Generalized Dual numbers

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

gdual_double

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.

abs(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Absolute value.

acos(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Arc cosine.

acosh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Inverse hyperbolic cosine.

arccos(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Arc cosine.

arccosh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Inverse hyperbolic cosine.

arcsin(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Arc sine.

arcsinh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Inverse hyperbolic sine.

arctan(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Arc tangent.

arctanh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Inverse hyperbolic arc tangent.

asin(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Arc sine.

asinh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Inverse hyperbolic sine.

atan(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Arc tangent.

atanh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Inverse hyperbolic arc tangent.

cbrt(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Cubic root.

property constant_cf

Constant term of the polynomial

cos(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Cosine.

cosh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Hyperbolic cosine.

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().

erf(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Error function.

erfc(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Error Function Complement

evaluate(self: pyaudi.core.gdual_double, arg0: dict) → float

Evaluates the Taylor polynomial

exp(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Exponential.

expm1(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

exp(x) - 1, inverse of log1p

extend_symbol_set(self: pyaudi.core.gdual_double, arg0: collections.abc.Sequence[str]) → None

Extends the symbol set

extract_terms(self: pyaudi.core.gdual_double, arg0: SupportsInt) → pyaudi.core.gdual_double

returns a new gdual containing only terms of a given order

find_cf(self: pyaudi.core.gdual_double, arg0: collections.abc.Sequence[SupportsInt]) → float

Find the coefficient of the Taylor expansion

get_derivative(*args, **kwargs)

Overloaded function.

  1. get_derivative(self: pyaudi.core.gdual_double, arg0: collections.abc.Sequence[typing.SupportsInt]) -> float

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

  1. get_derivative(self: pyaudi.core.gdual_double, arg0: collections.abc.Mapping[str, typing.SupportsInt]) -> float

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

info(self: pyaudi.core.gdual_double) → str

Full information on the gdual

integrate(self: pyaudi.core.gdual_double, arg0: str) → pyaudi.core.gdual_double

Integrate with respect to argument

is_zero(self: pyaudi.core.gdual_double, arg0: SupportsFloat) → bool

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

log(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Natural logarithm.

log10(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Base 10 Logarithm.

log1p(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

log(x)+1, inverse of expm1

log2(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Base 2 Logarithm.

property order

truncation order (>= degree)

partial(self: pyaudi.core.gdual_double, arg0: str) → pyaudi.core.gdual_double

Partial derivative with respect to argument

sin(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Sine.

sinh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Hyperbolic sine.

sqrt(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Square root.

subs(*args, **kwargs)

Overloaded function.

  1. subs(self: pyaudi.core.gdual_double, arg0: str, arg1: typing.SupportsFloat) -> pyaudi.core.gdual_double

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

  1. subs(self: pyaudi.core.gdual_double, arg0: str, arg1: pyaudi.core.gdual_double) -> pyaudi.core.gdual_double

Substitutes a symbol with a gdual

tan(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Tangent.

tanh(self: pyaudi.core.gdual_double) → pyaudi.core.gdual_double

Hyperbolic tangent.

trim(self: pyaudi.core.gdual_double, arg0: SupportsFloat) → pyaudi.core.gdual_double

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

gdual_vdouble

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.

abs(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Absolute value.

acos(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Arc cosine.

acosh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Inverse hyperbolic cosine.

arccos(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Arc cosine.

arccosh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Inverse hyperbolic cosine.

arcsin(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Arc sine.

arcsinh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Inverse hyperbolic sine.

arctan(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Arc tangent.

arctanh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Inverse hyperbolic arc tangent.

asin(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Arc sine.

asinh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Inverse hyperbolic sine.

atan(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Arc tangent.

atanh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Inverse hyperbolic arc tangent.

cbrt(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Cubic root.

property constant_cf

Constant term of the polynomial

cos(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Cosine.

cosh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Hyperbolic cosine.

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().

erf(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Error function.

erfc(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Error Function Complement

evaluate(self: pyaudi.core.gdual_vdouble, arg0: dict) → vectorized<double>

Evaluates the Taylor polynomial

exp(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Exponential.

expm1(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

exp(x) - 1, inverse of log1p

extend_symbol_set(self: pyaudi.core.gdual_vdouble, arg0: collections.abc.Sequence[str]) → None

Extends the symbol set

extract_terms(self: pyaudi.core.gdual_vdouble, arg0: SupportsInt) → pyaudi.core.gdual_vdouble

returns a new gdual containing only terms of a given order

find_cf(self: pyaudi.core.gdual_vdouble, arg0: collections.abc.Sequence[SupportsInt]) → vectorized<double>

Find the coefficient of the Taylor expansion

get_derivative(*args, **kwargs)

Overloaded function.

  1. get_derivative(self: pyaudi.core.gdual_vdouble, arg0: collections.abc.Sequence[typing.SupportsInt]) -> vectorized<double>

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

  1. get_derivative(self: pyaudi.core.gdual_vdouble, arg0: collections.abc.Mapping[str, typing.SupportsInt]) -> vectorized<double>

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

info(self: pyaudi.core.gdual_vdouble) → str

Full information on the gdual

integrate(self: pyaudi.core.gdual_vdouble, arg0: str) → pyaudi.core.gdual_vdouble

Integrate with respect to argument

is_zero(self: pyaudi.core.gdual_vdouble, arg0: SupportsFloat) → bool

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

log(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Natural logarithm.

log10(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Base 10 Logarithm.

log1p(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

log(x)+1, inverse of expm1

log2(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Base 2 Logarithm.

property order

truncation order (>= degree)

partial(self: pyaudi.core.gdual_vdouble, arg0: str) → pyaudi.core.gdual_vdouble

Partial derivative with respect to argument

sin(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Sine.

sinh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Hyperbolic sine.

sqrt(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Square root.

subs(*args, **kwargs)

Overloaded function.

  1. subs(self: pyaudi.core.gdual_vdouble, arg0: str, arg1: vectorized<double>) -> pyaudi.core.gdual_vdouble

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

  1. subs(self: pyaudi.core.gdual_vdouble, arg0: str, arg1: pyaudi.core.gdual_vdouble) -> pyaudi.core.gdual_vdouble

Substitutes a symbol with a gdual

tan(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Tangent.

tanh(self: pyaudi.core.gdual_vdouble) → pyaudi.core.gdual_vdouble

Hyperbolic tangent.

trim(self: pyaudi.core.gdual_vdouble, arg0: SupportsFloat) → pyaudi.core.gdual_vdouble

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