def rational_interpolate(data, degnum, X=symbols('x')):
"""
Returns a rational interpolation, where the data points are element of
any integral domain.
The first argument contains the data (as a list of coordinates). The
``degnum`` argument is the degree in the numerator of the rational
function. Setting it too high will decrease the maximal degree in the
denominator for the same amount of data.
Examples
========
>>> from sympy.polys.polyfuncs import rational_interpolate
>>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)]
>>> rational_interpolate(data, 2)
(105*x**2 - 525)/(x + 1)
Values do not need to be integers:
>>> from sympy import sympify
>>> x = [1, 2, 3, 4, 5, 6]
>>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]")
>>> rational_interpolate(zip(x, y), 2)
(3*x**2 - 7*x + 2)/(x + 1)
The symbol for the variable can be changed if needed:
>>> from sympy import symbols
>>> z = symbols('z')
>>> rational_interpolate(data, 2, X=z)
(105*z**2 - 525)/(z + 1)
References
==========
.. [1] Algorithm is adapted from:
http://axiom-wiki.newsynthesis.org/RationalInterpolation
"""
from sympy.matrices.dense import ones
xdata, ydata = list(zip(*data))
k = len(xdata) - degnum - 1
if k < 0:
raise OptionError("Too few values for the required degree.")
c = ones(degnum + k + 1, degnum + k + 2)
for j in range(max(degnum, k)):
for i in range(degnum + k + 1):
c[i, j + 1] = c[i, j] * xdata[i]
for j in range(k + 1):
for i in range(degnum + k + 1):
c[i, degnum + k + 1 - j] = -c[i, k - j] * ydata[i]
r = c.nullspace()[0]
return (sum(r[i] * X**i
for i in range(degnum + 1)) / sum(r[i + degnum + 1] * X**i
for i in range(k + 1)))
def rational_interpolate(data, degnum, X=symbols('x')):
"""
Returns a rational interpolation, where the data points are element of
any integral domain.
The first argument contains the data (as a list of coordinates). The
``degnum`` argument is the degree in the numerator of the rational
function. Setting it too high will decrease the maximal degree in the
denominator for the same amount of data.
Examples
========
>>> from sympy.polys.polyfuncs import rational_interpolate
>>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)]
>>> rational_interpolate(data, 2)
(105*x**2 - 525)/(x + 1)
Values do not need to be integers:
>>> from sympy import sympify
>>> x = [1, 2, 3, 4, 5, 6]
>>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]")
>>> rational_interpolate(zip(x, y), 2)
(3*x**2 - 7*x + 2)/(x + 1)
The symbol for the variable can be changed if needed:
>>> from sympy import symbols
>>> z = symbols('z')
>>> rational_interpolate(data, 2, X=z)
(105*z**2 - 525)/(z + 1)
References
==========
.. [1] Algorithm is adapted from:
http://axiom-wiki.newsynthesis.org/RationalInterpolation
"""
from sympy.matrices.dense import ones
xdata, ydata = list(zip(*data))
k = len(xdata) - degnum - 1
if k < 0:
raise OptionError("Too few values for the required degree.")
c = ones(degnum + k + 1, degnum + k + 2)
for j in range(max(degnum, k)):
for i in range(degnum + k + 1):
c[i, j + 1] = c[i, j]*xdata[i]
for j in range(k + 1):
for i in range(degnum + k + 1):
c[i, degnum + k + 1 - j] = -c[i, k - j]*ydata[i]
r = c.nullspace()[0]
return (sum(r[i] * X**i for i in range(degnum + 1))
/ sum(r[i + degnum + 1] * X**i for i in range(k + 1)))
def test_empty_Matrix():
sT(ones(0, 3), "MutableDenseMatrix(0, 3, [])")
sT(ones(4, 0), "MutableDenseMatrix(4, 0, [])")
sT(ones(0, 0), "MutableDenseMatrix([])")
def test_matrix_tensor_product():
if not np:
skip("numpy not installed.")
l1 = zeros(4)
for i in range(16):
l1[i] = 2**i
l2 = zeros(4)
for i in range(16):
l2[i] = i
l3 = zeros(2)
for i in range(4):
l3[i] = i
vec = Matrix([1, 2, 3])
#test for Matrix known 4x4 matricies
numpyl1 = np.array(l1.tolist())
numpyl2 = np.array(l2.tolist())
numpy_product = np.kron(numpyl1, numpyl2)
args = [l1, l2]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
numpy_product = np.kron(numpyl2, numpyl1)
args = [l2, l1]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
#test for other known matrix of different dimensions
numpyl2 = np.array(l3.tolist())
numpy_product = np.kron(numpyl1, numpyl2)
args = [l1, l3]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
numpy_product = np.kron(numpyl2, numpyl1)
args = [l3, l1]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
#test for non square matrix
numpyl2 = np.array(vec.tolist())
numpy_product = np.kron(numpyl1, numpyl2)
args = [l1, vec]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
numpy_product = np.kron(numpyl2, numpyl1)
args = [vec, l1]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
#test for random matrix with random values that are floats
random_matrix1 = np.random.rand(randint(1, 5), randint(1, 5))
random_matrix2 = np.random.rand(randint(1, 5), randint(1, 5))
numpy_product = np.kron(random_matrix1, random_matrix2)
args = [Matrix(random_matrix1.tolist()), Matrix(random_matrix2.tolist())]
sympy_product = matrix_tensor_product(*args)
assert not (sympy_product - Matrix(numpy_product.tolist())).tolist() > \
(ones(sympy_product.rows, sympy_product.cols)*epsilon).tolist()
#test for three matrix kronecker
sympy_product = matrix_tensor_product(l1, vec, l2)
numpy_product = np.kron(l1, np.kron(vec, l2))
assert numpy_product.tolist() == sympy_product.tolist()
def GeneralizedMultivariateLogGammaOmega(syms, omega, v, lamda, mu):
"""
Extends GeneralizedMultivariateLogGamma.
Parameters
==========
syms : list/tuple/set of symbols
For identifying each component
omega : A square matrix
Every element of square matrix must be absolute value of
square root of correlation coefficient
v : Positive real number
lamda : List of positive real numbers
mu : List of positive real numbers
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import density
>>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega
>>> from sympy import Matrix, symbols, S
>>> omega = Matrix([[1, S.Half, S.Half], [S.Half, 1, S.Half], [S.Half, S.Half, 1]])
>>> v = 1
>>> l, mu = [1, 1, 1], [1, 1, 1]
>>> G = GeneralizedMultivariateLogGammaOmega('G', omega, v, l, mu)
>>> y = symbols('y_1:4', positive=True)
>>> density(G)(y[0], y[1], y[2])
sqrt(2)*Sum((1 - sqrt(2)/2)**n*exp((n + 1)*(y_1 + y_2 + y_3) - exp(y_1) -
exp(y_2) - exp(y_3))/gamma(n + 1)**3, (n, 0, oo))/2
References
==========
.. [1] https://en.wikipedia.org/wiki/Generalized_multivariate_log-gamma_distribution
.. [2] https://www.researchgate.net/publication/234137346_On_a_multivariate_log-gamma_distribution_and_the_use_of_the_distribution_in_the_Bayesian_analysis
Notes
=====
If the GeneralizedMultivariateLogGammaOmega is too long to type use,
>>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega as GMVLGO
>>> G = GMVLGO('G', omega, v, l, mu)
"""
_value_check((omega.is_square, isinstance(omega, Matrix)), "omega must be a"
" square matrix")
for val in omega.values():
_value_check((val >= 0, val <= 1),
"all values in matrix must be between 0 and 1(both inclusive).")
_value_check(omega.diagonal().equals(ones(1, omega.shape[0])),
"all the elements of diagonal should be 1.")
_value_check((omega.shape[0] == len(lamda), len(lamda) == len(mu)),
"lamda, mu should be of same length and omega should "
" be of shape (length of lamda, length of mu)")
_value_check(len(lamda) > 1,"the distribution should have at least"
" two random variables.")
delta = Pow(Rational(omega.det()), Rational(1, len(lamda) - 1))
return GeneralizedMultivariateLogGamma(syms, delta, v, lamda, mu)
def test_banded():
raises(TypeError, lambda: banded())
raises(TypeError, lambda: banded(1))
raises(TypeError, lambda: banded(1, 2))
raises(TypeError, lambda: banded(1, 2, 3))
raises(TypeError, lambda: banded(1, 2, 3, 4))
raises(ValueError, lambda: banded({0: (1, 2)}, rows=1))
raises(ValueError, lambda: banded({0: (1, 2)}, cols=1))
raises(ValueError, lambda: banded(1, {0: (1, 2)}))
raises(ValueError, lambda: banded(2, 1, {0: (1, 2)}))
raises(ValueError, lambda: banded(1, 2, {0: (1, 2)}))
assert isinstance(banded(2, 4, {}), SparseMatrix)
assert banded(2, 4, {}) == zeros(2, 4)
assert banded({0: 0, 1: 0}) == zeros(0)
assert banded({0: Matrix([1, 2])}) == Matrix([1, 2])
assert banded({1: [1, 2, 3, 0], -1: [4, 5, 6]}) == \
banded({1: (1, 2, 3), -1: (4, 5, 6)}) == \
Matrix([
[0, 1, 0, 0],
[4, 0, 2, 0],
[0, 5, 0, 3],
[0, 0, 6, 0]])
assert banded(3, 4, {-1: 1, 0: 2, 1: 3}) == \
Matrix([
[2, 3, 0, 0],
[1, 2, 3, 0],
[0, 1, 2, 3]])
s = lambda d: (1 + d)**2
assert banded(5, {0: s, 2: s}) == \
Matrix([
[1, 0, 1, 0, 0],
[0, 4, 0, 4, 0],
[0, 0, 9, 0, 9],
[0, 0, 0, 16, 0],
[0, 0, 0, 0, 25]])
assert banded(2, {0: 1}) == \
Matrix([
[1, 0],
[0, 1]])
assert banded(2, 3, {0: 1}) == \
Matrix([
[1, 0, 0],
[0, 1, 0]])
vert = Matrix([1, 2, 3])
assert banded({0: vert}, cols=3) == \
Matrix([
[1, 0, 0],
[2, 1, 0],
[3, 2, 1],
[0, 3, 2],
[0, 0, 3]])
assert banded(4, {0: ones(2)}) == \
Matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 1],
[0, 0, 1, 1]])
raises(ValueError, lambda: banded({0: 2, 1: ones(2)}, rows=5))
assert banded({0: 2, 2: (ones(2),)*3}) == \
Matrix([
[2, 0, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 1, 0, 0, 0, 0],
[0, 0, 2, 0, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 0, 2, 0, 1, 1],
[0, 0, 0, 0, 0, 2, 1, 1]])
raises(ValueError, lambda: banded({0: (2, ) * 5, 1: (ones(2), ) * 3}))
u2 = Matrix([[1, 1], [0, 1]])
assert banded({0: (2,)*5, 1: (u2,)*3}) == \
Matrix([
[2, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 0, 0, 0, 0],
[0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 0, 0],
[0, 0, 0, 0, 2, 1, 1],
[0, 0, 0, 0, 0, 0, 1]])
assert banded({0:(0, ones(2)), 2: 2}) == \
Matrix([
[0, 0, 2],
[0, 1, 1],
[0, 1, 1]])
raises(ValueError, lambda: banded({0: (0, ones(2)), 1: 2}))
assert banded({0: 1}, cols=3) == banded({0: 1}, rows=3) == eye(3)
assert banded({1: 1}, rows=3) == Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 0]])
