ndmap 0.1.7

Higher order partial derivatives computation with respect to one or several tensor-like variables, application to nonlinear dynamics

ndmap, 2022-2023

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.

Install & build

$ pip install git+https://github.com/i-a-morozov/ndmap.git@main

or

$ pip install ndmap -U

Documentation

https://i-a-morozov.github.io/ndmap/

Derivative (composable jacobian)

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.]])]]

Derivative (gradient)

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.)}

Desription

>>> import ndmap
>>> ndmap.__about__

Animations

Stable and unstable invariant manifolds approximation

Collision of fixed points

Reduce real part of a hyperbolic fixed point