DerivativeStructure (Apache Commons Math 3.3 API)

org.apache.commons.math3.analysis.differentiation

## Class DerivativeStructure

- java.lang.Object
- - org.apache.commons.math3.analysis.differentiation.DerivativeStructure

- All Implemented Interfaces:
    Serializable, FieldElement<DerivativeStructure>, RealFieldElement<DerivativeStructure>

      public class DerivativeStructure
      extends Object
      implements RealFieldElement<DerivativeStructure>, Serializable

  Class representing both the value and the differentials of a function.

  This class is the workhorse of the differentiation package.

  This class is an implementation of the extension to Rall's numbers described in Dan Kalman's paper Doubly Recursive Multivariate Automatic Differentiation, 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.

  `DerivativeStructure` instances can be used directly thanks to the arithmetic operators to the mathematical functions provided as methods by this class (+, -, *, /, %, sin, cos ...).

  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 no 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 `differentiators` to create the `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 `finite differences differentiator`.

  Instances of this class are guaranteed to be immutable.

  Since:
    3.1
  Version:
    $Id: DerivativeStructure.java 1517789 2013-08-27 11:15:50Z luc $
  See Also:
    `DSCompiler`, Serialized Form

- - ### Constructor Summary

    Constructors
    Constructor and Description
    `DerivativeStructure(double a1, DerivativeStructure ds1, double a2, DerivativeStructure ds2)` Linear combination constructor.
    `DerivativeStructure(double a1, DerivativeStructure ds1, double a2, DerivativeStructure ds2, double a3, DerivativeStructure ds3)` Linear combination constructor.
    `DerivativeStructure(double a1, DerivativeStructure ds1, double a2, DerivativeStructure ds2, double a3, DerivativeStructure ds3, double a4, DerivativeStructure ds4)` Linear combination constructor.
    `DerivativeStructure(int parameters, int order)` Build an instance with all values and derivatives set to 0.
    `DerivativeStructure(int parameters, int order, double... derivatives)` Build an instance from all its derivatives.
    `DerivativeStructure(int parameters, int order, double value)` Build an instance representing a constant value.
    `DerivativeStructure(int parameters, int order, int index, double value)` Build an instance representing a variable.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `DerivativeStructure` | `abs()` absolute value.
    `DerivativeStructure` | `acos()` Arc cosine operation.
    `DerivativeStructure` | `acosh()` Inverse hyperbolic cosine operation.
    `DerivativeStructure` | `add(DerivativeStructure a)` Compute this + a.
    `DerivativeStructure` | `add(double a)` '+' operator.
    `DerivativeStructure` | `asin()` Arc sine operation.
    `DerivativeStructure` | `asinh()` Inverse hyperbolic sine operation.
    `DerivativeStructure` | `atan()` Arc tangent operation.
    `DerivativeStructure` | `atan2(DerivativeStructure x)` Two arguments arc tangent operation.
    `static DerivativeStructure` | `atan2(DerivativeStructure y, DerivativeStructure x)` Two arguments arc tangent operation.
    `DerivativeStructure` | `atanh()` Inverse hyperbolic tangent operation.
    `DerivativeStructure` | `cbrt()` Cubic root.
    `DerivativeStructure` | `ceil()` Get the smallest whole number larger than instance.
    `DerivativeStructure` | `compose(double... f)` Compute composition of the instance by a univariate function.
    `DerivativeStructure` | `copySign(DerivativeStructure sign)` Returns the instance with the sign of the argument.
    `DerivativeStructure` | `copySign(double sign)` Returns the instance with the sign of the argument.
    `DerivativeStructure` | `cos()` Cosine operation.
    `DerivativeStructure` | `cosh()` Hyperbolic cosine operation.
    `DerivativeStructure` | `createConstant(double c)` Create a constant compatible with instance order and number of parameters.
    `DerivativeStructure` | `divide(DerivativeStructure a)` Compute this ÷ a.
    `DerivativeStructure` | `divide(double a)` '÷' operator.
    `boolean` | `equals(Object other)` Test for the equality of two derivative structures.
    `DerivativeStructure` | `exp()` Exponential.
    `DerivativeStructure` | `expm1()` Exponential minus 1.
    `DerivativeStructure` | `floor()` Get the largest whole number smaller than instance.
    `double[]` | `getAllDerivatives()` Get all partial derivatives.
    `int` | `getExponent()` Return the exponent of the instance value, removing the bias.
    `Field<DerivativeStructure>` | `getField()` Get the `Field` to which the instance belongs.
    `int` | `getFreeParameters()` Get the number of free parameters.
    `int` | `getOrder()` Get the derivation order.
    `double` | `getPartialDerivative(int... orders)` Get a partial derivative.
    `double` | `getReal()` Get the real value of the number.
    `double` | `getValue()` Get the value part of the derivative structure.
    `int` | `hashCode()` Get a hashCode for the derivative structure.
    `DerivativeStructure` | `hypot(DerivativeStructure y)` Returns the hypotenuse of a triangle with sides `this` and `y` - sqrt(this2 +y2) avoiding intermediate overflow or underflow.
    `static DerivativeStructure` | `hypot(DerivativeStructure x, DerivativeStructure y)` Returns the hypotenuse of a triangle with sides `x` and `y` - sqrt(x2 +y2) avoiding intermediate overflow or underflow.
    `DerivativeStructure` | `linearCombination(DerivativeStructure[] a, DerivativeStructure[] b)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(DerivativeStructure a1, DerivativeStructure b1, DerivativeStructure a2, DerivativeStructure b2)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(DerivativeStructure a1, DerivativeStructure b1, DerivativeStructure a2, DerivativeStructure b2, DerivativeStructure a3, DerivativeStructure b3)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(DerivativeStructure a1, DerivativeStructure b1, DerivativeStructure a2, DerivativeStructure b2, DerivativeStructure a3, DerivativeStructure b3, DerivativeStructure a4, DerivativeStructure b4)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(double[] a, DerivativeStructure[] b)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(double a1, DerivativeStructure b1, double a2, DerivativeStructure b2)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(double a1, DerivativeStructure b1, double a2, DerivativeStructure b2, double a3, DerivativeStructure b3)` Compute a linear combination.
    `DerivativeStructure` | `linearCombination(double a1, DerivativeStructure b1, double a2, DerivativeStructure b2, double a3, DerivativeStructure b3, double a4, DerivativeStructure b4)` Compute a linear combination.
    `DerivativeStructure` | `log()` Natural logarithm.
    `DerivativeStructure` | `log10()` Base 10 logarithm.
    `DerivativeStructure` | `log1p()` Shifted natural logarithm.
    `DerivativeStructure` | `multiply(DerivativeStructure a)` Compute this × a.
    `DerivativeStructure` | `multiply(double a)` '×' operator.
    `DerivativeStructure` | `multiply(int n)` Compute n × this.
    `DerivativeStructure` | `negate()` Returns the additive inverse of `this` element.
    `DerivativeStructure` | `pow(DerivativeStructure e)` Power operation.
    `DerivativeStructure` | `pow(double p)` Power operation.
    `static DerivativeStructure` | `pow(double a, DerivativeStructure x)` Compute ax where a is a double and x a `DerivativeStructure`
    `DerivativeStructure` | `pow(int n)` Integer power operation.
    `DerivativeStructure` | `reciprocal()` Returns the multiplicative inverse of `this` element.
    `DerivativeStructure` | `remainder(DerivativeStructure a)` IEEE remainder operator.
    `DerivativeStructure` | `remainder(double a)` IEEE remainder operator.
    `DerivativeStructure` | `rint()` Get the whole number that is the nearest to the instance, or the even one if x is exactly half way between two integers.
    `DerivativeStructure` | `rootN(int n)` Nth root.
    `long` | `round()` Get the closest long to instance value.
    `DerivativeStructure` | `scalb(int n)` Multiply the instance by a power of 2.
    `DerivativeStructure` | `signum()` Compute the signum of the instance.
    `DerivativeStructure` | `sin()` Sine operation.
    `DerivativeStructure` | `sinh()` Hyperbolic sine operation.
    `DerivativeStructure` | `sqrt()` Square root.
    `DerivativeStructure` | `subtract(DerivativeStructure a)` Compute this - a.
    `DerivativeStructure` | `subtract(double a)` '-' operator.
    `DerivativeStructure` | `tan()` Tangent operation.
    `DerivativeStructure` | `tanh()` Hyperbolic tangent operation.
    `double` | `taylor(double... delta)` Evaluate Taylor expansion a derivative structure.
    `DerivativeStructure` | `toDegrees()` Convert radians to degrees, with error of less than 0.5 ULP
    `DerivativeStructure` | `toRadians()` Convert degrees to radians, with error of less than 0.5 ULP

    - ### Methods inherited from class java.lang.Object

      `clone, finalize, getClass, notify, notifyAll, toString, wait, wait, wait`

- - ### Constructor Detail

    - #### DerivativeStructure

          public DerivativeStructure(int parameters,
                             int order)
                              throws NumberIsTooLargeException

      Build an instance with all values and derivatives set to 0.
      Parameters:
        `parameters` - number of free parameters
        `order` - derivation order
      Throws:
        `NumberIsTooLargeException` - if order is too large

    - #### DerivativeStructure

          public DerivativeStructure(int parameters,
                             int order,
                             double value)
                              throws NumberIsTooLargeException

      Build an instance representing a constant value.
      Parameters:
        `parameters` - number of free parameters
        `order` - derivation order
        `value` - value of the constant
      Throws:
        `NumberIsTooLargeException` - if order is too large
      See Also:
        `DerivativeStructure(int, int, int, double)`

    - #### DerivativeStructure

          public DerivativeStructure(int parameters,
                             int order,
                             int index,
                             double value)
                              throws NumberIsTooLargeException

      Build an instance representing a variable.

      Instances built using this constructor are considered to be the free variables with respect to which differentials are computed. As such, their differential with respect to themselves is +1.

      Parameters:
        `parameters` - number of free parameters
        `order` - derivation order
        `index` - index of the variable (from 0 to `parameters - 1`)
        `value` - value of the variable
      Throws:
        `NumberIsTooLargeException` - if `index >= parameters`.
      See Also:
        `DerivativeStructure(int, int, double)`

    - #### DerivativeStructure

          public DerivativeStructure(double a1,
                             DerivativeStructure ds1,
                             double a2,
                             DerivativeStructure ds2)
                              throws DimensionMismatchException

      Linear combination constructor. The derivative structure built will be a1 * ds1 + a2 * ds2
      Parameters:
        `a1` - first scale factor
        `ds1` - first base (unscaled) derivative structure
        `a2` - second scale factor
        `ds2` - second base (unscaled) derivative structure
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders are inconsistent

    - #### DerivativeStructure

          public DerivativeStructure(double a1,
                             DerivativeStructure ds1,
                             double a2,
                             DerivativeStructure ds2,
                             double a3,
                             DerivativeStructure ds3)
                              throws DimensionMismatchException

      Linear combination constructor. The derivative structure built will be a1 * ds1 + a2 * ds2 + a3 * ds3
      Parameters:
        `a1` - first scale factor
        `ds1` - first base (unscaled) derivative structure
        `a2` - second scale factor
        `ds2` - second base (unscaled) derivative structure
        `a3` - third scale factor
        `ds3` - third base (unscaled) derivative structure
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders are inconsistent

    - #### DerivativeStructure

          public DerivativeStructure(double a1,
                             DerivativeStructure ds1,
                             double a2,
                             DerivativeStructure ds2,
                             double a3,
                             DerivativeStructure ds3,
                             double a4,
                             DerivativeStructure ds4)
                              throws DimensionMismatchException

      Linear combination constructor. The derivative structure built will be a1 * ds1 + a2 * ds2 + a3 * ds3 + a4 * ds4
      Parameters:
        `a1` - first scale factor
        `ds1` - first base (unscaled) derivative structure
        `a2` - second scale factor
        `ds2` - second base (unscaled) derivative structure
        `a3` - third scale factor
        `ds3` - third base (unscaled) derivative structure
        `a4` - fourth scale factor
        `ds4` - fourth base (unscaled) derivative structure
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders are inconsistent

    - #### DerivativeStructure

          public DerivativeStructure(int parameters,
                             int order,
                             double... derivatives)
                              throws DimensionMismatchException,
                                     NumberIsTooLargeException

      Build an instance from all its derivatives.
      Parameters:
        `parameters` - number of free parameters
        `order` - derivation order
        `derivatives` - derivatives sorted according to `DSCompiler.getPartialDerivativeIndex(int...)`
      Throws:
        `DimensionMismatchException` - if derivatives array does not match the `size` expected by the compiler
        `NumberIsTooLargeException` - if order is too large
      See Also:
        `getAllDerivatives()`

  - ### Method Detail

    - #### getFreeParameters

          public int getFreeParameters()

      Get the number of free parameters.
      Returns:
        number of free parameters

    - #### getOrder

          public int getOrder()

      Get the derivation order.
      Returns:
        derivation order

    - #### createConstant

          public DerivativeStructure createConstant(double c)

      Create a constant compatible with instance order and number of parameters.

      This method is a convenience factory method, it simply calls `new DerivativeStructure(getFreeParameters(), getOrder(), c)`

      Parameters:
        `c` - value of the constant
      Returns:
        a constant compatible with instance order and number of parameters
      Since:
        3.3
      See Also:
        `DerivativeStructure(int, int, double)`

    - #### getReal

          public double getReal()

      Get the real value of the number.
      Specified by:
        `getReal` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        real value
      Since:
        3.2

    - #### getValue

          public double getValue()

      Get the value part of the derivative structure.
      Returns:
        value part of the derivative structure
      See Also:
        `getPartialDerivative(int...)`

    - #### getPartialDerivative

          public double getPartialDerivative(int... orders)
                                      throws DimensionMismatchException,
                                             NumberIsTooLargeException

      Get a partial derivative.
      Parameters:
        `orders` - derivation orders with respect to each variable (if all orders are 0, the value is returned)
      Returns:
        partial derivative
      Throws:
        `DimensionMismatchException` - if the numbers of variables does not match the instance
        `NumberIsTooLargeException` - if sum of derivation orders is larger than the instance limits
      See Also:
        `getValue()`

    - #### getAllDerivatives

          public double[] getAllDerivatives()

      Get all partial derivatives.
      Returns:
        a fresh copy of partial derivatives, in an array sorted according to `DSCompiler.getPartialDerivativeIndex(int...)`

    - #### add

          public DerivativeStructure add(double a)

      '+' operator.
      Specified by:
        `add` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this+a
      Since:
        3.2

    - #### add

          public DerivativeStructure add(DerivativeStructure a)
                                  throws DimensionMismatchException

      Compute this + a.
      Specified by:
        `add` in interface `FieldElement<DerivativeStructure>`
      Parameters:
        `a` - element to add
      Returns:
        a new element representing this + a
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match

    - #### subtract

          public DerivativeStructure subtract(double a)

      '-' operator.
      Specified by:
        `subtract` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this-a
      Since:
        3.2

    - #### subtract

          public DerivativeStructure subtract(DerivativeStructure a)
                                       throws DimensionMismatchException

      Compute this - a.
      Specified by:
        `subtract` in interface `FieldElement<DerivativeStructure>`
      Parameters:
        `a` - element to subtract
      Returns:
        a new element representing this - a
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match

    - #### multiply

          public DerivativeStructure multiply(int n)

      Compute n × this. Multiplication by an integer number is defined as the following sum
      n × this = ∑i=1n this.
      Specified by:
        `multiply` in interface `FieldElement<DerivativeStructure>`
      Parameters:
        `n` - Number of times `this` must be added to itself.
      Returns:
        A new element representing n × this.

    - #### multiply

          public DerivativeStructure multiply(double a)

      '×' operator.
      Specified by:
        `multiply` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this×a
      Since:
        3.2

    - #### multiply

          public DerivativeStructure multiply(DerivativeStructure a)
                                       throws DimensionMismatchException

      Compute this × a.
      Specified by:
        `multiply` in interface `FieldElement<DerivativeStructure>`
      Parameters:
        `a` - element to multiply
      Returns:
        a new element representing this × a
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match

    - #### divide

          public DerivativeStructure divide(double a)

      '÷' operator.
      Specified by:
        `divide` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this÷s;a
      Since:
        3.2

    - #### divide

          public DerivativeStructure divide(DerivativeStructure a)
                                     throws DimensionMismatchException

      Compute this ÷ a.
      Specified by:
        `divide` in interface `FieldElement<DerivativeStructure>`
      Parameters:
        `a` - element to add
      Returns:
        a new element representing this ÷ a
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match

    - #### remainder

          public DerivativeStructure remainder(double a)

      IEEE remainder operator.
      Specified by:
        `remainder` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this - n × a where n is the closest integer to this/a (the even integer is chosen for n if this/a is halfway between two integers)

    - #### remainder

          public DerivativeStructure remainder(DerivativeStructure a)
                                        throws DimensionMismatchException

      IEEE remainder operator.
      Specified by:
        `remainder` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this - n × a where n is the closest integer to this/a (the even integer is chosen for n if this/a is halfway between two integers)
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### negate

          public DerivativeStructure negate()

      Returns the additive inverse of `this` element.
      Specified by:
        `negate` in interface `FieldElement<DerivativeStructure>`
      Returns:
        the opposite of `this`.

    - #### abs

          public DerivativeStructure abs()

      absolute value.
      Specified by:
        `abs` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        abs(this)
      Since:
        3.2

    - #### ceil

          public DerivativeStructure ceil()

      Get the smallest whole number larger than instance.
      Specified by:
        `ceil` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        ceil(this)
      Since:
        3.2

    - #### floor

          public DerivativeStructure floor()

      Get the largest whole number smaller than instance.
      Specified by:
        `floor` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        floor(this)
      Since:
        3.2

    - #### rint

          public DerivativeStructure rint()

      Get the whole number that is the nearest to the instance, or the even one if x is exactly half way between two integers.
      Specified by:
        `rint` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        a double number r such that r is an integer r - 0.5 <= this <= r + 0.5
      Since:
        3.2

    - #### round

          public long round()

      Get the closest long to instance value.
      Specified by:
        `round` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        closest long to `RealFieldElement.getReal()`

    - #### signum

          public DerivativeStructure signum()

      Compute the signum of the instance. The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise
      Specified by:
        `signum` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a
      Since:
        3.2

    - #### copySign

          public DerivativeStructure copySign(DerivativeStructure sign)

      Returns the instance with the sign of the argument. A NaN `sign` argument is treated as positive.
      Specified by:
        `copySign` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `sign` - the sign for the returned value
      Returns:
        the instance with the same sign as the `sign` argument
      Since:
        3.2

    - #### copySign

          public DerivativeStructure copySign(double sign)

      Returns the instance with the sign of the argument. A NaN `sign` argument is treated as positive.
      Specified by:
        `copySign` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `sign` - the sign for the returned value
      Returns:
        the instance with the same sign as the `sign` argument
      Since:
        3.2

    - #### getExponent

          public int getExponent()

      Return the exponent of the instance value, removing the bias.

      For double numbers of the form 2x, the unbiased exponent is exactly x.

      Returns:
        exponent for instance in IEEE754 representation, without bias

    - #### scalb

          public DerivativeStructure scalb(int n)

      Multiply the instance by a power of 2.
      Specified by:
        `scalb` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `n` - power of 2
      Returns:
        this × 2n
      Since:
        3.2

    - #### hypot

          public DerivativeStructure hypot(DerivativeStructure y)
                                    throws DimensionMismatchException

      Returns the hypotenuse of a triangle with sides `this` and `y` - sqrt(this2 +y2)
      avoiding intermediate overflow or underflow.
      - If either argument is infinite, then the result is positive infinity.
      - else, if either argument is NaN then the result is NaN.
      Specified by:
        `hypot` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `y` - a value
      Returns:
        sqrt(this2 +y2)
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### hypot

          public static DerivativeStructure hypot(DerivativeStructure x,
                                  DerivativeStructure y)
                                           throws DimensionMismatchException

      Returns the hypotenuse of a triangle with sides `x` and `y` - sqrt(x2 +y2)
      avoiding intermediate overflow or underflow.
      - If either argument is infinite, then the result is positive infinity.
      - else, if either argument is NaN then the result is NaN.
      Parameters:
        `x` - a value
        `y` - a value
      Returns:
        sqrt(x2 +y2)
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### compose

          public DerivativeStructure compose(double... f)
                                      throws DimensionMismatchException

      Compute composition of the instance by a univariate function.
      Parameters:
        `f` - array of value and derivatives of the function at the current point (i.e. [f(`getValue()`), f'(`getValue()`), f''(`getValue()`)...]).
      Returns:
        f(this)
      Throws:
        `DimensionMismatchException` - if the number of derivatives in the array is not equal to `order` + 1

    - #### reciprocal

          public DerivativeStructure reciprocal()

      Returns the multiplicative inverse of `this` element.
      Specified by:
        `reciprocal` in interface `FieldElement<DerivativeStructure>`
      Specified by:
        `reciprocal` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        the inverse of `this`.

    - #### sqrt

          public DerivativeStructure sqrt()

      Square root.
      Specified by:
        `sqrt` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        square root of the instance
      Since:
        3.2

    - #### cbrt

          public DerivativeStructure cbrt()

      Cubic root.
      Specified by:
        `cbrt` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        cubic root of the instance
      Since:
        3.2

    - #### rootN

          public DerivativeStructure rootN(int n)

      Nth root.
      Specified by:
        `rootN` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `n` - order of the root
      Returns:
        nth root of the instance
      Since:
        3.2

    - #### getField

          public Field<DerivativeStructure> getField()

      Get the `Field` to which the instance belongs.
      Specified by:
        `getField` in interface `FieldElement<DerivativeStructure>`
      Returns:
        `Field` to which the instance belongs

    - #### pow

          public static DerivativeStructure pow(double a,
                                DerivativeStructure x)

      Compute ax where a is a double and x a `DerivativeStructure`
      Parameters:
        `a` - number to exponentiate
        `x` - power to apply
      Returns:
        ax
      Since:
        3.3

    - #### pow

          public DerivativeStructure pow(double p)

      Power operation.
      Specified by:
        `pow` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `p` - power to apply
      Returns:
        thisp
      Since:
        3.2

    - #### pow

          public DerivativeStructure pow(int n)

      Integer power operation.
      Specified by:
        `pow` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `n` - power to apply
      Returns:
        thisn
      Since:
        3.2

    - #### pow

          public DerivativeStructure pow(DerivativeStructure e)
                                  throws DimensionMismatchException

      Power operation.
      Specified by:
        `pow` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `e` - exponent
      Returns:
        thise
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### exp

          public DerivativeStructure exp()

      Exponential.
      Specified by:
        `exp` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        exponential of the instance
      Since:
        3.2

    - #### expm1

          public DerivativeStructure expm1()

      Exponential minus 1.
      Specified by:
        `expm1` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        exponential minus one of the instance
      Since:
        3.2

    - #### log

          public DerivativeStructure log()

      Natural logarithm.
      Specified by:
        `log` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        logarithm of the instance
      Since:
        3.2

    - #### log1p

          public DerivativeStructure log1p()

      Shifted natural logarithm.
      Specified by:
        `log1p` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        logarithm of one plus the instance
      Since:
        3.2

    - #### log10

          public DerivativeStructure log10()

      Base 10 logarithm.
      Returns:
        base 10 logarithm of the instance

    - #### cos

          public DerivativeStructure cos()

      Cosine operation.
      Specified by:
        `cos` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        cos(this)
      Since:
        3.2

    - #### sin

          public DerivativeStructure sin()

      Sine operation.
      Specified by:
        `sin` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        sin(this)
      Since:
        3.2

    - #### tan

          public DerivativeStructure tan()

      Tangent operation.
      Specified by:
        `tan` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        tan(this)
      Since:
        3.2

    - #### acos

          public DerivativeStructure acos()

      Arc cosine operation.
      Specified by:
        `acos` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        acos(this)
      Since:
        3.2

    - #### asin

          public DerivativeStructure asin()

      Arc sine operation.
      Specified by:
        `asin` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        asin(this)
      Since:
        3.2

    - #### atan

          public DerivativeStructure atan()

      Arc tangent operation.
      Specified by:
        `atan` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        atan(this)
      Since:
        3.2

    - #### atan2

          public DerivativeStructure atan2(DerivativeStructure x)
                                    throws DimensionMismatchException

      Two arguments arc tangent operation.
      Specified by:
        `atan2` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `x` - second argument of the arc tangent
      Returns:
        atan2(this, x)
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders are inconsistent
      Since:
        3.2

    - #### atan2

          public static DerivativeStructure atan2(DerivativeStructure y,
                                  DerivativeStructure x)
                                           throws DimensionMismatchException

      Two arguments arc tangent operation.
      Parameters:
        `y` - first argument of the arc tangent
        `x` - second argument of the arc tangent
      Returns:
        atan2(y, x)
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### cosh

          public DerivativeStructure cosh()

      Hyperbolic cosine operation.
      Specified by:
        `cosh` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        cosh(this)
      Since:
        3.2

    - #### sinh

          public DerivativeStructure sinh()

      Hyperbolic sine operation.
      Specified by:
        `sinh` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        sinh(this)
      Since:
        3.2

    - #### tanh

          public DerivativeStructure tanh()

      Hyperbolic tangent operation.
      Specified by:
        `tanh` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        tanh(this)
      Since:
        3.2

    - #### acosh

          public DerivativeStructure acosh()

      Inverse hyperbolic cosine operation.
      Specified by:
        `acosh` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        acosh(this)
      Since:
        3.2

    - #### asinh

          public DerivativeStructure asinh()

      Inverse hyperbolic sine operation.
      Specified by:
        `asinh` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        asin(this)
      Since:
        3.2

    - #### atanh

          public DerivativeStructure atanh()

      Inverse hyperbolic tangent operation.
      Specified by:
        `atanh` in interface `RealFieldElement<DerivativeStructure>`
      Returns:
        atanh(this)
      Since:
        3.2

    - #### toDegrees

          public DerivativeStructure toDegrees()

      Convert radians to degrees, with error of less than 0.5 ULP
      Returns:
        instance converted into degrees

    - #### toRadians

          public DerivativeStructure toRadians()

      Convert degrees to radians, with error of less than 0.5 ULP
      Returns:
        instance converted into radians

    - #### taylor

          public double taylor(double... delta)
                        throws MathArithmeticException

      Evaluate Taylor expansion a derivative structure.
      Parameters:
        `delta` - parameters offsets (Δx, Δy, ...)
      Returns:
        value of the Taylor expansion at x + Δx, y + Δy, ...
      Throws:
        `MathArithmeticException` - if factorials becomes too large

    - #### linearCombination

          public DerivativeStructure linearCombination(DerivativeStructure[] a,
                                              DerivativeStructure[] b)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - Factors.
        `b` - Factors.
      Returns:
        `Σi ai bi`.
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### linearCombination

          public DerivativeStructure linearCombination(double[] a,
                                              DerivativeStructure[] b)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a` - Factors.
        `b` - Factors.
      Returns:
        `Σi ai bi`.
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2

    - #### linearCombination

          public DerivativeStructure linearCombination(DerivativeStructure a1,
                                              DerivativeStructure b1,
                                              DerivativeStructure a2,
                                              DerivativeStructure b2)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
      Returns:
        a1×b1 + a2×b2
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object)`, `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object, Object, Object)`

    - #### linearCombination

          public DerivativeStructure linearCombination(double a1,
                                              DerivativeStructure b1,
                                              double a2,
                                              DerivativeStructure b2)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
      Returns:
        a1×b1 + a2×b2
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(double, Object, double, Object, double, Object)`, `RealFieldElement.linearCombination(double, Object, double, Object, double, Object, double, Object)`

    - #### linearCombination

          public DerivativeStructure linearCombination(DerivativeStructure a1,
                                              DerivativeStructure b1,
                                              DerivativeStructure a2,
                                              DerivativeStructure b2,
                                              DerivativeStructure a3,
                                              DerivativeStructure b3)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(Object, Object, Object, Object)`, `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object, Object, Object)`

    - #### linearCombination

          public DerivativeStructure linearCombination(double a1,
                                              DerivativeStructure b1,
                                              double a2,
                                              DerivativeStructure b2,
                                              double a3,
                                              DerivativeStructure b3)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(double, Object, double, Object)`, `RealFieldElement.linearCombination(double, Object, double, Object, double, Object, double, Object)`

    - #### linearCombination

          public DerivativeStructure linearCombination(DerivativeStructure a1,
                                              DerivativeStructure b1,
                                              DerivativeStructure a2,
                                              DerivativeStructure b2,
                                              DerivativeStructure a3,
                                              DerivativeStructure b3,
                                              DerivativeStructure a4,
                                              DerivativeStructure b4)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
        `a4` - first factor of the third term
        `b4` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3 + a4×b4
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(Object, Object, Object, Object)`, `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object)`

    - #### linearCombination

          public DerivativeStructure linearCombination(double a1,
                                              DerivativeStructure b1,
                                              double a2,
                                              DerivativeStructure b2,
                                              double a3,
                                              DerivativeStructure b3,
                                              double a4,
                                              DerivativeStructure b4)
                                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<DerivativeStructure>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
        `a4` - first factor of the third term
        `b4` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3 + a4×b4
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders do not match
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(double, Object, double, Object)`, `RealFieldElement.linearCombination(double, Object, double, Object, double, Object)`

    - #### equals

          public boolean equals(Object other)

      Test for the equality of two derivative structures.

      Derivative structures are considered equal if they have the same number of free parameters, the same derivation order, and the same derivatives.

      Overrides:
        `equals` in class `Object`
      Parameters:
        `other` - Object to test for equality to this
      Returns:
        true if two derivative structures are equal
      Since:
        3.2

    - #### hashCode

          public int hashCode()

      Get a hashCode for the derivative structure.
      Overrides:
        `hashCode` in class `Object`
      Returns:
        a hash code value for this object
      Since:
        3.2

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