image-handler/image-handler10.0/venv/lib/site-packages/sympy/concrete/delta.py

"""
This module implements sums and products containing the Kronecker Delta function.

References
==========

.. [1] https://mathworld.wolfram.com/KroneckerDelta.html

"""
from .products import product
from .summations import Sum, summation
from sympy.core import Add, Mul, S, Dummy
from sympy.core.cache import cacheit
from sympy.core.sorting import default_sort_key
from sympy.functions import KroneckerDelta, Piecewise, piecewise_fold
from sympy.polys.polytools import factor
from sympy.sets.sets import Interval
from sympy.solvers.solvers import solve


@cacheit
def _expand_delta(expr, index):
    """
    Expand the first Add containing a simple KroneckerDelta.
    """
    if not expr.is_Mul:
        return expr
    delta = None
    func = Add
    terms = [S.One]
    for h in expr.args:
        if delta is None and h.is_Add and _has_simple_delta(h, index):
            delta = True
            func = h.func
            terms = [terms[0]*t for t in h.args]
        else:
            terms = [t*h for t in terms]
    return func(*terms)


@cacheit
def _extract_delta(expr, index):
    """
    Extract a simple KroneckerDelta from the expression.

    Explanation
    ===========

    Returns the tuple ``(delta, newexpr)`` where:

      - ``delta`` is a simple KroneckerDelta expression if one was found,
        or ``None`` if no simple KroneckerDelta expression was found.

      - ``newexpr`` is a Mul containing the remaining terms; ``expr`` is
        returned unchanged if no simple KroneckerDelta expression was found.

    Examples
    ========

    >>> from sympy import KroneckerDelta
    >>> from sympy.concrete.delta import _extract_delta
    >>> from sympy.abc import x, y, i, j, k
    >>> _extract_delta(4*x*y*KroneckerDelta(i, j), i)
    (KroneckerDelta(i, j), 4*x*y)
    >>> _extract_delta(4*x*y*KroneckerDelta(i, j), k)
    (None, 4*x*y*KroneckerDelta(i, j))

    See Also
    ========

    sympy.functions.special.tensor_functions.KroneckerDelta
    deltaproduct
    deltasummation
    """
    if not _has_simple_delta(expr, index):
        return (None, expr)
    if isinstance(expr, KroneckerDelta):
        return (expr, S.One)
    if not expr.is_Mul:
        raise ValueError("Incorrect expr")
    delta = None
    terms = []

    for arg in expr.args:
        if delta is None and _is_simple_delta(arg, index):
            delta = arg
        else:
            terms.append(arg)
    return (delta, expr.func(*terms))


@cacheit
def _has_simple_delta(expr, index):
    """
    Returns True if ``expr`` is an expression that contains a KroneckerDelta
    that is simple in the index ``index``, meaning that this KroneckerDelta
    is nonzero for a single value of the index ``index``.
    """
    if expr.has(KroneckerDelta):
        if _is_simple_delta(expr, index):
            return True
        if expr.is_Add or expr.is_Mul:
            for arg in expr.args:
                if _has_simple_delta(arg, index):
                    return True
    return False


@cacheit
def _is_simple_delta(delta, index):
    """
    Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single
    value of the index ``index``.
    """
    if isinstance(delta, KroneckerDelta) and delta.has(index):
        p = (delta.args[0] - delta.args[1]).as_poly(index)
        if p:
            return p.degree() == 1
    return False


@cacheit
def _remove_multiple_delta(expr):
    """
    Evaluate products of KroneckerDelta's.
    """
    if expr.is_Add:
        return expr.func(*list(map(_remove_multiple_delta, expr.args)))
    if not expr.is_Mul:
        return expr
    eqs = []
    newargs = []
    for arg in expr.args:
        if isinstance(arg, KroneckerDelta):
            eqs.append(arg.args[0] - arg.args[1])
        else:
            newargs.append(arg)
    if not eqs:
        return expr
    solns = solve(eqs, dict=True)
    if len(solns) == 0:
        return S.Zero
    elif len(solns) == 1:
        for key in solns[0].keys():
            newargs.append(KroneckerDelta(key, solns[0][key]))
        expr2 = expr.func(*newargs)
        if expr != expr2:
            return _remove_multiple_delta(expr2)
    return expr


@cacheit
def _simplify_delta(expr):
    """
    Rewrite a KroneckerDelta's indices in its simplest form.
    """
    if isinstance(expr, KroneckerDelta):
        try:
            slns = solve(expr.args[0] - expr.args[1], dict=True)
            if slns and len(slns) == 1:
                return Mul(*[KroneckerDelta(*(key, value))
                            for key, value in slns[0].items()])
        except NotImplementedError:
            pass
    return expr


@cacheit
def deltaproduct(f, limit):
    """
    Handle products containing a KroneckerDelta.

    See Also
    ========

    deltasummation
    sympy.functions.special.tensor_functions.KroneckerDelta
    sympy.concrete.products.product
    """
    if ((limit[2] - limit[1]) < 0) == True:
        return S.One

    if not f.has(KroneckerDelta):
        return product(f, limit)

    if f.is_Add:
        # Identify the term in the Add that has a simple KroneckerDelta
        delta = None
        terms = []
        for arg in sorted(f.args, key=default_sort_key):
            if delta is None and _has_simple_delta(arg, limit[0]):
                delta = arg
            else:
                terms.append(arg)
        newexpr = f.func(*terms)
        k = Dummy("kprime", integer=True)
        if isinstance(limit[1], int) and isinstance(limit[2], int):
            result = deltaproduct(newexpr, limit) + sum([
                deltaproduct(newexpr, (limit[0], limit[1], ik - 1)) *
                delta.subs(limit[0], ik) *
                deltaproduct(newexpr, (limit[0], ik + 1, limit[2])) for ik in range(int(limit[1]), int(limit[2] + 1))]
            )
        else:
            result = deltaproduct(newexpr, limit) + deltasummation(
                deltaproduct(newexpr, (limit[0], limit[1], k - 1)) *
                delta.subs(limit[0], k) *
                deltaproduct(newexpr, (limit[0], k + 1, limit[2])),
                (k, limit[1], limit[2]),
                no_piecewise=_has_simple_delta(newexpr, limit[0])
            )
        return _remove_multiple_delta(result)

    delta, _ = _extract_delta(f, limit[0])

    if not delta:
        g = _expand_delta(f, limit[0])
        if f != g:
            try:
                return factor(deltaproduct(g, limit))
            except AssertionError:
                return deltaproduct(g, limit)
        return product(f, limit)

    return _remove_multiple_delta(f.subs(limit[0], limit[1])*KroneckerDelta(limit[2], limit[1])) + \
        S.One*_simplify_delta(KroneckerDelta(limit[2], limit[1] - 1))


@cacheit
def deltasummation(f, limit, no_piecewise=False):
    """
    Handle summations containing a KroneckerDelta.

    Explanation
    ===========

    The idea for summation is the following:

    - If we are dealing with a KroneckerDelta expression, i.e. KroneckerDelta(g(x), j),
      we try to simplify it.

      If we could simplify it, then we sum the resulting expression.
      We already know we can sum a simplified expression, because only
      simple KroneckerDelta expressions are involved.

      If we could not simplify it, there are two cases:

      1) The expression is a simple expression: we return the summation,
         taking care if we are dealing with a Derivative or with a proper
         KroneckerDelta.

      2) The expression is not simple (i.e. KroneckerDelta(cos(x))): we can do
         nothing at all.

    - If the expr is a multiplication expr having a KroneckerDelta term:

      First we expand it.

      If the expansion did work, then we try to sum the expansion.

      If not, we try to extract a simple KroneckerDelta term, then we have two
      cases:

      1) We have a simple KroneckerDelta term, so we return the summation.

      2) We did not have a simple term, but we do have an expression with
         simplified KroneckerDelta terms, so we sum this expression.

    Examples
    ========

    >>> from sympy import oo, symbols
    >>> from sympy.abc import k
    >>> i, j = symbols('i, j', integer=True, finite=True)
    >>> from sympy.concrete.delta import deltasummation
    >>> from sympy import KroneckerDelta
    >>> deltasummation(KroneckerDelta(i, k), (k, -oo, oo))
    1
    >>> deltasummation(KroneckerDelta(i, k), (k, 0, oo))
    Piecewise((1, i >= 0), (0, True))
    >>> deltasummation(KroneckerDelta(i, k), (k, 1, 3))
    Piecewise((1, (i >= 1) & (i <= 3)), (0, True))
    >>> deltasummation(k*KroneckerDelta(i, j)*KroneckerDelta(j, k), (k, -oo, oo))
    j*KroneckerDelta(i, j)
    >>> deltasummation(j*KroneckerDelta(i, j), (j, -oo, oo))
    i
    >>> deltasummation(i*KroneckerDelta(i, j), (i, -oo, oo))
    j

    See Also
    ========

    deltaproduct
    sympy.functions.special.tensor_functions.KroneckerDelta
    sympy.concrete.sums.summation
    """
    if ((limit[2] - limit[1]) < 0) == True:
        return S.Zero

    if not f.has(KroneckerDelta):
        return summation(f, limit)

    x = limit[0]

    g = _expand_delta(f, x)
    if g.is_Add:
        return piecewise_fold(
            g.func(*[deltasummation(h, limit, no_piecewise) for h in g.args]))

    # try to extract a simple KroneckerDelta term
    delta, expr = _extract_delta(g, x)

    if (delta is not None) and (delta.delta_range is not None):
        dinf, dsup = delta.delta_range
        if (limit[1] - dinf <= 0) == True and (limit[2] - dsup >= 0) == True:
            no_piecewise = True

    if not delta:
        return summation(f, limit)

    solns = solve(delta.args[0] - delta.args[1], x)
    if len(solns) == 0:
        return S.Zero
    elif len(solns) != 1:
        return Sum(f, limit)
    value = solns[0]
    if no_piecewise:
        return expr.subs(x, value)
    return Piecewise(
        (expr.subs(x, value), Interval(*limit[1:3]).as_relational(value)),
        (S.Zero, True)
    )