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Statistical sampling library for use in virtdata libraries, based
on apache commons math 4
/*
* 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.math4.analysis.differentiation;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.concurrent.atomic.AtomicReference;
import org.apache.commons.numbers.arrays.LinearCombination;
import org.apache.commons.math4.exception.DimensionMismatchException;
import org.apache.commons.math4.exception.MathArithmeticException;
import org.apache.commons.math4.exception.MathInternalError;
import org.apache.commons.math4.exception.NotPositiveException;
import org.apache.commons.math4.exception.NumberIsTooLargeException;
import org.apache.commons.numbers.combinatorics.FactorialDouble;
import org.apache.commons.math4.util.FastMath;
/** 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 their 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 {
/** Cache for factorials. */
private static FactorialDouble FACTORIAL = FactorialDouble.create().withCache(30);
/** 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 length 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] =
LinearCombination.value(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] =
LinearCombination.value(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] =
LinearCombination.value(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) {
if (p == 0) {
// special case, x^0 = 1 for all x
result[resultOffset] = 1.0;
Arrays.fill(result, resultOffset + 1, resultOffset + getSize(), 0);
return;
}
if (operand[operandOffset] == 0) {
// special case, 0^p = 0 for all p
Arrays.fill(result, resultOffset, resultOffset + getSize(), 0);
return;
}
// 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]) / FACTORIAL.value(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);
}
}
}