MLPY/Lib/site-packages/sympy/matrices/expressions/tests/test_indexing.py

from sympy.concrete.summations import Sum
from sympy.core.symbol import symbols, Symbol, Dummy
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.dense import eye
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions.hadamard import HadamardPower
from sympy.matrices.expressions.matexpr import (MatrixSymbol,
    MatrixExpr, MatrixElement)
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions.special import (ZeroMatrix, Identity,
    OneMatrix)
from sympy.matrices.expressions.trace import Trace, trace
from sympy.matrices.immutable import ImmutableMatrix
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.testing.pytest import XFAIL, raises

k, l, m, n = symbols('k l m n', integer=True)
i, j = symbols('i j', integer=True)

W = MatrixSymbol('W', k, l)
X = MatrixSymbol('X', l, m)
Y = MatrixSymbol('Y', l, m)
Z = MatrixSymbol('Z', m, n)

X1 = MatrixSymbol('X1', m, m)
X2 = MatrixSymbol('X2', m, m)
X3 = MatrixSymbol('X3', m, m)
X4 = MatrixSymbol('X4', m, m)

A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
x = MatrixSymbol('x', 1, 2)
y = MatrixSymbol('x', 2, 1)


def test_symbolic_indexing():
    x12 = X[1, 2]
    assert all(s in str(x12) for s in ['1', '2', X.name])
    # We don't care about the exact form of this.  We do want to make sure
    # that all of these features are present


def test_add_index():
    assert (X + Y)[i, j] == X[i, j] + Y[i, j]


def test_mul_index():
    assert (A*y)[0, 0] == A[0, 0]*y[0, 0] + A[0, 1]*y[1, 0]
    assert (A*B).as_mutable() == (A.as_mutable() * B.as_mutable())
    X = MatrixSymbol('X', n, m)
    Y = MatrixSymbol('Y', m, k)

    result = (X*Y)[4,2]
    expected = Sum(X[4, i]*Y[i, 2], (i, 0, m - 1))
    assert result.args[0].dummy_eq(expected.args[0], i)
    assert result.args[1][1:] == expected.args[1][1:]


def test_pow_index():
    Q = MatPow(A, 2)
    assert Q[0, 0] == A[0, 0]**2 + A[0, 1]*A[1, 0]
    n = symbols("n")
    Q2 = A**n
    assert Q2[0, 0] == 2*(
            -sqrt((A[0, 0] + A[1, 1])**2 - 4*A[0, 0]*A[1, 1] +
            4*A[0, 1]*A[1, 0])/2 + A[0, 0]/2 + A[1, 1]/2
        )**n * \
        A[0, 1]*A[1, 0]/(
            (sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] +
                  A[1, 1]**2) + A[0, 0] - A[1, 1])*
            sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)
        ) - 2*(
            sqrt((A[0, 0] + A[1, 1])**2 - 4*A[0, 0]*A[1, 1] +
            4*A[0, 1]*A[1, 0])/2 + A[0, 0]/2 + A[1, 1]/2
        )**n * A[0, 1]*A[1, 0]/(
            (-sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] +
            A[1, 1]**2) + A[0, 0] - A[1, 1])*
            sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)
        )


def test_transpose_index():
    assert X.T[i, j] == X[j, i]


def test_Identity_index():
    I = Identity(3)
    assert I[0, 0] == I[1, 1] == I[2, 2] == 1
    assert I[1, 0] == I[0, 1] == I[2, 1] == 0
    assert I[i, 0].delta_range == (0, 2)
    raises(IndexError, lambda: I[3, 3])


def test_block_index():
    I = Identity(3)
    Z = ZeroMatrix(3, 3)
    B = BlockMatrix([[I, I], [I, I]])
    e3 = ImmutableMatrix(eye(3))
    BB = BlockMatrix([[e3, e3], [e3, e3]])
    assert B[0, 0] == B[3, 0] == B[0, 3] == B[3, 3] == 1
    assert B[4, 3] == B[5, 1] == 0

    BB = BlockMatrix([[e3, e3], [e3, e3]])
    assert B.as_explicit() == BB.as_explicit()

    BI = BlockMatrix([[I, Z], [Z, I]])

    assert BI.as_explicit().equals(eye(6))


def test_block_index_symbolic():
    # Note that these matrices may be zero-sized and indices may be negative, which causes
    # all naive simplifications given in the comments to be invalid
    A1 = MatrixSymbol('A1', n, k)
    A2 = MatrixSymbol('A2', n, l)
    A3 = MatrixSymbol('A3', m, k)
    A4 = MatrixSymbol('A4', m, l)
    A = BlockMatrix([[A1, A2], [A3, A4]])
    assert A[0, 0] == MatrixElement(A, 0, 0)  # Cannot be A1[0, 0]
    assert A[n - 1, k - 1] == A1[n - 1, k - 1]
    assert A[n, k] == A4[0, 0]
    assert A[n + m - 1, 0] == MatrixElement(A, n + m - 1, 0)  # Cannot be A3[m - 1, 0]
    assert A[0, k + l - 1] == MatrixElement(A, 0, k + l - 1)  # Cannot be A2[0, l - 1]
    assert A[n + m - 1, k + l - 1] == MatrixElement(A, n + m - 1, k + l - 1)  # Cannot be A4[m - 1, l - 1]
    assert A[i, j] == MatrixElement(A, i, j)
    assert A[n + i, k + j] == MatrixElement(A, n + i, k + j)  # Cannot be A4[i, j]
    assert A[n - i - 1, k - j - 1] == MatrixElement(A, n - i - 1, k - j - 1)  # Cannot be A1[n - i - 1, k - j - 1]


def test_block_index_symbolic_nonzero():
    # All invalid simplifications from test_block_index_symbolic() that become valid if all
    # matrices have nonzero size and all indices are nonnegative
    k, l, m, n = symbols('k l m n', integer=True, positive=True)
    i, j = symbols('i j', integer=True, nonnegative=True)
    A1 = MatrixSymbol('A1', n, k)
    A2 = MatrixSymbol('A2', n, l)
    A3 = MatrixSymbol('A3', m, k)
    A4 = MatrixSymbol('A4', m, l)
    A = BlockMatrix([[A1, A2], [A3, A4]])
    assert A[0, 0] == A1[0, 0]
    assert A[n + m - 1, 0] == A3[m - 1, 0]
    assert A[0, k + l - 1] == A2[0, l - 1]
    assert A[n + m - 1, k + l - 1] == A4[m - 1, l - 1]
    assert A[i, j] == MatrixElement(A, i, j)
    assert A[n + i, k + j] == A4[i, j]
    assert A[n - i - 1, k - j - 1] == A1[n - i - 1, k - j - 1]
    assert A[2 * n, 2 * k] == A4[n, k]


def test_block_index_large():
    n, m, k = symbols('n m k', integer=True, positive=True)
    i = symbols('i', integer=True, nonnegative=True)
    A1 = MatrixSymbol('A1', n, n)
    A2 = MatrixSymbol('A2', n, m)
    A3 = MatrixSymbol('A3', n, k)
    A4 = MatrixSymbol('A4', m, n)
    A5 = MatrixSymbol('A5', m, m)
    A6 = MatrixSymbol('A6', m, k)
    A7 = MatrixSymbol('A7', k, n)
    A8 = MatrixSymbol('A8', k, m)
    A9 = MatrixSymbol('A9', k, k)
    A = BlockMatrix([[A1, A2, A3], [A4, A5, A6], [A7, A8, A9]])
    assert A[n + i, n + i] == MatrixElement(A, n + i, n + i)


@XFAIL
def test_block_index_symbolic_fail():
    # To make this work, symbolic matrix dimensions would need to be somehow assumed nonnegative
    # even if the symbols aren't specified as such.  Then 2 * n < n would correctly evaluate to
    # False in BlockMatrix._entry()
    A1 = MatrixSymbol('A1', n, 1)
    A2 = MatrixSymbol('A2', m, 1)
    A = BlockMatrix([[A1], [A2]])
    assert A[2 * n, 0] == A2[n, 0]


def test_slicing():
    A.as_explicit()[0, :]  # does not raise an error


def test_errors():
    raises(IndexError, lambda: Identity(2)[1, 2, 3, 4, 5])
    raises(IndexError, lambda: Identity(2)[[1, 2, 3, 4, 5]])


def test_matrix_expression_to_indices():
    i, j = symbols("i, j")
    i1, i2, i3 = symbols("i_1:4")

    def replace_dummies(expr):
        repl = {i: Symbol(i.name) for i in expr.atoms(Dummy)}
        return expr.xreplace(repl)

    expr = W*X*Z
    assert replace_dummies(expr._entry(i, j)) == \
        Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr

    expr = Z.T*X.T*W.T
    assert replace_dummies(expr._entry(i, j)) == \
        Sum(W[j, i2]*X[i2, i1]*Z[i1, i], (i1, 0, m-1), (i2, 0, l-1))
    assert MatrixExpr.from_index_summation(expr._entry(i, j), i) == expr

    expr = W*X*Z + W*Y*Z
    assert replace_dummies(expr._entry(i, j)) == \
        Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
        Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr

    expr = 2*W*X*Z + 3*W*Y*Z
    assert replace_dummies(expr._entry(i, j)) == \
        2*Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
        3*Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr

    expr = W*(X + Y)*Z
    assert replace_dummies(expr._entry(i, j)) == \
            Sum(W[i, i1]*(X[i1, i2] + Y[i1, i2])*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr

    expr = A*B**2*A
    #assert replace_dummies(expr._entry(i, j)) == \
    #        Sum(A[i, i1]*B[i1, i2]*B[i2, i3]*A[i3, j], (i1, 0, 1), (i2, 0, 1), (i3, 0, 1))

    # Check that different dummies are used in sub-multiplications:
    expr = (X1*X2 + X2*X1)*X3
    assert replace_dummies(expr._entry(i, j)) == \
           Sum((Sum(X1[i, i2] * X2[i2, i1], (i2, 0, m - 1)) + Sum(X1[i3, i1] * X2[i, i3], (i3, 0, m - 1))) * X3[
               i1, j], (i1, 0, m - 1))


def test_matrix_expression_from_index_summation():
    from sympy.abc import a,b,c,d
    A = MatrixSymbol("A", k, k)
    B = MatrixSymbol("B", k, k)
    C = MatrixSymbol("C", k, k)
    w1 = MatrixSymbol("w1", k, 1)

    i0, i1, i2, i3, i4 = symbols("i0:5", cls=Dummy)

    expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
    assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
    expr = Sum(W.T[b,a]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
    assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
    expr = Sum(A[b, a]*B[b, c]*C[c, d], (b, 0, k-1), (c, 0, k-1))
    assert MatrixSymbol.from_index_summation(expr, a) == A.T*B*C
    expr = Sum(A[b, a]*B[c, b]*C[c, d], (b, 0, k-1), (c, 0, k-1))
    assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
    expr = Sum(C[c, d]*A[b, a]*B[c, b], (b, 0, k-1), (c, 0, k-1))
    assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
    expr = Sum(A[a, b] + B[a, b], (a, 0, k-1), (b, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == OneMatrix(1, k)*A*OneMatrix(k, 1) + OneMatrix(1, k)*B*OneMatrix(k, 1)
    expr = Sum(A[a, b]**2, (a, 0, k - 1), (b, 0, k - 1))
    assert MatrixExpr.from_index_summation(expr, a) == Trace(A * A.T)
    expr = Sum(A[a, b]**3, (a, 0, k - 1), (b, 0, k - 1))
    assert MatrixExpr.from_index_summation(expr, a) == Trace(HadamardPower(A.T, 2) * A)
    expr = Sum((A[a, b] + B[a, b])*C[b, c], (b, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == (A+B)*C
    expr = Sum((A[a, b] + B[b, a])*C[b, c], (b, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == (A+B.T)*C
    expr = Sum(A[a, b]*A[b, c]*A[c, d], (b, 0, k-1), (c, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == A**3
    expr = Sum(A[a, b]*A[b, c]*B[c, d], (b, 0, k-1), (c, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == A**2*B

    # Parse the trace of a matrix:

    expr = Sum(A[a, a], (a, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, None) == trace(A)
    expr = Sum(A[a, a]*B[b, c]*C[c, d], (a, 0, k-1), (c, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, b) == trace(A)*B*C

    # Check wrong sum ranges (should raise an exception):

    ## Case 1: 0 to m instead of 0 to m-1
    expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m))
    raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
    ## Case 2: 1 to m-1 instead of 0 to m-1
    expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 1, m-1))
    raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))

    # Parse nested sums:
    expr = Sum(A[a, b]*Sum(B[b, c]*C[c, d], (c, 0, k-1)), (b, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == A*B*C

    # Test Kronecker delta:
    expr = Sum(A[a, b]*KroneckerDelta(b, c)*B[c, d], (b, 0, k-1), (c, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, a) == A*B

    expr = Sum(KroneckerDelta(i1, m)*KroneckerDelta(i2, n)*A[i, i1]*A[j, i2], (i1, 0, k-1), (i2, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, m) == ArrayTensorProduct(A.T, A)

    # Test numbered indices:
    expr = Sum(A[i1, i2]*w1[i2, 0], (i2, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*w1, i1, 0)

    expr = Sum(A[i1, i2]*B[i2, 0], (i2, 0, k-1))
    assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*B, i1, 0)