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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
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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.
 * 

This class implements the computation rules described in Dan Kalman's paper Doubly * Recursive Multivariate Automatic Differentiation, 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.

*

* 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. *

*

* The arrays on which compilers operate contain only the partial derivatives together * with the 0th 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). *

*

* 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). *

*

* 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. *

*

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:

 *   // 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
 *
 *   }
 * 
*

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:

 *   // 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)];
 * 
* @see DerivativeStructure * @since 3.1 */ public class DSCompiler { /** Array of all compilers created so far. */ private static AtomicReference compilers = new AtomicReference(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. *

* This indirection array contains the indices of all elements * except derivatives for last derivation order. *

* @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. *

* 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)}). *

* @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 row = new ArrayList(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 combined = new ArrayList(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. *

* 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)}). *

* @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 row = new ArrayList(); 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 combined = new ArrayList(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. *

* If all orders are set to 0, then the 0th order derivative * is returned, which is the value of the function. *

*

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: *

*
    *
  • the index of 0th order derivative is always 0
  • *
  • 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, d2f/dp2 at index 2 ... * dkf/dpk at index k),
  • *
  • 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/dx1 * at index 1, df/dx2 at index 2 ... df/dxk at index k),
  • *
  • all other cases are not publicly specified
  • *
*

* This method is the inverse of method {@link #getPartialDerivativeOrders(int)} *

* @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 same 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. *

* This method is the inverse of {@link #getPartialDerivativeIndex(int...)}. *

* @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. *

* This number includes the single 0 order derivative element, which is * guaranteed to be stored in the first element of the array. *

* @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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 nth 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 * nth root the result array cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot * 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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 cannot 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); } } }




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