Higher order partial derivatives computation with respect to one or several tensor-like variables. Taylor series function approximation (derivative table and series function representation). Parametric fixed point computation.
$ pip install git+https://github.com/i-a-morozov/ndmap.git@main
or
$ pip install ndmap -U
https://i-a-morozov.github.io/ndmap/
Compute higher order function (partial) derivatives.
>>> from ndmap.derivative import derivative
>>> def fn(x):
... return 1 + x + x**2 + x**3 + x**4 + x**5
...
>>> import torch
>>> x = torch.tensor(0.0)
>>> derivative(5, fn, x)
[tensor(1.), tensor(1.), tensor(2.), tensor(6.), tensor(24.), tensor(120.)]
>>> from ndmap.derivative import derivative
>>> def fn(x):
... x1, x2 = x
... return x1**2 + x1*x2 + x2**2
...
>>> import torch
>>> x = torch.tensor([0.0, 0.0])
>>> derivative(2, fn, x, intermediate=False)
tensor([[2., 1.],
[1., 2.]])
>>> from ndmap.derivative import derivative
>>> def fn(x, y):
... x1, x2 = x
... return x1**2*(1 + y) + x2**2*(1 - y)
...
>>> import torch
>>> x = torch.tensor([0.0, 0.0])
>>> y = torch.tensor(0.0)
>>> derivative((2, 1), fn, x, y)
[[tensor(0.), tensor(0.)], [tensor([0., 0.]), tensor([0., 0.])], [tensor([[2., 0.],
[0., 2.]]), tensor([[ 2., 0.],
[ 0., -2.]])]]
Compute higher order function (partial) derivatives.
>>> from ndmap.gradient import series
>>> def fn(x):
... return 1 + x + x**2 + x**3 + x**4 + x**5
...
>>> import torch
>>> x = torch.tensor([0.0])
>>> series((5, ), fn, x, retain=False, series=False)
{(0,): tensor([1.]),
(1,): tensor([1.]),
(2,): tensor([2.]),
(3,): tensor([6.]),
(4,): tensor([24.]),
(5,): tensor([120.])}
>>> from ndmap.gradient import series
>>> def fn(x):
... x1, x2 = x
... return x1**2 + x1*x2 + x2**2
...
>>> import torch
>>> x = torch.tensor([0.0, 0.0])
>>> series((2, ), fn, x, intermediate=False, retain=False, series=False)
{(2, 0): tensor(2.), (1, 1): tensor(1.), (0, 2): tensor(2.)}
>>> from ndmap.gradient import series
>>> def fn(x, y):
... x1, x2 = x
... y1, = y
... return x1**2*(1 + y1) + x2**2*(1 - y1)
...
>>> import torch
>>> x = torch.tensor([0.0, 0.0])
>>> y = torch.tensor([0.0])
>>> series((2, 1), fn, x, y, retain=False, series=False)
{(0, 0, 0): tensor(0.),
(0, 0, 1): tensor(0.),
(1, 0, 0): tensor(0.),
(0, 1, 0): tensor(0.),
(1, 0, 1): tensor(0.),
(0, 1, 1): tensor(-0.),
(2, 0, 0): tensor(2.),
(1, 1, 0): tensor(0.),
(0, 2, 0): tensor(2.),
(2, 0, 1): tensor(2.),
(1, 1, 1): tensor(0.),
(0, 2, 1): tensor(-2.)}
>>> import ndmap
>>> ndmap.__about__
Stable and unstable invariant manifolds approximation
Collision of fixed points
Reduce real part of a hyperbolic fixed point