.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "examples_generated/python/numeric-arrays.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_examples_generated_python_numeric-arrays.py: .. _python_numeric_arrays: ======================== Numeric arrays in Python ======================== Examples demonstrating NumPy usage in ASE documentation. Links to NumPy's webpage: * `Numpy and Scipy Documentation`_ * `Numpy user guide `_ .. _Numpy and Scipy Documentation: https://docs.scipy.org/doc/ 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. Some examples from everyday ASE-life here ... .. GENERATED FROM PYTHON SOURCE LINES 24-34 .. code-block:: Python import numpy as np a = np.zeros((3, 2)) a[:, 1] = 1.0 a[1] = 2.0 print(a) print(a.shape) print(a.ndim) .. rst-class:: sphx-glr-script-out .. code-block:: none [[0. 1.] [2. 2.] [0. 1.]] (3, 2) 2 .. GENERATED FROM PYTHON SOURCE LINES 35-36 The conventions of numpy's linear algebra package: .. GENERATED FROM PYTHON SOURCE LINES 36-71 .. code-block:: Python # --- Basic array example --- a = np.zeros((3, 2)) a[:, 1] = 1.0 a[1] = 2.0 print('Array a:\n', a) print('Shape:', a.shape) print('Number of dimensions:', a.ndim) # --- Linear algebra example --- # Make a random Hermitian matrix H = np.random.rand(6, 6) + 1.0j * np.random.rand(6, 6) H = H + H.T.conj() # Eigenvalues and eigenvectors eps, U = np.linalg.eigh(H) # Sort eigenvalues and corresponding eigenvectors sorted_indices = eps.real.argsort() eps = eps[sorted_indices] U = U[:, sorted_indices] # Verify diagonalization print( 'Check diagonalization:\n', np.dot(np.dot(U.T.conj(), H), U) - np.diag(eps) ) print('All close?', np.allclose(np.dot(np.dot(U.T.conj(), H), U), np.diag(eps))) # Check eigenvectors individually print( 'Eigenvector check (one column):', np.allclose(np.dot(H, U[:, 3]), eps[3] * U[:, 3]), ) print('Eigenvector check (all):', np.allclose(np.dot(H, U), eps * U)) .. rst-class:: sphx-glr-script-out .. code-block:: none Array a: [[0. 1.] [2. 2.] [0. 1.]] Shape: (3, 2) Number of dimensions: 2 Check diagonalization: [[ 1.11022302e-15+1.44894341e-18j -2.35599530e-16-1.08991545e-16j 2.15247194e-16-1.84939907e-16j 2.25308034e-16-1.86499276e-16j 1.45716772e-15+1.94289029e-16j 1.94289029e-16+3.33066907e-16j] [-5.94269791e-17+4.57634519e-17j -2.22044605e-16+2.02624258e-17j 2.34455306e-16+2.19082359e-17j 1.03926732e-16-1.34262206e-16j -1.70002901e-16-1.04083409e-16j -8.05236954e-16+5.11743425e-17j] [ 1.53568128e-16+2.18596734e-16j 2.35568706e-16-1.37956210e-16j 6.66133815e-16+3.05133306e-17j 2.82192971e-16+2.37356884e-17j -9.88792381e-16+2.08166817e-16j -1.54216917e-15-2.57606436e-16j] [ 2.69178123e-16+2.77909551e-16j 8.15520040e-17+1.88963145e-16j 4.05903470e-16-4.90948052e-17j 0.00000000e+00-6.39266223e-17j -1.87350135e-16-1.11022302e-16j -4.30211422e-16-6.10622664e-16j] [ 1.41223655e-15-1.89893813e-16j -1.25252537e-16+1.38518657e-16j -8.20448623e-16-1.81470222e-16j -9.88939952e-17-1.64818228e-17j 4.44089210e-16+0.00000000e+00j 9.29811783e-16+4.99600361e-16j] [ 5.33433016e-16-3.80977345e-16j -5.53329270e-16-7.91486795e-17j -1.38934790e-15+2.37212393e-16j -8.07603808e-16+5.67252387e-16j 9.99200722e-16-5.55111512e-16j 3.55271368e-15+1.11022302e-16j]] All close? True Eigenvector check (one column): True Eigenvector check (all): True .. GENERATED FROM PYTHON SOURCE LINES 72-85 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 :math:`m` *row* vector can be multiplied with an :math:`n \times m` matrix, producing the same result as if replaced by a matrix with :math:`n` copies of the vector as rows. * A length :math:`n` *column* vector can be multiplied with an :math:`n \times m` matrix, producing the same result as if replaced by a matrix with :math:`m` copies of the vector as columns. Thus, for the arrays below: .. GENERATED FROM PYTHON SOURCE LINES 85-93 .. code-block:: Python # --- 1D vs 2D multiplication rules --- M = np.arange(5 * 6).reshape(5, 6) # A matrix of shape (5, 6) v5 = np.arange(5) + 10 # A vector of length 5 v51 = v5[:, None] # Column vector (5, 1) v6 = np.arange(6) - 12 # A vector of length 6 v16 = v6[None, :] # Row vector (1, 6) .. GENERATED FROM PYTHON SOURCE LINES 94-101 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 same rules apply for adding and subtracting 1D arrays to from 2D arrays. .. GENERATED FROM PYTHON SOURCE LINES 101-106 .. code-block:: Python # Identities print('v6 * M == M * v6?', np.allclose(v6 * M, M * v6)) print('v16 * M == M * v16?', np.allclose(v16 * M, M * v16)) print('v51 * M == M * v51?', np.allclose(v51 * M, M * v51)) .. rst-class:: sphx-glr-script-out .. code-block:: none v6 * M == M * v6? True v16 * M == M * v16? True v51 * M == M * v51? True .. _sphx_glr_download_examples_generated_python_numeric-arrays.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: numeric-arrays.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: numeric-arrays.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: numeric-arrays.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_