.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "11_demos\python_packages\numpy\demo_rank_nullspace.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_11_demos_python_packages_numpy_demo_rank_nullspace.py: Rank and nullspace demo ========================== This demo shows an example on how to use ``numpy`` in ``itom``. .. GENERATED FROM PYTHON SOURCE LINES 6-12 .. code-block:: Python import numpy as np from numpy.linalg import svd from numpy.typing import ArrayLike .. GENERATED FROM PYTHON SOURCE LINES 14-15 Function to estimate the rank (i.e. the dimension of the nullspace) of a matrix. .. GENERATED FROM PYTHON SOURCE LINES 15-49 .. code-block:: Python def rank(A: ArrayLike, atol: float = 1e-13, rtol: int = 0) -> int: """Estimate the rank (i.e. the dimension of the nullspace) of a matrix. The algorithm used by this function is based on the singular value decomposition of `A`. If both `atol` and `rtol` are positive, the combined tolerance is the maximum of the two; that is:: tol = max(atol, rtol * smax) Singular values smaller than `tol` are considered to be zero. Args: A (ArrayLike): A should be at most 2-D. A 1-D array with length n will be treated as a 2-D with shape (1, n) atol (float, optional): The absolute tolerance for a zero singular value. Singular values smaller than `atol` are considered to be zero. rtol (int, optional): The relative tolerance. Singular values less than rtol*smax are considered to be zero, where smax is the largest singular value. Returns: int: The estimated rank of the matrix. See also -------- numpy.linalg.matrix_rank matrix_rank is basically the same as this function, but it does not provide the option of the absolute tolerance.""" A = np.atleast_2d(A) s = svd(A, compute_uv=False) tol = max(atol, rtol * s[0]) rank = int((s >= tol).sum()) return rank .. GENERATED FROM PYTHON SOURCE LINES 50-51 Function to compute an approximate basis for the nullspace of A. .. GENERATED FROM PYTHON SOURCE LINES 51-86 .. code-block:: Python def nullspace(A: ArrayLike, atol: float = 1e-13, rtol: int = 0) -> ArrayLike: """Compute an approximate basis for the nullspace of A. The algorithm used by this function is based on the singular value decomposition of `A`. If both `atol` and `rtol` are positive, the combined tolerance is the maximum of the two; that is:: tol = max(atol, rtol * smax) Singular values smaller than `tol` are considered to be zero. Args: A (ArrayLike): A should be at most 2-D. A 1-D array with length k will be treated as a 2-D with shape (1, k) atol (float, optional): The absolute tolerance for a zero singular value. Singular values smaller than `atol` are considered to be zero. rtol (int, optional): The relative tolerance. Singular values less than rtol*smax are considered to be zero, where smax is the largest singular value. Returns: ArrayLike: If `A` is an array with shape (m, k), then `ns` will be an array with shape (k, n), where n is the estimated dimension of the nullspace of `A`. The columns of `ns` are a basis for the nullspace; each element in numpy.dot(A, ns) will be approximately zero. """ A = np.atleast_2d(A) u, s, vh = svd(A) tol = max(atol, rtol * s[0]) nnz = (s >= tol).sum() ns = vh[nnz:].conj().T return ns .. GENERATED FROM PYTHON SOURCE LINES 87-88 Function to check rank and nullspace of the matrix. .. GENERATED FROM PYTHON SOURCE LINES 88-139 .. code-block:: Python def checkit(mat): """This method calculates the rank and nullspace of matrix mat. The results are printed.""" print("mat:") print(mat) r = rank(mat) print("rank is", r) ns = nullspace(mat) print("nullspace:") print(ns) if ns.size > 0: res = np.abs(np.dot(mat, ns)).max() print("max residual is", res) print("-" * 25) # checks rank and nulllspace of a. The rank is limited since the 3rd row # is equal to 2x the 2nd row minus 1x the 1st row. a = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) checkit(a) b = 2 print("-" * 25) # full rank matrix a = np.array([[0.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) checkit(a) print("-" * 25) # the rank is 2 since the matrix has only two rows a = np.array([[0.0, 1.0, 2.0, 4.0], [1.0, 2.0, 3.0, 4.0]]) checkit(a) print("-" * 25) # check the rank and nullspace of a complex matrix a = np.array( [ [1.0, 1.0j, 2.0 + 2.0j], [1.0j, -1.0, -2.0 + 2.0j], [0.5, 0.5j, 1.0 + 1.0j], ] ) checkit(a) print("-" * 25) .. rst-class:: sphx-glr-script-out .. code-block:: none ------------------------- mat: [[1. 2. 3.] [4. 5. 6.] [7. 8. 9.]] rank is 2 nullspace: [[-0.40824829] [ 0.81649658] [-0.40824829]] max residual is 9.43689570931383e-16 ------------------------- mat: [[0. 2. 3.] [4. 5. 6.] [7. 8. 9.]] rank is 3 nullspace: [] ------------------------- mat: [[0. 1. 2. 4.] [1. 2. 3. 4.]] rank is 2 nullspace: [[-0.48666474 -0.61177492] [-0.27946883 0.76717915] [ 0.76613356 -0.15540423] [-0.31319957 -0.11409267]] max residual is 8.881784197001252e-16 ------------------------- mat: [[ 1. +0.j 0. +1.j 2. +2.j ] [ 0. +1.j -1. +0.j -2. +2.j ] [ 0.5+0.j 0. +0.5j 1. +1.j ]] rank is 1 nullspace: [[ 0. -0.j -0.9486833 -0.j ] [ 0.13333333+0.93333333j 0. -0.10540926j] [ 0.2 -0.26666667j 0.21081851-0.21081851j]] max residual is 9.930136612989092e-16 ------------------------- .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.010 seconds) .. _sphx_glr_download_11_demos_python_packages_numpy_demo_rank_nullspace.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: demo_rank_nullspace.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: demo_rank_nullspace.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: demo_rank_nullspace.zip `