/*
    * 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
    *
    *      https://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.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  package org.hipparchus.analysis.differentiation;
  
  import java.io.Serializable;
  
  import org.hipparchus.Field;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathRuntimeException;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.FieldSinhCosh;
  import org.hipparchus.util.MathArrays;
  import org.hipparchus.util.MathUtils;
  
  /** Class representing both the value and the differentials of a function.
   * <p>This class is the workhorse of the differentiation package.</p>
   * <p>This class is an implementation of the extension to Rall's
   * numbers described in Dan Kalman's paper <a
   * href="http://www.dankalman.net/AUhome/pdffiles/mmgautodiff.pdf">Doubly
   * Recursive Multivariate Automatic Differentiation</a>, Mathematics Magazine, vol. 75,
   * no. 3, June 2002. Rall's numbers are an extension to the real numbers used
   * throughout mathematical expressions; they hold the derivative together with the
   * value of a function. Dan Kalman's derivative structures hold all partial derivatives
   * up to any specified order, with respect to any number of free parameters. Rall's
   * numbers therefore can be seen as derivative structures for order one derivative and
   * one free parameter, and real numbers can be seen as derivative structures with zero
   * order derivative and no free parameters.</p>
   * <p>{@link DerivativeStructure} instances can be used directly thanks to
   * the arithmetic operators to the mathematical functions provided as
   * methods by this class (+, -, *, /, %, sin, cos ...).</p>
   * <p>Implementing complex expressions by hand using these classes is
   * a tedious and error-prone task but has the advantage of having no limitation
   * on the derivation order despite not requiring users to compute the derivatives by
   * themselves. Implementing complex expression can also be done by developing computation
   * code using standard primitive double values and to use {@link
   * UnivariateFunctionDifferentiator differentiators} to create the {@link
   * DerivativeStructure}-based instances. This method is simpler but may be limited in
   * the accuracy and derivation orders and may be computationally intensive (this is
   * typically the case for {@link FiniteDifferencesDifferentiator finite differences
   * differentiator}.</p>
   * <p>Instances of this class are guaranteed to be immutable.</p>
   * @see DSCompiler
   * @see FieldDerivativeStructure
   */
  public class DerivativeStructure implements Derivative<DerivativeStructure>, Serializable {
  
      /** Serializable UID. */
      private static final long serialVersionUID = 20161220L;
  
      /** Factory that built the instance. */
      private final DSFactory factory;
  
      /** Combined array holding all values. */
      private final double[] data;
  
      /** Build an instance with all values and derivatives set to 0.
       * @param factory factory that built the instance
       * @param data combined array holding all values
       */
      DerivativeStructure(final DSFactory factory, final double[] data) {
          this.factory = factory;
          this.data    = data.clone();
      }
  
      /** Build an instance with all values and derivatives set to 0.
       * @param factory factory that built the instance
       * @since 1.4
       */
      DerivativeStructure(final DSFactory factory) {
          this.factory = factory;
          this.data    = new double[factory.getCompiler().getSize()];
      }
  
      /** {@inheritDoc} */
      @Override
      public DerivativeStructure newInstance(final double value) {
          return factory.constant(value);
      }
  
     /** {@inheritDoc} */
     @Override
     public DerivativeStructure withValue(final double value) {
         final DerivativeStructure ds = factory.build();
         System.arraycopy(data, 1, ds.data, 1, data.length - 1);
         ds.data[0] = value;
         return ds;
     }
 
     /** Get the factory that built the instance.
      * @return factory that built the instance
      */
     public DSFactory getFactory() {
         return factory;
     }
 
     /** {@inheritDoc} */
     @Override
     public int getFreeParameters() {
         return getFactory().getCompiler().getFreeParameters();
     }
 
     /** {@inheritDoc} */
     @Override
     public int getOrder() {
         return getFactory().getCompiler().getOrder();
     }
 
     /** Set a derivative component.
      * <p>
      * This method is package-private (no modifier specified), as it is intended
      * to be used only by Hipparchus classes since it relied on the ordering of
      * derivatives within the class. This allows avoiding checks on the index,
      * for performance reasons.
      * </p>
      * @param index index of the derivative
      * @param value of the derivative to set
      * @since 1.4
      */
     void setDerivativeComponent(final int index, final double value) {
         data[index] = value;
     }
 
     /** Get a derivative component.
      * <p>
      * This method is package-private (no modifier specified), as it is intended
      * to be used only by Hipparchus classes since it relied on the ordering of
      * derivatives within the class. This allows avoiding checks on the index,
      * for performance reasons.
      * </p>
      * @param index index of the derivative
      * @return value of the derivative
      * @since 2.2
      */
     double getDerivativeComponent(final int index) {
         return data[index];
     }
 
     /** Get the value part of the derivative structure.
      * @return value part of the derivative structure
      * @see #getPartialDerivative(int...)
      */
     @Override
     public double getValue() {
         return data[0];
     }
 
     /** {@inheritDoc} */
     @Override
     public double getPartialDerivative(final int ... orders)
         throws MathIllegalArgumentException {
         return data[getFactory().getCompiler().getPartialDerivativeIndex(orders)];
     }
 
     /** Get all partial derivatives.
      * @return a fresh copy of partial derivatives, in an array sorted according to
      * {@link DSCompiler#getPartialDerivativeIndex(int...)}
      */
     public double[] getAllDerivatives() {
         return data.clone();
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure add(final DerivativeStructure a)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(a.factory);
         final DerivativeStructure ds = factory.build();
         factory.getCompiler().add(data, 0, a.data, 0, ds.data, 0);
         return ds;
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure subtract(final DerivativeStructure a)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(a.factory);
         final DerivativeStructure ds = factory.build();
         factory.getCompiler().subtract(data, 0, a.data, 0, ds.data, 0);
         return ds;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure multiply(final double a) {
         final DerivativeStructure ds = factory.build();
         for (int i = 0; i < ds.data.length; ++i) {
             ds.data[i] = data[i] * a;
         }
         return ds;
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure multiply(final DerivativeStructure a)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(a.factory);
         final DerivativeStructure result = factory.build();
         factory.getCompiler().multiply(data, 0, a.data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc} */
     @Override
     public DerivativeStructure square() {
         return multiply(this);
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure divide(final double a) {
         final DerivativeStructure ds = factory.build();
         final double inv = 1.0 / a;
         for (int i = 0; i < ds.data.length; ++i) {
             ds.data[i] = data[i] * inv;
         }
         return ds;
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure divide(final DerivativeStructure a)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(a.factory);
         final DerivativeStructure result = factory.build();
         factory.getCompiler().divide(data, 0, a.data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure remainder(final DerivativeStructure a)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(a.factory);
         final DerivativeStructure result = factory.build();
         factory.getCompiler().remainder(data, 0, a.data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc} */
     @Override
     public DerivativeStructure negate() {
         final DerivativeStructure ds = factory.build();
         for (int i = 0; i < ds.data.length; ++i) {
             ds.data[i] = -data[i];
         }
         return ds;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure abs() {
         if (Double.doubleToLongBits(data[0]) < 0) {
             // we use the bits representation to also handle -0.0
             return negate();
         } else {
             return this;
         }
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure copySign(final DerivativeStructure sign) {
         long m = Double.doubleToLongBits(data[0]);
         long s = Double.doubleToLongBits(sign.data[0]);
         if ((m >= 0 && s >= 0) || (m < 0 && s < 0)) { // Sign is currently OK
             return this;
         }
         return negate(); // flip sign
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure copySign(final double sign) {
         long m = Double.doubleToLongBits(data[0]);
         long s = Double.doubleToLongBits(sign);
         if ((m >= 0 && s >= 0) || (m < 0 && s < 0)) { // Sign is currently OK
             return this;
         }
         return negate(); // flip sign
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure scalb(final int n) {
         final DerivativeStructure ds = factory.build();
         for (int i = 0; i < ds.data.length; ++i) {
             ds.data[i] = FastMath.scalb(data[i], n);
         }
         return ds;
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure hypot(final DerivativeStructure y)
         throws MathIllegalArgumentException {
 
         factory.checkCompatibility(y.factory);
 
         if (Double.isInfinite(data[0]) || Double.isInfinite(y.data[0])) {
             return factory.constant(Double.POSITIVE_INFINITY);
         } else if (Double.isNaN(data[0]) || Double.isNaN(y.data[0])) {
             return factory.constant(Double.NaN);
         } else {
 
             final int expX = getExponent();
             final int expY = y.getExponent();
             if (expX > expY + 27) {
                 // y is negligible with respect to x
                 return abs();
             } else if (expY > expX + 27) {
                 // x is negligible with respect to y
                 return y.abs();
             } else {
 
                 // find an intermediate scale to avoid both overflow and underflow
                 final int middleExp = (expX + expY) / 2;
 
                 // scale parameters without losing precision
                 final DerivativeStructure scaledX = scalb(-middleExp);
                 final DerivativeStructure scaledY = y.scalb(-middleExp);
 
                 // compute scaled hypotenuse
                 final DerivativeStructure scaledH =
                         scaledX.multiply(scaledX).add(scaledY.multiply(scaledY)).sqrt();
 
                 // remove scaling
                 return scaledH.scalb(middleExp);
 
             }
 
         }
     }
 
     /**
      * Returns the hypotenuse of a triangle with sides {@code x} and {@code y}
      * - sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
      * avoiding intermediate overflow or underflow.
      *
      * <ul>
      * <li> If either argument is infinite, then the result is positive infinity.</li>
      * <li> else, if either argument is NaN then the result is NaN.</li>
      * </ul>
      *
      * @param x a value
      * @param y a value
      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     public static DerivativeStructure hypot(final DerivativeStructure x, final DerivativeStructure y)
         throws MathIllegalArgumentException {
         return x.hypot(y);
     }
 
     /** Compute composition of the instance by a univariate function.
      * @param f array of value and derivatives of the function at
      * the current point (i.e. [f({@link #getValue()}),
      * f'({@link #getValue()}), f''({@link #getValue()})...]).
      * @return f(this)
      * @exception MathIllegalArgumentException if the number of derivatives
      * in the array is not equal to {@link #getOrder() order} + 1
      */
     @Override
     public DerivativeStructure compose(final double ... f)
         throws MathIllegalArgumentException {
 
         MathUtils.checkDimension(f.length, getOrder() + 1);
         final DerivativeStructure result = factory.build();
         factory.getCompiler().compose(data, 0, f, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc} */
     @Override
     public DerivativeStructure reciprocal() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().reciprocal(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure sqrt() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().sqrt(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure rootN(final int n) {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().rootN(data, 0, n, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc} */
     @Override
     public Field<DerivativeStructure> getField() {
         return factory.getDerivativeField();
     }
 
     /** Compute a<sup>x</sup> where a is a double and x a {@link DerivativeStructure}
      * @param a number to exponentiate
      * @param x power to apply
      * @return a<sup>x</sup>
      */
     public static DerivativeStructure pow(final double a, final DerivativeStructure x) {
         final DerivativeStructure result = x.factory.build();
         x.factory.getCompiler().pow(a, x.data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure pow(final double p) {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().pow(data, 0, p, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure pow(final int n) {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().pow(data, 0, n, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure pow(final DerivativeStructure e)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(e.factory);
         final DerivativeStructure result = factory.build();
         factory.getCompiler().pow(data, 0, e.data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure exp() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().exp(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure expm1() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().expm1(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure log() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().log(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure log1p() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().log1p(data, 0, result.data, 0);
         return result;
     }
 
     /** Base 10 logarithm.
      * @return base 10 logarithm of the instance
      */
     @Override
     public DerivativeStructure log10() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().log10(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure cos() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().cos(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure sin() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().sin(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public FieldSinCos<DerivativeStructure> sinCos() {
         final DerivativeStructure sin = factory.build();
         final DerivativeStructure cos = factory.build();
         factory.getCompiler().sinCos(data, 0, sin.data, 0, cos.data, 0);
         return new FieldSinCos<>(sin, cos);
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure tan() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().tan(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure acos() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().acos(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure asin() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().asin(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure atan() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().atan(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure atan2(final DerivativeStructure x)
         throws MathIllegalArgumentException {
         factory.checkCompatibility(x.factory);
         final DerivativeStructure result = factory.build();
         factory.getCompiler().atan2(data, 0, x.data, 0, result.data, 0);
         return result;
     }
 
     /** Two arguments arc tangent operation.
      * @param y first argument of the arc tangent
      * @param x second argument of the arc tangent
      * @return atan2(y, x)
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     public static DerivativeStructure atan2(final DerivativeStructure y, final DerivativeStructure x)
         throws MathIllegalArgumentException {
         return y.atan2(x);
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure cosh() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().cosh(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure sinh() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().sinh(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public FieldSinhCosh<DerivativeStructure> sinhCosh() {
         final DerivativeStructure sinh = factory.build();
         final DerivativeStructure cosh = factory.build();
         factory.getCompiler().sinhCosh(data, 0, sinh.data, 0, cosh.data, 0);
         return new FieldSinhCosh<>(sinh, cosh);
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure tanh() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().tanh(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure acosh() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().acosh(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure asinh() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().asinh(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure atanh() {
         final DerivativeStructure result = factory.build();
         factory.getCompiler().atanh(data, 0, result.data, 0);
         return result;
     }
 
     /** {@inheritDoc} */
     @Override
     public DerivativeStructure toDegrees() {
         final DerivativeStructure ds = factory.build();
         for (int i = 0; i < ds.data.length; ++i) {
             ds.data[i] = FastMath.toDegrees(data[i]);
         }
         return ds;
     }
 
     /** {@inheritDoc} */
     @Override
     public DerivativeStructure toRadians() {
         final DerivativeStructure ds = factory.build();
         for (int i = 0; i < ds.data.length; ++i) {
             ds.data[i] = FastMath.toRadians(data[i]);
         }
         return ds;
     }
 
     /** Integrate w.r.t. one independent variable.
      * <p>
      * Rigorously, if the derivatives of a function are known up to
      * order N, the ones of its M-th integral w.r.t. a given variable
      * (seen as a function itself) are actually known up to order N+M.
      * However, this method still casts the output as a DerivativeStructure
      * of order N. The integration constants are systematically set to zero.
      * </p>
      * @param varIndex Index of independent variable w.r.t. which integration is done.
      * @param integrationOrder Number of times the integration operator must be applied. If non-positive, call the
      *                         differentiation operator.
      * @return DerivativeStructure on which integration operator has been applied a certain number of times.
      * @since 2.2
      */
     public DerivativeStructure integrate(final int varIndex, final int integrationOrder) {
 
         // Deal first with trivial case
         if (integrationOrder > getOrder()) {
             return factory.constant(0.);
         } else if (integrationOrder == 0) {
             return factory.build(data);
         }
 
         // Call 'inverse' (not rigorously) operation if necessary
         if (integrationOrder < 0) {
             return differentiate(varIndex, -integrationOrder);
         }
 
         final double[] newData = new double[data.length];
         final DSCompiler dsCompiler = factory.getCompiler();
         for (int i = 0; i < newData.length; i++) {
             if (data[i] != 0.) {
                 final int[] orders = dsCompiler.getPartialDerivativeOrders(i);
                 int sum = 0;
                 for (int order : orders) {
                     sum += order;
                 }
                 if (sum + integrationOrder <= getOrder()) {
                     final int saved = orders[varIndex];
                     orders[varIndex] += integrationOrder;
                     final int index = dsCompiler.getPartialDerivativeIndex(orders);
                     orders[varIndex] = saved;
                     newData[index] = data[i];
                 }
             }
         }
 
         return factory.build(newData);
     }
 
     /** Differentiate w.r.t. one independent variable.
      * <p>
      * Rigorously, if the derivatives of a function are known up to
      * order N, the ones of its M-th derivative w.r.t. a given variable
      * (seen as a function itself) are only known up to order N-M.
      * However, this method still casts the output as a DerivativeStructure
      * of order N with zeroes for the higher order terms.
      * </p>
      * @param varIndex Index of independent variable w.r.t. which differentiation is done.
      * @param differentiationOrder Number of times the differentiation operator must be applied. If non-positive, call
      *                             the integration operator instead.
      * @return DerivativeStructure on which differentiation operator has been applied a certain number of times
      * @since 2.2
      */
     public DerivativeStructure differentiate(final int varIndex, final int differentiationOrder) {
 
         // Deal first with trivial case
         if (differentiationOrder > getOrder()) {
             return factory.constant(0.);
         } else if (differentiationOrder == 0) {
             return factory.build(data);
         }
 
         // Call 'inverse' (not rigorously) operation if necessary
         if (differentiationOrder < 0) {
             return integrate(varIndex, -differentiationOrder);
         }
 
         final double[] newData = new double[data.length];
         final DSCompiler dsCompiler = factory.getCompiler();
         for (int i = 0; i < newData.length; i++) {
             if (data[i] != 0.) {
                 final int[] orders = dsCompiler.getPartialDerivativeOrders(i);
                 if (orders[varIndex] - differentiationOrder >= 0) {
                     final int saved = orders[varIndex];
                     orders[varIndex] -= differentiationOrder;
                     final int index = dsCompiler.getPartialDerivativeIndex(orders);
                     orders[varIndex] = saved;
                     newData[index] = data[i];
                 }
             }
         }
 
         return factory.build(newData);
     }
 
     /** Evaluate Taylor expansion a derivative structure.
      * @param delta parameters offsets (&Delta;x, &Delta;y, ...)
      * @return value of the Taylor expansion at x + &Delta;x, y + &Delta;y, ...
      * @throws MathRuntimeException if factorials becomes too large
      */
     public double taylor(final double ... delta) throws MathRuntimeException {
         return factory.getCompiler().taylor(data, 0, delta);
     }
 
     /** Rebase instance with respect to low level parameter functions.
      * <p>
      * The instance is considered to be a function of {@link #getFreeParameters()
      * n free parameters} up to order {@link #getOrder() o} \(f(p_0, p_1, \ldots p_{n-1})\).
      * Its {@link #getPartialDerivative(int...) partial derivatives} are therefore
      * \(f, \frac{\partial f}{\partial p_0}, \frac{\partial f}{\partial p_1}, \ldots
      * \frac{\partial^2 f}{\partial p_0^2}, \frac{\partial^2 f}{\partial p_0 p_1},
      * \ldots \frac{\partial^o f}{\partial p_{n-1}^o}\). The free parameters
      * \(p_0, p_1, \ldots p_{n-1}\) are considered to be functions of \(m\) lower
      * level other parameters \(q_0, q_1, \ldots q_{m-1}\).
      * </p>
      * \( \begin{align}
      * p_0 &amp; = p_0(q_0, q_1, \ldots q_{m-1})\\
      * p_1 &amp; = p_1(q_0, q_1, \ldots q_{m-1})\\
      * p_{n-1} &amp; = p_{n-1}(q_0, q_1, \ldots q_{m-1})
      * \end{align}\)
      * <p>
      * This method compute the composition of the partial derivatives of \(f\)
      * and the partial derivatives of \(p_0, p_1, \ldots p_{n-1}\), i.e. the
      * {@link #getPartialDerivative(int...) partial derivatives} of the value
      * returned will be
      * \(f, \frac{\partial f}{\partial q_0}, \frac{\partial f}{\partial q_1}, \ldots
      * \frac{\partial^2 f}{\partial q_0^2}, \frac{\partial^2 f}{\partial q_0 q_1},
      * \ldots \frac{\partial^o f}{\partial q_{m-1}^o}\).
      * </p>
      * <p>
      * The number of parameters must match {@link #getFreeParameters()} and the
      * derivation orders of the instance and parameters must also match.
      * </p>
      * @param p base parameters with respect to which partial derivatives
      * were computed in the instance
      * @return derivative structure with partial derivatives computed
      * with respect to the lower level parameters used in the \(p_i\)
      * @since 2.2
      */
     public DerivativeStructure rebase(final DerivativeStructure... p) {
 
         MathUtils.checkDimension(getFreeParameters(), p.length);
 
         // handle special case of no variables at all
         if (p.length == 0) {
             return this;
         }
 
         final int pSize = p[0].getFactory().getCompiler().getSize();
         final double[] pData = new double[p.length * pSize];
         for (int i = 0; i < p.length; ++i) {
             MathUtils.checkDimension(getOrder(), p[i].getOrder());
             MathUtils.checkDimension(p[0].getFreeParameters(), p[i].getFreeParameters());
             System.arraycopy(p[i].data, 0, pData, i * pSize, pSize);
         }
 
         final DerivativeStructure result = p[0].factory.build();
         factory.getCompiler().rebase(data, 0, p[0].factory.getCompiler(), pData, result.data, 0);
         return result;
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final DerivativeStructure[] a, final DerivativeStructure[] b)
         throws MathIllegalArgumentException {
 
         // compute an accurate value, taking care of cancellations
         final double[] aDouble = new double[a.length];
         for (int i = 0; i < a.length; ++i) {
             aDouble[i] = a[i].getValue();
         }
         final double[] bDouble = new double[b.length];
         for (int i = 0; i < b.length; ++i) {
             bDouble[i] = b[i].getValue();
         }
         final double accurateValue = MathArrays.linearCombination(aDouble, bDouble);
 
         // compute a simple value, with all partial derivatives
         DerivativeStructure simpleValue = a[0].getField().getZero();
         for (int i = 0; i < a.length; ++i) {
             simpleValue = simpleValue.add(a[i].multiply(b[i]));
         }
 
         // create a result with accurate value and all derivatives (not necessarily as accurate as the value)
         final double[] all = simpleValue.getAllDerivatives();
         all[0] = accurateValue;
         return factory.build(all);
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final double[] a, final DerivativeStructure[] b)
         throws MathIllegalArgumentException {
 
         // compute an accurate value, taking care of cancellations
         final double[] bDouble = new double[b.length];
         for (int i = 0; i < b.length; ++i) {
             bDouble[i] = b[i].getValue();
         }
         final double accurateValue = MathArrays.linearCombination(a, bDouble);
 
         // compute a simple value, with all partial derivatives
         DerivativeStructure simpleValue = b[0].getField().getZero();
         for (int i = 0; i < a.length; ++i) {
             simpleValue = simpleValue.add(b[i].multiply(a[i]));
         }
 
         // create a result with accurate value and all derivatives (not necessarily as accurate as the value)
         final double[] all = simpleValue.getAllDerivatives();
         all[0] = accurateValue;
         return factory.build(all);
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final DerivativeStructure a1, final DerivativeStructure b1,
                                                  final DerivativeStructure a2, final DerivativeStructure b2)
         throws MathIllegalArgumentException {
 
         // compute an accurate value, taking care of cancellations
         final double accurateValue = MathArrays.linearCombination(a1.getValue(), b1.getValue(),
                                                                   a2.getValue(), b2.getValue());
 
         // compute a simple value, with all partial derivatives
         final DerivativeStructure simpleValue = a1.multiply(b1).add(a2.multiply(b2));
 
         // create a result with accurate value and all derivatives (not necessarily as accurate as the value)
         final double[] all = simpleValue.getAllDerivatives();
         all[0] = accurateValue;
         return factory.build(all);
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final double a1, final DerivativeStructure b1,
                                                  final double a2, final DerivativeStructure b2)
         throws MathIllegalArgumentException {
 
         factory.checkCompatibility(b1.factory);
         factory.checkCompatibility(b2.factory);
 
         final DerivativeStructure ds = factory.build();
         factory.getCompiler().linearCombination(a1, b1.data, 0,
                                                 a2, b2.data, 0,
                                                 ds.data, 0);
 
         return ds;
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final DerivativeStructure a1, final DerivativeStructure b1,
                                                  final DerivativeStructure a2, final DerivativeStructure b2,
                                                  final DerivativeStructure a3, final DerivativeStructure b3)
         throws MathIllegalArgumentException {
 
         // compute an accurate value, taking care of cancellations
         final double accurateValue = MathArrays.linearCombination(a1.getValue(), b1.getValue(),
                                                                   a2.getValue(), b2.getValue(),
                                                                   a3.getValue(), b3.getValue());
 
         // compute a simple value, with all partial derivatives
         final DerivativeStructure simpleValue = a1.multiply(b1).add(a2.multiply(b2)).add(a3.multiply(b3));
 
         // create a result with accurate value and all derivatives (not necessarily as accurate as the value)
         final double[] all = simpleValue.getAllDerivatives();
         all[0] = accurateValue;
         return factory.build(all);
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final double a1, final DerivativeStructure b1,
                                                  final double a2, final DerivativeStructure b2,
                                                  final double a3, final DerivativeStructure b3)
         throws MathIllegalArgumentException {
 
         factory.checkCompatibility(b1.factory);
         factory.checkCompatibility(b2.factory);
         factory.checkCompatibility(b3.factory);
 
         final DerivativeStructure ds = factory.build();
         factory.getCompiler().linearCombination(a1, b1.data, 0,
                                                 a2, b2.data, 0,
                                                 a3, b3.data, 0,
                                                 ds.data, 0);
 
         return ds;
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final DerivativeStructure a1, final DerivativeStructure b1,
                                                  final DerivativeStructure a2, final DerivativeStructure b2,
                                                  final DerivativeStructure a3, final DerivativeStructure b3,
                                                  final DerivativeStructure a4, final DerivativeStructure b4)
         throws MathIllegalArgumentException {
 
         // compute an accurate value, taking care of cancellations
         final double accurateValue = MathArrays.linearCombination(a1.getValue(), b1.getValue(),
                                                                   a2.getValue(), b2.getValue(),
                                                                   a3.getValue(), b3.getValue(),
                                                                   a4.getValue(), b4.getValue());
 
         // compute a simple value, with all partial derivatives
         final DerivativeStructure simpleValue = a1.multiply(b1).add(a2.multiply(b2)).add(a3.multiply(b3)).add(a4.multiply(b4));
 
         // create a result with accurate value and all derivatives (not necessarily as accurate as the value)
         final double[] all = simpleValue.getAllDerivatives();
         all[0] = accurateValue;
         return factory.build(all);
 
     }
 
     /** {@inheritDoc}
      * @exception MathIllegalArgumentException if number of free parameters
      * or orders do not match
      */
     @Override
     public DerivativeStructure linearCombination(final double a1, final DerivativeStructure b1,
                                                  final double a2, final DerivativeStructure b2,
                                                  final double a3, final DerivativeStructure b3,
                                                  final double a4, final DerivativeStructure b4)
         throws MathIllegalArgumentException {
 
         factory.checkCompatibility(b1.factory);
         factory.checkCompatibility(b2.factory);
         factory.checkCompatibility(b3.factory);
         factory.checkCompatibility(b4.factory);
 
         final DerivativeStructure ds = factory.build();
         factory.getCompiler().linearCombination(a1, b1.data, 0,
                                                 a2, b2.data, 0,
                                                 a3, b3.data, 0,
                                                 a4, b4.data, 0,
                                                 ds.data, 0);
 
         return ds;
 
     }
 
     /** {@inheritDoc}
      */
     @Override
     public DerivativeStructure getPi() {
         return factory.getDerivativeField().getPi();
     }
 
     /**
      * Test for the equality of two derivative structures.
      * <p>
      * Derivative structures are considered equal if they have the same number
      * of free parameters, the same derivation order, and the same derivatives.
      * </p>
      * @param other Object to test for equality to this
      * @return true if two derivative structures are equal
      */
     @Override
     public boolean equals(Object other) {
 
         if (this == other) {
             return true;
         }
 
         if (other instanceof DerivativeStructure) {
             final DerivativeStructure rhs = (DerivativeStructure)other;
             return (getFreeParameters() == rhs.getFreeParameters()) &&
                    (getOrder() == rhs.getOrder()) &&
                    MathArrays.equals(data, rhs.data);
         }
 
         return false;
 
     }
 
     /**
      * Get a hashCode for the derivative structure.
      * @return a hash code value for this object
      */
     @Override
     public int hashCode() {
         return 227 + 229 * getFreeParameters() + 233 * getOrder() + 239 * MathUtils.hash(data);
     }
 
 }