.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "examples_generated/python/02_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_02_numeric-arrays.py: .. _numpy: ======================== 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: [[-4.44089210e-16-9.71445147e-17j 6.93889390e-17+3.64291930e-17j -2.22044605e-16+9.68192540e-17j -6.93889390e-17-1.38777878e-16j 8.40886624e-17+1.12462534e-16j -9.37380762e-17+1.53212577e-16j] [ 1.38777878e-17-6.24500451e-17j 4.44089210e-16-1.11022302e-16j -1.94289029e-16-1.38777878e-17j 1.94289029e-16+6.93889390e-17j -1.00734689e-16-2.34477481e-16j 1.31628004e-16-1.06641647e-16j] [-3.05311332e-16-1.18991188e-17j -3.81639165e-16+5.89805982e-17j 1.22124533e-15+7.69783542e-17j 2.70616862e-16+1.11022302e-16j 6.07751260e-17+6.62434216e-17j 8.41473714e-16-3.56139637e-16j] [-1.06685494e-16+2.34187669e-16j 8.23993651e-17+9.97465999e-18j 2.53269627e-16-4.07660017e-17j -5.62050406e-16-3.81639165e-17j 1.28747465e-15+5.11947027e-17j 1.35940163e-15-1.62204892e-16j] [ 0.00000000e+00-2.77555756e-17j -1.38777878e-16+1.94289029e-16j 3.46944695e-18-7.63278329e-17j 1.22124533e-15-8.32667268e-17j 2.22044605e-15-1.18422638e-18j -1.01124311e-17+2.26402636e-18j] [-3.01841885e-16+2.49800181e-16j -1.11022302e-16+1.11022302e-16j 8.04911693e-16-1.90819582e-16j 1.22124533e-15+1.11022302e-16j -7.71717665e-17-9.15529403e-18j 0.00000000e+00-7.09529922e-18j]] 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_02_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: 02_numeric-arrays.ipynb <02_numeric-arrays.ipynb>` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 02_numeric-arrays.py <02_numeric-arrays.py>` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: 02_numeric-arrays.zip <02_numeric-arrays.zip>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_