Package org.hipparchus.analysis.differentiation

Class UnivariateDerivative1

java.lang.Object

org.hipparchus.analysis.differentiation.UnivariateDerivative<UnivariateDerivative1>

org.hipparchus.analysis.differentiation.UnivariateDerivative1

All Implemented Interfaces:

Serializable, Comparable<UnivariateDerivative1>, Derivative<UnivariateDerivative1>, Derivative1<UnivariateDerivative1>, DifferentialAlgebra, CalculusFieldElement<UnivariateDerivative1>, FieldElement<UnivariateDerivative1>

public class UnivariateDerivative1
extends UnivariateDerivative<UnivariateDerivative1>
implements Derivative1<UnivariateDerivative1>

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

This class is a stripped-down version of DerivativeStructure with only one free parameter and derivation order also limited to one. It should have less overhead than DerivativeStructure in its domain.

This class is an implementation of Rall's numbers. 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.

UnivariateDerivative1 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 Derivative-based classes (or in fact any CalculusFieldElement class) is a tedious and error-prone task but has the advantage of not requiring users to compute the derivatives by themselves and allowing to switch for one derivative implementation to another as they all share the same filed API.

Instances of this class are guaranteed to be immutable.

Since:

1.7

See Also:

• DerivativeStructure
• UnivariateDerivative2
• Gradient
• FieldDerivativeStructure
• FieldUnivariateDerivative1
• FieldUnivariateDerivative2
• FieldGradient
• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
static final UnivariateDerivative1	PI	The constant value of π as a UnivariateDerivative1.

Constructor Summary

Constructors

Constructor	Description
UnivariateDerivative1(double f0, double f1)	Build an instance with values and derivative.
UnivariateDerivative1(DerivativeStructure ds)	Build an instance from a DerivativeStructure.

Method Summary

Modifier and Type	Method	Description
UnivariateDerivative1	abs()	absolute value.
UnivariateDerivative1	add(UnivariateDerivative1 a)	Compute this + a.
UnivariateDerivative1	atan2(UnivariateDerivative1 x)	Two arguments arc tangent operation.
int	compareTo(UnivariateDerivative1 o)
UnivariateDerivative1	compose(double... f)	Compute composition of the instance by a univariate function.
UnivariateDerivative1	compose(double ff0, double ff1)	Compute composition of the instance by a univariate function differentiable at order 1.
UnivariateDerivative1	copySign(double sign)	Returns the instance with the sign of the argument.
UnivariateDerivative1	copySign(UnivariateDerivative1 sign)	Returns the instance with the sign of the argument.
UnivariateDerivative1	divide(double a)	'÷' operator.
UnivariateDerivative1	divide(UnivariateDerivative1 a)	Compute this ÷ a.
boolean	equals(Object other)	Test for the equality of two univariate derivatives.
UnivariateDerivative1	getAddendum()	Get the addendum to the real value of the number.
double	getDerivative(int n)	Get a derivative from the univariate derivative.
UnivariateDerivative1Field	getField()	Get the Field to which the instance belongs.
double	getFirstDerivative()	Get the first derivative.
UnivariateDerivative1	getPi()	Get the Archimedes constant π.
double	getValue()	Get the value part of the function.
int	hashCode()	Get a hashCode for the univariate derivative.
UnivariateDerivative1	hypot(UnivariateDerivative1 y)	Returns the hypotenuse of a triangle with sides this and y - sqrt(this2 +y2) avoiding intermediate overflow or underflow.
UnivariateDerivative1	linearCombination(double[] a, UnivariateDerivative1[] b)	Compute a linear combination.
UnivariateDerivative1	linearCombination(double a1, UnivariateDerivative1 b1, double a2, UnivariateDerivative1 b2)	Compute a linear combination.
UnivariateDerivative1	linearCombination(double a1, UnivariateDerivative1 b1, double a2, UnivariateDerivative1 b2, double a3, UnivariateDerivative1 b3)	Compute a linear combination.
UnivariateDerivative1	linearCombination(double a1, UnivariateDerivative1 b1, double a2, UnivariateDerivative1 b2, double a3, UnivariateDerivative1 b3, double a4, UnivariateDerivative1 b4)	Compute a linear combination.
UnivariateDerivative1	linearCombination(UnivariateDerivative1[] a, UnivariateDerivative1[] b)	Compute a linear combination.
UnivariateDerivative1	linearCombination(UnivariateDerivative1 a1, UnivariateDerivative1 b1, UnivariateDerivative1 a2, UnivariateDerivative1 b2)	Compute a linear combination.
UnivariateDerivative1	linearCombination(UnivariateDerivative1 a1, UnivariateDerivative1 b1, UnivariateDerivative1 a2, UnivariateDerivative1 b2, UnivariateDerivative1 a3, UnivariateDerivative1 b3)	Compute a linear combination.
UnivariateDerivative1	linearCombination(UnivariateDerivative1 a1, UnivariateDerivative1 b1, UnivariateDerivative1 a2, UnivariateDerivative1 b2, UnivariateDerivative1 a3, UnivariateDerivative1 b3, UnivariateDerivative1 a4, UnivariateDerivative1 b4)	Compute a linear combination.
UnivariateDerivative1	multiply(double a)	'×' operator.
UnivariateDerivative1	multiply(int n)	Compute n × this.
UnivariateDerivative1	multiply(UnivariateDerivative1 a)	Compute this × a.
UnivariateDerivative1	negate()	Returns the additive inverse of this element.
UnivariateDerivative1	newInstance(double value)	Create an instance corresponding to a constant real value.
UnivariateDerivative1	pow(double p)	Power operation.
static UnivariateDerivative1	pow(double a, UnivariateDerivative1 x)	Compute ax where a is a double and x a UnivariateDerivative1
UnivariateDerivative1	pow(int n)	Integer power operation.
UnivariateDerivative1	remainder(UnivariateDerivative1 a)	IEEE remainder operator.
UnivariateDerivative1	scalb(int n)	Multiply the instance by a power of 2.
UnivariateDerivative1	subtract(UnivariateDerivative1 a)	Compute this - a.
double	taylor(double delta)	Evaluate Taylor expansion a univariate derivative.
UnivariateDerivative1	toDegrees()	Convert radians to degrees, with error of less than 0.5 ULP
DerivativeStructure	toDerivativeStructure()	Convert the instance to a DerivativeStructure.
UnivariateDerivative1	toRadians()	Convert degrees to radians, with error of less than 0.5 ULP
UnivariateDerivative1	withValue(double value)	Create a new object with new value (zeroth-order derivative, as passed as input) and same derivatives of order one and above.

Methods inherited from class org.hipparchus.analysis.differentiation.UnivariateDerivative

getFreeParameters, getPartialDerivative

Methods inherited from class java.lang.Object

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

Methods inherited from interface org.hipparchus.CalculusFieldElement

ceil, floor, isFinite, isInfinite, isNaN, norm, rint, round, sign, ulp

Methods inherited from interface org.hipparchus.analysis.differentiation.Derivative

add, getExponent, getPartialDerivative, getReal, pow, remainder, subtract

Methods inherited from interface org.hipparchus.analysis.differentiation.Derivative1

acos, acosh, asin, asinh, atan, atanh, cbrt, cos, cosh, exp, expm1, getOrder, log, log10, log1p, reciprocal, rootN, sin, sinCos, sinh, sinhCosh, sqrt, square, tan, tanh

Methods inherited from interface org.hipparchus.analysis.differentiation.DifferentialAlgebra

getFreeParameters

Methods inherited from interface org.hipparchus.FieldElement

isZero

Field Details

PI

public static final UnivariateDerivative1 PI

The constant value of π as a UnivariateDerivative1.

Since:

2.0

Constructor Details

UnivariateDerivative1

public UnivariateDerivative1(double f0,
 double f1)

Build an instance with values and derivative.

Parameters:

f0 - value of the function

f1 - first derivative of the function

UnivariateDerivative1

public UnivariateDerivative1(DerivativeStructure ds)
                      throws MathIllegalArgumentException

Build an instance from a DerivativeStructure.

Parameters:

ds - derivative structure

Throws:

MathIllegalArgumentException - if either ds parameters is not 1 or ds order is not 1

Method Details

newInstance

public UnivariateDerivative1 newInstance(double value)

Create an instance corresponding to a constant real value.

Specified by:

newInstance in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

value - constant real value

Returns:

instance corresponding to a constant real value

withValue

public UnivariateDerivative1 withValue(double value)

Create a new object with new value (zeroth-order derivative, as passed as input) and same derivatives of order one and above.

This default implementation is there so that no API gets broken by the next release, which is not a major one. Custom inheritors should probably overwrite it.

Specified by:

withValue in interface Derivative<UnivariateDerivative1>

Parameters:

value - zeroth-order derivative of new represented function

Returns:

new object with changed value

getAddendum

public UnivariateDerivative1 getAddendum()

Get the addendum to the real value of the number.

The addendum is considered to be the part that when added back to the real part recovers the instance. This means that when e.getReal() is finite (i.e. neither infinite nor NaN), then e.getAddendum().add(e.getReal()) is e and e.subtract(e.getReal()) is e.getAddendum(). Beware that for non-finite numbers, these two equalities may not hold. The first equality (with the addition), always holds even for infinity and NaNs if the real part is independent of the addendum (this is the case for all derivatives types, as well as for complex and Dfp, but it is not the case for Tuple and FieldTuple). The second equality (with the subtraction), generally doesn't hold for non-finite numbers, because the subtraction generates NaNs.

Specified by:

getAddendum in interface CalculusFieldElement<UnivariateDerivative1>

Returns:

real value

getValue

public double getValue()

Get the value part of the function.

Specified by:

getValue in interface Derivative<UnivariateDerivative1>

Returns:

value part of the value of the function

getDerivative

public double getDerivative(int n)

Get a derivative from the univariate derivative.

Specified by:

getDerivative in class UnivariateDerivative<UnivariateDerivative1>

Parameters:

n - derivation order (must be between 0 and DifferentialAlgebra.getOrder(), both inclusive)

Returns:

n^th derivative

getFirstDerivative

public double getFirstDerivative()

Get the first derivative.

Returns:

first derivative

See Also:

• getValue()

toDerivativeStructure

public DerivativeStructure toDerivativeStructure()

Convert the instance to a DerivativeStructure.

Specified by:

toDerivativeStructure in class UnivariateDerivative<UnivariateDerivative1>

Returns:

derivative structure with same value and derivative as the instance

add

public UnivariateDerivative1 add(UnivariateDerivative1 a)

Compute this + a.

Specified by:

add in interface FieldElement<UnivariateDerivative1>

Parameters:

a - element to add

Returns:

a new element representing this + a

subtract

public UnivariateDerivative1 subtract(UnivariateDerivative1 a)

Compute this - a.

Specified by:

subtract in interface CalculusFieldElement<UnivariateDerivative1>

Specified by:

subtract in interface FieldElement<UnivariateDerivative1>

Parameters:

a - element to subtract

Returns:

a new element representing this - a

multiply

public UnivariateDerivative1 multiply(int n)

Compute n × this. Multiplication by an integer number is defined as the following sum \[ n \times \mathrm{this} = \sum_{i=1}^n \mathrm{this} \]

Specified by:

multiply in interface CalculusFieldElement<UnivariateDerivative1>

Specified by:

multiply in interface FieldElement<UnivariateDerivative1>

Parameters:

n - Number of times this must be added to itself.

Returns:

A new element representing n × this.

multiply

public UnivariateDerivative1 multiply(double a)

'×' operator.

Specified by:

multiply in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

a - right hand side parameter of the operator

Returns:

this×a

multiply

public UnivariateDerivative1 multiply(UnivariateDerivative1 a)

Compute this × a.

Specified by:

multiply in interface FieldElement<UnivariateDerivative1>

Parameters:

a - element to multiply

Returns:

a new element representing this × a

divide

public UnivariateDerivative1 divide(double a)

'÷' operator.

Specified by:

divide in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

a - right hand side parameter of the operator

Returns:

this÷a

divide

public UnivariateDerivative1 divide(UnivariateDerivative1 a)

Compute this ÷ a.

Specified by:

divide in interface CalculusFieldElement<UnivariateDerivative1>

Specified by:

divide in interface FieldElement<UnivariateDerivative1>

Parameters:

a - element to divide by

Returns:

a new element representing this ÷ a

remainder

public UnivariateDerivative1 remainder(UnivariateDerivative1 a)

IEEE remainder operator.

Specified by:

remainder in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

a - right hand side parameter of the operator

Returns:

this - n × a where n is the closest integer to this/a

negate

public UnivariateDerivative1 negate()

Returns the additive inverse of this element.

Specified by:

negate in interface FieldElement<UnivariateDerivative1>

Returns:

the opposite of this.

abs

public UnivariateDerivative1 abs()

absolute value.

Specified by:

abs in interface CalculusFieldElement<UnivariateDerivative1>

Returns:

abs(this)

copySign

public UnivariateDerivative1 copySign(UnivariateDerivative1 sign)

Returns the instance with the sign of the argument. A NaN sign argument is treated as positive.

Specified by:

copySign in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

sign - the sign for the returned value

Returns:

the instance with the same sign as the sign argument

copySign

public UnivariateDerivative1 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 CalculusFieldElement<UnivariateDerivative1>

Parameters:

sign - the sign for the returned value

Returns:

the instance with the same sign as the sign argument

scalb

public UnivariateDerivative1 scalb(int n)

Multiply the instance by a power of 2.

Specified by:

scalb in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

n - power of 2

Returns:

this × 2ⁿ

hypot

public UnivariateDerivative1 hypot(UnivariateDerivative1 y)

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 CalculusFieldElement<UnivariateDerivative1>

Parameters:

y - a value

Returns:

sqrt(this² +y²)

compose

public UnivariateDerivative1 compose(double... f)

Compute composition of the instance by a univariate function.

Specified by:

compose in interface Derivative<UnivariateDerivative1>

Parameters:

f - array of value and derivatives of the function at the current point (i.e. [f(Derivative.getValue()), f'(Derivative.getValue()), f''(Derivative.getValue())...]).

Returns:

f(this)

compose

public UnivariateDerivative1 compose(double ff0,
 double ff1)

Compute composition of the instance by a univariate function differentiable at order 1.

Specified by:

compose in interface Derivative1<UnivariateDerivative1>

Parameters:

ff0 - value of function

ff1 - first-order derivative

Returns:

f(this)

getField

public UnivariateDerivative1Field getField()

Get the Field to which the instance belongs.

Specified by:

getField in interface FieldElement<UnivariateDerivative1>

Returns:

Field to which the instance belongs

pow

public static UnivariateDerivative1 pow(double a,
 UnivariateDerivative1 x)

Compute a^x where a is a double and x a UnivariateDerivative1

Parameters:

a - number to exponentiate

x - power to apply

Returns:

a^x

pow

public UnivariateDerivative1 pow(double p)

Power operation.

Specified by:

pow in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

p - power to apply

Returns:

this^p

pow

public UnivariateDerivative1 pow(int n)

Integer power operation.

Specified by:

pow in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

n - power to apply

Returns:

thisⁿ

atan2

public UnivariateDerivative1 atan2(UnivariateDerivative1 x)

Two arguments arc tangent operation.

Beware of the order or arguments! As this is based on a two-arguments functions, in order to be consistent with arguments order, the instance is the first argument and the single provided argument is the second argument. In order to be consistent with programming languages atan2, this method computes atan2(this, x), i.e. the instance represents the y argument and the x argument is the one passed as a single argument. This may seem confusing especially for users of Wolfram alpha, as this site is not consistent with programming languages atan2 two-arguments arc tangent and puts x as its first argument.

Specified by:

atan2 in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

x - second argument of the arc tangent

Returns:

atan2(this, x)

toDegrees

public UnivariateDerivative1 toDegrees()

Convert radians to degrees, with error of less than 0.5 ULP

Specified by:

toDegrees in interface CalculusFieldElement<UnivariateDerivative1>

Returns:

instance converted into degrees

toRadians

public UnivariateDerivative1 toRadians()

Convert degrees to radians, with error of less than 0.5 ULP

Specified by:

toRadians in interface CalculusFieldElement<UnivariateDerivative1>

Returns:

instance converted into radians

taylor

public double taylor(double delta)

Evaluate Taylor expansion a univariate derivative.

Parameters:

delta - parameter offset Δx

Returns:

value of the Taylor expansion at x + Δx

linearCombination

public UnivariateDerivative1 linearCombination(UnivariateDerivative1[] a,
 UnivariateDerivative1[] b)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

a - Factors.

b - Factors.

Returns:

Σi ai bi.

linearCombination

public UnivariateDerivative1 linearCombination(double[] a,
 UnivariateDerivative1[] b)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

Parameters:

a - Factors.

b - Factors.

Returns:

Σi ai bi.

linearCombination

public UnivariateDerivative1 linearCombination(UnivariateDerivative1 a1,
 UnivariateDerivative1 b1,
 UnivariateDerivative1 a2,
 UnivariateDerivative1 b2)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

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

See Also:

• CalculusFieldElement.linearCombination(FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement)
• CalculusFieldElement.linearCombination(FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement)

linearCombination

public UnivariateDerivative1 linearCombination(double a1,
 UnivariateDerivative1 b1,
 double a2,
 UnivariateDerivative1 b2)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

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

See Also:

• CalculusFieldElement.linearCombination(double, FieldElement, double, FieldElement, double, FieldElement)
• CalculusFieldElement.linearCombination(double, FieldElement, double, FieldElement, double, FieldElement, double, FieldElement)

linearCombination

public UnivariateDerivative1 linearCombination(UnivariateDerivative1 a1,
 UnivariateDerivative1 b1,
 UnivariateDerivative1 a2,
 UnivariateDerivative1 b2,
 UnivariateDerivative1 a3,
 UnivariateDerivative1 b3)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

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

See Also:

• CalculusFieldElement.linearCombination(FieldElement, FieldElement, FieldElement, FieldElement)
• CalculusFieldElement.linearCombination(FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement)

linearCombination

public UnivariateDerivative1 linearCombination(double a1,
 UnivariateDerivative1 b1,
 double a2,
 UnivariateDerivative1 b2,
 double a3,
 UnivariateDerivative1 b3)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

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

See Also:

• CalculusFieldElement.linearCombination(double, FieldElement, double, FieldElement)
• CalculusFieldElement.linearCombination(double, FieldElement, double, FieldElement, double, FieldElement, double, FieldElement)

linearCombination

public UnivariateDerivative1 linearCombination(UnivariateDerivative1 a1,
 UnivariateDerivative1 b1,
 UnivariateDerivative1 a2,
 UnivariateDerivative1 b2,
 UnivariateDerivative1 a3,
 UnivariateDerivative1 b3,
 UnivariateDerivative1 a4,
 UnivariateDerivative1 b4)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

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 fourth term

b4 - second factor of the fourth term

Returns:

a₁×b₁ + a₂×b₂ + a₃×b₃ + a₄×b₄

See Also:

• CalculusFieldElement.linearCombination(FieldElement, FieldElement, FieldElement, FieldElement)
• CalculusFieldElement.linearCombination(FieldElement, FieldElement, FieldElement, FieldElement, FieldElement, FieldElement)

linearCombination

public UnivariateDerivative1 linearCombination(double a1,
 UnivariateDerivative1 b1,
 double a2,
 UnivariateDerivative1 b2,
 double a3,
 UnivariateDerivative1 b3,
 double a4,
 UnivariateDerivative1 b4)

Compute a linear combination.

Specified by:

linearCombination in interface CalculusFieldElement<UnivariateDerivative1>

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 fourth term

b4 - second factor of the fourth term

Returns:

a₁×b₁ + a₂×b₂ + a₃×b₃ + a₄×b₄

See Also:

• CalculusFieldElement.linearCombination(double, FieldElement, double, FieldElement)
• CalculusFieldElement.linearCombination(double, FieldElement, double, FieldElement, double, FieldElement)

getPi

public UnivariateDerivative1 getPi()

Get the Archimedes constant π.

Archimedes constant is the ratio of a circle's circumference to its diameter.

Specified by:

getPi in interface CalculusFieldElement<UnivariateDerivative1>

Returns:

Archimedes constant π

equals

public boolean equals(Object other)

Test for the equality of two univariate derivatives.

univariate derivatives are considered equal if they have the same derivatives.

Overrides:

equals in class Object

Parameters:

other - Object to test for equality to this

Returns:

true if two univariate derivatives are equal

hashCode

public int hashCode()

Get a hashCode for the univariate derivative.

Overrides:

hashCode in class Object

Returns:

a hash code value for this object

compareTo

public int compareTo(UnivariateDerivative1 o)

Comparison performed considering that derivatives are intrinsically linked to monomials in the corresponding Taylor expansion and that the higher the degree, the smaller the term.

Specified by:

compareTo in interface Comparable<UnivariateDerivative1>

Since:

3.0