Generalized Dual numbers

The following instances of generalized dual numbers (and thus the corresponding algebra) are made available in the python module pyaudi:

class pyaudi.gdual_double

The generalized dual number class

This class represents a generalized dual number, in a nutshell, a truncated multivariate Taylor polynomial. A gdual is defined by its truncation order \(m\) as well as on its expansion point \(\mathbf a\). All arithmetic operators +,*,/,-,** are overloaded so that the Taylor expansion of arithmetic computations is obtained.

A basic example is:

gdual_double: \(m = 2\), \(\mathbf a = [1.2, -0.1]\), \(f = \frac{x1+x2}{x1-x2}\)

>>> from pyaudi import gdual_double as gdual
>>> x1 = gdual(1.2, "x1", 2)
>>> x2 = gdual(-0.1, "x2", 2)
>>> f = (x1+x2) / (x1-x2)
>>> print(f) 
1.42012*dx2+0.118343*dx1+1.0924*dx2**2+0.846154-1.00137*dx1*dx2-0.0910332*dx1**2

gdual_vdouble: \(m = 2\), \(\mathbf a = [[1.2, 1.1], [-0.1, -0.2]]\), \(f = \frac{x1+x2}{x1-x2}\)

>>> from pyaudi import gdual_vdouble as gdual
>>> x1 = gdual([1.2, 1.1], "x1", 2)
>>> x2 = gdual([-0.1, -0.2], "x2", 2)
>>> f = (x1+x2) / (x1-x2)
>>> print(f) 
[1.42012, 1.30178]*dx2+[0.118343, 0.236686]*dx1+[1.0924, 1.00137]*dx2**2+[0.846154, 0.692308]+[-1.00137, -0.819299]*dx1*dx2+[-0.0910332, -0.182066]*dx1**2

gdual_real128: \(m = 2\), \(\mathbf a = [1.2, -0.1]\), \(f = \frac{x1+x2}{x1-x2}\)

>>> from pyaudi import gdual_real128 as gdual
>>> x1 = gdual("1.2", "x1", 2)
>>> x2 = gdual("-0.1", "x2", 2)
>>> f = (x1+x2) / (x1-x2)
>>> print(f) 
1.42011834319526627218934911242603510e+00*dx2+1.18343195266272189349112426035503101e-01*dx1+1.09239872553482020937642239417387321e+00*dx2**2+8.46153846153846153846153846153845954e-01-1.00136549840691852526172052799271705e+00*dx1*dx2-9.10332271279016841147018661811561892e-02*dx1**2

Note

A gdual can operate on doubles (gdual_double), on vectorized doubles (gdual_vdouble) or on quadruple precision doubles (gdual_real128). The second case is immensely more efficient when applicable.

See also the docs of the C++ class gdual.

property constant_cf

Constant term of the polynomial

property degree

The degree of the generalized dual number

Note

The degree and the order of a gdual can be different, the degree is always smaller or equal ot the order.

Returns

The degree of the underlying Taylor polynomial

>>> from pyaudi import gdual_double as gdual
>>> x1 = gdual(1.2, "x1", 5)
>>> f = x1**2
>>> print(f.degree)
2

See also the docs of the C++ method of the gdual degree().

evaluate((gdual_double)arg1, (dict)arg2) → float :

Evaluates the Taylor polynomial

C++ signature :

double evaluate(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::dict)

extend_symbol_set((gdual_double)arg1, (object)arg2) → None :

Extends the symbol set

C++ signature :

void extend_symbol_set(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},boost::python::api::object)

extract_terms((gdual_double)arg1, (int)arg2) → gdual_double :

returns a new gdual containing only terms of a given order

C++ signature :

audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > extract_terms(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},unsigned int)

find_cf((gdual_double)arg1, (object)arg2) → float :

Finds the coefficient of the Taylor expansion

C++ signature :

double find_cf(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::api::object)

get_derivative((gdual_double)arg1, (object)arg2) → float :

Finds the derivative (i.e. the coefficient of the Taylor expansion discounted by the factorial factor

C++ signature :

double get_derivative(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::api::object)

get_derivative( (gdual_double)arg1, (dict)arg2) -> float :

Finds the derivative (i.e. the coefficient of the Taylor expansion discounted by the factorial factor

C++ signature :

double get_derivative(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::dict)

integrate((gdual_double)arg1, (str)arg2) → gdual_double :

Integrate with respect to argument

C++ signature :

audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > integrate(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >)

is_zero((gdual_double)arg1, (float)arg2) → bool :

checks if all coefficients of the gdual are zero within a tolerance

C++ signature :

bool is_zero(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},double)

property order

truncation order (>= degree)

partial((gdual_double)arg1, (str)arg2) → gdual_double :

Partial derivative with respect to argument

C++ signature :

audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > partial(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >)

subs((gdual_double)arg1, (str)arg2, (float)arg3) → gdual_double :

Substitutes a symbol with a value (does not remove symbol from symbol set)

C++ signature :

audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > subs(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >,double)

subs( (gdual_double)arg1, (str)arg2, (gdual_double)arg3) -> gdual_double :

Substitutes a symbol with a gdual

C++ signature :

audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > subs(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >,audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >)

trim((gdual_double)arg1, (float)arg2) → gdual_double :

returns a new gdual removing all coefficients that are smaller than a tolerance

C++ signature :

audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > trim(audi::gdual<double, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},double)

class pyaudi.gdual_vdouble

The generalized dual number class

This class represents a generalized dual number, in a nutshell, a truncated multivariate Taylor polynomial. A gdual is defined by its truncation order \(m\) as well as on its expansion point \(\mathbf a\). All arithmetic operators +,*,/,-,** are overloaded so that the Taylor expansion of arithmetic computations is obtained.

A basic example is:

gdual_double: \(m = 2\), \(\mathbf a = [1.2, -0.1]\), \(f = \frac{x1+x2}{x1-x2}\)

>>> from pyaudi import gdual_double as gdual
>>> x1 = gdual(1.2, "x1", 2)
>>> x2 = gdual(-0.1, "x2", 2)
>>> f = (x1+x2) / (x1-x2)
>>> print(f) 
1.42012*dx2+0.118343*dx1+1.0924*dx2**2+0.846154-1.00137*dx1*dx2-0.0910332*dx1**2

gdual_vdouble: \(m = 2\), \(\mathbf a = [[1.2, 1.1], [-0.1, -0.2]]\), \(f = \frac{x1+x2}{x1-x2}\)

>>> from pyaudi import gdual_vdouble as gdual
>>> x1 = gdual([1.2, 1.1], "x1", 2)
>>> x2 = gdual([-0.1, -0.2], "x2", 2)
>>> f = (x1+x2) / (x1-x2)
>>> print(f) 
[1.42012, 1.30178]*dx2+[0.118343, 0.236686]*dx1+[1.0924, 1.00137]*dx2**2+[0.846154, 0.692308]+[-1.00137, -0.819299]*dx1*dx2+[-0.0910332, -0.182066]*dx1**2

gdual_real128: \(m = 2\), \(\mathbf a = [1.2, -0.1]\), \(f = \frac{x1+x2}{x1-x2}\)

>>> from pyaudi import gdual_real128 as gdual
>>> x1 = gdual("1.2", "x1", 2)
>>> x2 = gdual("-0.1", "x2", 2)
>>> f = (x1+x2) / (x1-x2)
>>> print(f) 
1.42011834319526627218934911242603510e+00*dx2+1.18343195266272189349112426035503101e-01*dx1+1.09239872553482020937642239417387321e+00*dx2**2+8.46153846153846153846153846153845954e-01-1.00136549840691852526172052799271705e+00*dx1*dx2-9.10332271279016841147018661811561892e-02*dx1**2

Note

A gdual can operate on doubles (gdual_double), on vectorized doubles (gdual_vdouble) or on quadruple precision doubles (gdual_real128). The second case is immensely more efficient when applicable.

See also the docs of the C++ class gdual.

property constant_cf

Constant term of the polynomial

property degree

The degree of the generalized dual number

Note

The degree and the order of a gdual can be different, the degree is always smaller or equal ot the order.

Returns

The degree of the underlying Taylor polynomial

>>> from pyaudi import gdual_double as gdual
>>> x1 = gdual(1.2, "x1", 5)
>>> f = x1**2
>>> print(f.degree)
2

See also the docs of the C++ method of the gdual degree().

evaluate((gdual_vdouble)arg1, (dict)arg2) → object :

Evaluates the Taylor polynomial

C++ signature :

audi::vectorized<double> evaluate(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::dict)

extend_symbol_set((gdual_vdouble)arg1, (object)arg2) → None :

Extends the symbol set

C++ signature :

void extend_symbol_set(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},boost::python::api::object)

extract_terms((gdual_vdouble)arg1, (int)arg2) → gdual_vdouble :

returns a new gdual containing only terms of a given order

C++ signature :

audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > extract_terms(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},unsigned int)

find_cf((gdual_vdouble)arg1, (object)arg2) → object :

Finds the coefficient of the Taylor expansion

C++ signature :

audi::vectorized<double> find_cf(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::api::object)

get_derivative((gdual_vdouble)arg1, (object)arg2) → object :

Finds the derivative (i.e. the coefficient of the Taylor expansion discounted by the factorial factor

C++ signature :

audi::vectorized<double> get_derivative(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::api::object)

get_derivative( (gdual_vdouble)arg1, (dict)arg2) -> object :

Finds the derivative (i.e. the coefficient of the Taylor expansion discounted by the factorial factor

C++ signature :

audi::vectorized<double> get_derivative(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >,boost::python::dict)

integrate((gdual_vdouble)arg1, (str)arg2) → gdual_vdouble :

Integrate with respect to argument

C++ signature :

audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > integrate(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >)

is_zero((gdual_vdouble)arg1, (float)arg2) → bool :

checks if all coefficients of the gdual are zero within a tolerance

C++ signature :

bool is_zero(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},double)

property order

truncation order (>= degree)

partial((gdual_vdouble)arg1, (str)arg2) → gdual_vdouble :

Partial derivative with respect to argument

C++ signature :

audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > partial(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >)

subs((gdual_vdouble)arg1, (str)arg2, (object)arg3) → gdual_vdouble :

Substitutes a symbol with a value (does not remove symbol from symbol set)

C++ signature :

audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > subs(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >,boost::python::api::object)

subs( (gdual_vdouble)arg1, (str)arg2, (gdual_vdouble)arg3) -> gdual_vdouble :

Substitutes a symbol with a gdual

C++ signature :

audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > subs(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},std::__1::basic_string<char, std::__1::char_traits<char>, std::__1::allocator<char> >,audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> >)

trim((gdual_vdouble)arg1, (float)arg2) → gdual_vdouble :

returns a new gdual removing all coefficients that are smaller than a tolerance

C++ signature :

audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > trim(audi::gdual<audi::vectorized<double>, obake::polynomials::d_packed_monomial<unsigned long long, 8u, void> > {lvalue},double)