764 lines
20 KiB
Python
764 lines
20 KiB
Python
import random
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from sympy.core.basic import Basic
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from sympy.core.compatibility import is_sequence
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from sympy.core.symbol import Symbol
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from sympy.core.sympify import sympify
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from sympy.functions.elementary.trigonometric import cos, sin
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from sympy.simplify.simplify import simplify as _simplify
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from sympy.utilities.decorator import doctest_depends_on
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from sympy.utilities.exceptions import SymPyDeprecationWarning
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from .common import ShapeError
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from .decompositions import _cholesky, _LDLdecomposition
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from .matrices import MatrixBase
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from .repmatrix import MutableRepMatrix, RepMatrix
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from .solvers import _lower_triangular_solve, _upper_triangular_solve
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def _iszero(x):
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"""Returns True if x is zero."""
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return x.is_zero
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class DenseMatrix(RepMatrix):
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"""Matrix implementation based on DomainMatrix as the internal representation"""
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#
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# DenseMatrix is a superclass for both MutableDenseMatrix and
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# ImmutableDenseMatrix. Methods shared by both classes but not for the
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# Sparse classes should be implemented here.
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#
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is_MatrixExpr = False # type: bool
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_op_priority = 10.01
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_class_priority = 4
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@property
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def _mat(self):
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SymPyDeprecationWarning(
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feature="The private _mat attribute of Matrix",
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useinstead="the .flat() method",
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issue=21715,
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deprecated_since_version="1.9").warn()
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return self.flat()
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def _eval_inverse(self, **kwargs):
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return self.inv(method=kwargs.get('method', 'GE'),
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iszerofunc=kwargs.get('iszerofunc', _iszero),
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try_block_diag=kwargs.get('try_block_diag', False))
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def as_immutable(self):
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"""Returns an Immutable version of this Matrix
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"""
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from .immutable import ImmutableDenseMatrix as cls
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return cls._fromrep(self._rep.copy())
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def as_mutable(self):
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"""Returns a mutable version of this matrix
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Examples
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========
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>>> from sympy import ImmutableMatrix
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>>> X = ImmutableMatrix([[1, 2], [3, 4]])
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>>> Y = X.as_mutable()
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>>> Y[1, 1] = 5 # Can set values in Y
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>>> Y
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Matrix([
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[1, 2],
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[3, 5]])
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"""
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return Matrix(self)
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def cholesky(self, hermitian=True):
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return _cholesky(self, hermitian=hermitian)
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def LDLdecomposition(self, hermitian=True):
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return _LDLdecomposition(self, hermitian=hermitian)
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def lower_triangular_solve(self, rhs):
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return _lower_triangular_solve(self, rhs)
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def upper_triangular_solve(self, rhs):
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return _upper_triangular_solve(self, rhs)
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cholesky.__doc__ = _cholesky.__doc__
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LDLdecomposition.__doc__ = _LDLdecomposition.__doc__
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lower_triangular_solve.__doc__ = _lower_triangular_solve.__doc__
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upper_triangular_solve.__doc__ = _upper_triangular_solve.__doc__
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def _force_mutable(x):
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"""Return a matrix as a Matrix, otherwise return x."""
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if getattr(x, 'is_Matrix', False):
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return x.as_mutable()
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elif isinstance(x, Basic):
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return x
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elif hasattr(x, '__array__'):
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a = x.__array__()
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if len(a.shape) == 0:
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return sympify(a)
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return Matrix(x)
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return x
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class MutableDenseMatrix(DenseMatrix, MutableRepMatrix):
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def simplify(self, **kwargs):
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"""Applies simplify to the elements of a matrix in place.
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This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))
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See Also
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========
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sympy.simplify.simplify.simplify
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"""
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for (i, j), element in self.todok().items():
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self[i, j] = _simplify(element, **kwargs)
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MutableMatrix = Matrix = MutableDenseMatrix
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###########
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# Numpy Utility Functions:
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# list2numpy, matrix2numpy, symmarray, rot_axis[123]
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###########
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def list2numpy(l, dtype=object): # pragma: no cover
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"""Converts python list of SymPy expressions to a NumPy array.
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See Also
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========
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matrix2numpy
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"""
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from numpy import empty
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a = empty(len(l), dtype)
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for i, s in enumerate(l):
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a[i] = s
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return a
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def matrix2numpy(m, dtype=object): # pragma: no cover
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"""Converts SymPy's matrix to a NumPy array.
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See Also
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========
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list2numpy
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"""
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from numpy import empty
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a = empty(m.shape, dtype)
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for i in range(m.rows):
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for j in range(m.cols):
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a[i, j] = m[i, j]
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return a
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def rot_axis3(theta):
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"""Returns a rotation matrix for a rotation of theta (in radians) about
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the 3-axis.
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Examples
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========
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>>> from sympy import pi
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>>> from sympy.matrices import rot_axis3
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A rotation of pi/3 (60 degrees):
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>>> theta = pi/3
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>>> rot_axis3(theta)
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Matrix([
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[ 1/2, sqrt(3)/2, 0],
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[-sqrt(3)/2, 1/2, 0],
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[ 0, 0, 1]])
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If we rotate by pi/2 (90 degrees):
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>>> rot_axis3(pi/2)
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Matrix([
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[ 0, 1, 0],
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[-1, 0, 0],
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[ 0, 0, 1]])
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See Also
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========
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rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
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about the 1-axis
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rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
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about the 2-axis
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"""
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ct = cos(theta)
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st = sin(theta)
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lil = ((ct, st, 0),
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(-st, ct, 0),
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(0, 0, 1))
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return Matrix(lil)
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def rot_axis2(theta):
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"""Returns a rotation matrix for a rotation of theta (in radians) about
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the 2-axis.
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Examples
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========
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>>> from sympy import pi
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>>> from sympy.matrices import rot_axis2
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A rotation of pi/3 (60 degrees):
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>>> theta = pi/3
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>>> rot_axis2(theta)
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Matrix([
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[ 1/2, 0, -sqrt(3)/2],
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[ 0, 1, 0],
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[sqrt(3)/2, 0, 1/2]])
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If we rotate by pi/2 (90 degrees):
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>>> rot_axis2(pi/2)
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Matrix([
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[0, 0, -1],
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[0, 1, 0],
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[1, 0, 0]])
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See Also
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========
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rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
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about the 1-axis
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rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
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about the 3-axis
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"""
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ct = cos(theta)
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st = sin(theta)
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lil = ((ct, 0, -st),
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(0, 1, 0),
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(st, 0, ct))
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return Matrix(lil)
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def rot_axis1(theta):
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"""Returns a rotation matrix for a rotation of theta (in radians) about
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the 1-axis.
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Examples
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========
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>>> from sympy import pi
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>>> from sympy.matrices import rot_axis1
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A rotation of pi/3 (60 degrees):
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>>> theta = pi/3
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>>> rot_axis1(theta)
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Matrix([
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[1, 0, 0],
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[0, 1/2, sqrt(3)/2],
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[0, -sqrt(3)/2, 1/2]])
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If we rotate by pi/2 (90 degrees):
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>>> rot_axis1(pi/2)
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Matrix([
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[1, 0, 0],
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[0, 0, 1],
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[0, -1, 0]])
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See Also
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========
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rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
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about the 2-axis
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rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
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about the 3-axis
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"""
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ct = cos(theta)
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st = sin(theta)
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lil = ((1, 0, 0),
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(0, ct, st),
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(0, -st, ct))
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return Matrix(lil)
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@doctest_depends_on(modules=('numpy',))
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def symarray(prefix, shape, **kwargs): # pragma: no cover
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r"""Create a numpy ndarray of symbols (as an object array).
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The created symbols are named ``prefix_i1_i2_``... You should thus provide a
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non-empty prefix if you want your symbols to be unique for different output
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arrays, as SymPy symbols with identical names are the same object.
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Parameters
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----------
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prefix : string
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A prefix prepended to the name of every symbol.
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shape : int or tuple
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Shape of the created array. If an int, the array is one-dimensional; for
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more than one dimension the shape must be a tuple.
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\*\*kwargs : dict
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keyword arguments passed on to Symbol
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Examples
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========
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These doctests require numpy.
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>>> from sympy import symarray
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>>> symarray('', 3)
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[_0 _1 _2]
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If you want multiple symarrays to contain distinct symbols, you *must*
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provide unique prefixes:
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>>> a = symarray('', 3)
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>>> b = symarray('', 3)
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>>> a[0] == b[0]
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True
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>>> a = symarray('a', 3)
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>>> b = symarray('b', 3)
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>>> a[0] == b[0]
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False
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Creating symarrays with a prefix:
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>>> symarray('a', 3)
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[a_0 a_1 a_2]
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For more than one dimension, the shape must be given as a tuple:
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>>> symarray('a', (2, 3))
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[[a_0_0 a_0_1 a_0_2]
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[a_1_0 a_1_1 a_1_2]]
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>>> symarray('a', (2, 3, 2))
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[[[a_0_0_0 a_0_0_1]
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[a_0_1_0 a_0_1_1]
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[a_0_2_0 a_0_2_1]]
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<BLANKLINE>
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[[a_1_0_0 a_1_0_1]
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[a_1_1_0 a_1_1_1]
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[a_1_2_0 a_1_2_1]]]
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For setting assumptions of the underlying Symbols:
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>>> [s.is_real for s in symarray('a', 2, real=True)]
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[True, True]
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"""
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from numpy import empty, ndindex
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arr = empty(shape, dtype=object)
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for index in ndindex(shape):
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arr[index] = Symbol('%s_%s' % (prefix, '_'.join(map(str, index))),
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**kwargs)
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return arr
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###############
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# Functions
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###############
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def casoratian(seqs, n, zero=True):
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"""Given linear difference operator L of order 'k' and homogeneous
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equation Ly = 0 we want to compute kernel of L, which is a set
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of 'k' sequences: a(n), b(n), ... z(n).
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Solutions of L are linearly independent iff their Casoratian,
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denoted as C(a, b, ..., z), do not vanish for n = 0.
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Casoratian is defined by k x k determinant::
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+ a(n) b(n) . . . z(n) +
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| a(n+1) b(n+1) . . . z(n+1) |
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| . . . . |
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| . . . . |
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| . . . . |
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+ a(n+k-1) b(n+k-1) . . . z(n+k-1) +
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It proves very useful in rsolve_hyper() where it is applied
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to a generating set of a recurrence to factor out linearly
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dependent solutions and return a basis:
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>>> from sympy import Symbol, casoratian, factorial
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>>> n = Symbol('n', integer=True)
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Exponential and factorial are linearly independent:
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>>> casoratian([2**n, factorial(n)], n) != 0
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True
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"""
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seqs = list(map(sympify, seqs))
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if not zero:
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f = lambda i, j: seqs[j].subs(n, n + i)
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else:
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f = lambda i, j: seqs[j].subs(n, i)
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k = len(seqs)
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return Matrix(k, k, f).det()
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def eye(*args, **kwargs):
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"""Create square identity matrix n x n
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See Also
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========
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diag
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zeros
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ones
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"""
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return Matrix.eye(*args, **kwargs)
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def diag(*values, strict=True, unpack=False, **kwargs):
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"""Returns a matrix with the provided values placed on the
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diagonal. If non-square matrices are included, they will
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produce a block-diagonal matrix.
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Examples
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========
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This version of diag is a thin wrapper to Matrix.diag that differs
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in that it treats all lists like matrices -- even when a single list
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is given. If this is not desired, either put a `*` before the list or
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set `unpack=True`.
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>>> from sympy import diag
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>>> diag([1, 2, 3], unpack=True) # = diag(1,2,3) or diag(*[1,2,3])
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Matrix([
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[1, 0, 0],
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[0, 2, 0],
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[0, 0, 3]])
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>>> diag([1, 2, 3]) # a column vector
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Matrix([
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[1],
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[2],
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[3]])
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See Also
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========
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.common.MatrixCommon.eye
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.common.MatrixCommon.diagonal - to extract a diagonal
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.common.MatrixCommon.diag
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.expressions.blockmatrix.BlockMatrix
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"""
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return Matrix.diag(*values, strict=strict, unpack=unpack, **kwargs)
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def GramSchmidt(vlist, orthonormal=False):
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"""Apply the Gram-Schmidt process to a set of vectors.
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Parameters
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==========
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vlist : List of Matrix
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Vectors to be orthogonalized for.
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orthonormal : Bool, optional
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If true, return an orthonormal basis.
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Returns
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=======
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vlist : List of Matrix
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Orthogonalized vectors
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Notes
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=====
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This routine is mostly duplicate from ``Matrix.orthogonalize``,
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except for some difference that this always raises error when
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linearly dependent vectors are found, and the keyword ``normalize``
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has been named as ``orthonormal`` in this function.
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See Also
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========
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.matrices.MatrixSubspaces.orthogonalize
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
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"""
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return MutableDenseMatrix.orthogonalize(
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*vlist, normalize=orthonormal, rankcheck=True
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)
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def hessian(f, varlist, constraints=[]):
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"""Compute Hessian matrix for a function f wrt parameters in varlist
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which may be given as a sequence or a row/column vector. A list of
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constraints may optionally be given.
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Examples
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========
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>>> from sympy import Function, hessian, pprint
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>>> from sympy.abc import x, y
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>>> f = Function('f')(x, y)
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>>> g1 = Function('g')(x, y)
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>>> g2 = x**2 + 3*y
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>>> pprint(hessian(f, (x, y), [g1, g2]))
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[ d d ]
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[ 0 0 --(g(x, y)) --(g(x, y)) ]
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[ dx dy ]
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[ ]
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[ 0 0 2*x 3 ]
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[ ]
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[ 2 2 ]
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[d d d ]
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[--(g(x, y)) 2*x ---(f(x, y)) -----(f(x, y))]
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[dx 2 dy dx ]
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[ dx ]
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[ ]
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[ 2 2 ]
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[d d d ]
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[--(g(x, y)) 3 -----(f(x, y)) ---(f(x, y)) ]
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[dy dy dx 2 ]
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[ dy ]
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References
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==========
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https://en.wikipedia.org/wiki/Hessian_matrix
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See Also
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========
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sympy.matrices.matrices.MatrixCalculus.jacobian
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wronskian
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"""
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# f is the expression representing a function f, return regular matrix
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if isinstance(varlist, MatrixBase):
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if 1 not in varlist.shape:
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raise ShapeError("`varlist` must be a column or row vector.")
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if varlist.cols == 1:
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varlist = varlist.T
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varlist = varlist.tolist()[0]
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if is_sequence(varlist):
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n = len(varlist)
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if not n:
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raise ShapeError("`len(varlist)` must not be zero.")
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else:
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raise ValueError("Improper variable list in hessian function")
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if not getattr(f, 'diff'):
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# check differentiability
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raise ValueError("Function `f` (%s) is not differentiable" % f)
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m = len(constraints)
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N = m + n
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out = zeros(N)
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for k, g in enumerate(constraints):
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if not getattr(g, 'diff'):
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# check differentiability
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raise ValueError("Function `f` (%s) is not differentiable" % f)
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for i in range(n):
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out[k, i + m] = g.diff(varlist[i])
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for i in range(n):
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for j in range(i, n):
|
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out[i + m, j + m] = f.diff(varlist[i]).diff(varlist[j])
|
|
for i in range(N):
|
|
for j in range(i + 1, N):
|
|
out[j, i] = out[i, j]
|
|
return out
|
|
|
|
|
|
def jordan_cell(eigenval, n):
|
|
"""
|
|
Create a Jordan block:
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.matrices import jordan_cell
|
|
>>> from sympy.abc import x
|
|
>>> jordan_cell(x, 4)
|
|
Matrix([
|
|
[x, 1, 0, 0],
|
|
[0, x, 1, 0],
|
|
[0, 0, x, 1],
|
|
[0, 0, 0, x]])
|
|
"""
|
|
|
|
return Matrix.jordan_block(size=n, eigenvalue=eigenval)
|
|
|
|
|
|
def matrix_multiply_elementwise(A, B):
|
|
"""Return the Hadamard product (elementwise product) of A and B
|
|
|
|
>>> from sympy.matrices import matrix_multiply_elementwise
|
|
>>> from sympy.matrices import Matrix
|
|
>>> A = Matrix([[0, 1, 2], [3, 4, 5]])
|
|
>>> B = Matrix([[1, 10, 100], [100, 10, 1]])
|
|
>>> matrix_multiply_elementwise(A, B)
|
|
Matrix([
|
|
[ 0, 10, 200],
|
|
[300, 40, 5]])
|
|
|
|
See Also
|
|
========
|
|
|
|
sympy.matrices.common.MatrixCommon.__mul__
|
|
"""
|
|
return A.multiply_elementwise(B)
|
|
|
|
|
|
def ones(*args, **kwargs):
|
|
"""Returns a matrix of ones with ``rows`` rows and ``cols`` columns;
|
|
if ``cols`` is omitted a square matrix will be returned.
|
|
|
|
See Also
|
|
========
|
|
|
|
zeros
|
|
eye
|
|
diag
|
|
"""
|
|
|
|
if 'c' in kwargs:
|
|
kwargs['cols'] = kwargs.pop('c')
|
|
|
|
return Matrix.ones(*args, **kwargs)
|
|
|
|
|
|
def randMatrix(r, c=None, min=0, max=99, seed=None, symmetric=False,
|
|
percent=100, prng=None):
|
|
"""Create random matrix with dimensions ``r`` x ``c``. If ``c`` is omitted
|
|
the matrix will be square. If ``symmetric`` is True the matrix must be
|
|
square. If ``percent`` is less than 100 then only approximately the given
|
|
percentage of elements will be non-zero.
|
|
|
|
The pseudo-random number generator used to generate matrix is chosen in the
|
|
following way.
|
|
|
|
* If ``prng`` is supplied, it will be used as random number generator.
|
|
It should be an instance of ``random.Random``, or at least have
|
|
``randint`` and ``shuffle`` methods with same signatures.
|
|
* if ``prng`` is not supplied but ``seed`` is supplied, then new
|
|
``random.Random`` with given ``seed`` will be created;
|
|
* otherwise, a new ``random.Random`` with default seed will be used.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.matrices import randMatrix
|
|
>>> randMatrix(3) # doctest:+SKIP
|
|
[25, 45, 27]
|
|
[44, 54, 9]
|
|
[23, 96, 46]
|
|
>>> randMatrix(3, 2) # doctest:+SKIP
|
|
[87, 29]
|
|
[23, 37]
|
|
[90, 26]
|
|
>>> randMatrix(3, 3, 0, 2) # doctest:+SKIP
|
|
[0, 2, 0]
|
|
[2, 0, 1]
|
|
[0, 0, 1]
|
|
>>> randMatrix(3, symmetric=True) # doctest:+SKIP
|
|
[85, 26, 29]
|
|
[26, 71, 43]
|
|
[29, 43, 57]
|
|
>>> A = randMatrix(3, seed=1)
|
|
>>> B = randMatrix(3, seed=2)
|
|
>>> A == B
|
|
False
|
|
>>> A == randMatrix(3, seed=1)
|
|
True
|
|
>>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP
|
|
[77, 70, 0],
|
|
[70, 0, 0],
|
|
[ 0, 0, 88]
|
|
"""
|
|
# Note that ``Random()`` is equivalent to ``Random(None)``
|
|
prng = prng or random.Random(seed)
|
|
|
|
if c is None:
|
|
c = r
|
|
|
|
if symmetric and r != c:
|
|
raise ValueError('For symmetric matrices, r must equal c, but %i != %i' % (r, c))
|
|
|
|
ij = range(r * c)
|
|
if percent != 100:
|
|
ij = prng.sample(ij, int(len(ij)*percent // 100))
|
|
|
|
m = zeros(r, c)
|
|
|
|
if not symmetric:
|
|
for ijk in ij:
|
|
i, j = divmod(ijk, c)
|
|
m[i, j] = prng.randint(min, max)
|
|
else:
|
|
for ijk in ij:
|
|
i, j = divmod(ijk, c)
|
|
if i <= j:
|
|
m[i, j] = m[j, i] = prng.randint(min, max)
|
|
|
|
return m
|
|
|
|
|
|
def wronskian(functions, var, method='bareiss'):
|
|
"""
|
|
Compute Wronskian for [] of functions
|
|
|
|
::
|
|
|
|
| f1 f2 ... fn |
|
|
| f1' f2' ... fn' |
|
|
| . . . . |
|
|
W(f1, ..., fn) = | . . . . |
|
|
| . . . . |
|
|
| (n) (n) (n) |
|
|
| D (f1) D (f2) ... D (fn) |
|
|
|
|
see: https://en.wikipedia.org/wiki/Wronskian
|
|
|
|
See Also
|
|
========
|
|
|
|
sympy.matrices.matrices.MatrixCalculus.jacobian
|
|
hessian
|
|
"""
|
|
|
|
for index in range(0, len(functions)):
|
|
functions[index] = sympify(functions[index])
|
|
n = len(functions)
|
|
if n == 0:
|
|
return 1
|
|
W = Matrix(n, n, lambda i, j: functions[i].diff(var, j))
|
|
return W.det(method)
|
|
|
|
|
|
def zeros(*args, **kwargs):
|
|
"""Returns a matrix of zeros with ``rows`` rows and ``cols`` columns;
|
|
if ``cols`` is omitted a square matrix will be returned.
|
|
|
|
See Also
|
|
========
|
|
|
|
ones
|
|
eye
|
|
diag
|
|
"""
|
|
|
|
if 'c' in kwargs:
|
|
kwargs['cols'] = kwargs.pop('c')
|
|
|
|
return Matrix.zeros(*args, **kwargs)
|