Package org.hipparchus.analysis.differentiation

Class UnivariateDerivative2

java.lang.Object

org.hipparchus.analysis.differentiation.UnivariateDerivative<UnivariateDerivative2>

org.hipparchus.analysis.differentiation.UnivariateDerivative2

All Implemented Interfaces:

Serializable, Comparable<UnivariateDerivative2>, Derivative<UnivariateDerivative2>, DifferentialAlgebra, CalculusFieldElement<UnivariateDerivative2>, FieldElement<UnivariateDerivative2>

public class UnivariateDerivative2
extends UnivariateDerivative<UnivariateDerivative2>

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 two. 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.

UnivariateDerivative2 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 not requiring users to compute the derivatives by themselves.

Instances of this class are guaranteed to be immutable.

Since:

1.7

See Also:

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

Field Summary

Fields

Modifier and Type	Field	Description
static final UnivariateDerivative2	PI	The constant value of π as a UnivariateDerivative2.

Constructor Summary

Constructors

Constructor	Description
UnivariateDerivative2(double f0, double f1, double f2)	Build an instance with values and derivative.
UnivariateDerivative2(DerivativeStructure ds)	Build an instance from a DerivativeStructure.

Method Summary

Modifier and Type	Method	Description
UnivariateDerivative2	abs()	absolute value.
UnivariateDerivative2	acos()	Arc cosine operation.
UnivariateDerivative2	acosh()	Inverse hyperbolic cosine operation.
UnivariateDerivative2	add(UnivariateDerivative2 a)	Compute this + a.
UnivariateDerivative2	asin()	Arc sine operation.
UnivariateDerivative2	asinh()	Inverse hyperbolic sine operation.
UnivariateDerivative2	atan()	Arc tangent operation.
UnivariateDerivative2	atan2(UnivariateDerivative2 x)	Two arguments arc tangent operation.
UnivariateDerivative2	atanh()	Inverse hyperbolic tangent operation.
UnivariateDerivative2	cbrt()	Cubic root.
int	compareTo(UnivariateDerivative2 o)
UnivariateDerivative2	compose(double... f)	Compute composition of the instance by a univariate function.
UnivariateDerivative2	copySign(double sign)	Returns the instance with the sign of the argument.
UnivariateDerivative2	copySign(UnivariateDerivative2 sign)	Returns the instance with the sign of the argument.
UnivariateDerivative2	cos()	Cosine operation.
UnivariateDerivative2	cosh()	Hyperbolic cosine operation.
UnivariateDerivative2	divide(double a)	'÷' operator.
UnivariateDerivative2	divide(UnivariateDerivative2 a)	Compute this ÷ a.
boolean	equals(Object other)	Test for the equality of two univariate derivatives.
UnivariateDerivative2	exp()	Exponential.
UnivariateDerivative2	expm1()	Exponential minus 1.
double	getDerivative(int n)	Get a derivative from the univariate derivative.
UnivariateDerivative2Field	getField()	Get the Field to which the instance belongs.
double	getFirstDerivative()	Get the first derivative.
int	getOrder()	Get the maximum derivation order.
UnivariateDerivative2	getPi()	Get the Archimedes constant π.
double	getSecondDerivative()	Get the second derivative.
double	getValue()	Get the value part of the function.
int	hashCode()	Get a hashCode for the univariate derivative.
UnivariateDerivative2	hypot(UnivariateDerivative2 y)	Returns the hypotenuse of a triangle with sides this and y - sqrt(this2 +y2) avoiding intermediate overflow or underflow.
UnivariateDerivative2	linearCombination(double[] a, UnivariateDerivative2[] b)	Compute a linear combination.
UnivariateDerivative2	linearCombination(double a1, UnivariateDerivative2 b1, double a2, UnivariateDerivative2 b2)	Compute a linear combination.
UnivariateDerivative2	linearCombination(double a1, UnivariateDerivative2 b1, double a2, UnivariateDerivative2 b2, double a3, UnivariateDerivative2 b3)	Compute a linear combination.
UnivariateDerivative2	linearCombination(double a1, UnivariateDerivative2 b1, double a2, UnivariateDerivative2 b2, double a3, UnivariateDerivative2 b3, double a4, UnivariateDerivative2 b4)	Compute a linear combination.
UnivariateDerivative2	linearCombination(UnivariateDerivative2[] a, UnivariateDerivative2[] b)	Compute a linear combination.
UnivariateDerivative2	linearCombination(UnivariateDerivative2 a1, UnivariateDerivative2 b1, UnivariateDerivative2 a2, UnivariateDerivative2 b2)	Compute a linear combination.
UnivariateDerivative2	linearCombination(UnivariateDerivative2 a1, UnivariateDerivative2 b1, UnivariateDerivative2 a2, UnivariateDerivative2 b2, UnivariateDerivative2 a3, UnivariateDerivative2 b3)	Compute a linear combination.
UnivariateDerivative2	linearCombination(UnivariateDerivative2 a1, UnivariateDerivative2 b1, UnivariateDerivative2 a2, UnivariateDerivative2 b2, UnivariateDerivative2 a3, UnivariateDerivative2 b3, UnivariateDerivative2 a4, UnivariateDerivative2 b4)	Compute a linear combination.
UnivariateDerivative2	log()	Natural logarithm.
UnivariateDerivative2	log10()	Base 10 logarithm.
UnivariateDerivative2	log1p()	Shifted natural logarithm.
UnivariateDerivative2	multiply(double a)	'×' operator.
UnivariateDerivative2	multiply(int n)	Compute n × this.
UnivariateDerivative2	multiply(UnivariateDerivative2 a)	Compute this × a.
UnivariateDerivative2	negate()	Returns the additive inverse of this element.
UnivariateDerivative2	newInstance(double value)	Create an instance corresponding to a constant real value.
UnivariateDerivative2	pow(double p)	Power operation.
static UnivariateDerivative2	pow(double a, UnivariateDerivative2 x)	Compute ax where a is a double and x a UnivariateDerivative2
UnivariateDerivative2	pow(int n)	Integer power operation.
UnivariateDerivative2	reciprocal()	Returns the multiplicative inverse of this element.
UnivariateDerivative2	remainder(UnivariateDerivative2 a)	IEEE remainder operator.
UnivariateDerivative2	rootN(int n)	Nth root.
UnivariateDerivative2	scalb(int n)	Multiply the instance by a power of 2.
UnivariateDerivative2	sin()	Sine operation.
FieldSinCos<UnivariateDerivative2>	sinCos()	Combined Sine and Cosine operation.
UnivariateDerivative2	sinh()	Hyperbolic sine operation.
FieldSinhCosh<UnivariateDerivative2>	sinhCosh()	Combined hyperbolic sine and cosine operation.
UnivariateDerivative2	sqrt()	Square root.
UnivariateDerivative2	square()	Compute this × this.
UnivariateDerivative2	subtract(UnivariateDerivative2 a)	Compute this - a.
UnivariateDerivative2	tan()	Tangent operation.
UnivariateDerivative2	tanh()	Hyperbolic tangent operation.
double	taylor(double delta)	Evaluate Taylor expansion a univariate derivative.
UnivariateDerivative2	toDegrees()	Convert radians to degrees, with error of less than 0.5 ULP
DerivativeStructure	toDerivativeStructure()	Convert the instance to a DerivativeStructure.
UnivariateDerivative2	toRadians()	Convert degrees to radians, with error of less than 0.5 ULP
UnivariateDerivative2	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, getReal, pow, remainder, subtract

Methods inherited from interface org.hipparchus.FieldElement

isZero

Field Details

PI

public static final UnivariateDerivative2 PI

The constant value of π as a UnivariateDerivative2.

Since:

2.0

Constructor Details

UnivariateDerivative2

public UnivariateDerivative2(double f0,
 double f1,
 double f2)

Build an instance with values and derivative.

Parameters:

f0 - value of the function

f1 - first derivative of the function

f2 - second derivative of the function

UnivariateDerivative2

public UnivariateDerivative2(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 2

Method Details

newInstance

public UnivariateDerivative2 newInstance(double value)

Create an instance corresponding to a constant real value.

Parameters:

value - constant real value

Returns:

instance corresponding to a constant real value

withValue

public UnivariateDerivative2 withValue(double value)

Description copied from interface: Derivative

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.

Parameters:

value - zeroth-order derivative of new represented function

Returns:

new object with changed value

getValue

public double getValue()

Get the value part of the function.

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<UnivariateDerivative2>

Parameters:

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

Returns:

n^th derivative

getOrder

public int getOrder()

Get the maximum derivation order.

Returns:

maximum derivation order

getFirstDerivative

public double getFirstDerivative()

Get the first derivative.

Returns:

first derivative

See Also:

• getValue()
• getSecondDerivative()

getSecondDerivative

public double getSecondDerivative()

Get the second derivative.

Returns:

second derivative

See Also:

• getValue()
• getFirstDerivative()

toDerivativeStructure

public DerivativeStructure toDerivativeStructure()

Convert the instance to a DerivativeStructure.

Specified by:

toDerivativeStructure in class UnivariateDerivative<UnivariateDerivative2>

Returns:

derivative structure with same value and derivative as the instance

add

public UnivariateDerivative2 add(UnivariateDerivative2 a)

Compute this + a.

Parameters:

a - element to add

Returns:

a new element representing this + a

subtract

public UnivariateDerivative2 subtract(UnivariateDerivative2 a)

Compute this - a.

Parameters:

a - element to subtract

Returns:

a new element representing this - a

multiply

public UnivariateDerivative2 multiply(int n)

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

Parameters:

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

Returns:

A new element representing n × this.

multiply

public UnivariateDerivative2 multiply(double a)

'×' operator.

Parameters:

a - right hand side parameter of the operator

Returns:

this×a

multiply

public UnivariateDerivative2 multiply(UnivariateDerivative2 a)

Compute this × a.

Parameters:

a - element to multiply

Returns:

a new element representing this × a

square

public UnivariateDerivative2 square()

Compute this × this.

Returns:

a new element representing this × this

divide

public UnivariateDerivative2 divide(double a)

'÷' operator.

Parameters:

a - right hand side parameter of the operator

Returns:

this÷a

divide

public UnivariateDerivative2 divide(UnivariateDerivative2 a)

Compute this ÷ a.

Parameters:

a - element to divide by

Returns:

a new element representing this ÷ a

remainder

public UnivariateDerivative2 remainder(UnivariateDerivative2 a)

IEEE remainder operator.

Parameters:

a - right hand side parameter of the operator

Returns:

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

negate

public UnivariateDerivative2 negate()

Returns the additive inverse of this element.

Returns:

the opposite of this.

abs

public UnivariateDerivative2 abs()

absolute value.

Returns:

abs(this)

copySign

public UnivariateDerivative2 copySign(UnivariateDerivative2 sign)

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

Parameters:

sign - the sign for the returned value

Returns:

the instance with the same sign as the sign argument

copySign

public UnivariateDerivative2 copySign(double sign)

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

Parameters:

sign - the sign for the returned value

Returns:

the instance with the same sign as the sign argument

scalb

public UnivariateDerivative2 scalb(int n)

Multiply the instance by a power of 2.

Parameters:

n - power of 2

Returns:

this × 2ⁿ

hypot

public UnivariateDerivative2 hypot(UnivariateDerivative2 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.

Parameters:

y - a value

Returns:

sqrt(this² +y²)

reciprocal

public UnivariateDerivative2 reciprocal()

Returns the multiplicative inverse of this element.

Returns:

the inverse of this.

compose

public UnivariateDerivative2 compose(double... f)

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(Derivative.getValue()), f'(Derivative.getValue()), f''(Derivative.getValue())...]).

Returns:

f(this)

sqrt

public UnivariateDerivative2 sqrt()

Square root.

Returns:

square root of the instance

cbrt

public UnivariateDerivative2 cbrt()

Cubic root.

Returns:

cubic root of the instance

rootN

public UnivariateDerivative2 rootN(int n)

N^th root.

Parameters:

n - order of the root

Returns:

n^th root of the instance

getField

public UnivariateDerivative2Field getField()

Get the Field to which the instance belongs.

Returns:

Field to which the instance belongs

pow

public static UnivariateDerivative2 pow(double a,
 UnivariateDerivative2 x)

Compute a^x where a is a double and x a UnivariateDerivative2

Parameters:

a - number to exponentiate

x - power to apply

Returns:

a^x

pow

public UnivariateDerivative2 pow(double p)

Power operation.

Parameters:

p - power to apply

Returns:

this^p

pow

public UnivariateDerivative2 pow(int n)

Integer power operation.

Parameters:

n - power to apply

Returns:

thisⁿ

exp

public UnivariateDerivative2 exp()

Exponential.

Returns:

exponential of the instance

expm1

public UnivariateDerivative2 expm1()

Exponential minus 1.

Returns:

exponential minus one of the instance

log

public UnivariateDerivative2 log()

Natural logarithm.

Returns:

logarithm of the instance

log1p

public UnivariateDerivative2 log1p()

Shifted natural logarithm.

Returns:

logarithm of one plus the instance

log10

public UnivariateDerivative2 log10()

Base 10 logarithm.

Returns:

base 10 logarithm of the instance

cos

public UnivariateDerivative2 cos()

Cosine operation.

Returns:

cos(this)

sin

public UnivariateDerivative2 sin()

Sine operation.

Returns:

sin(this)

sinCos

public FieldSinCos<UnivariateDerivative2> sinCos()

Combined Sine and Cosine operation.

Returns:

[sin(this), cos(this)]

tan

public UnivariateDerivative2 tan()

Tangent operation.

Returns:

tan(this)

acos

public UnivariateDerivative2 acos()

Arc cosine operation.

Returns:

acos(this)

asin

public UnivariateDerivative2 asin()

Arc sine operation.

Returns:

asin(this)

atan

public UnivariateDerivative2 atan()

Arc tangent operation.

Returns:

atan(this)

atan2

public UnivariateDerivative2 atan2(UnivariateDerivative2 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.

Parameters:

x - second argument of the arc tangent

Returns:

atan2(this, x)

cosh

public UnivariateDerivative2 cosh()

Hyperbolic cosine operation.

Returns:

cosh(this)

sinh

public UnivariateDerivative2 sinh()

Hyperbolic sine operation.

Returns:

sinh(this)

sinhCosh

public FieldSinhCosh<UnivariateDerivative2> sinhCosh()

Combined hyperbolic sine and cosine operation.

Returns:

[sinh(this), cosh(this)]

tanh

public UnivariateDerivative2 tanh()

Hyperbolic tangent operation.

Returns:

tanh(this)

acosh

public UnivariateDerivative2 acosh()

Inverse hyperbolic cosine operation.

Returns:

acosh(this)

asinh

public UnivariateDerivative2 asinh()

Inverse hyperbolic sine operation.

Returns:

asin(this)

atanh

public UnivariateDerivative2 atanh()

Inverse hyperbolic tangent operation.

Returns:

atanh(this)

toDegrees

public UnivariateDerivative2 toDegrees()

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

Returns:

instance converted into degrees

toRadians

public UnivariateDerivative2 toRadians()

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

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 UnivariateDerivative2 linearCombination(UnivariateDerivative2[] a,
 UnivariateDerivative2[] b)

Compute a linear combination.

Parameters:

a - Factors.

b - Factors.

Returns:

Σi ai bi.

linearCombination

public UnivariateDerivative2 linearCombination(double[] a,
 UnivariateDerivative2[] b)

Compute a linear combination.

Parameters:

a - Factors.

b - Factors.

Returns:

Σi ai bi.

linearCombination

public UnivariateDerivative2 linearCombination(UnivariateDerivative2 a1,
 UnivariateDerivative2 b1,
 UnivariateDerivative2 a2,
 UnivariateDerivative2 b2)

Compute a linear combination.

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 UnivariateDerivative2 linearCombination(double a1,
 UnivariateDerivative2 b1,
 double a2,
 UnivariateDerivative2 b2)

Compute a linear combination.

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 UnivariateDerivative2 linearCombination(UnivariateDerivative2 a1,
 UnivariateDerivative2 b1,
 UnivariateDerivative2 a2,
 UnivariateDerivative2 b2,
 UnivariateDerivative2 a3,
 UnivariateDerivative2 b3)

Compute a linear combination.

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 UnivariateDerivative2 linearCombination(double a1,
 UnivariateDerivative2 b1,
 double a2,
 UnivariateDerivative2 b2,
 double a3,
 UnivariateDerivative2 b3)

Compute a linear combination.

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 UnivariateDerivative2 linearCombination(UnivariateDerivative2 a1,
 UnivariateDerivative2 b1,
 UnivariateDerivative2 a2,
 UnivariateDerivative2 b2,
 UnivariateDerivative2 a3,
 UnivariateDerivative2 b3,
 UnivariateDerivative2 a4,
 UnivariateDerivative2 b4)

Compute a linear combination.

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 UnivariateDerivative2 linearCombination(double a1,
 UnivariateDerivative2 b1,
 double a2,
 UnivariateDerivative2 b2,
 double a3,
 UnivariateDerivative2 b3,
 double a4,
 UnivariateDerivative2 b4)

Compute a linear combination.

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 UnivariateDerivative2 getPi()

Get the Archimedes constant π.

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

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(UnivariateDerivative2 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.

Since:

3.0