def _test_orthogonal_unitary(predicate):
assert ask(predicate(X), predicate(X))
assert ask(predicate(X.T), predicate(X)) is True
assert ask(predicate(X.I), predicate(X)) is True
assert ask(predicate(Y)) is False
assert ask(predicate(X)) is None
assert ask(predicate(X * Z * X), predicate(X) & predicate(Z)) is True
assert ask(predicate(Identity(3))) is True
assert ask(predicate(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), predicate(X))
assert not ask(predicate(X + Z), predicate(X) & predicate(Z))
def test_issue_9817_simplify():
# simplify on trace of substituted explicit quadratic form of matrix
# expressions (a scalar) should return without errors (AttributeError)
# See issue #9817 and #9190 for the original bug more discussion on this
from sympy.matrices.expressions import Identity, trace
v = MatrixSymbol('v', 3, 1)
A = MatrixSymbol('A', 3, 3)
x = Matrix([i + 1 for i in range(3)])
X = Identity(3)
quadratic = v.T * A * v
assert simplify((trace(quadratic.as_explicit())).xreplace({v:x, A:X})) == 14
def test_invertible():
assert ask(Q.invertible(X), Q.invertible(X))
assert ask(Q.invertible(Y)) is False
assert ask(Q.invertible(X*Y), Q.invertible(X)) is False
assert ask(Q.invertible(X*Z), Q.invertible(X)) is None
assert ask(Q.invertible(X*Z), Q.invertible(X) & Q.invertible(Z)) is True
assert ask(Q.invertible(X.T)) is None
assert ask(Q.invertible(X.T), Q.invertible(X)) is True
assert ask(Q.invertible(X.I)) is True
assert ask(Q.invertible(Identity(3))) is True
assert ask(Q.invertible(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
def test_fullrank():
assert ask(Q.fullrank(X), Q.fullrank(X))
assert ask(Q.fullrank(X**2), Q.fullrank(X))
assert ask(Q.fullrank(X.T), Q.fullrank(X)) is True
assert ask(Q.fullrank(X)) is None
assert ask(Q.fullrank(Y)) is None
assert ask(Q.fullrank(X * Z), Q.fullrank(X) & Q.fullrank(Z)) is True
assert ask(Q.fullrank(Identity(3))) is True
assert ask(Q.fullrank(ZeroMatrix(3, 3))) is False
assert ask(Q.fullrank(OneMatrix(1, 1))) is True
assert ask(Q.fullrank(OneMatrix(3, 3))) is False
assert ask(Q.invertible(X), ~Q.fullrank(X)) == False
def test_diagonal():
assert ask(Q.diagonal(X + Z.T + Identity(2)),
Q.diagonal(X) & Q.diagonal(Z)) is True
assert ask(Q.diagonal(ZeroMatrix(3, 3)))
assert ask(Q.diagonal(OneMatrix(1, 1))) is True
assert ask(Q.diagonal(OneMatrix(3, 3))) is False
assert ask(Q.lower_triangular(X) & Q.upper_triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(X), Q.lower_triangular(X) & Q.upper_triangular(X))
assert ask(Q.symmetric(X), Q.diagonal(X))
assert ask(Q.triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(C0x0))
assert ask(Q.diagonal(A1x1))
assert ask(Q.diagonal(A1x1 + B1x1))
assert ask(Q.diagonal(A1x1 * B1x1))
assert ask(Q.diagonal(V1.T * V2))
assert ask(Q.diagonal(V1.T * (X + Z) * V1))
assert ask(Q.diagonal(MatrixSlice(Y, (0, 1), (1, 2)))) is True
assert ask(Q.diagonal(V1.T * (V1 + V2))) is True
assert ask(Q.diagonal(X**3), Q.diagonal(X))
assert ask(Q.diagonal(Identity(3)))
assert ask(Q.diagonal(DiagMatrix(V1)))
assert ask(Q.diagonal(DiagonalMatrix(X)))
def test_issue_21866():
n = 10
I = Identity(n)
O = ZeroMatrix(n, n)
A = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ I, O, O, I ]])
Ainv = block_collapse(A.inv())
AinvT = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ -I, O, O, I ]])
assert Ainv == AinvT
def test_positive_definite():
assert ask(Q.positive_definite(X), Q.positive_definite(X))
assert ask(Q.positive_definite(X.T), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(X.I), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(Y)) is False
assert ask(Q.positive_definite(X)) is None
assert ask(Q.positive_definite(X*Z*X),
Q.positive_definite(X) & Q.positive_definite(Z)) is True
assert ask(Q.positive_definite(X), Q.orthogonal(X))
assert ask(Q.positive_definite(Y.T*X*Y),
Q.positive_definite(X) & Q.fullrank(Y)) is True
assert not ask(Q.positive_definite(Y.T*X*Y), Q.positive_definite(X))
assert ask(Q.positive_definite(Identity(3))) is True
assert ask(Q.positive_definite(ZeroMatrix(3, 3))) is False
assert ask(Q.positive_definite(X + Z), Q.positive_definite(X) &
Q.positive_definite(Z)) is True
assert not ask(Q.positive_definite(-X), Q.positive_definite(X))
assert ask(Q.positive(X[1, 1]), Q.positive_definite(X))
def test_symmetric():
assert ask(Q.symmetric(X), Q.symmetric(X))
assert ask(Q.symmetric(X * Z), Q.symmetric(X)) is None
assert ask(Q.symmetric(X * Z), Q.symmetric(X) & Q.symmetric(Z)) is True
assert ask(Q.symmetric(X + Z), Q.symmetric(X) & Q.symmetric(Z)) is True
assert ask(Q.symmetric(Y)) is False
assert ask(Q.symmetric(Y * Y.T)) is True
assert ask(Q.symmetric(Y.T * X * Y)) is None
assert ask(Q.symmetric(Y.T * X * Y), Q.symmetric(X)) is True
assert ask(Q.symmetric(X**10), Q.symmetric(X)) is True
assert ask(Q.symmetric(A1x1)) is True
assert ask(Q.symmetric(A1x1 + B1x1)) is True
assert ask(Q.symmetric(A1x1 * B1x1)) is True
assert ask(Q.symmetric(V1.T * V1)) is True
assert ask(Q.symmetric(V1.T * (V1 + V2))) is True
assert ask(Q.symmetric(V1.T * (V1 + V2) + A1x1)) is True
assert ask(Q.symmetric(MatrixSlice(Y, (0, 1), (1, 2)))) is True
assert ask(Q.symmetric(Identity(3))) is True
assert ask(Q.symmetric(ZeroMatrix(3, 3))) is True
assert ask(Q.symmetric(OneMatrix(3, 3))) is True
def __init__(self, *args, **kwargs):
"""
Initialises a PointLight object.
To initialize a point light, we need to supply
a name(optional), a reference frame, and a point.
Parameters
==========
name : str, optional
Name assigned to VisualizationFrame, default is unnamed
reference_frame : ReferenceFrame
A reference_frame with respect to which all orientations of the
shape takes place, during visualizations/animations.
origin : Point
A point with respect to which all the translations of the shape
takes place, during visualizations/animations.
rigidbody : RigidBody
A rigidbody whose reference frame and mass center are to be
assigned as reference_frame and origin of the
VisualizationFrame.
particle : Particle
A particle whose point is assigned as origin of the
VisualizationFrame.
Examples
========
>>> from pydy.viz import PointLight
>>> from sympy.physics.mechanics import \
ReferenceFrame, Point, RigidBody, \
Particle, inertia
>>> from sympy import symbols
>>> I = ReferenceFrame('I')
>>> O = Point('O')
>>> #initializing with reference frame, point
>>> light = PointLight('light', I, O)
>>> Ixx, Iyy, Izz, mass = symbols('Ixx Iyy Izz mass')
>>> i = inertia(I, Ixx, Iyy, Izz)
>>> rbody = RigidBody('rbody', O, I, mass, (inertia, O))
>>> # Initializing with a rigidbody ..
>>> light = PointLight('frame2', rbody)
>>> Pa = Particle('Pa', O, mass)
>>> #initializing with Particle, reference_frame ...
>>> light = PointLight('frame3', I, Pa)
"""
try:
self._color = kwargs['color']
except KeyError:
self._color = 'white'
#Now we use same approach as in VisualizationFrame
#for setting reference_frame and origin
i = 0
#If first arg is not str, name the visualization frame 'unnamed'
if isinstance(args[i], str):
self._name = args[i]
i += 1
else:
self._name = 'unnamed'
try:
self._reference_frame = args[i].get_frame()
self._origin = args[i].get_masscenter()
except AttributeError:
#It is not a rigidbody, hence this arg should be a
#reference frame
try:
dcm = args[i]._dcm_dict
self._reference_frame = args[i]
i += 1
except AttributeError:
raise TypeError(''' A ReferenceFrame is to be supplied
before a Particle/Point. ''')
#Now next arg can either be a Particle or point
try:
self._origin = args[i].get_point()
except AttributeError:
self._origin = args[i]
#basic thing required, transform matrix
self._transform = Identity(4).as_mutable()
def test_eval_determinant():
assert det(Identity(n)) == 1
assert det(ZeroMatrix(n, n)) == 0
assert det(Transpose(A)) == det(A)
def test_invertible_BlockDiagMatrix():
assert ask(Q.invertible(BlockDiagMatrix(Identity(3), Identity(5)))) == True
assert ask(Q.invertible(BlockDiagMatrix(ZeroMatrix(3, 3),
Identity(5)))) == False
assert ask(Q.invertible(BlockDiagMatrix(Identity(3),
OneMatrix(5, 5)))) == False
def __init__(self, *args, **kwargs):
"""
Initialises a PerspectiveCamera object.
To initialize a visualization frame, we need to supply
a name(optional), a reference frame, a point,
field of view(fov) (optional), near plane distance(optional)
and far plane distance(optional).
Examples
========
>>> from pydy.viz import VisualizationFrame, Shape
>>> from sympy.physics.mechanics import \
ReferenceFrame, Point, RigidBody, \
Particle, inertia
>>> from sympy import symbols
>>> I = ReferenceFrame('I')
>>> O = Point('O')
>>> shape = Shape()
>>> #initializing with reference frame, point
>>> camera1 = PerspectiveCamera('frame1', I, O)
>>> Ixx, Iyy, Izz, mass = symbols('Ixx Iyy Izz mass')
>>> i = inertia(I, Ixx, Iyy, Izz)
>>> rbody = RigidBody('rbody', O, I, mass, (inertia, O))
>>> # Initializing with a rigidbody ..
>>> camera2 = PerspectiveCamera('frame2', rbody)
>>> Pa = Particle('Pa', O, mass)
>>> #initializing with Particle, reference_frame ...
>>> camera3 = PerspectiveCamera('frame3', I, Pa)
"""
try:
self._fov = kwargs['fov']
except KeyError:
self._fov = 45
try:
self._near = kwargs['near']
except KeyError:
self._near = 1
try:
self._far = kwargs['far']
except KeyError:
self._far = 1000
#Now we use same approach as in VisualizationFrame
#for setting reference_frame and origin
i = 0
#If first arg is not str, name the visualization frame 'unnamed'
if isinstance(args[i], str):
self._name = args[i]
i += 1
else:
self._name = 'unnamed'
try:
self._reference_frame = args[i].get_frame()
self._origin = args[i].get_masscenter()
except AttributeError:
#It is not a rigidbody, hence this arg should be a
#reference frame
try:
dcm = args[i]._dcm_dict
self._reference_frame = args[i]
i += 1
except AttributeError:
raise TypeError(''' A ReferenceFrame is to be supplied
before a Particle/Point. ''')
#Now next arg can either be a Particle or point
try:
self._origin = args[i].get_point()
except AttributeError:
self._origin = args[i]
#basic thing required, transform matrix
self._transform = Identity(4).as_mutable()
def __init__(self, *args, **kwargs):
"""
Initialises an OrthoGraphicCamera object. To initialize a
visualization frame, we need to supply a name (optional), a
reference frame, a point, near plane distance (optional) and far
plane distance (optional).
Examples
========
>>> from pydy.viz import OrthoGraphicCamera
>>> from sympy.physics.mechanics import (ReferenceFrame, Point,
... RigidBody, Particle,
... inertia)
>>> from sympy import symbols
>>> I = ReferenceFrame('I')
>>> O = Point('O')
>>> shape = Shape()
>>> # Initializing with ReferenceFrame, Point
>>> camera1 = OrthoGraphicCamera('frame1', I, O)
>>> Ixx, Iyy, Izz, mass = symbols('Ixx Iyy Izz mass')
>>> i = inertia(I, Ixx, Iyy, Izz)
>>> rbody = RigidBody('rbody', O, I, mass, (inertia, O))
>>> # Initializing with a Rigidbody
>>> camera2 = OrthoGraphicCamera('frame2', rbody)
>>> Pa = Particle('Pa', O, mass)
>>> # Initializing with Particle, ReferenceFrame
>>> camera3 = OrthoGraphicCamera('frame3', I, Pa)
"""
try:
self._near = kwargs['near']
except KeyError:
self._near = 1
try:
self._far = kwargs['far']
except KeyError:
self._far = 1000
# Now we use same approach as in VisualizationFrame for setting
# reference_frame and origin
i = 0
# If first arg is not str, name the visualization frame 'unnamed'
if isinstance(args[i], str):
self._name = args[i]
i += 1
else:
self._name = 'unnamed'
try:
self._reference_frame = args[i].get_frame()
self._origin = args[i].get_masscenter()
except AttributeError:
# It is not a rigidbody, hence this arg should be a reference
# frame.
self._reference_frame = args[i]
i += 1
# Now next arg can either be a Particle or point
try:
self._origin = args[i].get_point()
except AttributeError:
self._origin = args[i]
# basic thing required, transform matrix
self._transform = Identity(4).as_mutable()
