Class Gradient
- All Implemented Interfaces:
Serializable
,Derivative<Gradient>
,Derivative1<Gradient>
,DifferentialAlgebra
,CalculusFieldElement<Gradient>
,FieldElement<Gradient>
This class is a stripped-down version of DerivativeStructure
with derivation order
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.
Gradient
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.
-
Constructor Summary
ConstructorDescriptionGradient
(double value, double... gradient) Build an instance with values and derivative.Build an instance from aDerivativeStructure
. -
Method Summary
Modifier and TypeMethodDescriptionabs()
absolute value.Compute this + a.Two arguments arc tangent operation.compose
(double... f) Compute composition of the instance by a univariate function.compose
(double f0, double f1) Compute composition of the instance by a univariate function differentiable at order 1.static Gradient
constant
(int freeParameters, double value) Build an instance corresponding to a constant value.copySign
(double sign) Returns the instance with the sign of the argument.Returns the instance with the sign of the argument.divide
(double a) '÷' operator.Compute this ÷ a.boolean
Test for the equality of two univariate derivatives.getField()
Get theField
to which the instance belongs.int
Get the number of free parameters.double[]
Get the gradient part of the function.double
getPartialDerivative
(int n) Get the partial derivative with respect to one parameter.double
getPartialDerivative
(int... orders) Get a partial derivative.double
getValue()
Get the value part of the function.int
hashCode()
Get a hashCode for the univariate derivative.Returns the hypotenuse of a triangle with sidesthis
andy
- sqrt(this2 +y2) avoiding intermediate overflow or underflow.linearCombination
(double[] a, Gradient[] b) Compute a linear combination.linearCombination
(double a1, Gradient b1, double a2, Gradient b2) Compute a linear combination.linearCombination
(double a1, Gradient b1, double a2, Gradient b2, double a3, Gradient b3) Compute a linear combination.linearCombination
(double a1, Gradient b1, double a2, Gradient b2, double a3, Gradient b3, double a4, Gradient b4) Compute a linear combination.linearCombination
(Gradient[] a, Gradient[] b) Compute a linear combination.linearCombination
(Gradient a1, Gradient b1, Gradient a2, Gradient b2) Compute a linear combination.Compute a linear combination.linearCombination
(Gradient a1, Gradient b1, Gradient a2, Gradient b2, Gradient a3, Gradient b3, Gradient a4, Gradient b4) Compute a linear combination.multiply
(double a) '×' operator.multiply
(int n) Compute n × this.Compute this × a.negate()
Returns the additive inverse ofthis
element.newInstance
(double c) Create an instance corresponding to a constant real value.pow
(double p) Power operation.static Gradient
Compute ax where a is a double and x aGradient
pow
(int n) Integer power operation.IEEE remainder operator.scalb
(int n) Multiply the instance by a power of 2.sinCos()
Combined Sine and Cosine operation.sinhCosh()
Combined hyperbolic sine and cosine operation.Compute this - a.double
taylor
(double... delta) Evaluate Taylor expansion a derivative structure.Convert radians to degrees, with error of less than 0.5 ULPConvert the instance to aDerivativeStructure
.Convert degrees to radians, with error of less than 0.5 ULPstatic Gradient
variable
(int freeParameters, int index, double value) Build aGradient
representing a variable.withValue
(double v) 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 java.lang.Object
clone, finalize, getClass, notify, notifyAll, toString, wait, wait, wait
Methods inherited from interface org.hipparchus.CalculusFieldElement
ceil, floor, getPi, 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.analysis.differentiation.Derivative1
acos, acosh, asin, asinh, atan, atanh, cbrt, cos, cosh, exp, expm1, getOrder, log, log10, log1p, reciprocal, rootN, sin, sinh, sqrt, square, tan, tanh
Methods inherited from interface org.hipparchus.FieldElement
isZero
-
Constructor Details
-
Gradient
public Gradient(double value, double... gradient) Build an instance with values and derivative.- Parameters:
value
- value of the functiongradient
- gradient of the function
-
Gradient
Build an instance from aDerivativeStructure
.- Parameters:
ds
- derivative structure- Throws:
MathIllegalArgumentException
- ifds
order is not 1
-
-
Method Details
-
constant
Build an instance corresponding to a constant value.- Parameters:
freeParameters
- number of free parameters (i.e. dimension of the gradient)value
- constant value of the function- Returns:
- a
Gradient
with a constant value and all derivatives set to 0.0
-
variable
Build aGradient
representing a variable.Instances built using this method 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:
freeParameters
- number of free parameters (i.e. dimension of the gradient)index
- index of the variable (from 0 togetFreeParameters()
- 1)value
- value of the variable- Returns:
- a
Gradient
with a constant value and all derivatives set to 0.0 except the one atindex
which will be set to 1.0
-
newInstance
Create an instance corresponding to a constant real value.- Specified by:
newInstance
in interfaceCalculusFieldElement<Gradient>
- Parameters:
c
- constant real value- Returns:
- instance corresponding to a constant real value
-
withValue
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 interfaceDerivative<Gradient>
- Parameters:
v
- zeroth-order derivative of new represented function- Returns:
- new object with changed value
-
getValue
public double getValue()Get the value part of the function.- Specified by:
getValue
in interfaceDerivative<Gradient>
- Returns:
- value part of the value of the function
-
getGradient
public double[] getGradient()Get the gradient part of the function.- Returns:
- gradient part of the value of the function
- See Also:
-
getFreeParameters
public int getFreeParameters()Get the number of free parameters.- Specified by:
getFreeParameters
in interfaceDifferentialAlgebra
- Returns:
- number of free parameters
-
getPartialDerivative
Get a partial derivative.- Specified by:
getPartialDerivative
in interfaceDerivative<Gradient>
- Parameters:
orders
- derivation orders with respect to each variable (if all orders are 0, the value is returned)- Returns:
- partial derivative
- Throws:
MathIllegalArgumentException
- if the numbers of variables does not match the instance- See Also:
-
getPartialDerivative
Get the partial derivative with respect to one parameter.- Parameters:
n
- index of the parameter (counting from 0)- Returns:
- partial derivative with respect to the nth parameter
- Throws:
MathIllegalArgumentException
- if n is either negative or larger or equal togetFreeParameters()
-
toDerivativeStructure
Convert the instance to aDerivativeStructure
.- Returns:
- derivative structure with same value and derivative as the instance
-
add
Compute this + a.- Specified by:
add
in interfaceFieldElement<Gradient>
- Parameters:
a
- element to add- Returns:
- a new element representing this + a
-
subtract
Compute this - a.- Specified by:
subtract
in interfaceCalculusFieldElement<Gradient>
- Specified by:
subtract
in interfaceFieldElement<Gradient>
- Parameters:
a
- element to subtract- Returns:
- a new element representing this - a
-
multiply
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 interfaceCalculusFieldElement<Gradient>
- Specified by:
multiply
in interfaceFieldElement<Gradient>
- Parameters:
n
- Number of timesthis
must be added to itself.- Returns:
- A new element representing n × this.
-
multiply
'×' operator.- Specified by:
multiply
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a
- right hand side parameter of the operator- Returns:
- this×a
-
multiply
Compute this × a.- Specified by:
multiply
in interfaceFieldElement<Gradient>
- Parameters:
a
- element to multiply- Returns:
- a new element representing this × a
-
divide
'÷' operator.- Specified by:
divide
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a
- right hand side parameter of the operator- Returns:
- this÷a
-
divide
Compute this ÷ a.- Specified by:
divide
in interfaceCalculusFieldElement<Gradient>
- Specified by:
divide
in interfaceFieldElement<Gradient>
- Parameters:
a
- element to divide by- Returns:
- a new element representing this ÷ a
-
remainder
IEEE remainder operator.- Specified by:
remainder
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a
- right hand side parameter of the operator- Returns:
- this - n × a where n is the closest integer to this/a
-
negate
Returns the additive inverse ofthis
element.- Specified by:
negate
in interfaceFieldElement<Gradient>
- Returns:
- the opposite of
this
.
-
abs
absolute value.- Specified by:
abs
in interfaceCalculusFieldElement<Gradient>
- Returns:
- abs(this)
-
copySign
Returns the instance with the sign of the argument. A NaNsign
argument is treated as positive.- Specified by:
copySign
in interfaceCalculusFieldElement<Gradient>
- Parameters:
sign
- the sign for the returned value- Returns:
- the instance with the same sign as the
sign
argument
-
copySign
Returns the instance with the sign of the argument. A NaNsign
argument is treated as positive.- Specified by:
copySign
in interfaceCalculusFieldElement<Gradient>
- Parameters:
sign
- the sign for the returned value- Returns:
- the instance with the same sign as the
sign
argument
-
scalb
Multiply the instance by a power of 2.- Specified by:
scalb
in interfaceCalculusFieldElement<Gradient>
- Parameters:
n
- power of 2- Returns:
- this × 2n
-
hypot
Returns the hypotenuse of a triangle with sidesthis
andy
- 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 interfaceCalculusFieldElement<Gradient>
- Parameters:
y
- a value- Returns:
- sqrt(this2 +y2)
-
compose
Compute composition of the instance by a univariate function.- Specified by:
compose
in interfaceDerivative<Gradient>
- 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
Compute composition of the instance by a univariate function differentiable at order 1.- Specified by:
compose
in interfaceDerivative1<Gradient>
- Parameters:
f0
- value of functionf1
- first-order derivative- Returns:
- f(this)
-
getField
Get theField
to which the instance belongs.- Specified by:
getField
in interfaceFieldElement<Gradient>
- Returns:
Field
to which the instance belongs
-
pow
Compute ax where a is a double and x aGradient
- Parameters:
a
- number to exponentiatex
- power to apply- Returns:
- ax
-
pow
Power operation.- Specified by:
pow
in interfaceCalculusFieldElement<Gradient>
- Parameters:
p
- power to apply- Returns:
- thisp
-
pow
Integer power operation.- Specified by:
pow
in interfaceCalculusFieldElement<Gradient>
- Parameters:
n
- power to apply- Returns:
- thisn
-
sinCos
Combined Sine and Cosine operation.- Specified by:
sinCos
in interfaceCalculusFieldElement<Gradient>
- Specified by:
sinCos
in interfaceDerivative1<Gradient>
- Returns:
- [sin(this), cos(this)]
-
atan2
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 computesatan2(this, x)
, i.e. the instance represents they
argument and thex
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 languagesatan2
two-arguments arc tangent and putsx
as its first argument.- Specified by:
atan2
in interfaceCalculusFieldElement<Gradient>
- Parameters:
x
- second argument of the arc tangent- Returns:
- atan2(this, x)
-
sinhCosh
Combined hyperbolic sine and cosine operation.- Specified by:
sinhCosh
in interfaceCalculusFieldElement<Gradient>
- Specified by:
sinhCosh
in interfaceDerivative1<Gradient>
- Returns:
- [sinh(this), cosh(this)]
-
toDegrees
Convert radians to degrees, with error of less than 0.5 ULP- Specified by:
toDegrees
in interfaceCalculusFieldElement<Gradient>
- Returns:
- instance converted into degrees
-
toRadians
Convert degrees to radians, with error of less than 0.5 ULP- Specified by:
toRadians
in interfaceCalculusFieldElement<Gradient>
- Returns:
- instance converted into radians
-
taylor
public double taylor(double... delta) Evaluate Taylor expansion a derivative structure.- Parameters:
delta
- parameters offsets (Δx, Δy, ...)- Returns:
- value of the Taylor expansion at x + Δx, y + Δy, ...
-
linearCombination
Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a
- Factors.b
- Factors.- Returns:
Σi ai bi
.
-
linearCombination
Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a
- Factors.b
- Factors.- Returns:
Σi ai bi
.
-
linearCombination
Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a1
- first factor of the first termb1
- second factor of the first terma2
- first factor of the second termb2
- second factor of the second term- Returns:
- a1×b1 + a2×b2
- See Also:
-
linearCombination
Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a1
- first factor of the first termb1
- second factor of the first terma2
- first factor of the second termb2
- second factor of the second term- Returns:
- a1×b1 + a2×b2
- See Also:
-
linearCombination
public Gradient linearCombination(Gradient a1, Gradient b1, Gradient a2, Gradient b2, Gradient a3, Gradient b3) Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a1
- first factor of the first termb1
- second factor of the first terma2
- first factor of the second termb2
- second factor of the second terma3
- first factor of the third termb3
- second factor of the third term- Returns:
- a1×b1 + a2×b2 + a3×b3
- See Also:
-
linearCombination
public Gradient linearCombination(double a1, Gradient b1, double a2, Gradient b2, double a3, Gradient b3) Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a1
- first factor of the first termb1
- second factor of the first terma2
- first factor of the second termb2
- second factor of the second terma3
- first factor of the third termb3
- second factor of the third term- Returns:
- a1×b1 + a2×b2 + a3×b3
- See Also:
-
linearCombination
public Gradient linearCombination(Gradient a1, Gradient b1, Gradient a2, Gradient b2, Gradient a3, Gradient b3, Gradient a4, Gradient b4) Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a1
- first factor of the first termb1
- second factor of the first terma2
- first factor of the second termb2
- second factor of the second terma3
- first factor of the third termb3
- second factor of the third terma4
- first factor of the fourth termb4
- second factor of the fourth term- Returns:
- a1×b1 + a2×b2 + a3×b3 + a4×b4
- See Also:
-
linearCombination
public Gradient linearCombination(double a1, Gradient b1, double a2, Gradient b2, double a3, Gradient b3, double a4, Gradient b4) Compute a linear combination.- Specified by:
linearCombination
in interfaceCalculusFieldElement<Gradient>
- Parameters:
a1
- first factor of the first termb1
- second factor of the first terma2
- first factor of the second termb2
- second factor of the second terma3
- first factor of the third termb3
- second factor of the third terma4
- first factor of the fourth termb4
- second factor of the fourth term- Returns:
- a1×b1 + a2×b2 + a3×b3 + a4×b4
- See Also:
-
equals
Test for the equality of two univariate derivatives.univariate derivatives are considered equal if they have the same derivatives.
-
hashCode
public int hashCode()Get a hashCode for the univariate derivative.
-