heurisch.py

from functools import reduce
from itertools import permutations

from ..core import Add, Basic, Dummy, E, Eq, Mul, S, Wild, pi, sympify
from ..core.compatibility import ordered
from ..functions import (Ei, LambertW, Piecewise, acosh, asin, asinh, atan,
                         cos, cosh, cot, coth, erf, erfi, exp, li, log, root,
                         sin, sinh, sqrt, tan, tanh)
from ..logic import And
from ..polys import PolynomialError, cancel, factor, gcd, lcm, quo
from ..polys.constructor import construct_domain
from ..polys.monomials import itermonomials
from ..polys.polyroots import root_factors
from ..polys.solvers import solve_lin_sys
from ..utilities.iterables import uniq


def components(f, x):
    """
    Returns a set of all functional components of the given expression
    which includes symbols, function applications and compositions and
    non-integer powers. Fractional powers are collected with with
    minimal, positive exponents.

    >>> components(sin(x)*cos(x)**2, x)
    {x, sin(x), cos(x)}

    See Also
    ========

    diofant.integrals.heurisch.heurisch
    """
    result = set()

    if x in f.free_symbols:
        if f.is_Symbol:
            result.add(f)
        elif f.is_Function or f.is_Derivative:
            for g in f.args:
                result |= components(g, x)

            result.add(f)
        elif f.is_Pow:
            result |= components(f.base, x)

            if not f.exp.is_Integer:
                if f.exp.is_Rational:
                    result.add(root(f.base, f.exp.denominator))
                else:
                    result |= components(f.exp, x) | {f}
        else:
            for g in f.args:
                result |= components(g, x)

    return result


# name -> [] of symbols
_symbols_cache = {}


# NB @cacheit is not convenient here
def _symbols(name, n):
    """get vector of symbols local to this module"""
    try:
        lsyms = _symbols_cache[name]
    except KeyError:
        lsyms = []
        _symbols_cache[name] = lsyms

    while len(lsyms) < n:
        lsyms.append( Dummy('%s%i' % (name, len(lsyms))) )

    return lsyms[:n]


def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
                     degree_offset=0, unnecessary_permutations=None):
    """
    A wrapper around the heurisch integration algorithm.

    This method takes the result from heurisch and checks for poles in the
    denominator. For each of these poles, the integral is reevaluated, and
    the final integration result is given in terms of a Piecewise.

    Examples
    ========

    >>> heurisch(cos(n*x), x)
    sin(n*x)/n
    >>> heurisch_wrapper(cos(n*x), x)
    Piecewise((x, Eq(n, 0)), (sin(n*x)/n, true))

    See Also
    ========

    diofant.integrals.heurisch.heurisch
    """
    from ..solvers.solvers import solve, denoms
    f = sympify(f)
    if x not in f.free_symbols:
        return f*x

    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
                   unnecessary_permutations)
    if not isinstance(res, Basic):
        return res
    # We consider each denominator in the expression, and try to find
    # cases where one or more symbolic denominator might be zero. The
    # conditions for these cases are stored in the list slns.
    slns = []
    for d in denoms(res):
        try:
            ds = list(ordered(d.free_symbols - {x}))
            if ds:
                slns += solve(d, *ds)
        except NotImplementedError:
            pass
    if not slns:
        return res
    slns = list(uniq(slns))
    # Remove the solutions corresponding to poles in the original expression.
    slns0 = []
    for d in denoms(f):
        try:
            ds = list(ordered(d.free_symbols - {x}))
            if ds:
                slns0 += solve(d, *ds)
        except NotImplementedError:
            pass
    slns = [s for s in slns if s not in slns0]
    if not slns:
        return res
    if len(slns) > 1:
        eqs = []
        for sub_dict in slns:
            eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
        slns = solve(eqs, *ordered(set().union(*[e.free_symbols
                                                 for e in eqs]) - {x})) + slns
    # For each case listed in the list slns, we reevaluate the integral.
    pairs = []
    for sub_dict in slns:
        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
                        degree_offset, unnecessary_permutations)
        cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
        pairs.append((expr, cond))
    pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
                           degree_offset, unnecessary_permutations), True))
    return Piecewise(*pairs)


def heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3,
             degree_offset=0, unnecessary_permutations=None):
    """
    Compute indefinite integral using heuristic Risch algorithm.

    This is a heuristic approach to indefinite integration in finite
    terms using the extended heuristic (parallel) Risch algorithm, based
    on Manuel Bronstein's "Poor Man's Integrator".

    The algorithm supports various classes of functions including
    transcendental elementary or special functions like Airy,
    Bessel, Whittaker and Lambert.

    Note that this algorithm is not a decision procedure. If it isn't
    able to compute the antiderivative for a given function, then this is
    not a proof that such a functions does not exist.  One should use
    recursive Risch algorithm in such case.  It's an open question if
    this algorithm can be made a full decision procedure.

    This is an internal integrator procedure. You should use toplevel
    'integrate' function in most cases,  as this procedure needs some
    preprocessing steps and otherwise may fail.

    Parameters
    ==========

    f : Expr
        expression
    x : Symbol
        variable

    rewrite : Boolean, optional
        force rewrite 'f' in terms of 'tan' and 'tanh', default False.
    hints : None or list
        a list of functions that may appear in anti-derivate.  If
        None (default) - no suggestions at all, if empty list - try
        to figure out.

    Examples
    ========

    >>> heurisch(y*tan(x), x)
    y*log(tan(x)**2 + 1)/2

    References
    ==========

    * Manuel Bronstein's "Poor Man's Integrator",
      http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html

    See Also
    ========

    diofant.integrals.integrals.Integral.doit
    diofant.integrals.integrals.Integral
    diofant.integrals.heurisch.components
    """
    f = sympify(f)
    if x not in f.free_symbols:
        return f*x

    if not f.is_Add:
        indep, f = f.as_independent(x)
    else:
        indep = S.One

    rewritables = {
        (sin, cos, cot): tan,
        (sinh, cosh, coth): tanh,
    }

    if rewrite:
        for candidates, rule in rewritables.items():
            f = f.rewrite(candidates, rule)
    else:
        for candidates in rewritables:
            if f.has(*candidates):
                break
        else:
            rewrite = True

    terms = components(f, x)

    if hints is not None:
        if not hints:
            a = Wild('a', exclude=[x])
            b = Wild('b', exclude=[x])
            c = Wild('c', exclude=[x])

            for g in set(terms):  # using copy of terms
                if g.is_Function:
                    if isinstance(g, li):
                        M = g.args[0].match(a*x**b)

                        if M is not None:
                            terms.add( x*(li(M[a]*x**M[b]) - (M[a]*x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )

                elif g.is_Pow:
                    if g.base is E:
                        M = g.exp.match(a*x**2)

                        if M is not None:
                            if M[a].is_positive:
                                terms.add(erfi(sqrt(M[a])*x))
                            else:  # M[a].is_negative or unknown
                                terms.add(erf(sqrt(-M[a])*x))

                        M = g.exp.match(a*x**2 + b*x + c)

                        if M is not None:
                            if M[a].is_positive:
                                terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a])) *
                                          erfi(sqrt(M[a])*x + M[b]/(2*sqrt(M[a]))))
                            elif M[a].is_negative:
                                terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a])) *
                                          erf(sqrt(-M[a])*x - M[b]/(2*sqrt(-M[a]))))

                        M = g.exp.match(a*log(x)**2)

                        if M is not None:
                            if M[a].is_positive:
                                terms.add(erfi(sqrt(M[a])*log(x) + 1/(2*sqrt(M[a]))))
                            if M[a].is_negative:
                                terms.add(erf(sqrt(-M[a])*log(x) - 1/(2*sqrt(-M[a]))))

                    elif g.exp.is_Rational and g.exp.denominator == 2:
                        M = g.base.match(a*x**2 + b)

                        if M is not None and M[b].is_positive:
                            if M[a].is_positive:
                                terms.add(asinh(sqrt(M[a]/M[b])*x))
                            elif M[a].is_negative:
                                terms.add(asin(sqrt(-M[a]/M[b])*x))

                        M = g.base.match(a*x**2 - b)

                        if M is not None and M[b].is_positive:
                            if M[a].is_positive:
                                terms.add(acosh(sqrt(M[a]/M[b])*x))
                            elif M[a].is_negative:
                                terms.add((-M[b]/2*sqrt(-M[a]) *
                                           atan(sqrt(-M[a])*x/sqrt(M[a]*x**2 - M[b]))))

        else:
            terms |= set(hints)

    for g in set(terms):  # using copy of terms
        terms |= components(cancel(g.diff(x)), x)

    # TODO: caching is significant factor for why permutations work at all. Change this.
    V = _symbols('x', len(terms))

    # sort mapping expressions from largest to smallest (last is always x).
    mapping = list(reversed(list(zip(*ordered(                           #
        [(a[0].as_independent(x)[1], a) for a in zip(terms, V)])))[1]))  #
    rev_mapping = {v: k for k, v in mapping}                             #
    if mappings is None:                                                 #
        # optimizing the number of permutations of mapping               #
        assert mapping[-1][0] == x  # if not, find it and correct this comment
        unnecessary_permutations = [mapping.pop(-1)]
        mappings = permutations(mapping)
    else:
        unnecessary_permutations = unnecessary_permutations or []

    def _substitute(expr):
        return expr.subs(mapping)

    for mapping in mappings:
        mapping = list(mapping)
        mapping = mapping + unnecessary_permutations
        diffs = [ _substitute(cancel(g.diff(x))) for g in terms ]
        denoms = [ g.as_numer_denom()[1] for g in diffs ]
        if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V):
            denom = reduce(lambda p, q: lcm(p, q, *V), denoms)
            break
    else:
        if not rewrite:
            result = heurisch(f, x, rewrite=True, hints=hints,
                              unnecessary_permutations=unnecessary_permutations)

            if result is not None:
                return indep*result
        return

    numers = [ cancel(denom*g) for g in diffs ]

    def _derivation(h):
        return Add(*[ d * h.diff(v) for d, v in zip(numers, V) ])

    def _deflation(p):
        for y in V:
            if not p.has(y):
                continue

            if _derivation(p) is not S.Zero:
                c, q = p.as_poly(y).primitive()
                return _deflation(c)*gcd(q, q.diff(y)).as_expr()
        else:
            return p

    def _splitter(p):
        for y in V:
            if not p.has(y):
                continue

            if _derivation(y) is not S.Zero:
                c, q = p.as_poly(y).primitive()

                q = q.as_expr()

                h = gcd(q, _derivation(q), y)
                s = quo(h, gcd(q, q.diff(y), y), y)

                c_split = _splitter(c)

                if s.as_poly(y).degree() == 0:
                    return c_split[0], q*c_split[1]

                q_split = _splitter(cancel(q / s))

                return c_split[0]*q_split[0]*s, c_split[1]*q_split[1]
        else:
            return S.One, p

    special = {}

    for term in terms:
        if term.is_Function:
            if isinstance(term, tan):
                special[1 + _substitute(term)**2] = False
            elif isinstance(term, tanh):
                special[1 + _substitute(term)] = False
                special[1 - _substitute(term)] = False
            elif isinstance(term, LambertW):
                special[_substitute(term)] = True

    F = _substitute(f)

    P, Q = F.as_numer_denom()

    u_split = _splitter(denom)
    v_split = _splitter(Q)

    polys = set(list(v_split) + [u_split[0]] + list(special))

    s = u_split[0] * Mul(*[ k for k, v in special.items() if v ])
    polified = [ p.as_poly(*V) for p in [s, P, Q] ]

    if None in polified:
        return

    # --- definitions for _integrate ---
    a, b, c = [ p.total_degree() for p in polified ]

    poly_denom = (s * v_split[0] * _deflation(v_split[1])).as_expr()

    def _exponent(g):
        if g.is_Pow:
            if g.exp.is_Rational and g.exp.denominator != 1:
                if g.exp.numerator > 0:
                    return g.exp.numerator + g.exp.denominator - 1
                else:
                    return abs(g.exp.numerator + g.exp.denominator)
            else:
                return 1
        elif not g.is_Atom and g.args:
            return max(_exponent(h) for h in g.args)
        else:
            return 1

    A, B = _exponent(f), a + max(b, c)

    if A > 1 and B > 1:
        monoms = itermonomials(V, A + B - 1 + degree_offset)
    else:
        monoms = itermonomials(V, A + B + degree_offset)

    poly_coeffs = _symbols('A', len(monoms))

    poly_part = Add(*[ poly_coeffs[i]*monomial
                       for i, monomial in enumerate(ordered(monoms)) ])

    reducibles = set()

    for poly in polys:
        if poly.has(*V):
            try:
                factorization = factor(poly, greedy=True)
            except PolynomialError:
                factorization = poly
            factorization = poly

            if factorization.is_Mul:
                reducibles |= set(factorization.args)
            else:
                reducibles.add(factorization)

    def _integrate(field=None):
        irreducibles = set()

        for poly in reducibles:
            for z in poly.free_symbols:
                if z in V:
                    break  # should this be: `irreducibles |= \
            else:          # set(root_factors(poly, z, filter=field))`
                continue   # and the line below deleted?
                #                          |
                #                          V
            irreducibles |= set(root_factors(poly, z, filter=field))

        log_part = []
        B = _symbols('B', len(irreducibles))

        # Note: the ordering matters here
        for poly, b in reversed(list(ordered(zip(irreducibles, B)))):
            if poly.has(*V):
                poly_coeffs.append(b)
                log_part.append(b * log(poly))

        # TODO: Currently it's better to use symbolic expressions here instead
        # of rational functions, because it's simpler and FracElement doesn't
        # give big speed improvement yet. This is because cancellation is slow
        # due to slow polynomial GCD algorithms. If this gets improved then
        # revise this code.
        candidate = poly_part/poly_denom + Add(*log_part)
        h = F - _derivation(candidate) / denom
        raw_numer = h.as_numer_denom()[0]

        # Rewrite raw_numer as a polynomial in K[coeffs][V] where K is a field
        # that we have to determine. We can't use simply atoms() because log(3),
        # sqrt(y) and similar expressions can appear, leading to non-trivial
        # domains.
        syms = set(poly_coeffs) | set(V)
        non_syms = set()

        def find_non_syms(expr):
            if expr.is_Integer or expr.is_Rational:
                pass  # ignore trivial numbers
            elif expr in syms:
                pass  # ignore variables
            elif not expr.has(*syms):
                non_syms.add(expr)
            elif expr.is_Add or expr.is_Mul or expr.is_Pow:
                list(map(find_non_syms, expr.args))
            else:
                # TODO: Non-polynomial expression. This should have been
                # filtered out at an earlier stage.
                raise PolynomialError

        try:
            find_non_syms(raw_numer)
        except PolynomialError:
            return
        else:
            ground, _ = construct_domain(non_syms, field=True)

        coeff_ring = ground.poly_ring(*poly_coeffs)
        ring = coeff_ring.poly_ring(*V)

        try:
            numer = ring.from_expr(raw_numer)
        except ValueError:
            raise PolynomialError

        solution = solve_lin_sys(numer.coeffs(), coeff_ring)

        if solution is None:
            return
        else:
            solution = [(coeff_ring.symbols[coeff_ring.index(k)],
                         v.as_expr()) for k, v in solution.items()]
            return candidate.subs(solution).subs(
                list(zip(poly_coeffs, [S.Zero]*len(poly_coeffs))))

    if not (F.free_symbols - set(V)):
        solution = _integrate('Q')

        if solution is None:
            solution = _integrate()
    else:
        solution = _integrate()

    if solution is not None:
        antideriv = solution.subs(rev_mapping)
        antideriv = cancel(antideriv).expand(force=True)

        if antideriv.is_Add:
            antideriv = antideriv.as_independent(x)[1]

        return indep*antideriv
    else:
        if retries >= 0:
            result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations)

            if result is not None:
                return indep*result

        return