def test_derivative_by_array():
from sympy.abc import i, j, t, x, y, z
bexpr = x*y**2*exp(z)*log(t)
sexpr = sin(bexpr)
cexpr = cos(bexpr)
a = Array([sexpr])
assert derive_by_array(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
assert derive_by_array(sexpr, [x, y, z]) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
assert derive_by_array(a, [x, y, z]) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])
assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
assert derive_by_array(a, [[x, y], [z, t]]) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
assert derive_by_array([[x, y], [z, t]], [[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
[[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
assert diff(sexpr, Array([x, y, z])) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
assert diff(a, Array([x, y, z])) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])
assert diff(sexpr, Array([[x, y], [z, t]])) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
assert diff(a, Array([[x, y], [z, t]])) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
assert diff(Array([[x, y], [z, t]]), Array([x, y])) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
assert diff(Array([[x, y], [z, t]]), Array([[x, y], [z, t]])) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
[[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
# test for large scale sparse array
for SparseArrayType in [ImmutableSparseNDimArray, MutableSparseNDimArray]:
b = MutableSparseNDimArray({0:i, 1:j}, (10000, 20000))
assert derive_by_array(b, i) == ImmutableSparseNDimArray({0: 1}, (10000, 20000))
assert derive_by_array(b, (i, j)) == ImmutableSparseNDimArray({0: 1, 200000001: 1}, (2, 10000, 20000))
def test_tensorproduct():
x, y, z, t = symbols("x y z t")
from sympy.abc import a, b, c, d
assert tensorproduct() == 1
assert tensorproduct([x]) == Array([x])
assert tensorproduct([x], [y]) == Array([[x * y]])
assert tensorproduct([x], [y], [z]) == Array([[[x * y * z]]])
assert tensorproduct([x], [y], [z], [t]) == Array([[[[x * y * z * t]]]])
assert tensorproduct(x) == x
assert tensorproduct(x, y) == x * y
assert tensorproduct(x, y, z) == x * y * z
assert tensorproduct(x, y, z, t) == x * y * z * t
for ArrayType in [ImmutableDenseNDimArray, ImmutableSparseNDimArray]:
A = ArrayType([x, y])
B = ArrayType([1, 2, 3])
C = ArrayType([a, b, c, d])
assert tensorproduct(A, B, C) == ArrayType([
[
[a * x, b * x, c * x, d * x],
[2 * a * x, 2 * b * x, 2 * c * x, 2 * d * x],
[3 * a * x, 3 * b * x, 3 * c * x, 3 * d * x],
],
[
[a * y, b * y, c * y, d * y],
[2 * a * y, 2 * b * y, 2 * c * y, 2 * d * y],
[3 * a * y, 3 * b * y, 3 * c * y, 3 * d * y],
],
])
assert tensorproduct([x, y], [1, 2, 3]) == tensorproduct(A, B)
assert tensorproduct(A, 2) == ArrayType([2 * x, 2 * y])
assert tensorproduct(A, [2]) == ArrayType([[2 * x], [2 * y]])
assert tensorproduct([2], A) == ArrayType([[2 * x, 2 * y]])
assert tensorproduct(a, A) == ArrayType([a * x, a * y])
assert tensorproduct(a, A,
B) == ArrayType([[a * x, 2 * a * x, 3 * a * x],
[a * y, 2 * a * y, 3 * a * y]])
assert tensorproduct(A, B,
a) == ArrayType([[a * x, 2 * a * x, 3 * a * x],
[a * y, 2 * a * y, 3 * a * y]])
assert tensorproduct(B, a, A) == ArrayType([[a * x, a * y],
[2 * a * x, 2 * a * y],
[3 * a * x, 3 * a * y]])
# tests for large scale sparse array
for SparseArrayType in [ImmutableSparseNDimArray, MutableSparseNDimArray]:
a = SparseArrayType({1: 2, 3: 4}, (1000, 2000))
b = SparseArrayType({1: 2, 3: 4}, (1000, 2000))
assert tensorproduct(a, b) == ImmutableSparseNDimArray(
{
2000001: 4,
2000003: 8,
6000001: 8,
6000003: 16
}, (1000, 2000, 1000, 2000))
def tensorproduct(*args):
"""
Tensor product among scalars or array-like objects.
Examples
========
>>> from sympy.tensor.array import tensorproduct, Array
>>> from sympy.abc import x, y, z, t
>>> A = Array([[1, 2], [3, 4]])
>>> B = Array([x, y])
>>> tensorproduct(A, B)
[[[x, y], [2*x, 2*y]], [[3*x, 3*y], [4*x, 4*y]]]
>>> tensorproduct(A, x)
[[x, 2*x], [3*x, 4*x]]
>>> tensorproduct(A, B, B)
[[[[x**2, x*y], [x*y, y**2]], [[2*x**2, 2*x*y], [2*x*y, 2*y**2]]], [[[3*x**2, 3*x*y], [3*x*y, 3*y**2]], [[4*x**2, 4*x*y], [4*x*y, 4*y**2]]]]
Applying this function on two matrices will result in a rank 4 array.
>>> from sympy import Matrix, eye
>>> m = Matrix([[x, y], [z, t]])
>>> p = tensorproduct(eye(3), m)
>>> p
[[[[x, y], [z, t]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[x, y], [z, t]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[x, y], [z, t]]]]
"""
from sympy.tensor.array import SparseNDimArray, ImmutableSparseNDimArray
if len(args) == 0:
return S.One
if len(args) == 1:
return _arrayfy(args[0])
from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.tensor.array.expressions.array_expressions import _ArrayExpr
from sympy.matrices.expressions.matexpr import MatrixSymbol
if any(
isinstance(arg, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol))
for arg in args):
return ArrayTensorProduct(*args)
if len(args) > 2:
return tensorproduct(tensorproduct(args[0], args[1]), *args[2:])
# length of args is 2:
a, b = map(_arrayfy, args)
if not isinstance(a, NDimArray) or not isinstance(b, NDimArray):
return a * b
if isinstance(a, SparseNDimArray) and isinstance(b, SparseNDimArray):
lp = len(b)
new_array = {
k1 * lp + k2: v1 * v2
for k1, v1 in a._sparse_array.items()
for k2, v2 in b._sparse_array.items()
}
return ImmutableSparseNDimArray(new_array, a.shape + b.shape)
product_list = [i * j for i in Flatten(a) for j in Flatten(b)]
return ImmutableDenseNDimArray(product_list, a.shape + b.shape)
def test_NDim_array_conv():
MD = MutableDenseNDimArray([x, y, z])
MS = MutableSparseNDimArray([x, y, z])
ID = ImmutableDenseNDimArray([x, y, z])
IS = ImmutableSparseNDimArray([x, y, z])
assert MD.as_immutable() == ID
assert MD.as_mutable() == MD
assert MS.as_immutable() == IS
assert MS.as_mutable() == MS
assert ID.as_immutable() == ID
assert ID.as_mutable() == MD
assert IS.as_immutable() == IS
assert IS.as_mutable() == MS
def test_NDim_array_conv():
MD = MutableDenseNDimArray([x, y, z])
MS = MutableSparseNDimArray([x, y, z])
ID = ImmutableDenseNDimArray([x, y, z])
IS = ImmutableSparseNDimArray([x, y, z])
assert MD.as_immutable() == ID
assert MD.as_mutable() == MD
assert MS.as_immutable() == IS
assert MS.as_mutable() == MS
assert ID.as_immutable() == ID
assert ID.as_mutable() == MD
assert IS.as_immutable() == IS
assert IS.as_mutable() == MS
def test_NDArray():
from sympy.tensor.array import (
MutableDenseNDimArray,
ImmutableDenseNDimArray,
MutableSparseNDimArray,
ImmutableSparseNDimArray,
)
example = MutableDenseNDimArray(
[
[[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]],
[[13, 14, 15, 16], [17, 18, 19, 20], [21, 22, 23, 24]],
]
)
assert (
mcode(example) == "{{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}, "
"{{13, 14, 15, 16}, {17, 18, 19, 20}, {21, 22, 23, 24}}}"
)
example = ImmutableDenseNDimArray(example)
assert (
mcode(example) == "{{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}, "
"{{13, 14, 15, 16}, {17, 18, 19, 20}, {21, 22, 23, 24}}}"
)
example = MutableSparseNDimArray(example)
assert (
mcode(example) == "SparseArray[{"
"{1, 1, 1} -> 1, {1, 1, 2} -> 2, {1, 1, 3} -> 3, "
"{1, 1, 4} -> 4, {1, 2, 1} -> 5, {1, 2, 2} -> 6, "
"{1, 2, 3} -> 7, {1, 2, 4} -> 8, {1, 3, 1} -> 9, "
"{1, 3, 2} -> 10, {1, 3, 3} -> 11, {1, 3, 4} -> 12, "
"{2, 1, 1} -> 13, {2, 1, 2} -> 14, {2, 1, 3} -> 15, "
"{2, 1, 4} -> 16, {2, 2, 1} -> 17, {2, 2, 2} -> 18, "
"{2, 2, 3} -> 19, {2, 2, 4} -> 20, {2, 3, 1} -> 21, "
"{2, 3, 2} -> 22, {2, 3, 3} -> 23, {2, 3, 4} -> 24"
"}, {2, 3, 4}]"
)
example = ImmutableSparseNDimArray(example)
assert (
mcode(example) == "SparseArray[{"
"{1, 1, 1} -> 1, {1, 1, 2} -> 2, {1, 1, 3} -> 3, "
"{1, 1, 4} -> 4, {1, 2, 1} -> 5, {1, 2, 2} -> 6, "
"{1, 2, 3} -> 7, {1, 2, 4} -> 8, {1, 3, 1} -> 9, "
"{1, 3, 2} -> 10, {1, 3, 3} -> 11, {1, 3, 4} -> 12, "
"{2, 1, 1} -> 13, {2, 1, 2} -> 14, {2, 1, 3} -> 15, "
"{2, 1, 4} -> 16, {2, 2, 1} -> 17, {2, 2, 2} -> 18, "
"{2, 2, 3} -> 19, {2, 2, 4} -> 20, {2, 3, 1} -> 21, "
"{2, 3, 2} -> 22, {2, 3, 3} -> 23, {2, 3, 4} -> 24"
"}, {2, 3, 4}]"
)
def test_derivative_by_array():
from sympy.abc import a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z
from sympy.tensor.array import MutableSparseNDimArray, ImmutableSparseNDimArray
bexpr = x*y**2*exp(z)*log(t)
sexpr = sin(bexpr)
cexpr = cos(bexpr)
a = Array([sexpr])
assert derive_by_array(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
assert derive_by_array(sexpr, [x, y, z]) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
assert derive_by_array(a, [x, y, z]) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])
assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
assert derive_by_array(a, [[x, y], [z, t]]) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
assert derive_by_array([[x, y], [z, t]], [[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
[[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
assert diff(sexpr, Array([x, y, z])) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
assert diff(a, Array([x, y, z])) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])
assert diff(sexpr, Array([[x, y], [z, t]])) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
assert diff(a, Array([[x, y], [z, t]])) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
assert diff(Array([[x, y], [z, t]]), Array([x, y])) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
assert diff(Array([[x, y], [z, t]]), Array([[x, y], [z, t]])) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
[[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
# test for large scale sparse array
b = MutableSparseNDimArray.zeros(10000, 20000)
b[0, 0] = i
b[0, 1] = j
assert derive_by_array(b, i) == ImmutableSparseNDimArray({0: 1}, (10000, 20000))
assert derive_by_array(b, (i, j)) == ImmutableSparseNDimArray({0: 1, 200000001: 1}, (2, 10000, 20000))
