1; i <= order; ++i) {
- function[i] *= coefficient;
- coefficient *= p - i;
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute integer power of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param n power to apply
- * @param result array where result must be stored (for
- * power the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void pow(final double[] operand, final int operandOffset, final int n,
- final double[] result, final int resultOffset) {
-
- if (n == 0) {
- // special case, x^0 = 1 for all x
- result[resultOffset] = 1.0;
- Arrays.fill(result, resultOffset + 1, resultOffset + getSize(), 0);
- return;
- }
-
- // create the power function value and derivatives
- // [x^n, nx^(n-1), n(n-1)x^(n-2), ... ]
- double[] function = new double[1 + order];
-
- if (n > 0) {
- // strictly positive power
- final int maxOrder = FastMath.min(order, n);
- double xk = FastMath.pow(operand[operandOffset], n - maxOrder);
- for (int i = maxOrder; i > 0; --i) {
- function[i] = xk;
- xk *= operand[operandOffset];
- }
- function[0] = xk;
- } else {
- // strictly negative power
- final double inv = 1.0 / operand[operandOffset];
- double xk = FastMath.pow(inv, -n);
- for (int i = 0; i <= order; ++i) {
- function[i] = xk;
- xk *= inv;
- }
- }
-
- double coefficient = n;
- for (int i = 1; i <= order; ++i) {
- function[i] *= coefficient;
- coefficient *= n - i;
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute power of a derivative structure.
- * @param x array holding the base
- * @param xOffset offset of the base in its array
- * @param y array holding the exponent
- * @param yOffset offset of the exponent in its array
- * @param result array where result must be stored (for
- * power the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void pow(final double[] x, final int xOffset,
- final double[] y, final int yOffset,
- final double[] result, final int resultOffset) {
- final double[] logX = new double[getSize()];
- log(x, xOffset, logX, 0);
- final double[] yLogX = new double[getSize()];
- multiply(logX, 0, y, yOffset, yLogX, 0);
- exp(yLogX, 0, result, resultOffset);
- }
-
- /** Compute n<sup>th</sup> root of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param n order of the root
- * @param result array where result must be stored (for
- * n<sup>th</sup> root the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void rootN(final double[] operand, final int operandOffset, final int n,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- // [x^(1/n), (1/n)x^((1/n)-1), (1-n)/n^2x^((1/n)-2), ... ]
- double[] function = new double[1 + order];
- double xk;
- if (n == 2) {
- function[0] = FastMath.sqrt(operand[operandOffset]);
- xk = 0.5 / function[0];
- } else if (n == 3) {
- function[0] = FastMath.cbrt(operand[operandOffset]);
- xk = 1.0 / (3.0 * function[0] * function[0]);
- } else {
- function[0] = FastMath.pow(operand[operandOffset], 1.0 / n);
- xk = 1.0 / (n * FastMath.pow(function[0], n - 1));
- }
- final double nReciprocal = 1.0 / n;
- final double xReciprocal = 1.0 / operand[operandOffset];
- for (int i = 1; i <= order; ++i) {
- function[i] = xk;
- xk *= xReciprocal * (nReciprocal - i);
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute exponential of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * exponential the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void exp(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- Arrays.fill(function, FastMath.exp(operand[operandOffset]));
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute exp(x) - 1 of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * exponential the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void expm1(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.expm1(operand[operandOffset]);
- Arrays.fill(function, 1, 1 + order, FastMath.exp(operand[operandOffset]));
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute natural logarithm of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * logarithm the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void log(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.log(operand[operandOffset]);
- if (order > 0) {
- double inv = 1.0 / operand[operandOffset];
- double xk = inv;
- for (int i = 1; i <= order; ++i) {
- function[i] = xk;
- xk *= -i * inv;
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Computes shifted logarithm of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * shifted logarithm the result array <em>cannot</em> be the input array)
- * @param resultOffset offset of the result in its array
- */
- public void log1p(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.log1p(operand[operandOffset]);
- if (order > 0) {
- double inv = 1.0 / (1.0 + operand[operandOffset]);
- double xk = inv;
- for (int i = 1; i <= order; ++i) {
- function[i] = xk;
- xk *= -i * inv;
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Computes base 10 logarithm of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * base 10 logarithm the result array <em>cannot</em> be the input array)
- * @param resultOffset offset of the result in its array
- */
- public void log10(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.log10(operand[operandOffset]);
- if (order > 0) {
- double inv = 1.0 / operand[operandOffset];
- double xk = inv / FastMath.log(10.0);
- for (int i = 1; i <= order; ++i) {
- function[i] = xk;
- xk *= -i * inv;
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute cosine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * cosine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void cos(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.cos(operand[operandOffset]);
- if (order > 0) {
- function[1] = -FastMath.sin(operand[operandOffset]);
- for (int i = 2; i <= order; ++i) {
- function[i] = -function[i - 2];
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute sine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * sine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void sin(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.sin(operand[operandOffset]);
- if (order > 0) {
- function[1] = FastMath.cos(operand[operandOffset]);
- for (int i = 2; i <= order; ++i) {
- function[i] = -function[i - 2];
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute tangent of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * tangent the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void tan(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- final double[] function = new double[1 + order];
- final double t = FastMath.tan(operand[operandOffset]);
- function[0] = t;
-
- if (order > 0) {
-
- // the nth order derivative of tan has the form:
- // dn(tan(x)/dxn = P_n(tan(x))
- // where P_n(t) is a degree n+1 polynomial with same parity as n+1
- // P_0(t) = t, P_1(t) = 1 + t^2, P_2(t) = 2 t (1 + t^2) ...
- // the general recurrence relation for P_n is:
- // P_n(x) = (1+t^2) P_(n-1)'(t)
- // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
- final double[] p = new double[order + 2];
- p[1] = 1;
- final double t2 = t * t;
- for (int n = 1; n <= order; ++n) {
-
- // update and evaluate polynomial P_n(t)
- double v = 0;
- p[n + 1] = n * p[n];
- for (int k = n + 1; k >= 0; k -= 2) {
- v = v * t2 + p[k];
- if (k > 2) {
- p[k - 2] = (k - 1) * p[k - 1] + (k - 3) * p[k - 3];
- } else if (k == 2) {
- p[0] = p[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= t;
- }
-
- function[n] = v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute arc cosine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * arc cosine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void acos(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- final double x = operand[operandOffset];
- function[0] = FastMath.acos(x);
- if (order > 0) {
- // the nth order derivative of acos has the form:
- // dn(acos(x)/dxn = P_n(x) / [1 - x^2]^((2n-1)/2)
- // where P_n(x) is a degree n-1 polynomial with same parity as n-1
- // P_1(x) = -1, P_2(x) = -x, P_3(x) = -2x^2 - 1 ...
- // the general recurrence relation for P_n is:
- // P_n(x) = (1-x^2) P_(n-1)'(x) + (2n-3) x P_(n-1)(x)
- // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
- final double[] p = new double[order];
- p[0] = -1;
- final double x2 = x * x;
- final double f = 1.0 / (1 - x2);
- double coeff = FastMath.sqrt(f);
- function[1] = coeff * p[0];
- for (int n = 2; n <= order; ++n) {
-
- // update and evaluate polynomial P_n(x)
- double v = 0;
- p[n - 1] = (n - 1) * p[n - 2];
- for (int k = n - 1; k >= 0; k -= 2) {
- v = v * x2 + p[k];
- if (k > 2) {
- p[k - 2] = (k - 1) * p[k - 1] + (2 * n - k) * p[k - 3];
- } else if (k == 2) {
- p[0] = p[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= x;
- }
-
- coeff *= f;
- function[n] = coeff * v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute arc sine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * arc sine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void asin(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- final double x = operand[operandOffset];
- function[0] = FastMath.asin(x);
- if (order > 0) {
- // the nth order derivative of asin has the form:
- // dn(asin(x)/dxn = P_n(x) / [1 - x^2]^((2n-1)/2)
- // where P_n(x) is a degree n-1 polynomial with same parity as n-1
- // P_1(x) = 1, P_2(x) = x, P_3(x) = 2x^2 + 1 ...
- // the general recurrence relation for P_n is:
- // P_n(x) = (1-x^2) P_(n-1)'(x) + (2n-3) x P_(n-1)(x)
- // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
- final double[] p = new double[order];
- p[0] = 1;
- final double x2 = x * x;
- final double f = 1.0 / (1 - x2);
- double coeff = FastMath.sqrt(f);
- function[1] = coeff * p[0];
- for (int n = 2; n <= order; ++n) {
-
- // update and evaluate polynomial P_n(x)
- double v = 0;
- p[n - 1] = (n - 1) * p[n - 2];
- for (int k = n - 1; k >= 0; k -= 2) {
- v = v * x2 + p[k];
- if (k > 2) {
- p[k - 2] = (k - 1) * p[k - 1] + (2 * n - k) * p[k - 3];
- } else if (k == 2) {
- p[0] = p[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= x;
- }
-
- coeff *= f;
- function[n] = coeff * v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute arc tangent of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * arc tangent the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void atan(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- final double x = operand[operandOffset];
- function[0] = FastMath.atan(x);
- if (order > 0) {
- // the nth order derivative of atan has the form:
- // dn(atan(x)/dxn = Q_n(x) / (1 + x^2)^n
- // where Q_n(x) is a degree n-1 polynomial with same parity as n-1
- // Q_1(x) = 1, Q_2(x) = -2x, Q_3(x) = 6x^2 - 2 ...
- // the general recurrence relation for Q_n is:
- // Q_n(x) = (1+x^2) Q_(n-1)'(x) - 2(n-1) x Q_(n-1)(x)
- // as per polynomial parity, we can store coefficients of both Q_(n-1) and Q_n in the same array
- final double[] q = new double[order];
- q[0] = 1;
- final double x2 = x * x;
- final double f = 1.0 / (1 + x2);
- double coeff = f;
- function[1] = coeff * q[0];
- for (int n = 2; n <= order; ++n) {
-
- // update and evaluate polynomial Q_n(x)
- double v = 0;
- q[n - 1] = -n * q[n - 2];
- for (int k = n - 1; k >= 0; k -= 2) {
- v = v * x2 + q[k];
- if (k > 2) {
- q[k - 2] = (k - 1) * q[k - 1] + (k - 1 - 2 * n) * q[k - 3];
- } else if (k == 2) {
- q[0] = q[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= x;
- }
-
- coeff *= f;
- function[n] = coeff * v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute two arguments arc tangent of a derivative structure.
- * @param y array holding the first operand
- * @param yOffset offset of the first operand in its array
- * @param x array holding the second operand
- * @param xOffset offset of the second operand in its array
- * @param result array where result must be stored (for
- * two arguments arc tangent the result array <em>cannot</em>
- * be the input array)
- * @param resultOffset offset of the result in its array
- */
- public void atan2(final double[] y, final int yOffset,
- final double[] x, final int xOffset,
- final double[] result, final int resultOffset) {
-
- // compute r = sqrt(x^2+y^2)
- double[] tmp1 = new double[getSize()];
- multiply(x, xOffset, x, xOffset, tmp1, 0); // x^2
- double[] tmp2 = new double[getSize()];
- multiply(y, yOffset, y, yOffset, tmp2, 0); // y^2
- add(tmp1, 0, tmp2, 0, tmp2, 0); // x^2 + y^2
- rootN(tmp2, 0, 2, tmp1, 0); // r = sqrt(x^2 + y^2)
-
- if (x[xOffset] >= 0) {
-
- // compute atan2(y, x) = 2 atan(y / (r + x))
- add(tmp1, 0, x, xOffset, tmp2, 0); // r + x
- divide(y, yOffset, tmp2, 0, tmp1, 0); // y /(r + x)
- atan(tmp1, 0, tmp2, 0); // atan(y / (r + x))
- for (int i = 0; i < tmp2.length; ++i) {
- result[resultOffset + i] = 2 * tmp2[i]; // 2 * atan(y / (r + x))
- }
-
- } else {
-
- // compute atan2(y, x) = +/- pi - 2 atan(y / (r - x))
- subtract(tmp1, 0, x, xOffset, tmp2, 0); // r - x
- divide(y, yOffset, tmp2, 0, tmp1, 0); // y /(r - x)
- atan(tmp1, 0, tmp2, 0); // atan(y / (r - x))
- result[resultOffset] =
- ((tmp2[0] <= 0) ? -FastMath.PI : FastMath.PI) - 2 * tmp2[0]; // +/-pi - 2 * atan(y / (r - x))
- for (int i = 1; i < tmp2.length; ++i) {
- result[resultOffset + i] = -2 * tmp2[i]; // +/-pi - 2 * atan(y / (r - x))
- }
-
- }
-
- // fix value to take special cases (+0/+0, +0/-0, -0/+0, -0/-0, +/-infinity) correctly
- result[resultOffset] = FastMath.atan2(y[yOffset], x[xOffset]);
-
- }
-
- /** Compute hyperbolic cosine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * hyperbolic cosine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void cosh(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.cosh(operand[operandOffset]);
- if (order > 0) {
- function[1] = FastMath.sinh(operand[operandOffset]);
- for (int i = 2; i <= order; ++i) {
- function[i] = function[i - 2];
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute hyperbolic sine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * hyperbolic sine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void sinh(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- function[0] = FastMath.sinh(operand[operandOffset]);
- if (order > 0) {
- function[1] = FastMath.cosh(operand[operandOffset]);
- for (int i = 2; i <= order; ++i) {
- function[i] = function[i - 2];
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute hyperbolic tangent of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * hyperbolic tangent the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void tanh(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- final double[] function = new double[1 + order];
- final double t = FastMath.tanh(operand[operandOffset]);
- function[0] = t;
-
- if (order > 0) {
-
- // the nth order derivative of tanh has the form:
- // dn(tanh(x)/dxn = P_n(tanh(x))
- // where P_n(t) is a degree n+1 polynomial with same parity as n+1
- // P_0(t) = t, P_1(t) = 1 - t^2, P_2(t) = -2 t (1 - t^2) ...
- // the general recurrence relation for P_n is:
- // P_n(x) = (1-t^2) P_(n-1)'(t)
- // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
- final double[] p = new double[order + 2];
- p[1] = 1;
- final double t2 = t * t;
- for (int n = 1; n <= order; ++n) {
-
- // update and evaluate polynomial P_n(t)
- double v = 0;
- p[n + 1] = -n * p[n];
- for (int k = n + 1; k >= 0; k -= 2) {
- v = v * t2 + p[k];
- if (k > 2) {
- p[k - 2] = (k - 1) * p[k - 1] - (k - 3) * p[k - 3];
- } else if (k == 2) {
- p[0] = p[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= t;
- }
-
- function[n] = v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute inverse hyperbolic cosine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * inverse hyperbolic cosine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void acosh(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- final double x = operand[operandOffset];
- function[0] = FastMath.acosh(x);
- if (order > 0) {
- // the nth order derivative of acosh has the form:
- // dn(acosh(x)/dxn = P_n(x) / [x^2 - 1]^((2n-1)/2)
- // where P_n(x) is a degree n-1 polynomial with same parity as n-1
- // P_1(x) = 1, P_2(x) = -x, P_3(x) = 2x^2 + 1 ...
- // the general recurrence relation for P_n is:
- // P_n(x) = (x^2-1) P_(n-1)'(x) - (2n-3) x P_(n-1)(x)
- // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
- final double[] p = new double[order];
- p[0] = 1;
- final double x2 = x * x;
- final double f = 1.0 / (x2 - 1);
- double coeff = FastMath.sqrt(f);
- function[1] = coeff * p[0];
- for (int n = 2; n <= order; ++n) {
-
- // update and evaluate polynomial P_n(x)
- double v = 0;
- p[n - 1] = (1 - n) * p[n - 2];
- for (int k = n - 1; k >= 0; k -= 2) {
- v = v * x2 + p[k];
- if (k > 2) {
- p[k - 2] = (1 - k) * p[k - 1] + (k - 2 * n) * p[k - 3];
- } else if (k == 2) {
- p[0] = -p[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= x;
- }
-
- coeff *= f;
- function[n] = coeff * v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute inverse hyperbolic sine of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * inverse hyperbolic sine the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void asinh(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- final double x = operand[operandOffset];
- function[0] = FastMath.asinh(x);
- if (order > 0) {
- // the nth order derivative of asinh has the form:
- // dn(asinh(x)/dxn = P_n(x) / [x^2 + 1]^((2n-1)/2)
- // where P_n(x) is a degree n-1 polynomial with same parity as n-1
- // P_1(x) = 1, P_2(x) = -x, P_3(x) = 2x^2 - 1 ...
- // the general recurrence relation for P_n is:
- // P_n(x) = (x^2+1) P_(n-1)'(x) - (2n-3) x P_(n-1)(x)
- // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
- final double[] p = new double[order];
- p[0] = 1;
- final double x2 = x * x;
- final double f = 1.0 / (1 + x2);
- double coeff = FastMath.sqrt(f);
- function[1] = coeff * p[0];
- for (int n = 2; n <= order; ++n) {
-
- // update and evaluate polynomial P_n(x)
- double v = 0;
- p[n - 1] = (1 - n) * p[n - 2];
- for (int k = n - 1; k >= 0; k -= 2) {
- v = v * x2 + p[k];
- if (k > 2) {
- p[k - 2] = (k - 1) * p[k - 1] + (k - 2 * n) * p[k - 3];
- } else if (k == 2) {
- p[0] = p[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= x;
- }
-
- coeff *= f;
- function[n] = coeff * v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute inverse hyperbolic tangent of a derivative structure.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param result array where result must be stored (for
- * inverse hyperbolic tangent the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void atanh(final double[] operand, final int operandOffset,
- final double[] result, final int resultOffset) {
-
- // create the function value and derivatives
- double[] function = new double[1 + order];
- final double x = operand[operandOffset];
- function[0] = FastMath.atanh(x);
- if (order > 0) {
- // the nth order derivative of atanh has the form:
- // dn(atanh(x)/dxn = Q_n(x) / (1 - x^2)^n
- // where Q_n(x) is a degree n-1 polynomial with same parity as n-1
- // Q_1(x) = 1, Q_2(x) = 2x, Q_3(x) = 6x^2 + 2 ...
- // the general recurrence relation for Q_n is:
- // Q_n(x) = (1-x^2) Q_(n-1)'(x) + 2(n-1) x Q_(n-1)(x)
- // as per polynomial parity, we can store coefficients of both Q_(n-1) and Q_n in the same array
- final double[] q = new double[order];
- q[0] = 1;
- final double x2 = x * x;
- final double f = 1.0 / (1 - x2);
- double coeff = f;
- function[1] = coeff * q[0];
- for (int n = 2; n <= order; ++n) {
-
- // update and evaluate polynomial Q_n(x)
- double v = 0;
- q[n - 1] = n * q[n - 2];
- for (int k = n - 1; k >= 0; k -= 2) {
- v = v * x2 + q[k];
- if (k > 2) {
- q[k - 2] = (k - 1) * q[k - 1] + (2 * n - k + 1) * q[k - 3];
- } else if (k == 2) {
- q[0] = q[1];
- }
- }
- if ((n & 0x1) == 0) {
- v *= x;
- }
-
- coeff *= f;
- function[n] = coeff * v;
-
- }
- }
-
- // apply function composition
- compose(operand, operandOffset, function, result, resultOffset);
-
- }
-
- /** Compute composition of a derivative structure by a function.
- * @param operand array holding the operand
- * @param operandOffset offset of the operand in its array
- * @param f array of value and derivatives of the function at
- * the current point (i.e. at {@code operand[operandOffset]}).
- * @param result array where result must be stored (for
- * composition the result array <em>cannot</em> be the input
- * array)
- * @param resultOffset offset of the result in its array
- */
- public void compose(final double[] operand, final int operandOffset, final double[] f,
- final double[] result, final int resultOffset) {
- for (int i = 0; i < compIndirection.length; ++i) {
- final int[][] mappingI = compIndirection[i];
- double r = 0;
- for (int j = 0; j < mappingI.length; ++j) {
- final int[] mappingIJ = mappingI[j];
- double product = mappingIJ[0] * f[mappingIJ[1]];
- for (int k = 2; k < mappingIJ.length; ++k) {
- product *= operand[operandOffset + mappingIJ[k]];
- }
- r += product;
- }
- result[resultOffset + i] = r;
- }
- }
-
- /** Evaluate Taylor expansion of a derivative structure.
- * @param ds array holding the derivative structure
- * @param dsOffset offset of the derivative structure in its array
- * @param delta parameters offsets (Δx, Δy, ...)
- * @return value of the Taylor expansion at x + Δx, y + Δy, ...
- * @throws MathArithmeticException if factorials becomes too large
- */
- public double taylor(final double[] ds, final int dsOffset, final double ... delta)
- throws MathArithmeticException {
- double value = 0;
- for (int i = getSize() - 1; i >= 0; --i) {
- final int[] orders = getPartialDerivativeOrders(i);
- double term = ds[dsOffset + i];
- for (int k = 0; k < orders.length; ++k) {
- if (orders[k] > 0) {
- try {
- term *= FastMath.pow(delta[k], orders[k]) /
- CombinatoricsUtils.factorial(orders[k]);
- } catch (NotPositiveException e) {
- // this cannot happen
- throw new MathInternalError(e);
- }
- }
- }
- value += term;
- }
- return value;
- }
-
- /** Check rules set compatibility.
- * @param compiler other compiler to check against instance
- * @exception DimensionMismatchException if number of free parameters or orders are inconsistent
- */
- public void checkCompatibility(final DSCompiler compiler)
- throws DimensionMismatchException {
- if (parameters != compiler.parameters) {
- throw new DimensionMismatchException(parameters, compiler.parameters);
- }
- if (order != compiler.order) {
- throw new DimensionMismatchException(order, compiler.order);
- }
- }
-
-}