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1006 lines
32 KiB
1006 lines
32 KiB
from ..libmp.backend import xrange
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import warnings
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# TODO: interpret list as vectors (for multiplication)
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rowsep = '\n'
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colsep = ' '
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class _matrix(object):
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"""
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Numerical matrix.
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Specify the dimensions or the data as a nested list.
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Elements default to zero.
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Use a flat list to create a column vector easily.
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The datatype of the context (mpf for mp, mpi for iv, and float for fp) is used to store the data.
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Creating matrices
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-----------------
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Matrices in mpmath are implemented using dictionaries. Only non-zero values
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are stored, so it is cheap to represent sparse matrices.
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The most basic way to create one is to use the ``matrix`` class directly.
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You can create an empty matrix specifying the dimensions:
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>>> from mpmath import *
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>>> mp.dps = 15
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>>> matrix(2)
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matrix(
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[['0.0', '0.0'],
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['0.0', '0.0']])
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>>> matrix(2, 3)
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matrix(
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[['0.0', '0.0', '0.0'],
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['0.0', '0.0', '0.0']])
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Calling ``matrix`` with one dimension will create a square matrix.
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To access the dimensions of a matrix, use the ``rows`` or ``cols`` keyword:
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>>> A = matrix(3, 2)
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>>> A
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matrix(
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[['0.0', '0.0'],
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['0.0', '0.0'],
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['0.0', '0.0']])
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>>> A.rows
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3
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>>> A.cols
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2
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You can also change the dimension of an existing matrix. This will set the
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new elements to 0. If the new dimension is smaller than before, the
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concerning elements are discarded:
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>>> A.rows = 2
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>>> A
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matrix(
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[['0.0', '0.0'],
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['0.0', '0.0']])
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Internally ``mpmathify`` is used every time an element is set. This
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is done using the syntax A[row,column], counting from 0:
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>>> A = matrix(2)
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>>> A[1,1] = 1 + 1j
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>>> A
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matrix(
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[['0.0', '0.0'],
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['0.0', mpc(real='1.0', imag='1.0')]])
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A more comfortable way to create a matrix lets you use nested lists:
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>>> matrix([[1, 2], [3, 4]])
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matrix(
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[['1.0', '2.0'],
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['3.0', '4.0']])
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Convenient advanced functions are available for creating various standard
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matrices, see ``zeros``, ``ones``, ``diag``, ``eye``, ``randmatrix`` and
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``hilbert``.
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Vectors
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.......
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Vectors may also be represented by the ``matrix`` class (with rows = 1 or cols = 1).
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For vectors there are some things which make life easier. A column vector can
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be created using a flat list, a row vectors using an almost flat nested list::
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>>> matrix([1, 2, 3])
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matrix(
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[['1.0'],
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['2.0'],
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['3.0']])
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>>> matrix([[1, 2, 3]])
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matrix(
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[['1.0', '2.0', '3.0']])
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Optionally vectors can be accessed like lists, using only a single index::
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>>> x = matrix([1, 2, 3])
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>>> x[1]
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mpf('2.0')
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>>> x[1,0]
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mpf('2.0')
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Other
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.....
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Like you probably expected, matrices can be printed::
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>>> print randmatrix(3) # doctest:+SKIP
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[ 0.782963853573023 0.802057689719883 0.427895717335467]
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[0.0541876859348597 0.708243266653103 0.615134039977379]
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[ 0.856151514955773 0.544759264818486 0.686210904770947]
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Use ``nstr`` or ``nprint`` to specify the number of digits to print::
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>>> nprint(randmatrix(5), 3) # doctest:+SKIP
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[2.07e-1 1.66e-1 5.06e-1 1.89e-1 8.29e-1]
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[6.62e-1 6.55e-1 4.47e-1 4.82e-1 2.06e-2]
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[4.33e-1 7.75e-1 6.93e-2 2.86e-1 5.71e-1]
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[1.01e-1 2.53e-1 6.13e-1 3.32e-1 2.59e-1]
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[1.56e-1 7.27e-2 6.05e-1 6.67e-2 2.79e-1]
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As matrices are mutable, you will need to copy them sometimes::
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>>> A = matrix(2)
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>>> A
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matrix(
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[['0.0', '0.0'],
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['0.0', '0.0']])
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>>> B = A.copy()
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>>> B[0,0] = 1
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>>> B
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matrix(
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[['1.0', '0.0'],
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['0.0', '0.0']])
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>>> A
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matrix(
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[['0.0', '0.0'],
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['0.0', '0.0']])
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Finally, it is possible to convert a matrix to a nested list. This is very useful,
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as most Python libraries involving matrices or arrays (namely NumPy or SymPy)
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support this format::
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>>> B.tolist()
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[[mpf('1.0'), mpf('0.0')], [mpf('0.0'), mpf('0.0')]]
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Matrix operations
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-----------------
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You can add and subtract matrices of compatible dimensions::
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>>> A = matrix([[1, 2], [3, 4]])
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>>> B = matrix([[-2, 4], [5, 9]])
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>>> A + B
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matrix(
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[['-1.0', '6.0'],
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['8.0', '13.0']])
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>>> A - B
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matrix(
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[['3.0', '-2.0'],
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['-2.0', '-5.0']])
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>>> A + ones(3) # doctest:+ELLIPSIS
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Traceback (most recent call last):
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...
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ValueError: incompatible dimensions for addition
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It is possible to multiply or add matrices and scalars. In the latter case the
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operation will be done element-wise::
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>>> A * 2
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matrix(
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[['2.0', '4.0'],
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['6.0', '8.0']])
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>>> A / 4
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matrix(
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[['0.25', '0.5'],
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['0.75', '1.0']])
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>>> A - 1
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matrix(
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[['0.0', '1.0'],
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['2.0', '3.0']])
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Of course you can perform matrix multiplication, if the dimensions are
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compatible, using ``@`` (for Python >= 3.5) or ``*``. For clarity, ``@`` is
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recommended (`PEP 465 <https://www.python.org/dev/peps/pep-0465/>`), because
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the meaning of ``*`` is different in many other Python libraries such as NumPy.
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>>> A @ B # doctest:+SKIP
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matrix(
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[['8.0', '22.0'],
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['14.0', '48.0']])
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>>> A * B # same as A @ B
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matrix(
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[['8.0', '22.0'],
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['14.0', '48.0']])
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>>> matrix([[1, 2, 3]]) * matrix([[-6], [7], [-2]])
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matrix(
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[['2.0']])
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..
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COMMENT: TODO: the above "doctest:+SKIP" may be removed as soon as we
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have dropped support for Python 3.5 and below.
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You can raise powers of square matrices::
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>>> A**2
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matrix(
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[['7.0', '10.0'],
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['15.0', '22.0']])
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Negative powers will calculate the inverse::
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>>> A**-1
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matrix(
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[['-2.0', '1.0'],
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['1.5', '-0.5']])
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>>> A * A**-1
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matrix(
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[['1.0', '1.0842021724855e-19'],
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['-2.16840434497101e-19', '1.0']])
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Matrix transposition is straightforward::
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>>> A = ones(2, 3)
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>>> A
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matrix(
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[['1.0', '1.0', '1.0'],
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['1.0', '1.0', '1.0']])
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>>> A.T
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matrix(
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[['1.0', '1.0'],
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['1.0', '1.0'],
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['1.0', '1.0']])
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Norms
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.....
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Sometimes you need to know how "large" a matrix or vector is. Due to their
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multidimensional nature it's not possible to compare them, but there are
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several functions to map a matrix or a vector to a positive real number, the
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so called norms.
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For vectors the p-norm is intended, usually the 1-, the 2- and the oo-norm are
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used.
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>>> x = matrix([-10, 2, 100])
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>>> norm(x, 1)
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mpf('112.0')
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>>> norm(x, 2)
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mpf('100.5186549850325')
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>>> norm(x, inf)
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mpf('100.0')
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Please note that the 2-norm is the most used one, though it is more expensive
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to calculate than the 1- or oo-norm.
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It is possible to generalize some vector norms to matrix norm::
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>>> A = matrix([[1, -1000], [100, 50]])
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>>> mnorm(A, 1)
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mpf('1050.0')
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>>> mnorm(A, inf)
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mpf('1001.0')
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>>> mnorm(A, 'F')
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mpf('1006.2310867787777')
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The last norm (the "Frobenius-norm") is an approximation for the 2-norm, which
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is hard to calculate and not available. The Frobenius-norm lacks some
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mathematical properties you might expect from a norm.
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"""
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def __init__(self, *args, **kwargs):
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self.__data = {}
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# LU decompostion cache, this is useful when solving the same system
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# multiple times, when calculating the inverse and when calculating the
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# determinant
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self._LU = None
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if "force_type" in kwargs:
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warnings.warn("The force_type argument was removed, it did not work"
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" properly anyway. If you want to force floating-point or"
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" interval computations, use the respective methods from `fp`"
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" or `mp` instead, e.g., `fp.matrix()` or `iv.matrix()`."
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" If you want to truncate values to integer, use .apply(int) instead.")
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if isinstance(args[0], (list, tuple)):
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if isinstance(args[0][0], (list, tuple)):
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# interpret nested list as matrix
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A = args[0]
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self.__rows = len(A)
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self.__cols = len(A[0])
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for i, row in enumerate(A):
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for j, a in enumerate(row):
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# note: this will call __setitem__ which will call self.ctx.convert() to convert the datatype.
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self[i, j] = a
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else:
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# interpret list as row vector
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v = args[0]
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self.__rows = len(v)
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self.__cols = 1
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for i, e in enumerate(v):
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self[i, 0] = e
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elif isinstance(args[0], int):
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# create empty matrix of given dimensions
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if len(args) == 1:
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self.__rows = self.__cols = args[0]
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else:
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if not isinstance(args[1], int):
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raise TypeError("expected int")
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self.__rows = args[0]
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self.__cols = args[1]
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elif isinstance(args[0], _matrix):
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A = args[0]
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self.__rows = A._matrix__rows
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self.__cols = A._matrix__cols
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for i in xrange(A.__rows):
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for j in xrange(A.__cols):
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self[i, j] = A[i, j]
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elif hasattr(args[0], 'tolist'):
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A = self.ctx.matrix(args[0].tolist())
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self.__data = A._matrix__data
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self.__rows = A._matrix__rows
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self.__cols = A._matrix__cols
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else:
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raise TypeError('could not interpret given arguments')
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def apply(self, f):
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"""
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Return a copy of self with the function `f` applied elementwise.
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"""
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new = self.ctx.matrix(self.__rows, self.__cols)
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for i in xrange(self.__rows):
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for j in xrange(self.__cols):
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new[i,j] = f(self[i,j])
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return new
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def __nstr__(self, n=None, **kwargs):
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# Build table of string representations of the elements
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res = []
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# Track per-column max lengths for pretty alignment
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maxlen = [0] * self.cols
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for i in range(self.rows):
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res.append([])
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for j in range(self.cols):
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if n:
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string = self.ctx.nstr(self[i,j], n, **kwargs)
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else:
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string = str(self[i,j])
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res[-1].append(string)
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maxlen[j] = max(len(string), maxlen[j])
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# Patch strings together
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for i, row in enumerate(res):
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for j, elem in enumerate(row):
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# Pad each element up to maxlen so the columns line up
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row[j] = elem.rjust(maxlen[j])
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res[i] = "[" + colsep.join(row) + "]"
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return rowsep.join(res)
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def __str__(self):
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return self.__nstr__()
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def _toliststr(self, avoid_type=False):
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"""
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Create a list string from a matrix.
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If avoid_type: avoid multiple 'mpf's.
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"""
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# XXX: should be something like self.ctx._types
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typ = self.ctx.mpf
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s = '['
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for i in xrange(self.__rows):
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s += '['
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for j in xrange(self.__cols):
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if not avoid_type or not isinstance(self[i,j], typ):
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a = repr(self[i,j])
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else:
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a = "'" + str(self[i,j]) + "'"
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s += a + ', '
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s = s[:-2]
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s += '],\n '
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s = s[:-3]
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s += ']'
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return s
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def tolist(self):
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"""
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Convert the matrix to a nested list.
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"""
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return [[self[i,j] for j in range(self.__cols)] for i in range(self.__rows)]
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def __repr__(self):
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if self.ctx.pretty:
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return self.__str__()
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s = 'matrix(\n'
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s += self._toliststr(avoid_type=True) + ')'
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return s
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def __get_element(self, key):
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'''
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Fast extraction of the i,j element from the matrix
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This function is for private use only because is unsafe:
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1. Does not check on the value of key it expects key to be a integer tuple (i,j)
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2. Does not check bounds
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'''
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if key in self.__data:
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return self.__data[key]
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else:
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return self.ctx.zero
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def __set_element(self, key, value):
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'''
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Fast assignment of the i,j element in the matrix
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This function is unsafe:
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1. Does not check on the value of key it expects key to be a integer tuple (i,j)
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2. Does not check bounds
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3. Does not check the value type
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4. Does not reset the LU cache
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'''
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if value: # only store non-zeros
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self.__data[key] = value
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elif key in self.__data:
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del self.__data[key]
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def __getitem__(self, key):
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'''
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Getitem function for mp matrix class with slice index enabled
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it allows the following assingments
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scalar to a slice of the matrix
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B = A[:,2:6]
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'''
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# Convert vector to matrix indexing
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if isinstance(key, int) or isinstance(key,slice):
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# only sufficent for vectors
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if self.__rows == 1:
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key = (0, key)
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elif self.__cols == 1:
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key = (key, 0)
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else:
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raise IndexError('insufficient indices for matrix')
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if isinstance(key[0],slice) or isinstance(key[1],slice):
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#Rows
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if isinstance(key[0],slice):
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#Check bounds
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if (key[0].start is None or key[0].start >= 0) and \
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(key[0].stop is None or key[0].stop <= self.__rows+1):
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# Generate indices
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rows = xrange(*key[0].indices(self.__rows))
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else:
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raise IndexError('Row index out of bounds')
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else:
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# Single row
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rows = [key[0]]
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# Columns
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if isinstance(key[1],slice):
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# Check bounds
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if (key[1].start is None or key[1].start >= 0) and \
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(key[1].stop is None or key[1].stop <= self.__cols+1):
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# Generate indices
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columns = xrange(*key[1].indices(self.__cols))
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else:
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raise IndexError('Column index out of bounds')
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else:
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# Single column
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columns = [key[1]]
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# Create matrix slice
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m = self.ctx.matrix(len(rows),len(columns))
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# Assign elements to the output matrix
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for i,x in enumerate(rows):
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for j,y in enumerate(columns):
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m.__set_element((i,j),self.__get_element((x,y)))
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return m
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else:
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# single element extraction
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if key[0] >= self.__rows or key[1] >= self.__cols:
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raise IndexError('matrix index out of range')
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if key in self.__data:
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return self.__data[key]
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else:
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return self.ctx.zero
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def __setitem__(self, key, value):
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# setitem function for mp matrix class with slice index enabled
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# it allows the following assingments
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# scalar to a slice of the matrix
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# A[:,2:6] = 2.5
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# submatrix to matrix (the value matrix should be the same size as the slice size)
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# A[3,:] = B where A is n x m and B is n x 1
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# Convert vector to matrix indexing
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if isinstance(key, int) or isinstance(key,slice):
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# only sufficent for vectors
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if self.__rows == 1:
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key = (0, key)
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elif self.__cols == 1:
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key = (key, 0)
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else:
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raise IndexError('insufficient indices for matrix')
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# Slice indexing
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if isinstance(key[0],slice) or isinstance(key[1],slice):
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# Rows
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if isinstance(key[0],slice):
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# Check bounds
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if (key[0].start is None or key[0].start >= 0) and \
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|
(key[0].stop is None or key[0].stop <= self.__rows+1):
|
|
# generate row indices
|
|
rows = xrange(*key[0].indices(self.__rows))
|
|
else:
|
|
raise IndexError('Row index out of bounds')
|
|
else:
|
|
# Single row
|
|
rows = [key[0]]
|
|
# Columns
|
|
if isinstance(key[1],slice):
|
|
# Check bounds
|
|
if (key[1].start is None or key[1].start >= 0) and \
|
|
(key[1].stop is None or key[1].stop <= self.__cols+1):
|
|
# Generate column indices
|
|
columns = xrange(*key[1].indices(self.__cols))
|
|
else:
|
|
raise IndexError('Column index out of bounds')
|
|
else:
|
|
# Single column
|
|
columns = [key[1]]
|
|
# Assign slice with a scalar
|
|
if isinstance(value,self.ctx.matrix):
|
|
# Assign elements to matrix if input and output dimensions match
|
|
if len(rows) == value.rows and len(columns) == value.cols:
|
|
for i,x in enumerate(rows):
|
|
for j,y in enumerate(columns):
|
|
self.__set_element((x,y), value.__get_element((i,j)))
|
|
else:
|
|
raise ValueError('Dimensions do not match')
|
|
else:
|
|
# Assign slice with scalars
|
|
value = self.ctx.convert(value)
|
|
for i in rows:
|
|
for j in columns:
|
|
self.__set_element((i,j), value)
|
|
else:
|
|
# Single element assingment
|
|
# Check bounds
|
|
if key[0] >= self.__rows or key[1] >= self.__cols:
|
|
raise IndexError('matrix index out of range')
|
|
# Convert and store value
|
|
value = self.ctx.convert(value)
|
|
if value: # only store non-zeros
|
|
self.__data[key] = value
|
|
elif key in self.__data:
|
|
del self.__data[key]
|
|
|
|
if self._LU:
|
|
self._LU = None
|
|
return
|
|
|
|
def __iter__(self):
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(self.__cols):
|
|
yield self[i,j]
|
|
|
|
def __mul__(self, other):
|
|
if isinstance(other, self.ctx.matrix):
|
|
# dot multiplication
|
|
if self.__cols != other.__rows:
|
|
raise ValueError('dimensions not compatible for multiplication')
|
|
new = self.ctx.matrix(self.__rows, other.__cols)
|
|
self_zero = self.ctx.zero
|
|
self_get = self.__data.get
|
|
other_zero = other.ctx.zero
|
|
other_get = other.__data.get
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(other.__cols):
|
|
new[i, j] = self.ctx.fdot((self_get((i,k), self_zero), other_get((k,j), other_zero))
|
|
for k in xrange(other.__rows))
|
|
return new
|
|
else:
|
|
# try scalar multiplication
|
|
new = self.ctx.matrix(self.__rows, self.__cols)
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(self.__cols):
|
|
new[i, j] = other * self[i, j]
|
|
return new
|
|
|
|
def __matmul__(self, other):
|
|
return self.__mul__(other)
|
|
|
|
def __rmul__(self, other):
|
|
# assume other is scalar and thus commutative
|
|
if isinstance(other, self.ctx.matrix):
|
|
raise TypeError("other should not be type of ctx.matrix")
|
|
return self.__mul__(other)
|
|
|
|
def __pow__(self, other):
|
|
# avoid cyclic import problems
|
|
#from linalg import inverse
|
|
if not isinstance(other, int):
|
|
raise ValueError('only integer exponents are supported')
|
|
if not self.__rows == self.__cols:
|
|
raise ValueError('only powers of square matrices are defined')
|
|
n = other
|
|
if n == 0:
|
|
return self.ctx.eye(self.__rows)
|
|
if n < 0:
|
|
n = -n
|
|
neg = True
|
|
else:
|
|
neg = False
|
|
i = n
|
|
y = 1
|
|
z = self.copy()
|
|
while i != 0:
|
|
if i % 2 == 1:
|
|
y = y * z
|
|
z = z*z
|
|
i = i // 2
|
|
if neg:
|
|
y = self.ctx.inverse(y)
|
|
return y
|
|
|
|
def __div__(self, other):
|
|
# assume other is scalar and do element-wise divison
|
|
assert not isinstance(other, self.ctx.matrix)
|
|
new = self.ctx.matrix(self.__rows, self.__cols)
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(self.__cols):
|
|
new[i,j] = self[i,j] / other
|
|
return new
|
|
|
|
__truediv__ = __div__
|
|
|
|
def __add__(self, other):
|
|
if isinstance(other, self.ctx.matrix):
|
|
if not (self.__rows == other.__rows and self.__cols == other.__cols):
|
|
raise ValueError('incompatible dimensions for addition')
|
|
new = self.ctx.matrix(self.__rows, self.__cols)
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(self.__cols):
|
|
new[i,j] = self[i,j] + other[i,j]
|
|
return new
|
|
else:
|
|
# assume other is scalar and add element-wise
|
|
new = self.ctx.matrix(self.__rows, self.__cols)
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(self.__cols):
|
|
new[i,j] += self[i,j] + other
|
|
return new
|
|
|
|
def __radd__(self, other):
|
|
return self.__add__(other)
|
|
|
|
def __sub__(self, other):
|
|
if isinstance(other, self.ctx.matrix) and not (self.__rows == other.__rows
|
|
and self.__cols == other.__cols):
|
|
raise ValueError('incompatible dimensions for subtraction')
|
|
return self.__add__(other * (-1))
|
|
|
|
def __pos__(self):
|
|
"""
|
|
+M returns a copy of M, rounded to current working precision.
|
|
"""
|
|
return (+1) * self
|
|
|
|
def __neg__(self):
|
|
return (-1) * self
|
|
|
|
def __rsub__(self, other):
|
|
return -self + other
|
|
|
|
def __eq__(self, other):
|
|
return self.__rows == other.__rows and self.__cols == other.__cols \
|
|
and self.__data == other.__data
|
|
|
|
def __len__(self):
|
|
if self.rows == 1:
|
|
return self.cols
|
|
elif self.cols == 1:
|
|
return self.rows
|
|
else:
|
|
return self.rows # do it like numpy
|
|
|
|
def __getrows(self):
|
|
return self.__rows
|
|
|
|
def __setrows(self, value):
|
|
for key in self.__data.copy():
|
|
if key[0] >= value:
|
|
del self.__data[key]
|
|
self.__rows = value
|
|
|
|
rows = property(__getrows, __setrows, doc='number of rows')
|
|
|
|
def __getcols(self):
|
|
return self.__cols
|
|
|
|
def __setcols(self, value):
|
|
for key in self.__data.copy():
|
|
if key[1] >= value:
|
|
del self.__data[key]
|
|
self.__cols = value
|
|
|
|
cols = property(__getcols, __setcols, doc='number of columns')
|
|
|
|
def transpose(self):
|
|
new = self.ctx.matrix(self.__cols, self.__rows)
|
|
for i in xrange(self.__rows):
|
|
for j in xrange(self.__cols):
|
|
new[j,i] = self[i,j]
|
|
return new
|
|
|
|
T = property(transpose)
|
|
|
|
def conjugate(self):
|
|
return self.apply(self.ctx.conj)
|
|
|
|
def transpose_conj(self):
|
|
return self.conjugate().transpose()
|
|
|
|
H = property(transpose_conj)
|
|
|
|
def copy(self):
|
|
new = self.ctx.matrix(self.__rows, self.__cols)
|
|
new.__data = self.__data.copy()
|
|
return new
|
|
|
|
__copy__ = copy
|
|
|
|
def column(self, n):
|
|
m = self.ctx.matrix(self.rows, 1)
|
|
for i in range(self.rows):
|
|
m[i] = self[i,n]
|
|
return m
|
|
|
|
class MatrixMethods(object):
|
|
|
|
def __init__(ctx):
|
|
# XXX: subclass
|
|
ctx.matrix = type('matrix', (_matrix,), {})
|
|
ctx.matrix.ctx = ctx
|
|
ctx.matrix.convert = ctx.convert
|
|
|
|
def eye(ctx, n, **kwargs):
|
|
"""
|
|
Create square identity matrix n x n.
|
|
"""
|
|
A = ctx.matrix(n, **kwargs)
|
|
for i in xrange(n):
|
|
A[i,i] = 1
|
|
return A
|
|
|
|
def diag(ctx, diagonal, **kwargs):
|
|
"""
|
|
Create square diagonal matrix using given list.
|
|
|
|
Example:
|
|
>>> from mpmath import diag, mp
|
|
>>> mp.pretty = False
|
|
>>> diag([1, 2, 3])
|
|
matrix(
|
|
[['1.0', '0.0', '0.0'],
|
|
['0.0', '2.0', '0.0'],
|
|
['0.0', '0.0', '3.0']])
|
|
"""
|
|
A = ctx.matrix(len(diagonal), **kwargs)
|
|
for i in xrange(len(diagonal)):
|
|
A[i,i] = diagonal[i]
|
|
return A
|
|
|
|
def zeros(ctx, *args, **kwargs):
|
|
"""
|
|
Create matrix m x n filled with zeros.
|
|
One given dimension will create square matrix n x n.
|
|
|
|
Example:
|
|
>>> from mpmath import zeros, mp
|
|
>>> mp.pretty = False
|
|
>>> zeros(2)
|
|
matrix(
|
|
[['0.0', '0.0'],
|
|
['0.0', '0.0']])
|
|
"""
|
|
if len(args) == 1:
|
|
m = n = args[0]
|
|
elif len(args) == 2:
|
|
m = args[0]
|
|
n = args[1]
|
|
else:
|
|
raise TypeError('zeros expected at most 2 arguments, got %i' % len(args))
|
|
A = ctx.matrix(m, n, **kwargs)
|
|
for i in xrange(m):
|
|
for j in xrange(n):
|
|
A[i,j] = 0
|
|
return A
|
|
|
|
def ones(ctx, *args, **kwargs):
|
|
"""
|
|
Create matrix m x n filled with ones.
|
|
One given dimension will create square matrix n x n.
|
|
|
|
Example:
|
|
>>> from mpmath import ones, mp
|
|
>>> mp.pretty = False
|
|
>>> ones(2)
|
|
matrix(
|
|
[['1.0', '1.0'],
|
|
['1.0', '1.0']])
|
|
"""
|
|
if len(args) == 1:
|
|
m = n = args[0]
|
|
elif len(args) == 2:
|
|
m = args[0]
|
|
n = args[1]
|
|
else:
|
|
raise TypeError('ones expected at most 2 arguments, got %i' % len(args))
|
|
A = ctx.matrix(m, n, **kwargs)
|
|
for i in xrange(m):
|
|
for j in xrange(n):
|
|
A[i,j] = 1
|
|
return A
|
|
|
|
def hilbert(ctx, m, n=None):
|
|
"""
|
|
Create (pseudo) hilbert matrix m x n.
|
|
One given dimension will create hilbert matrix n x n.
|
|
|
|
The matrix is very ill-conditioned and symmetric, positive definite if
|
|
square.
|
|
"""
|
|
if n is None:
|
|
n = m
|
|
A = ctx.matrix(m, n)
|
|
for i in xrange(m):
|
|
for j in xrange(n):
|
|
A[i,j] = ctx.one / (i + j + 1)
|
|
return A
|
|
|
|
def randmatrix(ctx, m, n=None, min=0, max=1, **kwargs):
|
|
"""
|
|
Create a random m x n matrix.
|
|
|
|
All values are >= min and <max.
|
|
n defaults to m.
|
|
|
|
Example:
|
|
>>> from mpmath import randmatrix
|
|
>>> randmatrix(2) # doctest:+SKIP
|
|
matrix(
|
|
[['0.53491598236191806', '0.57195669543302752'],
|
|
['0.85589992269513615', '0.82444367501382143']])
|
|
"""
|
|
if not n:
|
|
n = m
|
|
A = ctx.matrix(m, n, **kwargs)
|
|
for i in xrange(m):
|
|
for j in xrange(n):
|
|
A[i,j] = ctx.rand() * (max - min) + min
|
|
return A
|
|
|
|
def swap_row(ctx, A, i, j):
|
|
"""
|
|
Swap row i with row j.
|
|
"""
|
|
if i == j:
|
|
return
|
|
if isinstance(A, ctx.matrix):
|
|
for k in xrange(A.cols):
|
|
A[i,k], A[j,k] = A[j,k], A[i,k]
|
|
elif isinstance(A, list):
|
|
A[i], A[j] = A[j], A[i]
|
|
else:
|
|
raise TypeError('could not interpret type')
|
|
|
|
def extend(ctx, A, b):
|
|
"""
|
|
Extend matrix A with column b and return result.
|
|
"""
|
|
if not isinstance(A, ctx.matrix):
|
|
raise TypeError("A should be a type of ctx.matrix")
|
|
if A.rows != len(b):
|
|
raise ValueError("Value should be equal to len(b)")
|
|
A = A.copy()
|
|
A.cols += 1
|
|
for i in xrange(A.rows):
|
|
A[i, A.cols-1] = b[i]
|
|
return A
|
|
|
|
def norm(ctx, x, p=2):
|
|
r"""
|
|
Gives the entrywise `p`-norm of an iterable *x*, i.e. the vector norm
|
|
`\left(\sum_k |x_k|^p\right)^{1/p}`, for any given `1 \le p \le \infty`.
|
|
|
|
Special cases:
|
|
|
|
If *x* is not iterable, this just returns ``absmax(x)``.
|
|
|
|
``p=1`` gives the sum of absolute values.
|
|
|
|
``p=2`` is the standard Euclidean vector norm.
|
|
|
|
``p=inf`` gives the magnitude of the largest element.
|
|
|
|
For *x* a matrix, ``p=2`` is the Frobenius norm.
|
|
For operator matrix norms, use :func:`~mpmath.mnorm` instead.
|
|
|
|
You can use the string 'inf' as well as float('inf') or mpf('inf')
|
|
to specify the infinity norm.
|
|
|
|
**Examples**
|
|
|
|
>>> from mpmath import *
|
|
>>> mp.dps = 15; mp.pretty = False
|
|
>>> x = matrix([-10, 2, 100])
|
|
>>> norm(x, 1)
|
|
mpf('112.0')
|
|
>>> norm(x, 2)
|
|
mpf('100.5186549850325')
|
|
>>> norm(x, inf)
|
|
mpf('100.0')
|
|
|
|
"""
|
|
try:
|
|
iter(x)
|
|
except TypeError:
|
|
return ctx.absmax(x)
|
|
if type(p) is not int:
|
|
p = ctx.convert(p)
|
|
if p == ctx.inf:
|
|
return max(ctx.absmax(i) for i in x)
|
|
elif p == 1:
|
|
return ctx.fsum(x, absolute=1)
|
|
elif p == 2:
|
|
return ctx.sqrt(ctx.fsum(x, absolute=1, squared=1))
|
|
elif p > 1:
|
|
return ctx.nthroot(ctx.fsum(abs(i)**p for i in x), p)
|
|
else:
|
|
raise ValueError('p has to be >= 1')
|
|
|
|
def mnorm(ctx, A, p=1):
|
|
r"""
|
|
Gives the matrix (operator) `p`-norm of A. Currently ``p=1`` and ``p=inf``
|
|
are supported:
|
|
|
|
``p=1`` gives the 1-norm (maximal column sum)
|
|
|
|
``p=inf`` gives the `\infty`-norm (maximal row sum).
|
|
You can use the string 'inf' as well as float('inf') or mpf('inf')
|
|
|
|
``p=2`` (not implemented) for a square matrix is the usual spectral
|
|
matrix norm, i.e. the largest singular value.
|
|
|
|
``p='f'`` (or 'F', 'fro', 'Frobenius, 'frobenius') gives the
|
|
Frobenius norm, which is the elementwise 2-norm. The Frobenius norm is an
|
|
approximation of the spectral norm and satisfies
|
|
|
|
.. math ::
|
|
|
|
\frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F
|
|
|
|
The Frobenius norm lacks some mathematical properties that might
|
|
be expected of a norm.
|
|
|
|
For general elementwise `p`-norms, use :func:`~mpmath.norm` instead.
|
|
|
|
**Examples**
|
|
|
|
>>> from mpmath import *
|
|
>>> mp.dps = 15; mp.pretty = False
|
|
>>> A = matrix([[1, -1000], [100, 50]])
|
|
>>> mnorm(A, 1)
|
|
mpf('1050.0')
|
|
>>> mnorm(A, inf)
|
|
mpf('1001.0')
|
|
>>> mnorm(A, 'F')
|
|
mpf('1006.2310867787777')
|
|
|
|
"""
|
|
A = ctx.matrix(A)
|
|
if type(p) is not int:
|
|
if type(p) is str and 'frobenius'.startswith(p.lower()):
|
|
return ctx.norm(A, 2)
|
|
p = ctx.convert(p)
|
|
m, n = A.rows, A.cols
|
|
if p == 1:
|
|
return max(ctx.fsum((A[i,j] for i in xrange(m)), absolute=1) for j in xrange(n))
|
|
elif p == ctx.inf:
|
|
return max(ctx.fsum((A[i,j] for j in xrange(n)), absolute=1) for i in xrange(m))
|
|
else:
|
|
raise NotImplementedError("matrix p-norm for arbitrary p")
|
|
|
|
if __name__ == '__main__':
|
|
import doctest
|
|
doctest.testmod()
|