Numeric arrays in Python

Links to NumPy’s webpage:

ASE makes heavy use of an extension to Python called NumPy. The NumPy module defines an ndarray type that can hold large arrays of uniform multidimensional numeric data. An array is similar to a list or a tuple, but it is a lot more powerful and efficient.

XXX More examples from everyday ASE-life here …

>>> import numpy as np
>>> a = np.zeros((3, 2))
>>> a[:, 1] = 1.0
>>> a[1] = 2.0
>>> a
array([[ 0.,  1.],
       [ 2.,  2.],
       [ 0.,  1.]])
>>> a.shape
(3, 2)
>>> a.ndim
2

The conventions of numpy’s linear algebra package:

>>> import numpy as np
>>>
>>> # Make a random hermitian matrix, H
>>> H = np.random.rand(6, 6) + 1.j * np.random.rand(6, 6)
>>> H = H + H.T.conj()
>>>
>>> # Determine eigenvalues and rotation matrix
>>> eps, U = np.linalg.eigh(H)
>>>
>>> # Sort eigenvalues
>>> sorted_indices = eps.real.argsort()
>>> eps = eps[sorted_indices]
>>> U = U[:, sorted_indices]
>>>
>>> # Make print of numpy arrays less messy:
>>> np.set_printoptions(precision=3, suppress=True)
>>>
>>> # Check that U diagonalizes H:
>>> print(np.dot(np.dot(U.T.conj(), H), U) - np.diag(eps))
>>> print(np.allclose(np.dot(np.dot(U.T.conj(), H), U), np.diag(eps)))
>>>
>>> # The eigenvectors of H are the *coloumns* of U:
>>> np.allclose(np.dot(H, U[:, 3]), eps[3] * U[:, 3])
>>> np.allclose(np.dot(H, U), eps * U)

The rules for multiplying 1D arrays with 2D arrays:

  • 1D arrays and treated like shape (1, N) arrays (row vectors).
  • left and right multiplications are treated identically.
  • A length \(m\) row vector can be multiplied with an \(n \times m\) matrix, producing the same result as if replaced by a matrix with \(n\) copies of the vector as rows.
  • A length \(n\) column vector can be multiplied with an \(n \times m\) matrix, producing the same result as if replaced by a matrix with \(m\) copies of the vector as columns.

Thus, for the arrays below:

>>> M = np.arange(5 * 6).reshape(5, 6) # A matrix af shape (5, 6)
>>> v5 = np.arange(5) + 10             # A vector of length 5
>>> v51 = v5[:, None]                  # A length 5 column vector
>>> v6 = np.arange(6) - 12             # A vector of length 6
>>> v16 = v6[None, :]                  # A length 6 row vector

The following identities hold:

v6 * M == v16 * M == M * v6 == M * v16 == M * v16.repeat(5, 0)
v51 * M == M * v51 == M * v51.repeat(6, 1)

The exact same rules apply for adding and subtracting 1D arrays to / from 2D arrays.