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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 new file mode 100644 index 0000000..15fa499 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/analysis/differentiation/DSCompiler.java @@ -0,0 +1,1820 @@ +/* + * 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:</p> <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> + * <p>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:</p> <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)]; + * </pre> + * @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); + } + } + +} |