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.venv/lib/python3.11/site-packages/mpmath/calculus/differentiation.py

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+from ..libmp.backend import xrange
+from .calculus import defun
+
+try:
+    iteritems = dict.iteritems
+except AttributeError:
+    iteritems = dict.items
+
+#----------------------------------------------------------------------------#
+#                                Differentiation                             #
+#----------------------------------------------------------------------------#
+
+@defun
+def difference(ctx, s, n):
+    r"""
+    Given a sequence `(s_k)` containing at least `n+1` items, returns the
+    `n`-th forward difference,
+
+    .. math ::
+
+        \Delta^n = \sum_{k=0}^{\infty} (-1)^{k+n} {n \choose k} s_k.
+    """
+    n = int(n)
+    d = ctx.zero
+    b = (-1) ** (n & 1)
+    for k in xrange(n+1):
+        d += b * s[k]
+        b = (b * (k-n)) // (k+1)
+    return d
+
+def hsteps(ctx, f, x, n, prec, **options):
+    singular = options.get('singular')
+    addprec = options.get('addprec', 10)
+    direction = options.get('direction', 0)
+    workprec = (prec+2*addprec) * (n+1)
+    orig = ctx.prec
+    try:
+        ctx.prec = workprec
+        h = options.get('h')
+        if h is None:
+            if options.get('relative'):
+                hextramag = int(ctx.mag(x))
+            else:
+                hextramag = 0
+            h = ctx.ldexp(1, -prec-addprec-hextramag)
+        else:
+            h = ctx.convert(h)
+        # Directed: steps x, x+h, ... x+n*h
+        direction = options.get('direction', 0)
+        if direction:
+            h *= ctx.sign(direction)
+            steps = xrange(n+1)
+            norm = h
+        # Central: steps x-n*h, x-(n-2)*h ..., x, ..., x+(n-2)*h, x+n*h
+        else:
+            steps = xrange(-n, n+1, 2)
+            norm = (2*h)
+        # Perturb
+        if singular:
+            x += 0.5*h
+        values = [f(x+k*h) for k in steps]
+        return values, norm, workprec
+    finally:
+        ctx.prec = orig
+
+
+@defun
+def diff(ctx, f, x, n=1, **options):
+    r"""
+    Numerically computes the derivative of `f`, `f'(x)`, or generally for
+    an integer `n \ge 0`, the `n`-th derivative `f^{(n)}(x)`.
+    A few basic examples are::
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> diff(lambda x: x**2 + x, 1.0)
+        3.0
+        >>> diff(lambda x: x**2 + x, 1.0, 2)
+        2.0
+        >>> diff(lambda x: x**2 + x, 1.0, 3)
+        0.0
+        >>> nprint([diff(exp, 3, n) for n in range(5)])   # exp'(x) = exp(x)
+        [20.0855, 20.0855, 20.0855, 20.0855, 20.0855]
+
+    Even more generally, given a tuple of arguments `(x_1, \ldots, x_k)`
+    and order `(n_1, \ldots, n_k)`, the partial derivative
+    `f^{(n_1,\ldots,n_k)}(x_1,\ldots,x_k)` is evaluated. For example::
+
+        >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (0,1))
+        2.75
+        >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (1,1))
+        3.0
+
+    **Options**
+
+    The following optional keyword arguments are recognized:
+
+    ``method``
+        Supported methods are ``'step'`` or ``'quad'``: derivatives may be
+        computed using either a finite difference with a small step
+        size `h` (default), or numerical quadrature.
+    ``direction``
+        Direction of finite difference: can be -1 for a left
+        difference, 0 for a central difference (default), or +1
+        for a right difference; more generally can be any complex number.
+    ``addprec``
+        Extra precision for `h` used to account for the function's
+        sensitivity to perturbations (default = 10).
+    ``relative``
+        Choose `h` relative to the magnitude of `x`, rather than an
+        absolute value; useful for large or tiny `x` (default = False).
+    ``h``
+        As an alternative to ``addprec`` and ``relative``, manually
+        select the step size `h`.
+    ``singular``
+        If True, evaluation exactly at the point `x` is avoided; this is
+        useful for differentiating functions with removable singularities.
+        Default = False.
+    ``radius``
+        Radius of integration contour (with ``method = 'quad'``).
+        Default = 0.25. A larger radius typically is faster and more
+        accurate, but it must be chosen so that `f` has no
+        singularities within the radius from the evaluation point.
+
+    A finite difference requires `n+1` function evaluations and must be
+    performed at `(n+1)` times the target precision. Accordingly, `f` must
+    support fast evaluation at high precision.
+
+    With integration, a larger number of function evaluations is
+    required, but not much extra precision is required. For high order
+    derivatives, this method may thus be faster if f is very expensive to
+    evaluate at high precision.
+
+    **Further examples**
+
+    The direction option is useful for computing left- or right-sided
+    derivatives of nonsmooth functions::
+
+        >>> diff(abs, 0, direction=0)
+        0.0
+        >>> diff(abs, 0, direction=1)
+        1.0
+        >>> diff(abs, 0, direction=-1)
+        -1.0
+
+    More generally, if the direction is nonzero, a right difference
+    is computed where the step size is multiplied by sign(direction).
+    For example, with direction=+j, the derivative from the positive
+    imaginary direction will be computed::
+
+        >>> diff(abs, 0, direction=j)
+        (0.0 - 1.0j)
+
+    With integration, the result may have a small imaginary part
+    even even if the result is purely real::
+
+        >>> diff(sqrt, 1, method='quad')    # doctest:+ELLIPSIS
+        (0.5 - 4.59...e-26j)
+        >>> chop(_)
+        0.5
+
+    Adding precision to obtain an accurate value::
+
+        >>> diff(cos, 1e-30)
+        0.0
+        >>> diff(cos, 1e-30, h=0.0001)
+        -9.99999998328279e-31
+        >>> diff(cos, 1e-30, addprec=100)
+        -1.0e-30
+
+    """
+    partial = False
+    try:
+        orders = list(n)
+        x = list(x)
+        partial = True
+    except TypeError:
+        pass
+    if partial:
+        x = [ctx.convert(_) for _ in x]
+        return _partial_diff(ctx, f, x, orders, options)
+    method = options.get('method', 'step')
+    if n == 0 and method != 'quad' and not options.get('singular'):
+        return f(ctx.convert(x))
+    prec = ctx.prec
+    try:
+        if method == 'step':
+            values, norm, workprec = hsteps(ctx, f, x, n, prec, **options)
+            ctx.prec = workprec
+            v = ctx.difference(values, n) / norm**n
+        elif method == 'quad':
+            ctx.prec += 10
+            radius = ctx.convert(options.get('radius', 0.25))
+            def g(t):
+                rei = radius*ctx.expj(t)
+                z = x + rei
+                return f(z) / rei**n
+            d = ctx.quadts(g, [0, 2*ctx.pi])
+            v = d * ctx.factorial(n) / (2*ctx.pi)
+        else:
+            raise ValueError("unknown method: %r" % method)
+    finally:
+        ctx.prec = prec
+    return +v
+
+def _partial_diff(ctx, f, xs, orders, options):
+    if not orders:
+        return f()
+    if not sum(orders):
+        return f(*xs)
+    i = 0
+    for i in range(len(orders)):
+        if orders[i]:
+            break
+    order = orders[i]
+    def fdiff_inner(*f_args):
+        def inner(t):
+            return f(*(f_args[:i] + (t,) + f_args[i+1:]))
+        return ctx.diff(inner, f_args[i], order, **options)
+    orders[i] = 0
+    return _partial_diff(ctx, fdiff_inner, xs, orders, options)
+
+@defun
+def diffs(ctx, f, x, n=None, **options):
+    r"""
+    Returns a generator that yields the sequence of derivatives
+
+    .. math ::
+
+        f(x), f'(x), f''(x), \ldots, f^{(k)}(x), \ldots
+
+    With ``method='step'``, :func:`~mpmath.diffs` uses only `O(k)`
+    function evaluations to generate the first `k` derivatives,
+    rather than the roughly `O(k^2)` evaluations
+    required if one calls :func:`~mpmath.diff` `k` separate times.
+
+    With `n < \infty`, the generator stops as soon as the
+    `n`-th derivative has been generated. If the exact number of
+    needed derivatives is known in advance, this is further
+    slightly more efficient.
+
+    Options are the same as for :func:`~mpmath.diff`.
+
+    **Examples**
+
+        >>> from mpmath import *
+        >>> mp.dps = 15
+        >>> nprint(list(diffs(cos, 1, 5)))
+        [0.540302, -0.841471, -0.540302, 0.841471, 0.540302, -0.841471]
+        >>> for i, d in zip(range(6), diffs(cos, 1)):
+        ...     print("%s %s" % (i, d))
+        ...
+        0 0.54030230586814
+        1 -0.841470984807897
+        2 -0.54030230586814
+        3 0.841470984807897
+        4 0.54030230586814
+        5 -0.841470984807897
+
+    """
+    if n is None:
+        n = ctx.inf
+    else:
+        n = int(n)
+    if options.get('method', 'step') != 'step':
+        k = 0
+        while k < n + 1:
+            yield ctx.diff(f, x, k, **options)
+            k += 1
+        return
+    singular = options.get('singular')
+    if singular:
+        yield ctx.diff(f, x, 0, singular=True)
+    else:
+        yield f(ctx.convert(x))
+    if n < 1:
+        return
+    if n == ctx.inf:
+        A, B = 1, 2
+    else:
+        A, B = 1, n+1
+    while 1:
+        callprec = ctx.prec
+        y, norm, workprec = hsteps(ctx, f, x, B, callprec, **options)
+        for k in xrange(A, B):
+            try:
+                ctx.prec = workprec
+                d = ctx.difference(y, k) / norm**k
+            finally:
+                ctx.prec = callprec
+            yield +d
+            if k >= n:
+                return
+        A, B = B, int(A*1.4+1)
+        B = min(B, n)
+
+def iterable_to_function(gen):
+    gen = iter(gen)
+    data = []
+    def f(k):
+        for i in xrange(len(data), k+1):
+            data.append(next(gen))
+        return data[k]
+    return f
+
+@defun
+def diffs_prod(ctx, factors):
+    r"""
+    Given a list of `N` iterables or generators yielding
+    `f_k(x), f'_k(x), f''_k(x), \ldots` for `k = 1, \ldots, N`,
+    generate `g(x), g'(x), g''(x), \ldots` where
+    `g(x) = f_1(x) f_2(x) \cdots f_N(x)`.
+
+    At high precision and for large orders, this is typically more efficient
+    than numerical differentiation if the derivatives of each `f_k(x)`
+    admit direct computation.
+
+    Note: This function does not increase the working precision internally,
+    so guard digits may have to be added externally for full accuracy.
+
+    **Examples**
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> f = lambda x: exp(x)*cos(x)*sin(x)
+        >>> u = diffs(f, 1)
+        >>> v = mp.diffs_prod([diffs(exp,1), diffs(cos,1), diffs(sin,1)])
+        >>> next(u); next(v)
+        1.23586333600241
+        1.23586333600241
+        >>> next(u); next(v)
+        0.104658952245596
+        0.104658952245596
+        >>> next(u); next(v)
+        -5.96999877552086
+        -5.96999877552086
+        >>> next(u); next(v)
+        -12.4632923122697
+        -12.4632923122697
+
+    """
+    N = len(factors)
+    if N == 1:
+        for c in factors[0]:
+            yield c
+    else:
+        u = iterable_to_function(ctx.diffs_prod(factors[:N//2]))
+        v = iterable_to_function(ctx.diffs_prod(factors[N//2:]))
+        n = 0
+        while 1:
+            #yield sum(binomial(n,k)*u(n-k)*v(k) for k in xrange(n+1))
+            s = u(n) * v(0)
+            a = 1
+            for k in xrange(1,n+1):
+                a = a * (n-k+1) // k
+                s += a * u(n-k) * v(k)
+            yield s
+            n += 1
+
+def dpoly(n, _cache={}):
+    """
+    nth differentiation polynomial for exp (Faa di Bruno's formula).
+
+    TODO: most exponents are zero, so maybe a sparse representation
+    would be better.
+    """
+    if n in _cache:
+        return _cache[n]
+    if not _cache:
+        _cache[0] = {(0,):1}
+    R = dpoly(n-1)
+    R = dict((c+(0,),v) for (c,v) in iteritems(R))
+    Ra = {}
+    for powers, count in iteritems(R):
+        powers1 = (powers[0]+1,) + powers[1:]
+        if powers1 in Ra:
+            Ra[powers1] += count
+        else:
+            Ra[powers1] = count
+    for powers, count in iteritems(R):
+        if not sum(powers):
+            continue
+        for k,p in enumerate(powers):
+            if p:
+                powers2 = powers[:k] + (p-1,powers[k+1]+1) + powers[k+2:]
+                if powers2 in Ra:
+                    Ra[powers2] += p*count
+                else:
+                    Ra[powers2] = p*count
+    _cache[n] = Ra
+    return _cache[n]
+
+@defun
+def diffs_exp(ctx, fdiffs):
+    r"""
+    Given an iterable or generator yielding `f(x), f'(x), f''(x), \ldots`
+    generate `g(x), g'(x), g''(x), \ldots` where `g(x) = \exp(f(x))`.
+
+    At high precision and for large orders, this is typically more efficient
+    than numerical differentiation if the derivatives of `f(x)`
+    admit direct computation.
+
+    Note: This function does not increase the working precision internally,
+    so guard digits may have to be added externally for full accuracy.
+
+    **Examples**
+
+    The derivatives of the gamma function can be computed using
+    logarithmic differentiation::
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>>
+        >>> def diffs_loggamma(x):
+        ...     yield loggamma(x)
+        ...     i = 0
+        ...     while 1:
+        ...         yield psi(i,x)
+        ...         i += 1
+        ...
+        >>> u = diffs_exp(diffs_loggamma(3))
+        >>> v = diffs(gamma, 3)
+        >>> next(u); next(v)
+        2.0
+        2.0
+        >>> next(u); next(v)
+        1.84556867019693
+        1.84556867019693
+        >>> next(u); next(v)
+        2.49292999190269
+        2.49292999190269
+        >>> next(u); next(v)
+        3.44996501352367
+        3.44996501352367
+
+    """
+    fn = iterable_to_function(fdiffs)
+    f0 = ctx.exp(fn(0))
+    yield f0
+    i = 1
+    while 1:
+        s = ctx.mpf(0)
+        for powers, c in iteritems(dpoly(i)):
+            s += c*ctx.fprod(fn(k+1)**p for (k,p) in enumerate(powers) if p)
+        yield s * f0
+        i += 1
+
+@defun
+def differint(ctx, f, x, n=1, x0=0):
+    r"""
+    Calculates the Riemann-Liouville differintegral, or fractional
+    derivative, defined by
+
+    .. math ::
+
+        \,_{x_0}{\mathbb{D}}^n_xf(x) = \frac{1}{\Gamma(m-n)} \frac{d^m}{dx^m}
+        \int_{x_0}^{x}(x-t)^{m-n-1}f(t)dt
+
+    where `f` is a given (presumably well-behaved) function,
+    `x` is the evaluation point, `n` is the order, and `x_0` is
+    the reference point of integration (`m` is an arbitrary
+    parameter selected automatically).
+
+    With `n = 1`, this is just the standard derivative `f'(x)`; with `n = 2`,
+    the second derivative `f''(x)`, etc. With `n = -1`, it gives
+    `\int_{x_0}^x f(t) dt`, with `n = -2`
+    it gives `\int_{x_0}^x \left( \int_{x_0}^t f(u) du \right) dt`, etc.
+
+    As `n` is permitted to be any number, this operator generalizes
+    iterated differentiation and iterated integration to a single
+    operator with a continuous order parameter.
+
+    **Examples**
+
+    There is an exact formula for the fractional derivative of a
+    monomial `x^p`, which may be used as a reference. For example,
+    the following gives a half-derivative (order 0.5)::
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> x = mpf(3); p = 2; n = 0.5
+        >>> differint(lambda t: t**p, x, n)
+        7.81764019044672
+        >>> gamma(p+1)/gamma(p-n+1) * x**(p-n)
+        7.81764019044672
+
+    Another useful test function is the exponential function, whose
+    integration / differentiation formula easy generalizes
+    to arbitrary order. Here we first compute a third derivative,
+    and then a triply nested integral. (The reference point `x_0`
+    is set to `-\infty` to avoid nonzero endpoint terms.)::
+
+        >>> differint(lambda x: exp(pi*x), -1.5, 3)
+        0.278538406900792
+        >>> exp(pi*-1.5) * pi**3
+        0.278538406900792
+        >>> differint(lambda x: exp(pi*x), 3.5, -3, -inf)
+        1922.50563031149
+        >>> exp(pi*3.5) / pi**3
+        1922.50563031149
+
+    However, for noninteger `n`, the differentiation formula for the
+    exponential function must be modified to give the same result as the
+    Riemann-Liouville differintegral::
+
+        >>> x = mpf(3.5)
+        >>> c = pi
+        >>> n = 1+2*j
+        >>> differint(lambda x: exp(c*x), x, n)
+        (-123295.005390743 + 140955.117867654j)
+        >>> x**(-n) * exp(c)**x * (x*c)**n * gammainc(-n, 0, x*c) / gamma(-n)
+        (-123295.005390743 + 140955.117867654j)
+
+
+    """
+    m = max(int(ctx.ceil(ctx.re(n)))+1, 1)
+    r = m-n-1
+    g = lambda x: ctx.quad(lambda t: (x-t)**r * f(t), [x0, x])
+    return ctx.diff(g, x, m) / ctx.gamma(m-n)
+
+@defun
+def diffun(ctx, f, n=1, **options):
+    r"""
+    Given a function `f`, returns a function `g(x)` that evaluates the nth
+    derivative `f^{(n)}(x)`::
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> cos2 = diffun(sin)
+        >>> sin2 = diffun(sin, 4)
+        >>> cos(1.3), cos2(1.3)
+        (0.267498828624587, 0.267498828624587)
+        >>> sin(1.3), sin2(1.3)
+        (0.963558185417193, 0.963558185417193)
+
+    The function `f` must support arbitrary precision evaluation.
+    See :func:`~mpmath.diff` for additional details and supported
+    keyword options.
+    """
+    if n == 0:
+        return f
+    def g(x):
+        return ctx.diff(f, x, n, **options)
+    return g
+
+@defun
+def taylor(ctx, f, x, n, **options):
+    r"""
+    Produces a degree-`n` Taylor polynomial around the point `x` of the
+    given function `f`. The coefficients are returned as a list.
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> nprint(chop(taylor(sin, 0, 5)))
+        [0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333]
+
+    The coefficients are computed using high-order numerical
+    differentiation. The function must be possible to evaluate
+    to arbitrary precision. See :func:`~mpmath.diff` for additional details
+    and supported keyword options.
+
+    Note that to evaluate the Taylor polynomial as an approximation
+    of `f`, e.g. with :func:`~mpmath.polyval`, the coefficients must be reversed,
+    and the point of the Taylor expansion must be subtracted from
+    the argument:
+
+        >>> p = taylor(exp, 2.0, 10)
+        >>> polyval(p[::-1], 2.5 - 2.0)
+        12.1824939606092
+        >>> exp(2.5)
+        12.1824939607035
+
+    """
+    gen = enumerate(ctx.diffs(f, x, n, **options))
+    if options.get("chop", True):
+        return [ctx.chop(d)/ctx.factorial(i) for i, d in gen]
+    else:
+        return [d/ctx.factorial(i) for i, d in gen]
+
+@defun
+def pade(ctx, a, L, M):
+    r"""
+    Computes a Pade approximation of degree `(L, M)` to a function.
+    Given at least `L+M+1` Taylor coefficients `a` approximating
+    a function `A(x)`, :func:`~mpmath.pade` returns coefficients of
+    polynomials `P, Q` satisfying
+
+    .. math ::
+
+        P = \sum_{k=0}^L p_k x^k
+
+        Q = \sum_{k=0}^M q_k x^k
+
+        Q_0 = 1
+
+        A(x) Q(x) = P(x) + O(x^{L+M+1})
+
+    `P(x)/Q(x)` can provide a good approximation to an analytic function
+    beyond the radius of convergence of its Taylor series (example
+    from G.A. Baker 'Essentials of Pade Approximants' Academic Press,
+    Ch.1A)::
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> one = mpf(1)
+        >>> def f(x):
+        ...     return sqrt((one + 2*x)/(one + x))
+        ...
+        >>> a = taylor(f, 0, 6)
+        >>> p, q = pade(a, 3, 3)
+        >>> x = 10
+        >>> polyval(p[::-1], x)/polyval(q[::-1], x)
+        1.38169105566806
+        >>> f(x)
+        1.38169855941551
+
+    """
+    # To determine L+1 coefficients of P and M coefficients of Q
+    # L+M+1 coefficients of A must be provided
+    if len(a) < L+M+1:
+        raise ValueError("L+M+1 Coefficients should be provided")
+
+    if M == 0:
+        if L == 0:
+            return [ctx.one], [ctx.one]
+        else:
+            return a[:L+1], [ctx.one]
+
+    # Solve first
+    # a[L]*q[1] + ... + a[L-M+1]*q[M] = -a[L+1]
+    # ...
+    # a[L+M-1]*q[1] + ... + a[L]*q[M] = -a[L+M]
+    A = ctx.matrix(M)
+    for j in range(M):
+        for i in range(min(M, L+j+1)):
+            A[j, i] = a[L+j-i]
+    v = -ctx.matrix(a[(L+1):(L+M+1)])
+    x = ctx.lu_solve(A, v)
+    q = [ctx.one] + list(x)
+    # compute p
+    p = [0]*(L+1)
+    for i in range(L+1):
+        s = a[i]
+        for j in range(1, min(M,i) + 1):
+            s += q[j]*a[i-j]
+        p[i] = s
+    return p, q

.venv/lib/python3.11/site-packages/mpmath/calculus/odes.py

@@ -0,0 +1,288 @@
+from bisect import bisect
+from ..libmp.backend import xrange
+
+class ODEMethods(object):
+    pass
+
+def ode_taylor(ctx, derivs, x0, y0, tol_prec, n):
+    h = tol = ctx.ldexp(1, -tol_prec)
+    dim = len(y0)
+    xs = [x0]
+    ys = [y0]
+    x = x0
+    y = y0
+    orig = ctx.prec
+    try:
+        ctx.prec = orig*(1+n)
+        # Use n steps with Euler's method to get
+        # evaluation points for derivatives
+        for i in range(n):
+            fxy = derivs(x, y)
+            y = [y[i]+h*fxy[i] for i in xrange(len(y))]
+            x += h
+            xs.append(x)
+            ys.append(y)
+        # Compute derivatives
+        ser = [[] for d in range(dim)]
+        for j in range(n+1):
+            s = [0]*dim
+            b = (-1) ** (j & 1)
+            k = 1
+            for i in range(j+1):
+                for d in range(dim):
+                    s[d] += b * ys[i][d]
+                b = (b * (j-k+1)) // (-k)
+                k += 1
+            scale = h**(-j) / ctx.fac(j)
+            for d in range(dim):
+                s[d] = s[d] * scale
+                ser[d].append(s[d])
+    finally:
+        ctx.prec = orig
+    # Estimate radius for which we can get full accuracy.
+    # XXX: do this right for zeros
+    radius = ctx.one
+    for ts in ser:
+        if ts[-1]:
+            radius = min(radius, ctx.nthroot(tol/abs(ts[-1]), n))
+    radius /= 2  # XXX
+    return ser, x0+radius
+
+def odefun(ctx, F, x0, y0, tol=None, degree=None, method='taylor', verbose=False):
+    r"""
+    Returns a function `y(x) = [y_0(x), y_1(x), \ldots, y_n(x)]`
+    that is a numerical solution of the `n+1`-dimensional first-order
+    ordinary differential equation (ODE) system
+
+    .. math ::
+
+        y_0'(x) = F_0(x, [y_0(x), y_1(x), \ldots, y_n(x)])
+
+        y_1'(x) = F_1(x, [y_0(x), y_1(x), \ldots, y_n(x)])
+
+        \vdots
+
+        y_n'(x) = F_n(x, [y_0(x), y_1(x), \ldots, y_n(x)])
+
+    The derivatives are specified by the vector-valued function
+    *F* that evaluates
+    `[y_0', \ldots, y_n'] = F(x, [y_0, \ldots, y_n])`.
+    The initial point `x_0` is specified by the scalar argument *x0*,
+    and the initial value `y(x_0) =  [y_0(x_0), \ldots, y_n(x_0)]` is
+    specified by the vector argument *y0*.
+
+    For convenience, if the system is one-dimensional, you may optionally
+    provide just a scalar value for *y0*. In this case, *F* should accept
+    a scalar *y* argument and return a scalar. The solution function
+    *y* will return scalar values instead of length-1 vectors.
+
+    Evaluation of the solution function `y(x)` is permitted
+    for any `x \ge x_0`.
+
+    A high-order ODE can be solved by transforming it into first-order
+    vector form. This transformation is described in standard texts
+    on ODEs. Examples will also be given below.
+
+    **Options, speed and accuracy**
+
+    By default, :func:`~mpmath.odefun` uses a high-order Taylor series
+    method. For reasonably well-behaved problems, the solution will
+    be fully accurate to within the working precision. Note that
+    *F* must be possible to evaluate to very high precision
+    for the generation of Taylor series to work.
+
+    To get a faster but less accurate solution, you can set a large
+    value for *tol* (which defaults roughly to *eps*). If you just
+    want to plot the solution or perform a basic simulation,
+    *tol = 0.01* is likely sufficient.
+
+    The *degree* argument controls the degree of the solver (with
+    *method='taylor'*, this is the degree of the Taylor series
+    expansion). A higher degree means that a longer step can be taken
+    before a new local solution must be generated from *F*,
+    meaning that fewer steps are required to get from `x_0` to a given
+    `x_1`. On the other hand, a higher degree also means that each
+    local solution becomes more expensive (i.e., more evaluations of
+    *F* are required per step, and at higher precision).
+
+    The optimal setting therefore involves a tradeoff. Generally,
+    decreasing the *degree* for Taylor series is likely to give faster
+    solution at low precision, while increasing is likely to be better
+    at higher precision.
+
+    The function
+    object returned by :func:`~mpmath.odefun` caches the solutions at all step
+    points and uses polynomial interpolation between step points.
+    Therefore, once `y(x_1)` has been evaluated for some `x_1`,
+    `y(x)` can be evaluated very quickly for any `x_0 \le x \le x_1`.
+    and continuing the evaluation up to `x_2 > x_1` is also fast.
+
+    **Examples of first-order ODEs**
+
+    We will solve the standard test problem `y'(x) = y(x), y(0) = 1`
+    which has explicit solution `y(x) = \exp(x)`::
+
+        >>> from mpmath import *
+        >>> mp.dps = 15; mp.pretty = True
+        >>> f = odefun(lambda x, y: y, 0, 1)
+        >>> for x in [0, 1, 2.5]:
+        ...     print((f(x), exp(x)))
+        ...
+        (1.0, 1.0)
+        (2.71828182845905, 2.71828182845905)
+        (12.1824939607035, 12.1824939607035)
+
+    The solution with high precision::
+
+        >>> mp.dps = 50
+        >>> f = odefun(lambda x, y: y, 0, 1)
+        >>> f(1)
+        2.7182818284590452353602874713526624977572470937
+        >>> exp(1)
+        2.7182818284590452353602874713526624977572470937
+
+    Using the more general vectorized form, the test problem
+    can be input as (note that *f* returns a 1-element vector)::
+
+        >>> mp.dps = 15
+        >>> f = odefun(lambda x, y: [y[0]], 0, [1])
+        >>> f(1)
+        [2.71828182845905]
+
+    :func:`~mpmath.odefun` can solve nonlinear ODEs, which are generally
+    impossible (and at best difficult) to solve analytically. As
+    an example of a nonlinear ODE, we will solve `y'(x) = x \sin(y(x))`
+    for `y(0) = \pi/2`. An exact solution happens to be known
+    for this problem, and is given by
+    `y(x) = 2 \tan^{-1}\left(\exp\left(x^2/2\right)\right)`::
+
+        >>> f = odefun(lambda x, y: x*sin(y), 0, pi/2)
+        >>> for x in [2, 5, 10]:
+        ...     print((f(x), 2*atan(exp(mpf(x)**2/2))))
+        ...
+        (2.87255666284091, 2.87255666284091)
+        (3.14158520028345, 3.14158520028345)
+        (3.14159265358979, 3.14159265358979)
+
+    If `F` is independent of `y`, an ODE can be solved using direct
+    integration. We can therefore obtain a reference solution with
+    :func:`~mpmath.quad`::
+
+        >>> f = lambda x: (1+x**2)/(1+x**3)
+        >>> g = odefun(lambda x, y: f(x), pi, 0)
+        >>> g(2*pi)
+        0.72128263801696
+        >>> quad(f, [pi, 2*pi])
+        0.72128263801696
+
+    **Examples of second-order ODEs**
+
+    We will solve the harmonic oscillator equation `y''(x) + y(x) = 0`.
+    To do this, we introduce the helper functions `y_0 = y, y_1 = y_0'`
+    whereby the original equation can be written as `y_1' + y_0' = 0`. Put
+    together, we get the first-order, two-dimensional vector ODE
+
+    .. math ::
+
+        \begin{cases}
+        y_0' = y_1 \\
+        y_1' = -y_0
+        \end{cases}
+
+    To get a well-defined IVP, we need two initial values. With
+    `y(0) = y_0(0) = 1` and `-y'(0) = y_1(0) = 0`, the problem will of
+    course be solved by `y(x) = y_0(x) = \cos(x)` and
+    `-y'(x) = y_1(x) = \sin(x)`. We check this::
+
+        >>> f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
+        >>> for x in [0, 1, 2.5, 10]:
+        ...     nprint(f(x), 15)
+        ...     nprint([cos(x), sin(x)], 15)
+        ...     print("---")
+        ...
+        [1.0, 0.0]
+        [1.0, 0.0]
+        ---
+        [0.54030230586814, 0.841470984807897]
+        [0.54030230586814, 0.841470984807897]
+        ---
+        [-0.801143615546934, 0.598472144103957]
+        [-0.801143615546934, 0.598472144103957]
+        ---
+        [-0.839071529076452, -0.54402111088937]
+        [-0.839071529076452, -0.54402111088937]
+        ---
+
+    Note that we get both the sine and the cosine solutions
+    simultaneously.
+
+    **TODO**
+
+    * Better automatic choice of degree and step size
+    * Make determination of Taylor series convergence radius
+      more robust
+    * Allow solution for `x < x_0`
+    * Allow solution for complex `x`
+    * Test for difficult (ill-conditioned) problems
+    * Implement Runge-Kutta and other algorithms
+
+    """
+    if tol:
+        tol_prec = int(-ctx.log(tol, 2))+10
+    else:
+        tol_prec = ctx.prec+10
+    degree = degree or (3 + int(3*ctx.dps/2.))
+    workprec = ctx.prec + 40
+    try:
+        len(y0)
+        return_vector = True
+    except TypeError:
+        F_ = F
+        F = lambda x, y: [F_(x, y[0])]
+        y0 = [y0]
+        return_vector = False
+    ser, xb = ode_taylor(ctx, F, x0, y0, tol_prec, degree)
+    series_boundaries = [x0, xb]
+    series_data = [(ser, x0, xb)]
+    # We will be working with vectors of Taylor series
+    def mpolyval(ser, a):
+        return [ctx.polyval(s[::-1], a) for s in ser]
+    # Find nearest expansion point; compute if necessary
+    def get_series(x):
+        if x < x0:
+            raise ValueError
+        n = bisect(series_boundaries, x)
+        if n < len(series_boundaries):
+            return series_data[n-1]
+        while 1:
+            ser, xa, xb = series_data[-1]
+            if verbose:
+                print("Computing Taylor series for [%f, %f]" % (xa, xb))
+            y = mpolyval(ser, xb-xa)
+            xa = xb
+            ser, xb = ode_taylor(ctx, F, xb, y, tol_prec, degree)
+            series_boundaries.append(xb)
+            series_data.append((ser, xa, xb))
+            if x <= xb:
+                return series_data[-1]
+    # Evaluation function
+    def interpolant(x):
+        x = ctx.convert(x)
+        orig = ctx.prec
+        try:
+            ctx.prec = workprec
+            ser, xa, xb = get_series(x)
+            y = mpolyval(ser, x-xa)
+        finally:
+            ctx.prec = orig
+        if return_vector:
+            return [+yk for yk in y]
+        else:
+            return +y[0]
+    return interpolant
+
+ODEMethods.odefun = odefun
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()

.venv/lib/python3.11/site-packages/mpmath/calculus/optimization.py

@@ -0,0 +1,1102 @@
+from __future__ import print_function
+
+from copy import copy
+
+from ..libmp.backend import xrange
+
+class OptimizationMethods(object):
+    def __init__(ctx):
+        pass
+
+##############
+# 1D-SOLVERS #
+##############
+
+class Newton:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Needs starting points x0 close to the root.
+
+    Pro:
+
+    * converges fast
+    * sometimes more robust than secant with bad second starting point
+
+    Contra:
+
+    * converges slowly for multiple roots
+    * needs first derivative
+    * 2 function evaluations per iteration
+    """
+    maxsteps = 20
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if len(x0) == 1:
+            self.x0 = x0[0]
+        else:
+            raise ValueError('expected 1 starting point, got %i' % len(x0))
+        self.f = f
+        if not 'df' in kwargs:
+            def df(x):
+                return self.ctx.diff(f, x)
+        else:
+            df = kwargs['df']
+        self.df = df
+
+    def __iter__(self):
+        f = self.f
+        df = self.df
+        x0 = self.x0
+        while True:
+            x1 = x0 - f(x0) / df(x0)
+            error = abs(x1 - x0)
+            x0 = x1
+            yield (x1, error)
+
+class Secant:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Needs starting points x0 and x1 close to the root.
+    x1 defaults to x0 + 0.25.
+
+    Pro:
+
+    * converges fast
+
+    Contra:
+
+    * converges slowly for multiple roots
+    """
+    maxsteps = 30
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if len(x0) == 1:
+            self.x0 = x0[0]
+            self.x1 = self.x0 + 0.25
+        elif len(x0) == 2:
+            self.x0 = x0[0]
+            self.x1 = x0[1]
+        else:
+            raise ValueError('expected 1 or 2 starting points, got %i' % len(x0))
+        self.f = f
+
+    def __iter__(self):
+        f = self.f
+        x0 = self.x0
+        x1 = self.x1
+        f0 = f(x0)
+        while True:
+            f1 = f(x1)
+            l = x1 - x0
+            if not l:
+                break
+            s = (f1 - f0) / l
+            if not s:
+                break
+            x0, x1 = x1, x1 - f1/s
+            f0 = f1
+            yield x1, abs(l)
+
+class MNewton:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Needs starting point x0 close to the root.
+    Uses modified Newton's method that converges fast regardless of the
+    multiplicity of the root.
+
+    Pro:
+
+    * converges fast for multiple roots
+
+    Contra:
+
+    * needs first and second derivative of f
+    * 3 function evaluations per iteration
+    """
+    maxsteps = 20
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if not len(x0) == 1:
+            raise ValueError('expected 1 starting point, got %i' % len(x0))
+        self.x0 = x0[0]
+        self.f = f
+        if not 'df' in kwargs:
+            def df(x):
+                return self.ctx.diff(f, x)
+        else:
+            df = kwargs['df']
+        self.df = df
+        if not 'd2f' in kwargs:
+            def d2f(x):
+                return self.ctx.diff(df, x)
+        else:
+            d2f = kwargs['df']
+        self.d2f = d2f
+
+    def __iter__(self):
+        x = self.x0
+        f = self.f
+        df = self.df
+        d2f = self.d2f
+        while True:
+            prevx = x
+            fx = f(x)
+            if fx == 0:
+                break
+            dfx = df(x)
+            d2fx = d2f(x)
+            # x = x - F(x)/F'(x) with F(x) = f(x)/f'(x)
+            x -= fx / (dfx - fx * d2fx / dfx)
+            error = abs(x - prevx)
+            yield x, error
+
+class Halley:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Needs a starting point x0 close to the root.
+    Uses Halley's method with cubic convergence rate.
+
+    Pro:
+
+    * converges even faster the Newton's method
+    * useful when computing with *many* digits
+
+    Contra:
+
+    * needs first and second derivative of f
+    * 3 function evaluations per iteration
+    * converges slowly for multiple roots
+    """
+
+    maxsteps = 20
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if not len(x0) == 1:
+            raise ValueError('expected 1 starting point, got %i' % len(x0))
+        self.x0 = x0[0]
+        self.f = f
+        if not 'df' in kwargs:
+            def df(x):
+                return self.ctx.diff(f, x)
+        else:
+            df = kwargs['df']
+        self.df = df
+        if not 'd2f' in kwargs:
+            def d2f(x):
+                return self.ctx.diff(df, x)
+        else:
+            d2f = kwargs['df']
+        self.d2f = d2f
+
+    def __iter__(self):
+        x = self.x0
+        f = self.f
+        df = self.df
+        d2f = self.d2f
+        while True:
+            prevx = x
+            fx = f(x)
+            dfx = df(x)
+            d2fx = d2f(x)
+            x -=  2*fx*dfx / (2*dfx**2 - fx*d2fx)
+            error = abs(x - prevx)
+            yield x, error
+
+class Muller:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Needs starting points x0, x1 and x2 close to the root.
+    x1 defaults to x0 + 0.25; x2 to x1 + 0.25.
+    Uses Muller's method that converges towards complex roots.
+
+    Pro:
+
+    * converges fast (somewhat faster than secant)
+    * can find complex roots
+
+    Contra:
+
+    * converges slowly for multiple roots
+    * may have complex values for real starting points and real roots
+
+    http://en.wikipedia.org/wiki/Muller's_method
+    """
+    maxsteps = 30
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if len(x0) == 1:
+            self.x0 = x0[0]
+            self.x1 = self.x0 + 0.25
+            self.x2 = self.x1 + 0.25
+        elif len(x0) == 2:
+            self.x0 = x0[0]
+            self.x1 = x0[1]
+            self.x2 = self.x1 + 0.25
+        elif len(x0) == 3:
+            self.x0 = x0[0]
+            self.x1 = x0[1]
+            self.x2 = x0[2]
+        else:
+            raise ValueError('expected 1, 2 or 3 starting points, got %i'
+                             % len(x0))
+        self.f = f
+        self.verbose = kwargs['verbose']
+
+    def __iter__(self):
+        f = self.f
+        x0 = self.x0
+        x1 = self.x1
+        x2 = self.x2
+        fx0 = f(x0)
+        fx1 = f(x1)
+        fx2 = f(x2)
+        while True:
+            # TODO: maybe refactoring with function for divided differences
+            # calculate divided differences
+            fx2x1 = (fx1 - fx2) / (x1 - x2)
+            fx2x0 = (fx0 - fx2) / (x0 - x2)
+            fx1x0 = (fx0 - fx1) / (x0 - x1)
+            w = fx2x1 + fx2x0 - fx1x0
+            fx2x1x0 = (fx1x0 - fx2x1) / (x0 - x2)
+            if w == 0 and fx2x1x0 == 0:
+                if self.verbose:
+                    print('canceled with')
+                    print('x0 =', x0, ', x1 =', x1, 'and x2 =', x2)
+                break
+            x0 = x1
+            fx0 = fx1
+            x1 = x2
+            fx1 = fx2
+            # denominator should be as large as possible => choose sign
+            r = self.ctx.sqrt(w**2 - 4*fx2*fx2x1x0)
+            if abs(w - r) > abs(w + r):
+                r = -r
+            x2 -= 2*fx2 / (w + r)
+            fx2 = f(x2)
+            error = abs(x2 - x1)
+            yield x2, error
+
+# TODO: consider raising a ValueError when there's no sign change in a and b
+class Bisection:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Uses bisection method to find a root of f in [a, b].
+    Might fail for multiple roots (needs sign change).
+
+    Pro:
+
+    * robust and reliable
+
+    Contra:
+
+    * converges slowly
+    * needs sign change
+    """
+    maxsteps = 100
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if len(x0) != 2:
+            raise ValueError('expected interval of 2 points, got %i' % len(x0))
+        self.f = f
+        self.a = x0[0]
+        self.b = x0[1]
+
+    def __iter__(self):
+        f = self.f
+        a = self.a
+        b = self.b
+        l = b - a
+        fb = f(b)
+        while True:
+            m = self.ctx.ldexp(a + b, -1)
+            fm = f(m)
+            sign = fm * fb
+            if sign < 0:
+                a = m
+            elif sign > 0:
+                b = m
+                fb = fm
+            else:
+                yield m, self.ctx.zero
+            l /= 2
+            yield (a + b)/2, abs(l)
+
+def _getm(method):
+    """
+    Return a function to calculate m for Illinois-like methods.
+    """
+    if method == 'illinois':
+        def getm(fz, fb):
+            return 0.5
+    elif method == 'pegasus':
+        def getm(fz, fb):
+            return fb/(fb + fz)
+    elif method == 'anderson':
+        def getm(fz, fb):
+            m = 1 - fz/fb
+            if m > 0:
+                return m
+            else:
+                return 0.5
+    else:
+        raise ValueError("method '%s' not recognized" % method)
+    return getm
+
+class Illinois:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Uses Illinois method or similar to find a root of f in [a, b].
+    Might fail for multiple roots (needs sign change).
+    Combines bisect with secant (improved regula falsi).
+
+    The only difference between the methods is the scaling factor m, which is
+    used to ensure convergence (you can choose one using the 'method' keyword):
+
+    Illinois method ('illinois'):
+        m = 0.5
+
+    Pegasus method ('pegasus'):
+        m = fb/(fb + fz)
+
+    Anderson-Bjoerk method ('anderson'):
+        m = 1 - fz/fb if positive else 0.5
+
+    Pro:
+
+    * converges very fast
+
+    Contra:
+
+    * has problems with multiple roots
+    * needs sign change
+    """
+    maxsteps = 30
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if len(x0) != 2:
+            raise ValueError('expected interval of 2 points, got %i' % len(x0))
+        self.a = x0[0]
+        self.b = x0[1]
+        self.f = f
+        self.tol = kwargs['tol']
+        self.verbose = kwargs['verbose']
+        self.method = kwargs.get('method', 'illinois')
+        self.getm = _getm(self.method)
+        if self.verbose:
+            print('using %s method' % self.method)
+
+    def __iter__(self):
+        method = self.method
+        f = self.f
+        a = self.a
+        b = self.b
+        fa = f(a)
+        fb = f(b)
+        m = None
+        while True:
+            l = b - a
+            if l == 0:
+                break
+            s = (fb - fa) / l
+            z = a - fa/s
+            fz = f(z)
+            if abs(fz) < self.tol:
+                # TODO: better condition (when f is very flat)
+                if self.verbose:
+                    print('canceled with z =', z)
+                yield z, l
+                break
+            if fz * fb < 0: # root in [z, b]
+                a = b
+                fa = fb
+                b = z
+                fb = fz
+            else: # root in [a, z]
+                m = self.getm(fz, fb)
+                b = z
+                fb = fz
+                fa = m*fa # scale down to ensure convergence
+            if self.verbose and m and not method == 'illinois':
+                print('m:', m)
+            yield (a + b)/2, abs(l)
+
+def Pegasus(*args, **kwargs):
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Uses Pegasus method to find a root of f in [a, b].
+    Wrapper for illinois to use method='pegasus'.
+    """
+    kwargs['method'] = 'pegasus'
+    return Illinois(*args, **kwargs)
+
+def Anderson(*args, **kwargs):
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Uses Anderson-Bjoerk method to find a root of f in [a, b].
+    Wrapper for illinois to use method='pegasus'.
+    """
+    kwargs['method'] = 'anderson'
+    return Illinois(*args, **kwargs)
+
+# TODO: check whether it's possible to combine it with Illinois stuff
+class Ridder:
+    """
+    1d-solver generating pairs of approximative root and error.
+
+    Ridders' method to find a root of f in [a, b].
+    Is told to perform as well as Brent's method while being simpler.
+
+    Pro:
+
+    * very fast
+    * simpler than Brent's method
+
+    Contra:
+
+    * two function evaluations per step
+    * has problems with multiple roots
+    * needs sign change
+
+    http://en.wikipedia.org/wiki/Ridders'_method
+    """
+    maxsteps = 30
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        self.f = f
+        if len(x0) != 2:
+            raise ValueError('expected interval of 2 points, got %i' % len(x0))
+        self.x1 = x0[0]
+        self.x2 = x0[1]
+        self.verbose = kwargs['verbose']
+        self.tol = kwargs['tol']
+
+    def __iter__(self):
+        ctx = self.ctx
+        f = self.f
+        x1 = self.x1
+        fx1 = f(x1)
+        x2 = self.x2
+        fx2 = f(x2)
+        while True:
+            x3 = 0.5*(x1 + x2)
+            fx3 = f(x3)
+            x4 = x3 + (x3 - x1) * ctx.sign(fx1 - fx2) * fx3 / ctx.sqrt(fx3**2 - fx1*fx2)
+            fx4 = f(x4)
+            if abs(fx4) < self.tol:
+                # TODO: better condition (when f is very flat)
+                if self.verbose:
+                    print('canceled with f(x4) =', fx4)
+                yield x4, abs(x1 - x2)
+                break
+            if fx4 * fx2 < 0: # root in [x4, x2]
+                x1 = x4
+                fx1 = fx4
+            else: # root in [x1, x4]
+                x2 = x4
+                fx2 = fx4
+            error = abs(x1 - x2)
+            yield (x1 + x2)/2, error
+
+class ANewton:
+    """
+    EXPERIMENTAL 1d-solver generating pairs of approximative root and error.
+
+    Uses Newton's method modified to use Steffensens method when convergence is
+    slow. (I.e. for multiple roots.)
+    """
+    maxsteps = 20
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        if not len(x0) == 1:
+            raise ValueError('expected 1 starting point, got %i' % len(x0))
+        self.x0 = x0[0]
+        self.f = f
+        if not 'df' in kwargs:
+            def df(x):
+                return self.ctx.diff(f, x)
+        else:
+            df = kwargs['df']
+        self.df = df
+        def phi(x):
+            return x - f(x) / df(x)
+        self.phi = phi
+        self.verbose = kwargs['verbose']
+
+    def __iter__(self):
+        x0 = self.x0
+        f = self.f
+        df = self.df
+        phi = self.phi
+        error = 0
+        counter = 0
+        while True:
+            prevx = x0
+            try:
+                x0 = phi(x0)
+            except ZeroDivisionError:
+                if self.verbose:
+                    print('ZeroDivisionError: canceled with x =', x0)
+                break
+            preverror = error
+            error = abs(prevx - x0)
+            # TODO: decide not to use convergence acceleration
+            if error and abs(error - preverror) / error < 1:
+                if self.verbose:
+                    print('converging slowly')
+                counter += 1
+            if counter >= 3:
+                # accelerate convergence
+                phi = steffensen(phi)
+                counter = 0
+                if self.verbose:
+                    print('accelerating convergence')
+            yield x0, error
+
+# TODO: add Brent
+
+############################
+# MULTIDIMENSIONAL SOLVERS #
+############################
+
+def jacobian(ctx, f, x):
+    """
+    Calculate the Jacobian matrix of a function at the point x0.
+
+    This is the first derivative of a vectorial function:
+
+        f : R^m -> R^n with m >= n
+    """
+    x = ctx.matrix(x)
+    h = ctx.sqrt(ctx.eps)
+    fx = ctx.matrix(f(*x))
+    m = len(fx)
+    n = len(x)
+    J = ctx.matrix(m, n)
+    for j in xrange(n):
+        xj = x.copy()
+        xj[j] += h
+        Jj = (ctx.matrix(f(*xj)) - fx) / h
+        for i in xrange(m):
+            J[i,j] = Jj[i]
+    return J
+
+# TODO: test with user-specified jacobian matrix
+class MDNewton:
+    """
+    Find the root of a vector function numerically using Newton's method.
+
+    f is a vector function representing a nonlinear equation system.
+
+    x0 is the starting point close to the root.
+
+    J is a function returning the Jacobian matrix for a point.
+
+    Supports overdetermined systems.
+
+    Use the 'norm' keyword to specify which norm to use. Defaults to max-norm.
+    The function to calculate the Jacobian matrix can be given using the
+    keyword 'J'. Otherwise it will be calculated numerically.
+
+    Please note that this method converges only locally. Especially for high-
+    dimensional systems it is not trivial to find a good starting point being
+    close enough to the root.
+
+    It is recommended to use a faster, low-precision solver from SciPy [1] or
+    OpenOpt [2] to get an initial guess. Afterwards you can use this method for
+    root-polishing to any precision.
+
+    [1] http://scipy.org
+
+    [2] http://openopt.org/Welcome
+    """
+    maxsteps = 10
+
+    def __init__(self, ctx, f, x0, **kwargs):
+        self.ctx = ctx
+        self.f = f
+        if isinstance(x0, (tuple, list)):
+            x0 = ctx.matrix(x0)
+        assert x0.cols == 1, 'need a vector'
+        self.x0 = x0
+        if 'J' in kwargs:
+            self.J = kwargs['J']
+        else:
+            def J(*x):
+                return ctx.jacobian(f, x)
+            self.J = J
+        self.norm = kwargs['norm']
+        self.verbose = kwargs['verbose']
+
+    def __iter__(self):
+        f = self.f
+        x0 = self.x0
+        norm = self.norm
+        J = self.J
+        fx = self.ctx.matrix(f(*x0))
+        fxnorm = norm(fx)
+        cancel = False
+        while not cancel:
+            # get direction of descent
+            fxn = -fx
+            Jx = J(*x0)
+            s = self.ctx.lu_solve(Jx, fxn)
+            if self.verbose:
+                print('Jx:')
+                print(Jx)
+                print('s:', s)
+            # damping step size TODO: better strategy (hard task)
+            l = self.ctx.one
+            x1 = x0 + s
+            while True:
+                if x1 == x0:
+                    if self.verbose:
+                        print("canceled, won't get more excact")
+                    cancel = True
+                    break
+                fx = self.ctx.matrix(f(*x1))
+                newnorm = norm(fx)
+                if newnorm < fxnorm:
+                    # new x accepted
+                    fxnorm = newnorm
+                    x0 = x1
+                    break
+                l /= 2
+                x1 = x0 + l*s
+            yield (x0, fxnorm)
+
+#############
+# UTILITIES #
+#############
+
+str2solver = {'newton':Newton, 'secant':Secant, 'mnewton':MNewton,
+              'halley':Halley, 'muller':Muller, 'bisect':Bisection,
+              'illinois':Illinois, 'pegasus':Pegasus, 'anderson':Anderson,
+              'ridder':Ridder, 'anewton':ANewton, 'mdnewton':MDNewton}
+
+def findroot(ctx, f, x0, solver='secant', tol=None, verbose=False, verify=True, **kwargs):
+    r"""
+    Find an approximate solution to `f(x) = 0`, using *x0* as starting point or
+    interval for *x*.
+
+    Multidimensional overdetermined systems are supported.
+    You can specify them using a function or a list of functions.
+
+    Mathematically speaking, this function returns `x` such that
+    `|f(x)|^2 \leq \mathrm{tol}` is true within the current working precision.
+    If the computed value does not meet this criterion, an exception is raised.
+    This exception can be disabled with *verify=False*.
+
+    For interval arithmetic (``iv.findroot()``), please note that
+    the returned interval ``x`` is not guaranteed to contain `f(x)=0`!
+    It is only some `x` for which `|f(x)|^2 \leq \mathrm{tol}` certainly holds
+    regardless of numerical error. This may be improved in the future.
+
+    **Arguments**
+
+    *f*
+        one dimensional function
+    *x0*
+        starting point, several starting points or interval (depends on solver)
+    *tol*
+        the returned solution has an error smaller than this
+    *verbose*
+        print additional information for each iteration if true
+    *verify*
+        verify the solution and raise a ValueError if `|f(x)|^2 > \mathrm{tol}`
+    *solver*
+        a generator for *f* and *x0* returning approximative solution and error
+    *maxsteps*
+        after how many steps the solver will cancel
+    *df*
+        first derivative of *f* (used by some solvers)
+    *d2f*
+        second derivative of *f* (used by some solvers)
+    *multidimensional*
+        force multidimensional solving
+    *J*
+        Jacobian matrix of *f* (used by multidimensional solvers)
+    *norm*
+        used vector norm (used by multidimensional solvers)
+
+    solver has to be callable with ``(f, x0, **kwargs)`` and return an generator
+    yielding pairs of approximative solution and estimated error (which is
+    expected to be positive).
+    You can use the following string aliases:
+    'secant', 'mnewton', 'halley', 'muller', 'illinois', 'pegasus', 'anderson',
+    'ridder', 'anewton', 'bisect'
+
+    See mpmath.calculus.optimization for their documentation.
+
+    **Examples**
+
+    The function :func:`~mpmath.findroot` locates a root of a given function using the
+    secant method by default. A simple example use of the secant method is to
+    compute `\pi` as the root of `\sin x` closest to `x_0 = 3`::
+
+        >>> from mpmath import *
+        >>> mp.dps = 30; mp.pretty = True
+        >>> findroot(sin, 3)
+        3.14159265358979323846264338328
+
+    The secant method can be used to find complex roots of analytic functions,
+    although it must in that case generally be given a nonreal starting value
+    (or else it will never leave the real line)::
+
+        >>> mp.dps = 15
+        >>> findroot(lambda x: x**3 + 2*x + 1, j)
+        (0.226698825758202 + 1.46771150871022j)
+
+    A nice application is to compute nontrivial roots of the Riemann zeta
+    function with many digits (good initial values are needed for convergence)::
+
+        >>> mp.dps = 30
+        >>> findroot(zeta, 0.5+14j)
+        (0.5 + 14.1347251417346937904572519836j)
+
+    The secant method can also be used as an optimization algorithm, by passing
+    it a derivative of a function. The following example locates the positive
+    minimum of the gamma function::
+
+        >>> mp.dps = 20
+        >>> findroot(lambda x: diff(gamma, x), 1)
+        1.4616321449683623413
+
+    Finally, a useful application is to compute inverse functions, such as the
+    Lambert W function which is the inverse of `w e^w`, given the first
+    term of the solution's asymptotic expansion as the initial value. In basic
+    cases, this gives identical results to mpmath's built-in ``lambertw``
+    function::
+
+        >>> def lambert(x):
+        ...     return findroot(lambda w: w*exp(w) - x, log(1+x))
+        ...
+        >>> mp.dps = 15
+        >>> lambert(1); lambertw(1)
+        0.567143290409784
+        0.567143290409784
+        >>> lambert(1000); lambert(1000)
+        5.2496028524016
+        5.2496028524016
+
+    Multidimensional functions are also supported::
+
+        >>> f = [lambda x1, x2: x1**2 + x2,
+        ...      lambda x1, x2: 5*x1**2 - 3*x1 + 2*x2 - 3]
+        >>> findroot(f, (0, 0))
+        [-0.618033988749895]
+        [-0.381966011250105]
+        >>> findroot(f, (10, 10))
+        [ 1.61803398874989]
+        [-2.61803398874989]
+
+    You can verify this by solving the system manually.
+
+    Please note that the following (more general) syntax also works::
+
+        >>> def f(x1, x2):
+        ...     return x1**2 + x2, 5*x1**2 - 3*x1 + 2*x2 - 3
+        ...
+        >>> findroot(f, (0, 0))
+        [-0.618033988749895]
+        [-0.381966011250105]
+
+
+    **Multiple roots**
+
+    For multiple roots all methods of the Newtonian family (including secant)
+    converge slowly. Consider this example::
+
+        >>> f = lambda x: (x - 1)**99
+        >>> findroot(f, 0.9, verify=False)
+        0.918073542444929
+
+    Even for a very close starting point the secant method converges very
+    slowly. Use ``verbose=True`` to illustrate this.
+
+    It is possible to modify Newton's method to make it converge regardless of
+    the root's multiplicity::
+
+        >>> findroot(f, -10, solver='mnewton')
+        1.0
+
+    This variant uses the first and second derivative of the function, which is
+    not very efficient.
+
+    Alternatively you can use an experimental Newtonian solver that keeps track
+    of the speed of convergence and accelerates it using Steffensen's method if
+    necessary::
+
+        >>> findroot(f, -10, solver='anewton', verbose=True)
+        x:     -9.88888888888888888889
+        error: 0.111111111111111111111
+        converging slowly
+        x:     -9.77890011223344556678
+        error: 0.10998877665544332211
+        converging slowly
+        x:     -9.67002233332199662166
+        error: 0.108877778911448945119
+        converging slowly
+        accelerating convergence
+        x:     -9.5622443299551077669
+        error: 0.107778003366888854764
+        converging slowly
+        x:     0.99999999999999999214
+        error: 10.562244329955107759
+        x:     1.0
+        error: 7.8598304758094664213e-18
+        ZeroDivisionError: canceled with x = 1.0
+        1.0
+
+    **Complex roots**
+
+    For complex roots it's recommended to use Muller's method as it converges
+    even for real starting points very fast::
+
+        >>> findroot(lambda x: x**4 + x + 1, (0, 1, 2), solver='muller')
+        (0.727136084491197 + 0.934099289460529j)
+
+
+    **Intersection methods**
+
+    When you need to find a root in a known interval, it's highly recommended to
+    use an intersection-based solver like ``'anderson'`` or ``'ridder'``.
+    Usually they converge faster and more reliable. They have however problems
+    with multiple roots and usually need a sign change to find a root::
+
+        >>> findroot(lambda x: x**3, (-1, 1), solver='anderson')
+        0.0
+
+    Be careful with symmetric functions::
+
+        >>> findroot(lambda x: x**2, (-1, 1), solver='anderson') #doctest:+ELLIPSIS
+        Traceback (most recent call last):
+          ...
+        ZeroDivisionError
+
+    It fails even for better starting points, because there is no sign change::
+
+        >>> findroot(lambda x: x**2, (-1, .5), solver='anderson')
+        Traceback (most recent call last):
+          ...
+        ValueError: Could not find root within given tolerance. (1.0 > 2.16840434497100886801e-19)
+        Try another starting point or tweak arguments.
+
+    """
+    prec = ctx.prec
+    try:
+        ctx.prec += 20
+
+        # initialize arguments
+        if tol is None:
+            tol = ctx.eps * 2**10
+
+        kwargs['verbose'] = kwargs.get('verbose', verbose)
+
+        if 'd1f' in kwargs:
+            kwargs['df'] = kwargs['d1f']
+
+        kwargs['tol'] = tol
+        if isinstance(x0, (list, tuple)):
+            x0 = [ctx.convert(x) for x in x0]
+        else:
+            x0 = [ctx.convert(x0)]
+
+        if isinstance(solver, str):
+            try:
+                solver = str2solver[solver]
+            except KeyError:
+                raise ValueError('could not recognize solver')
+
+        # accept list of functions
+        if isinstance(f, (list, tuple)):
+            f2 = copy(f)
+            def tmp(*args):
+                return [fn(*args) for fn in f2]
+            f = tmp
+
+        # detect multidimensional functions
+        try:
+            fx = f(*x0)
+            multidimensional = isinstance(fx, (list, tuple, ctx.matrix))
+        except TypeError:
+            fx = f(x0[0])
+            multidimensional = False
+        if 'multidimensional' in kwargs:
+            multidimensional = kwargs['multidimensional']
+        if multidimensional:
+            # only one multidimensional solver available at the moment
+            solver = MDNewton
+            if not 'norm' in kwargs:
+                norm = lambda x: ctx.norm(x, 'inf')
+                kwargs['norm'] = norm
+            else:
+                norm = kwargs['norm']
+        else:
+            norm = abs
+
+        # happily return starting point if it's a root
+        if norm(fx) == 0:
+            if multidimensional:
+                return ctx.matrix(x0)
+            else:
+                return x0[0]
+
+        # use solver
+        iterations = solver(ctx, f, x0, **kwargs)
+        if 'maxsteps' in kwargs:
+            maxsteps = kwargs['maxsteps']
+        else:
+            maxsteps = iterations.maxsteps
+        i = 0
+        for x, error in iterations:
+            if verbose:
+                print('x:    ', x)
+                print('error:', error)
+            i += 1
+            if error < tol * max(1, norm(x)) or i >= maxsteps:
+                break
+        else:
+            if not i:
+                raise ValueError('Could not find root using the given solver.\n'
+                                 'Try another starting point or tweak arguments.')
+        if not isinstance(x, (list, tuple, ctx.matrix)):
+            xl = [x]
+        else:
+            xl = x
+        if verify and norm(f(*xl))**2 > tol: # TODO: better condition?
+            raise ValueError('Could not find root within given tolerance. '
+                             '(%s > %s)\n'
+                             'Try another starting point or tweak arguments.'
+                             % (norm(f(*xl))**2, tol))
+        return x
+    finally:
+        ctx.prec = prec
+
+
+def multiplicity(ctx, f, root, tol=None, maxsteps=10, **kwargs):
+    """
+    Return the multiplicity of a given root of f.
+
+    Internally, numerical derivatives are used. This might be inefficient for
+    higher order derviatives. Due to this, ``multiplicity`` cancels after
+    evaluating 10 derivatives by default. You can be specify the n-th derivative
+    using the dnf keyword.
+
+    >>> from mpmath import *
+    >>> multiplicity(lambda x: sin(x) - 1, pi/2)
+    2
+
+    """
+    if tol is None:
+        tol = ctx.eps ** 0.8
+    kwargs['d0f'] = f
+    for i in xrange(maxsteps):
+        dfstr = 'd' + str(i) + 'f'
+        if dfstr in kwargs:
+            df = kwargs[dfstr]
+        else:
+            df = lambda x: ctx.diff(f, x, i)
+        if not abs(df(root)) < tol:
+            break
+    return i
+
+def steffensen(f):
+    """
+    linear convergent function -> quadratic convergent function
+
+    Steffensen's method for quadratic convergence of a linear converging
+    sequence.
+    Don not use it for higher rates of convergence.
+    It may even work for divergent sequences.
+
+    Definition:
+    F(x) = (x*f(f(x)) - f(x)**2) / (f(f(x)) - 2*f(x) + x)
+
+    Example
+    .......
+
+    You can use Steffensen's method to accelerate a fixpoint iteration of linear
+    (or less) convergence.
+
+    x* is a fixpoint of the iteration x_{k+1} = phi(x_k) if x* = phi(x*). For
+    phi(x) = x**2 there are two fixpoints: 0 and 1.
+
+    Let's try Steffensen's method:
+
+    >>> f = lambda x: x**2
+    >>> from mpmath.calculus.optimization import steffensen
+    >>> F = steffensen(f)
+    >>> for x in [0.5, 0.9, 2.0]:
+    ...     fx = Fx = x
+    ...     for i in xrange(9):
+    ...         try:
+    ...             fx = f(fx)
+    ...         except OverflowError:
+    ...             pass
+    ...         try:
+    ...             Fx = F(Fx)
+    ...         except ZeroDivisionError:
+    ...             pass
+    ...         print('%20g  %20g' % (fx, Fx))
+                    0.25                  -0.5
+                  0.0625                   0.1
+              0.00390625            -0.0011236
+             1.52588e-05           1.41691e-09
+             2.32831e-10          -2.84465e-27
+             5.42101e-20           2.30189e-80
+             2.93874e-39          -1.2197e-239
+             8.63617e-78                     0
+            7.45834e-155                     0
+                    0.81               1.02676
+                  0.6561               1.00134
+                0.430467                     1
+                0.185302                     1
+               0.0343368                     1
+              0.00117902                     1
+             1.39008e-06                     1
+             1.93233e-12                     1
+             3.73392e-24                     1
+                       4                   1.6
+                      16                1.2962
+                     256               1.10194
+                   65536               1.01659
+             4.29497e+09               1.00053
+             1.84467e+19                     1
+             3.40282e+38                     1
+             1.15792e+77                     1
+            1.34078e+154                     1
+
+    Unmodified, the iteration converges only towards 0. Modified it converges
+    not only much faster, it converges even to the repelling fixpoint 1.
+    """
+    def F(x):
+        fx = f(x)
+        ffx = f(fx)
+        return (x*ffx - fx**2) / (ffx - 2*fx + x)
+    return F
+
+OptimizationMethods.jacobian = jacobian
+OptimizationMethods.findroot = findroot
+OptimizationMethods.multiplicity = multiplicity
+
+if __name__ == '__main__':
+    import doctest
+    doctest.testmod()

.venv/lib/python3.11/site-packages/mpmath/calculus/quadrature.py

@@ -0,0 +1,1115 @@
+import math
+
+from ..libmp.backend import xrange
+
+class QuadratureRule(object):
+    """
+    Quadrature rules are implemented using this class, in order to
+    simplify the code and provide a common infrastructure
+    for tasks such as error estimation and node caching.
+
+    You can implement a custom quadrature rule by subclassing
+    :class:`QuadratureRule` and implementing the appropriate
+    methods. The subclass can then be used by :func:`~mpmath.quad` by
+    passing it as the *method* argument.
+
+    :class:`QuadratureRule` instances are supposed to be singletons.
+    :class:`QuadratureRule` therefore implements instance caching
+    in :func:`~mpmath.__new__`.
+    """
+
+    def __init__(self, ctx):
+        self.ctx = ctx
+        self.standard_cache = {}
+        self.transformed_cache = {}
+        self.interval_count = {}
+
+    def clear(self):
+        """
+        Delete cached node data.
+        """
+        self.standard_cache = {}
+        self.transformed_cache = {}
+        self.interval_count = {}
+
+    def calc_nodes(self, degree, prec, verbose=False):
+        r"""
+        Compute nodes for the standard interval `[-1, 1]`. Subclasses
+        should probably implement only this method, and use
+        :func:`~mpmath.get_nodes` method to retrieve the nodes.
+        """
+        raise NotImplementedError
+
+    def get_nodes(self, a, b, degree, prec, verbose=False):
+        """
+        Return nodes for given interval, degree and precision. The
+        nodes are retrieved from a cache if already computed;
+        otherwise they are computed by calling :func:`~mpmath.calc_nodes`
+        and are then cached.
+
+        Subclasses should probably not implement this method,
+        but just implement :func:`~mpmath.calc_nodes` for the actual
+        node computation.
+        """
+        key = (a, b, degree, prec)
+        if key in self.transformed_cache:
+            return self.transformed_cache[key]
+        orig = self.ctx.prec
+        try:
+            self.ctx.prec = prec+20
+            # Get nodes on standard interval
+            if (degree, prec) in self.standard_cache:
+                nodes = self.standard_cache[degree, prec]
+            else:
+                nodes = self.calc_nodes(degree, prec, verbose)
+                self.standard_cache[degree, prec] = nodes
+            # Transform to general interval
+            nodes = self.transform_nodes(nodes, a, b, verbose)
+            if key in self.interval_count:
+                self.transformed_cache[key] = nodes
+            else:
+                self.interval_count[key] = True
+        finally:
+            self.ctx.prec = orig
+        return nodes
+
+    def transform_nodes(self, nodes, a, b, verbose=False):
+        r"""
+        Rescale standardized nodes (for `[-1, 1]`) to a general
+        interval `[a, b]`. For a finite interval, a simple linear
+        change of variables is used. Otherwise, the following
+        transformations are used:
+
+        .. math ::
+
+            \lbrack a, \infty \rbrack : t = \frac{1}{x} + (a-1)
+
+            \lbrack -\infty, b \rbrack : t = (b+1) - \frac{1}{x}
+
+            \lbrack -\infty, \infty \rbrack : t = \frac{x}{\sqrt{1-x^2}}
+
+        """
+        ctx = self.ctx
+        a = ctx.convert(a)
+        b = ctx.convert(b)
+        one = ctx.one
+        if (a, b) == (-one, one):
+            return nodes
+        half = ctx.mpf(0.5)
+        new_nodes = []
+        if ctx.isinf(a) or ctx.isinf(b):
+            if (a, b) == (ctx.ninf, ctx.inf):
+                p05 = -half
+                for x, w in nodes:
+                    x2 = x*x
+                    px1 = one-x2
+                    spx1 = px1**p05
+                    x = x*spx1
+                    w *= spx1/px1
+                    new_nodes.append((x, w))
+            elif a == ctx.ninf:
+                b1 = b+1
+                for x, w in nodes:
+                    u = 2/(x+one)
+                    x = b1-u
+                    w *= half*u**2
+                    new_nodes.append((x, w))
+            elif b == ctx.inf:
+                a1 = a-1
+                for x, w in nodes:
+                    u = 2/(x+one)
+                    x = a1+u
+                    w *= half*u**2
+                    new_nodes.append((x, w))
+            elif a == ctx.inf or b == ctx.ninf:
+                return [(x,-w) for (x,w) in self.transform_nodes(nodes, b, a, verbose)]
+            else:
+                raise NotImplementedError
+        else:
+            # Simple linear change of variables
+            C = (b-a)/2
+            D = (b+a)/2
+            for x, w in nodes:
+                new_nodes.append((D+C*x, C*w))
+        return new_nodes
+
+    def guess_degree(self, prec):
+        """
+        Given a desired precision `p` in bits, estimate the degree `m`
+        of the quadrature required to accomplish full accuracy for
+        typical integrals. By default, :func:`~mpmath.quad` will perform up
+        to `m` iterations. The value of `m` should be a slight
+        overestimate, so that "slightly bad" integrals can be dealt
+        with automatically using a few extra iterations. On the
+        other hand, it should not be too big, so :func:`~mpmath.quad` can
+        quit within a reasonable amount of time when it is given
+        an "unsolvable" integral.
+
+        The default formula used by :func:`~mpmath.guess_degree` is tuned
+        for both :class:`TanhSinh` and :class:`GaussLegendre`.
+        The output is roughly as follows:
+
+            +---------+---------+
+            | `p`     | `m`     |
+            +=========+=========+
+            | 50      | 6       |
+            +---------+---------+
+            | 100     | 7       |
+            +---------+---------+
+            | 500     | 10      |
+            +---------+---------+
+            | 3000    | 12      |
+            +---------+---------+
+
+        This formula is based purely on a limited amount of
+        experimentation and will sometimes be wrong.
+        """
+        # Expected degree
+        # XXX: use mag
+        g = int(4 + max(0, self.ctx.log(prec/30.0, 2)))
+        # Reasonable "worst case"
+        g += 2
+        return g
+
+    def estimate_error(self, results, prec, epsilon):
+        r"""
+        Given results from integrations `[I_1, I_2, \ldots, I_k]` done
+        with a quadrature of rule of degree `1, 2, \ldots, k`, estimate
+        the error of `I_k`.
+
+        For `k = 2`, we estimate  `|I_{\infty}-I_2|` as `|I_2-I_1|`.
+
+        For `k > 2`, we extrapolate `|I_{\infty}-I_k| \approx |I_{k+1}-I_k|`
+        from `|I_k-I_{k-1}|` and `|I_k-I_{k-2}|` under the assumption
+        that each degree increment roughly doubles the accuracy of
+        the quadrature rule (this is true for both :class:`TanhSinh`
+        and :class:`GaussLegendre`). The extrapolation formula is given
+        by Borwein, Bailey & Girgensohn. Although not very conservative,
+        this method seems to be very robust in practice.
+        """
+        if len(results) == 2:
+            return abs(results[0]-results[1])
+        try:
+            if results[-1] == results[-2] == results[-3]:
+                return self.ctx.zero
+            D1 = self.ctx.log(abs(results[-1]-results[-2]), 10)
+            D2 = self.ctx.log(abs(results[-1]-results[-3]), 10)
+        except ValueError:
+            return epsilon
+        D3 = -prec
+        D4 = min(0, max(D1**2/D2, 2*D1, D3))
+        return self.ctx.mpf(10) ** int(D4)
+
+    def summation(self, f, points, prec, epsilon, max_degree, verbose=False):
+        """
+        Main integration function. Computes the 1D integral over
+        the interval specified by *points*. For each subinterval,
+        performs quadrature of degree from 1 up to *max_degree*
+        until :func:`~mpmath.estimate_error` signals convergence.
+
+        :func:`~mpmath.summation` transforms each subintegration to
+        the standard interval and then calls :func:`~mpmath.sum_next`.
+        """
+        ctx = self.ctx
+        I = total_err = ctx.zero
+        for i in xrange(len(points)-1):
+            a, b = points[i], points[i+1]
+            if a == b:
+                continue
+            # XXX: we could use a single variable transformation,
+            # but this is not good in practice. We get better accuracy
+            # by having 0 as an endpoint.
+            if (a, b) == (ctx.ninf, ctx.inf):
+                _f = f
+                f = lambda x: _f(-x) + _f(x)
+                a, b = (ctx.zero, ctx.inf)
+            results = []
+            err = ctx.zero
+            for degree in xrange(1, max_degree+1):
+                nodes = self.get_nodes(a, b, degree, prec, verbose)
+                if verbose:
+                    print("Integrating from %s to %s (degree %s of %s)" % \
+                        (ctx.nstr(a), ctx.nstr(b), degree, max_degree))
+                result = self.sum_next(f, nodes, degree, prec, results, verbose)
+                results.append(result)
+                if degree > 1:
+                    err = self.estimate_error(results, prec, epsilon)
+                    if verbose:
+                        print("Estimated error:", ctx.nstr(err), " epsilon:", ctx.nstr(epsilon), " result: ", ctx.nstr(result))
+                    if err <= epsilon:
+                        break
+            I += results[-1]
+            total_err += err
+        if total_err > epsilon:
+            if verbose:
+                print("Failed to reach full accuracy. Estimated error:", ctx.nstr(total_err))
+        return I, total_err
+
+    def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
+        r"""
+        Evaluates the step sum `\sum w_k f(x_k)` where the *nodes* list
+        contains the `(w_k, x_k)` pairs.
+
+        :func:`~mpmath.summation` will supply the list *results* of
+        values computed by :func:`~mpmath.sum_next` at previous degrees, in
+        case the quadrature rule is able to reuse them.
+        """
+        return self.ctx.fdot((w, f(x)) for (x,w) in nodes)
+
+
+class TanhSinh(QuadratureRule):
+    r"""
+    This class implements "tanh-sinh" or "doubly exponential"
+    quadrature. This quadrature rule is based on the Euler-Maclaurin
+    integral formula. By performing a change of variables involving
+    nested exponentials / hyperbolic functions (hence the name), the
+    derivatives at the endpoints vanish rapidly. Since the error term
+    in the Euler-Maclaurin formula depends on the derivatives at the
+    endpoints, a simple step sum becomes extremely accurate. In
+    practice, this means that doubling the number of evaluation
+    points roughly doubles the number of accurate digits.
+
+    Comparison to Gauss-Legendre:
+      * Initial computation of nodes is usually faster
+      * Handles endpoint singularities better
+      * Handles infinite integration intervals better
+      * Is slower for smooth integrands once nodes have been computed
+
+    The implementation of the tanh-sinh algorithm is based on the
+    description given in Borwein, Bailey & Girgensohn, "Experimentation
+    in Mathematics - Computational Paths to Discovery", A K Peters,
+    2003, pages 312-313. In the present implementation, a few
+    improvements have been made:
+
+      * A more efficient scheme is used to compute nodes (exploiting
+        recurrence for the exponential function)
+      * The nodes are computed successively instead of all at once
+
+    **References**
+
+    * [Bailey]_
+    * http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf
+
+    """
+
+    def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
+        """
+        Step sum for tanh-sinh quadrature of degree `m`. We exploit the
+        fact that half of the abscissas at degree `m` are precisely the
+        abscissas from degree `m-1`. Thus reusing the result from
+        the previous level allows a 2x speedup.
+        """
+        h = self.ctx.mpf(2)**(-degree)
+        # Abscissas overlap, so reusing saves half of the time
+        if previous:
+            S = previous[-1]/(h*2)
+        else:
+            S = self.ctx.zero
+        S += self.ctx.fdot((w,f(x)) for (x,w) in nodes)
+        return h*S
+
+    def calc_nodes(self, degree, prec, verbose=False):
+        r"""
+        The abscissas and weights for tanh-sinh quadrature of degree
+        `m` are given by
+
+        .. math::
+
+            x_k = \tanh(\pi/2 \sinh(t_k))
+
+            w_k = \pi/2 \cosh(t_k) / \cosh(\pi/2 \sinh(t_k))^2
+
+        where `t_k = t_0 + hk` for a step length `h \sim 2^{-m}`. The
+        list of nodes is actually infinite, but the weights die off so
+        rapidly that only a few are needed.
+        """
+        ctx = self.ctx
+        nodes = []
+
+        extra = 20
+        ctx.prec += extra
+        tol = ctx.ldexp(1, -prec-10)
+        pi4 = ctx.pi/4
+
+        # For simplicity, we work in steps h = 1/2^n, with the first point
+        # offset so that we can reuse the sum from the previous degree
+
+        # We define degree 1 to include the "degree 0" steps, including
+        # the point x = 0. (It doesn't work well otherwise; not sure why.)
+        t0 = ctx.ldexp(1, -degree)
+        if degree == 1:
+            #nodes.append((mpf(0), pi4))
+            #nodes.append((-mpf(0), pi4))
+            nodes.append((ctx.zero, ctx.pi/2))
+            h = t0
+        else:
+            h = t0*2
+
+        # Since h is fixed, we can compute the next exponential
+        # by simply multiplying by exp(h)
+        expt0 = ctx.exp(t0)
+        a = pi4 * expt0
+        b = pi4 / expt0
+        udelta = ctx.exp(h)
+        urdelta = 1/udelta
+
+        for k in xrange(0, 20*2**degree+1):
+            # Reference implementation:
+            # t = t0 + k*h
+            # x = tanh(pi/2 * sinh(t))
+            # w = pi/2 * cosh(t) / cosh(pi/2 * sinh(t))**2
+
+            # Fast implementation. Note that c = exp(pi/2 * sinh(t))
+            c = ctx.exp(a-b)
+            d = 1/c
+            co = (c+d)/2
+            si = (c-d)/2
+            x = si / co
+            w = (a+b) / co**2
+            diff = abs(x-1)
+            if diff <= tol:
+                break
+
+            nodes.append((x, w))
+            nodes.append((-x, w))
+
+            a *= udelta
+            b *= urdelta
+
+            if verbose and k % 300 == 150:
+                # Note: the number displayed is rather arbitrary. Should
+                # figure out how to print something that looks more like a
+                # percentage
+                print("Calculating nodes:", ctx.nstr(-ctx.log(diff, 10) / prec))
+
+        ctx.prec -= extra
+        return nodes
+
+
+class GaussLegendre(QuadratureRule):
+    r"""
+    This class implements Gauss-Legendre quadrature, which is
+    exceptionally efficient for polynomials and polynomial-like (i.e.
+    very smooth) integrands.
+
+    The abscissas and weights are given by roots and values of
+    Legendre polynomials, which are the orthogonal polynomials
+    on `[-1, 1]` with respect to the unit weight
+    (see :func:`~mpmath.legendre`).
+
+    In this implementation, we take the "degree" `m` of the quadrature
+    to denote a Gauss-Legendre rule of degree `3 \cdot 2^m` (following
+    Borwein, Bailey & Girgensohn). This way we get quadratic, rather
+    than linear, convergence as the degree is incremented.
+
+    Comparison to tanh-sinh quadrature:
+      * Is faster for smooth integrands once nodes have been computed
+      * Initial computation of nodes is usually slower
+      * Handles endpoint singularities worse
+      * Handles infinite integration intervals worse
+
+    """
+
+    def calc_nodes(self, degree, prec, verbose=False):
+        r"""
+        Calculates the abscissas and weights for Gauss-Legendre
+        quadrature of degree of given degree (actually `3 \cdot 2^m`).
+        """
+        ctx = self.ctx
+        # It is important that the epsilon is set lower than the
+        # "real" epsilon
+        epsilon = ctx.ldexp(1, -prec-8)
+        # Fairly high precision might be required for accurate
+        # evaluation of the roots
+        orig = ctx.prec
+        ctx.prec = int(prec*1.5)
+        if degree == 1:
+            x = ctx.sqrt(ctx.mpf(3)/5)
+            w = ctx.mpf(5)/9
+            nodes = [(-x,w),(ctx.zero,ctx.mpf(8)/9),(x,w)]
+            ctx.prec = orig
+            return nodes
+        nodes = []
+        n = 3*2**(degree-1)
+        upto = n//2 + 1
+        for j in xrange(1, upto):
+            # Asymptotic formula for the roots
+            r = ctx.mpf(math.cos(math.pi*(j-0.25)/(n+0.5)))
+            # Newton iteration
+            while 1:
+                t1, t2 = 1, 0
+                # Evaluates the Legendre polynomial using its defining
+                # recurrence relation
+                for j1 in xrange(1,n+1):
+                    t3, t2, t1 = t2, t1, ((2*j1-1)*r*t1 - (j1-1)*t2)/j1
+                t4 = n*(r*t1-t2)/(r**2-1)
+                a = t1/t4
+                r = r - a
+                if abs(a) < epsilon:
+                    break
+            x = r
+            w = 2/((1-r**2)*t4**2)
+            if verbose  and j % 30 == 15:
+                print("Computing nodes (%i of %i)" % (j, upto))
+            nodes.append((x, w))
+            nodes.append((-x, w))
+        ctx.prec = orig
+        return nodes
+
+class QuadratureMethods(object):
+
+    def __init__(ctx, *args, **kwargs):
+        ctx._gauss_legendre = GaussLegendre(ctx)
+        ctx._tanh_sinh = TanhSinh(ctx)
+
+    def quad(ctx, f, *points, **kwargs):
+        r"""
+        Computes a single, double or triple integral over a given
+        1D interval, 2D rectangle, or 3D cuboid. A basic example::
+
+            >>> from mpmath import *
+            >>> mp.dps = 15; mp.pretty = True
+            >>> quad(sin, [0, pi])
+            2.0
+
+        A basic 2D integral::
+
+            >>> f = lambda x, y: cos(x+y/2)
+            >>> quad(f, [-pi/2, pi/2], [0, pi])
+            4.0
+
+        **Interval format**
+
+        The integration range for each dimension may be specified
+        using a list or tuple. Arguments are interpreted as follows:
+
+        ``quad(f, [x1, x2])`` -- calculates
+        `\int_{x_1}^{x_2} f(x) \, dx`
+
+        ``quad(f, [x1, x2], [y1, y2])`` -- calculates
+        `\int_{x_1}^{x_2} \int_{y_1}^{y_2} f(x,y) \, dy \, dx`
+
+        ``quad(f, [x1, x2], [y1, y2], [z1, z2])`` -- calculates
+        `\int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} f(x,y,z)
+        \, dz \, dy \, dx`
+
+        Endpoints may be finite or infinite. An interval descriptor
+        may also contain more than two points. In this
+        case, the integration is split into subintervals, between
+        each pair of consecutive points. This is useful for
+        dealing with mid-interval discontinuities, or integrating
+        over large intervals where the function is irregular or
+        oscillates.
+
+        **Options**
+
+        :func:`~mpmath.quad` recognizes the following keyword arguments:
+
+        *method*
+            Chooses integration algorithm (described below).
+        *error*
+            If set to true, :func:`~mpmath.quad` returns `(v, e)` where `v` is the
+            integral and `e` is the estimated error.
+        *maxdegree*
+            Maximum degree of the quadrature rule to try before
+            quitting.
+        *verbose*
+            Print details about progress.
+
+        **Algorithms**
+
+        Mpmath presently implements two integration algorithms: tanh-sinh
+        quadrature and Gauss-Legendre quadrature. These can be selected
+        using *method='tanh-sinh'* or *method='gauss-legendre'* or by
+        passing the classes *method=TanhSinh*, *method=GaussLegendre*.
+        The functions :func:`~mpmath.quadts` and :func:`~mpmath.quadgl` are also available
+        as shortcuts.
+
+        Both algorithms have the property that doubling the number of
+        evaluation points roughly doubles the accuracy, so both are ideal
+        for high precision quadrature (hundreds or thousands of digits).
+
+        At high precision, computing the nodes and weights for the
+        integration can be expensive (more expensive than computing the
+        function values). To make repeated integrations fast, nodes
+        are automatically cached.
+
+        The advantages of the tanh-sinh algorithm are that it tends to
+        handle endpoint singularities well, and that the nodes are cheap
+        to compute on the first run. For these reasons, it is used by
+        :func:`~mpmath.quad` as the default algorithm.
+
+        Gauss-Legendre quadrature often requires fewer function
+        evaluations, and is therefore often faster for repeated use, but
+        the algorithm does not handle endpoint singularities as well and
+        the nodes are more expensive to compute. Gauss-Legendre quadrature
+        can be a better choice if the integrand is smooth and repeated
+        integrations are required (e.g. for multiple integrals).
+
+        See the documentation for :class:`TanhSinh` and
+        :class:`GaussLegendre` for additional details.
+
+        **Examples of 1D integrals**
+
+        Intervals may be infinite or half-infinite. The following two
+        examples evaluate the limits of the inverse tangent function
+        (`\int 1/(1+x^2) = \tan^{-1} x`), and the Gaussian integral
+        `\int_{\infty}^{\infty} \exp(-x^2)\,dx = \sqrt{\pi}`::
+
+            >>> mp.dps = 15
+            >>> quad(lambda x: 2/(x**2+1), [0, inf])
+            3.14159265358979
+            >>> quad(lambda x: exp(-x**2), [-inf, inf])**2
+            3.14159265358979
+
+        Integrals can typically be resolved to high precision.
+        The following computes 50 digits of `\pi` by integrating the
+        area of the half-circle defined by `x^2 + y^2 \le 1`,
+        `-1 \le x \le 1`, `y \ge 0`::
+
+            >>> mp.dps = 50
+            >>> 2*quad(lambda x: sqrt(1-x**2), [-1, 1])
+            3.1415926535897932384626433832795028841971693993751
+
+        One can just as well compute 1000 digits (output truncated)::
+
+            >>> mp.dps = 1000
+            >>> 2*quad(lambda x: sqrt(1-x**2), [-1, 1])  #doctest:+ELLIPSIS
+            3.141592653589793238462643383279502884...216420199
+
+        Complex integrals are supported. The following computes
+        a residue at `z = 0` by integrating counterclockwise along the
+        diamond-shaped path from `1` to `+i` to `-1` to `-i` to `1`::
+
+            >>> mp.dps = 15
+            >>> chop(quad(lambda z: 1/z, [1,j,-1,-j,1]))
+            (0.0 + 6.28318530717959j)
+
+        **Examples of 2D and 3D integrals**
+
+        Here are several nice examples of analytically solvable
+        2D integrals (taken from MathWorld [1]) that can be evaluated
+        to high precision fairly rapidly by :func:`~mpmath.quad`::
+
+            >>> mp.dps = 30
+            >>> f = lambda x, y: (x-1)/((1-x*y)*log(x*y))
+            >>> quad(f, [0, 1], [0, 1])
+            0.577215664901532860606512090082
+            >>> +euler
+            0.577215664901532860606512090082
+
+            >>> f = lambda x, y: 1/sqrt(1+x**2+y**2)
+            >>> quad(f, [-1, 1], [-1, 1])
+            3.17343648530607134219175646705
+            >>> 4*log(2+sqrt(3))-2*pi/3
+            3.17343648530607134219175646705
+
+            >>> f = lambda x, y: 1/(1-x**2 * y**2)
+            >>> quad(f, [0, 1], [0, 1])
+            1.23370055013616982735431137498
+            >>> pi**2 / 8
+            1.23370055013616982735431137498
+
+            >>> quad(lambda x, y: 1/(1-x*y), [0, 1], [0, 1])
+            1.64493406684822643647241516665
+            >>> pi**2 / 6
+            1.64493406684822643647241516665
+
+        Multiple integrals may be done over infinite ranges::
+
+            >>> mp.dps = 15
+            >>> print(quad(lambda x,y: exp(-x-y), [0, inf], [1, inf]))
+            0.367879441171442
+            >>> print(1/e)
+            0.367879441171442
+
+        For nonrectangular areas, one can call :func:`~mpmath.quad` recursively.
+        For example, we can replicate the earlier example of calculating
+        `\pi` by integrating over the unit-circle, and actually use double
+        quadrature to actually measure the area circle::
+
+            >>> f = lambda x: quad(lambda y: 1, [-sqrt(1-x**2), sqrt(1-x**2)])
+            >>> quad(f, [-1, 1])
+            3.14159265358979
+
+        Here is a simple triple integral::
+
+            >>> mp.dps = 15
+            >>> f = lambda x,y,z: x*y/(1+z)
+            >>> quad(f, [0,1], [0,1], [1,2], method='gauss-legendre')
+            0.101366277027041
+            >>> (log(3)-log(2))/4
+            0.101366277027041
+
+        **Singularities**
+
+        Both tanh-sinh and Gauss-Legendre quadrature are designed to
+        integrate smooth (infinitely differentiable) functions. Neither
+        algorithm copes well with mid-interval singularities (such as
+        mid-interval discontinuities in `f(x)` or `f'(x)`).
+        The best solution is to split the integral into parts::
+
+            >>> mp.dps = 15
+            >>> quad(lambda x: abs(sin(x)), [0, 2*pi])   # Bad
+            3.99900894176779
+            >>> quad(lambda x: abs(sin(x)), [0, pi, 2*pi])  # Good
+            4.0
+
+        The tanh-sinh rule often works well for integrands having a
+        singularity at one or both endpoints::
+
+            >>> mp.dps = 15
+            >>> quad(log, [0, 1], method='tanh-sinh')  # Good
+            -1.0
+            >>> quad(log, [0, 1], method='gauss-legendre')  # Bad
+            -0.999932197413801
+
+        However, the result may still be inaccurate for some functions::
+
+            >>> quad(lambda x: 1/sqrt(x), [0, 1], method='tanh-sinh')
+            1.99999999946942
+
+        This problem is not due to the quadrature rule per se, but to
+        numerical amplification of errors in the nodes. The problem can be
+        circumvented by temporarily increasing the precision::
+
+            >>> mp.dps = 30
+            >>> a = quad(lambda x: 1/sqrt(x), [0, 1], method='tanh-sinh')
+            >>> mp.dps = 15
+            >>> +a
+            2.0
+
+        **Highly variable functions**
+
+        For functions that are smooth (in the sense of being infinitely
+        differentiable) but contain sharp mid-interval peaks or many
+        "bumps", :func:`~mpmath.quad` may fail to provide full accuracy. For
+        example, with default settings, :func:`~mpmath.quad` is able to integrate
+        `\sin(x)` accurately over an interval of length 100 but not over
+        length 1000::
+
+            >>> quad(sin, [0, 100]); 1-cos(100)   # Good
+            0.137681127712316
+            0.137681127712316
+            >>> quad(sin, [0, 1000]); 1-cos(1000)   # Bad
+            -37.8587612408485
+            0.437620923709297
+
+        One solution is to break the integration into 10 intervals of
+        length 100::
+
+            >>> quad(sin, linspace(0, 1000, 10))   # Good
+            0.437620923709297
+
+        Another is to increase the degree of the quadrature::
+
+            >>> quad(sin, [0, 1000], maxdegree=10)   # Also good
+            0.437620923709297
+
+        Whether splitting the interval or increasing the degree is
+        more efficient differs from case to case. Another example is the
+        function `1/(1+x^2)`, which has a sharp peak centered around
+        `x = 0`::
+
+            >>> f = lambda x: 1/(1+x**2)
+            >>> quad(f, [-100, 100])   # Bad
+            3.64804647105268
+            >>> quad(f, [-100, 100], maxdegree=10)   # Good
+            3.12159332021646
+            >>> quad(f, [-100, 0, 100])   # Also good
+            3.12159332021646
+
+        **References**
+
+        1. http://mathworld.wolfram.com/DoubleIntegral.html
+
+        """
+        rule = kwargs.get('method', 'tanh-sinh')
+        if type(rule) is str:
+            if rule == 'tanh-sinh':
+                rule = ctx._tanh_sinh
+            elif rule == 'gauss-legendre':
+                rule = ctx._gauss_legendre
+            else:
+                raise ValueError("unknown quadrature rule: %s" % rule)
+        else:
+            rule = rule(ctx)
+        verbose = kwargs.get('verbose')
+        dim = len(points)
+        orig = prec = ctx.prec
+        epsilon = ctx.eps/8
+        m = kwargs.get('maxdegree') or rule.guess_degree(prec)
+        points = [ctx._as_points(p) for p in points]
+        try:
+            ctx.prec += 20
+            if dim == 1:
+                v, err = rule.summation(f, points[0], prec, epsilon, m, verbose)
+            elif dim == 2:
+                v, err = rule.summation(lambda x: \
+                        rule.summation(lambda y: f(x,y), \
+                        points[1], prec, epsilon, m)[0],
+                    points[0], prec, epsilon, m, verbose)
+            elif dim == 3:
+                v, err = rule.summation(lambda x: \
+                        rule.summation(lambda y: \
+                            rule.summation(lambda z: f(x,y,z), \
+                            points[2], prec, epsilon, m)[0],
+                        points[1], prec, epsilon, m)[0],
+                    points[0], prec, epsilon, m, verbose)
+            else:
+                raise NotImplementedError("quadrature must have dim 1, 2 or 3")
+        finally:
+            ctx.prec = orig
+        if kwargs.get("error"):
+            return +v, err
+        return +v
+
+    def quadts(ctx, *args, **kwargs):
+        """
+        Performs tanh-sinh quadrature. The call
+
+            quadts(func, *points, ...)
+
+        is simply a shortcut for:
+
+            quad(func, *points, ..., method=TanhSinh)
+
+        For example, a single integral and a double integral:
+
+            quadts(lambda x: exp(cos(x)), [0, 1])
+            quadts(lambda x, y: exp(cos(x+y)), [0, 1], [0, 1])
+
+        See the documentation for quad for information about how points
+        arguments and keyword arguments are parsed.
+
+        See documentation for TanhSinh for algorithmic information about
+        tanh-sinh quadrature.
+        """
+        kwargs['method'] = 'tanh-sinh'
+        return ctx.quad(*args, **kwargs)
+
+    def quadgl(ctx, *args, **kwargs):
+        """
+        Performs Gauss-Legendre quadrature. The call
+
+            quadgl(func, *points, ...)
+
+        is simply a shortcut for:
+
+            quad(func, *points, ..., method=GaussLegendre)
+
+        For example, a single integral and a double integral:
+
+            quadgl(lambda x: exp(cos(x)), [0, 1])
+            quadgl(lambda x, y: exp(cos(x+y)), [0, 1], [0, 1])
+
+        See the documentation for quad for information about how points
+        arguments and keyword arguments are parsed.
+
+        See documentation for TanhSinh for algorithmic information about
+        tanh-sinh quadrature.
+        """
+        kwargs['method'] = 'gauss-legendre'
+        return ctx.quad(*args, **kwargs)
+
+    def quadosc(ctx, f, interval, omega=None, period=None, zeros=None):
+        r"""
+        Calculates
+
+        .. math ::
+
+            I = \int_a^b f(x) dx
+
+        where at least one of `a` and `b` is infinite and where
+        `f(x) = g(x) \cos(\omega x  + \phi)` for some slowly
+        decreasing function `g(x)`. With proper input, :func:`~mpmath.quadosc`
+        can also handle oscillatory integrals where the oscillation
+        rate is different from a pure sine or cosine wave.
+
+        In the standard case when `|a| < \infty, b = \infty`,
+        :func:`~mpmath.quadosc` works by evaluating the infinite series
+
+        .. math ::
+
+            I = \int_a^{x_1} f(x) dx +
+            \sum_{k=1}^{\infty} \int_{x_k}^{x_{k+1}} f(x) dx
+
+        where `x_k` are consecutive zeros (alternatively
+        some other periodic reference point) of `f(x)`.
+        Accordingly, :func:`~mpmath.quadosc` requires information about the
+        zeros of `f(x)`. For a periodic function, you can specify
+        the zeros by either providing the angular frequency `\omega`
+        (*omega*) or the *period* `2 \pi/\omega`. In general, you can
+        specify the `n`-th zero by providing the *zeros* arguments.
+        Below is an example of each::
+
+            >>> from mpmath import *
+            >>> mp.dps = 15; mp.pretty = True
+            >>> f = lambda x: sin(3*x)/(x**2+1)
+            >>> quadosc(f, [0,inf], omega=3)
+            0.37833007080198
+            >>> quadosc(f, [0,inf], period=2*pi/3)
+            0.37833007080198
+            >>> quadosc(f, [0,inf], zeros=lambda n: pi*n/3)
+            0.37833007080198
+            >>> (ei(3)*exp(-3)-exp(3)*ei(-3))/2  # Computed by Mathematica
+            0.37833007080198
+
+        Note that *zeros* was specified to multiply `n` by the
+        *half-period*, not the full period. In theory, it does not matter
+        whether each partial integral is done over a half period or a full
+        period. However, if done over half-periods, the infinite series
+        passed to :func:`~mpmath.nsum` becomes an *alternating series* and this
+        typically makes the extrapolation much more efficient.
+
+        Here is an example of an integration over the entire real line,
+        and a half-infinite integration starting at `-\infty`::
+
+            >>> quadosc(lambda x: cos(x)/(1+x**2), [-inf, inf], omega=1)
+            1.15572734979092
+            >>> pi/e
+            1.15572734979092
+            >>> quadosc(lambda x: cos(x)/x**2, [-inf, -1], period=2*pi)
+            -0.0844109505595739
+            >>> cos(1)+si(1)-pi/2
+            -0.0844109505595738
+
+        Of course, the integrand may contain a complex exponential just as
+        well as a real sine or cosine::
+
+            >>> quadosc(lambda x: exp(3*j*x)/(1+x**2), [-inf,inf], omega=3)
+            (0.156410688228254 + 0.0j)
+            >>> pi/e**3
+            0.156410688228254
+            >>> quadosc(lambda x: exp(3*j*x)/(2+x+x**2), [-inf,inf], omega=3)
+            (0.00317486988463794 - 0.0447701735209082j)
+            >>> 2*pi/sqrt(7)/exp(3*(j+sqrt(7))/2)
+            (0.00317486988463794 - 0.0447701735209082j)
+
+        **Non-periodic functions**
+
+        If `f(x) = g(x) h(x)` for some function `h(x)` that is not
+        strictly periodic, *omega* or *period* might not work, and it might
+        be necessary to use *zeros*.
+
+        A notable exception can be made for Bessel functions which, though not
+        periodic, are "asymptotically periodic" in a sufficiently strong sense
+        that the sum extrapolation will work out::
+
+            >>> quadosc(j0, [0, inf], period=2*pi)
+            1.0
+            >>> quadosc(j1, [0, inf], period=2*pi)
+            1.0
+
+        More properly, one should provide the exact Bessel function zeros::
+
+            >>> j0zero = lambda n: findroot(j0, pi*(n-0.25))
+            >>> quadosc(j0, [0, inf], zeros=j0zero)
+            1.0
+
+        For an example where *zeros* becomes necessary, consider the
+        complete Fresnel integrals
+
+        .. math ::
+
+            \int_0^{\infty} \cos x^2\,dx = \int_0^{\infty} \sin x^2\,dx
+            = \sqrt{\frac{\pi}{8}}.
+
+        Although the integrands do not decrease in magnitude as
+        `x \to \infty`, the integrals are convergent since the oscillation
+        rate increases (causing consecutive periods to asymptotically
+        cancel out). These integrals are virtually impossible to calculate
+        to any kind of accuracy using standard quadrature rules. However,
+        if one provides the correct asymptotic distribution of zeros
+        (`x_n \sim \sqrt{n}`), :func:`~mpmath.quadosc` works::
+
+            >>> mp.dps = 30
+            >>> f = lambda x: cos(x**2)
+            >>> quadosc(f, [0,inf], zeros=lambda n:sqrt(pi*n))
+            0.626657068657750125603941321203
+            >>> f = lambda x: sin(x**2)
+            >>> quadosc(f, [0,inf], zeros=lambda n:sqrt(pi*n))
+            0.626657068657750125603941321203
+            >>> sqrt(pi/8)
+            0.626657068657750125603941321203
+
+        (Interestingly, these integrals can still be evaluated if one
+        places some other constant than `\pi` in the square root sign.)
+
+        In general, if `f(x) \sim g(x) \cos(h(x))`, the zeros follow
+        the inverse-function distribution `h^{-1}(x)`::
+
+            >>> mp.dps = 15
+            >>> f = lambda x: sin(exp(x))
+            >>> quadosc(f, [1,inf], zeros=lambda n: log(n))
+            -0.25024394235267
+            >>> pi/2-si(e)
+            -0.250243942352671
+
+        **Non-alternating functions**
+
+        If the integrand oscillates around a positive value, without
+        alternating signs, the extrapolation might fail. A simple trick
+        that sometimes works is to multiply or divide the frequency by 2::
+
+            >>> f = lambda x: 1/x**2+sin(x)/x**4
+            >>> quadosc(f, [1,inf], omega=1)  # Bad
+            1.28642190869861
+            >>> quadosc(f, [1,inf], omega=0.5)  # Perfect
+            1.28652953559617
+            >>> 1+(cos(1)+ci(1)+sin(1))/6
+            1.28652953559617
+
+        **Fast decay**
+
+        :func:`~mpmath.quadosc` is primarily useful for slowly decaying
+        integrands. If the integrand decreases exponentially or faster,
+        :func:`~mpmath.quad` will likely handle it without trouble (and generally be
+        much faster than :func:`~mpmath.quadosc`)::
+
+            >>> quadosc(lambda x: cos(x)/exp(x), [0, inf], omega=1)
+            0.5
+            >>> quad(lambda x: cos(x)/exp(x), [0, inf])
+            0.5
+
+        """
+        a, b = ctx._as_points(interval)
+        a = ctx.convert(a)
+        b = ctx.convert(b)
+        if [omega, period, zeros].count(None) != 2:
+            raise ValueError( \
+                "must specify exactly one of omega, period, zeros")
+        if a == ctx.ninf and b == ctx.inf:
+            s1 = ctx.quadosc(f, [a, 0], omega=omega, zeros=zeros, period=period)
+            s2 = ctx.quadosc(f, [0, b], omega=omega, zeros=zeros, period=period)
+            return s1 + s2
+        if a == ctx.ninf:
+            if zeros:
+                return ctx.quadosc(lambda x:f(-x), [-b,-a], lambda n: zeros(-n))
+            else:
+                return ctx.quadosc(lambda x:f(-x), [-b,-a], omega=omega, period=period)
+        if b != ctx.inf:
+            raise ValueError("quadosc requires an infinite integration interval")
+        if not zeros:
+            if omega:
+                period = 2*ctx.pi/omega
+            zeros = lambda n: n*period/2
+        #for n in range(1,10):
+        #    p = zeros(n)
+        #    if p > a:
+        #        break
+        #if n >= 9:
+        #    raise ValueError("zeros do not appear to be correctly indexed")
+        n = 1
+        s = ctx.quadgl(f, [a, zeros(n)])
+        def term(k):
+            return ctx.quadgl(f, [zeros(k), zeros(k+1)])
+        s += ctx.nsum(term, [n, ctx.inf])
+        return s
+
+    def quadsubdiv(ctx, f, interval, tol=None, maxintervals=None, **kwargs):
+        """
+        Computes the integral of *f* over the interval or path specified
+        by *interval*, using :func:`~mpmath.quad` together with adaptive
+        subdivision of the interval.
+
+        This function gives an accurate answer for some integrals where
+        :func:`~mpmath.quad` fails::
+
+            >>> from mpmath import *
+            >>> mp.dps = 15; mp.pretty = True
+            >>> quad(lambda x: abs(sin(x)), [0, 2*pi])
+            3.99900894176779
+            >>> quadsubdiv(lambda x: abs(sin(x)), [0, 2*pi])
+            4.0
+            >>> quadsubdiv(sin, [0, 1000])
+            0.437620923709297
+            >>> quadsubdiv(lambda x: 1/(1+x**2), [-100, 100])
+            3.12159332021646
+            >>> quadsubdiv(lambda x: ceil(x), [0, 100])
+            5050.0
+            >>> quadsubdiv(lambda x: sin(x+exp(x)), [0,8])
+            0.347400172657248
+
+        The argument *maxintervals* can be set to limit the permissible
+        subdivision::
+
+            >>> quadsubdiv(lambda x: sin(x**2), [0,100], maxintervals=5, error=True)
+            (-5.40487904307774, 5.011)
+            >>> quadsubdiv(lambda x: sin(x**2), [0,100], maxintervals=100, error=True)
+            (0.631417921866934, 1.10101120134116e-17)
+
+        Subdivision does not guarantee a correct answer since, the error
+        estimate on subintervals may be inaccurate::
+
+            >>> quadsubdiv(lambda x: sech(10*x-2)**2 + sech(100*x-40)**4 + sech(1000*x-600)**6, [0,1], error=True)
+            (0.210802735500549, 1.0001111101e-17)
+            >>> mp.dps = 20
+            >>> quadsubdiv(lambda x: sech(10*x-2)**2 + sech(100*x-40)**4 + sech(1000*x-600)**6, [0,1], error=True)
+            (0.21080273550054927738, 2.200000001e-24)
+
+        The second answer is correct. We can get an accurate result at lower
+        precision by forcing a finer initial subdivision::
+
+            >>> mp.dps = 15
+            >>> quadsubdiv(lambda x: sech(10*x-2)**2 + sech(100*x-40)**4 + sech(1000*x-600)**6, linspace(0,1,5))
+            0.210802735500549
+
+        The following integral is too oscillatory for convergence, but we can get a
+        reasonable estimate::
+
+            >>> v, err = fp.quadsubdiv(lambda x: fp.sin(1/x), [0,1], error=True)
+            >>> round(v, 6), round(err, 6)
+            (0.504067, 1e-06)
+            >>> sin(1) - ci(1)
+            0.504067061906928
+
+        """
+        queue = []
+        for i in range(len(interval)-1):
+            queue.append((interval[i], interval[i+1]))
+        total = ctx.zero
+        total_error = ctx.zero
+        if maxintervals is None:
+            maxintervals = 10 * ctx.prec
+        count = 0
+        quad_args = kwargs.copy()
+        quad_args["verbose"] = False
+        quad_args["error"] = True
+        if tol is None:
+            tol = +ctx.eps
+        orig = ctx.prec
+        try:
+            ctx.prec += 5
+            while queue:
+                a, b = queue.pop()
+                s, err = ctx.quad(f, [a, b], **quad_args)
+                if kwargs.get("verbose"):
+                    print("subinterval", count, a, b, err)
+                if err < tol or count > maxintervals:
+                    total += s
+                    total_error += err
+                else:
+                    count += 1
+                    if count == maxintervals and kwargs.get("verbose"):
+                        print("warning: number of intervals exceeded maxintervals")
+                    if a == -ctx.inf and b == ctx.inf:
+                        m = 0
+                    elif a == -ctx.inf:
+                        m = min(b-1, 2*b)
+                    elif b == ctx.inf:
+                        m = max(a+1, 2*a)
+                    else:
+                        m = a + (b - a) / 2
+                    queue.append((a, m))
+                    queue.append((m, b))
+        finally:
+            ctx.prec = orig
+        if kwargs.get("error"):
+            return +total, +total_error
+        else:
+            return +total
+
+if __name__ == '__main__':
+    import doctest
+    doctest.testmod()

.venv/lib/python3.11/site-packages/mpmath/libmp/libelefun.py

@@ -0,0 +1,1428 @@
+"""
+This module implements computation of elementary transcendental
+functions (powers, logarithms, trigonometric and hyperbolic
+functions, inverse trigonometric and hyperbolic) for real
+floating-point numbers.
+
+For complex and interval implementations of the same functions,
+see libmpc and libmpi.
+
+"""
+
+import math
+from bisect import bisect
+
+from .backend import xrange
+from .backend import MPZ, MPZ_ZERO, MPZ_ONE, MPZ_TWO, MPZ_FIVE, BACKEND
+
+from .libmpf import (
+    round_floor, round_ceiling, round_down, round_up,
+    round_nearest, round_fast,
+    ComplexResult,
+    bitcount, bctable, lshift, rshift, giant_steps, sqrt_fixed,
+    from_int, to_int, from_man_exp, to_fixed, to_float, from_float,
+    from_rational, normalize,
+    fzero, fone, fnone, fhalf, finf, fninf, fnan,
+    mpf_cmp, mpf_sign, mpf_abs,
+    mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul, mpf_div, mpf_shift,
+    mpf_rdiv_int, mpf_pow_int, mpf_sqrt,
+    reciprocal_rnd, negative_rnd, mpf_perturb,
+    isqrt_fast
+)
+
+from .libintmath import ifib
+
+
+#-------------------------------------------------------------------------------
+# Tuning parameters
+#-------------------------------------------------------------------------------
+
+# Cutoff for computing exp from cosh+sinh. This reduces the
+# number of terms by half, but also requires a square root which
+# is expensive with the pure-Python square root code.
+if BACKEND == 'python':
+    EXP_COSH_CUTOFF = 600
+else:
+    EXP_COSH_CUTOFF = 400
+# Cutoff for using more than 2 series
+EXP_SERIES_U_CUTOFF = 1500
+
+# Also basically determined by sqrt
+if BACKEND == 'python':
+    COS_SIN_CACHE_PREC = 400
+else:
+    COS_SIN_CACHE_PREC = 200
+COS_SIN_CACHE_STEP = 8
+cos_sin_cache = {}
+
+# Number of integer logarithms to cache (for zeta sums)
+MAX_LOG_INT_CACHE = 2000
+log_int_cache = {}
+
+LOG_TAYLOR_PREC = 2500  # Use Taylor series with caching up to this prec
+LOG_TAYLOR_SHIFT = 9    # Cache log values in steps of size 2^-N
+log_taylor_cache = {}
+# prec/size ratio of x for fastest convergence in AGM formula
+LOG_AGM_MAG_PREC_RATIO = 20
+
+ATAN_TAYLOR_PREC = 3000  # Same as for log
+ATAN_TAYLOR_SHIFT = 7   # steps of size 2^-N
+atan_taylor_cache = {}
+
+
+# ~= next power of two + 20
+cache_prec_steps = [22,22]
+for k in xrange(1, bitcount(LOG_TAYLOR_PREC)+1):
+    cache_prec_steps += [min(2**k,LOG_TAYLOR_PREC)+20] * 2**(k-1)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                   Elementary mathematical constants                        #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+def constant_memo(f):
+    """
+    Decorator for caching computed values of mathematical
+    constants. This decorator should be applied to a
+    function taking a single argument prec as input and
+    returning a fixed-point value with the given precision.
+    """
+    f.memo_prec = -1
+    f.memo_val = None
+    def g(prec, **kwargs):
+        memo_prec = f.memo_prec
+        if prec <= memo_prec:
+            return f.memo_val >> (memo_prec-prec)
+        newprec = int(prec*1.05+10)
+        f.memo_val = f(newprec, **kwargs)
+        f.memo_prec = newprec
+        return f.memo_val >> (newprec-prec)
+    g.__name__ = f.__name__
+    g.__doc__ = f.__doc__
+    return g
+
+def def_mpf_constant(fixed):
+    """
+    Create a function that computes the mpf value for a mathematical
+    constant, given a function that computes the fixed-point value.
+
+    Assumptions: the constant is positive and has magnitude ~= 1;
+    the fixed-point function rounds to floor.
+    """
+    def f(prec, rnd=round_fast):
+        wp = prec + 20
+        v = fixed(wp)
+        if rnd in (round_up, round_ceiling):
+            v += 1
+        return normalize(0, v, -wp, bitcount(v), prec, rnd)
+    f.__doc__ = fixed.__doc__
+    return f
+
+def bsp_acot(q, a, b, hyperbolic):
+    if b - a == 1:
+        a1 = MPZ(2*a + 3)
+        if hyperbolic or a&1:
+            return MPZ_ONE, a1 * q**2, a1
+        else:
+            return -MPZ_ONE, a1 * q**2, a1
+    m = (a+b)//2
+    p1, q1, r1 = bsp_acot(q, a, m, hyperbolic)
+    p2, q2, r2 = bsp_acot(q, m, b, hyperbolic)
+    return q2*p1 + r1*p2, q1*q2, r1*r2
+
+# the acoth(x) series converges like the geometric series for x^2
+# N = ceil(p*log(2)/(2*log(x)))
+def acot_fixed(a, prec, hyperbolic):
+    """
+    Compute acot(a) or acoth(a) for an integer a with binary splitting; see
+    http://numbers.computation.free.fr/Constants/Algorithms/splitting.html
+    """
+    N = int(0.35 * prec/math.log(a) + 20)
+    p, q, r = bsp_acot(a, 0,N, hyperbolic)
+    return ((p+q)<<prec)//(q*a)
+
+def machin(coefs, prec, hyperbolic=False):
+    """
+    Evaluate a Machin-like formula, i.e., a linear combination of
+    acot(n) or acoth(n) for specific integer values of n, using fixed-
+    point arithmetic. The input should be a list [(c, n), ...], giving
+    c*acot[h](n) + ...
+    """
+    extraprec = 10
+    s = MPZ_ZERO
+    for a, b in coefs:
+        s += MPZ(a) * acot_fixed(MPZ(b), prec+extraprec, hyperbolic)
+    return (s >> extraprec)
+
+# Logarithms of integers are needed for various computations involving
+# logarithms, powers, radix conversion, etc
+
+@constant_memo
+def ln2_fixed(prec):
+    """
+    Computes ln(2). This is done with a hyperbolic Machin-type formula,
+    with binary splitting at high precision.
+    """
+    return machin([(18, 26), (-2, 4801), (8, 8749)], prec, True)
+
+@constant_memo
+def ln10_fixed(prec):
+    """
+    Computes ln(10). This is done with a hyperbolic Machin-type formula.
+    """
+    return machin([(46, 31), (34, 49), (20, 161)], prec, True)
+
+
+r"""
+For computation of pi, we use the Chudnovsky series:
+
+             oo
+             ___        k
+      1     \       (-1)  (6 k)! (A + B k)
+    ----- =  )     -----------------------
+    12 pi   /___               3  3k+3/2
+                    (3 k)! (k!)  C
+            k = 0
+
+where A, B, and C are certain integer constants. This series adds roughly
+14 digits per term. Note that C^(3/2) can be extracted so that the
+series contains only rational terms. This makes binary splitting very
+efficient.
+
+The recurrence formulas for the binary splitting were taken from
+ftp://ftp.gmplib.org/pub/src/gmp-chudnovsky.c
+
+Previously, Machin's formula was used at low precision and the AGM iteration
+was used at high precision. However, the Chudnovsky series is essentially as
+fast as the Machin formula at low precision and in practice about 3x faster
+than the AGM at high precision (despite theoretically having a worse
+asymptotic complexity), so there is no reason not to use it in all cases.
+
+"""
+
+# Constants in Chudnovsky's series
+CHUD_A = MPZ(13591409)
+CHUD_B = MPZ(545140134)
+CHUD_C = MPZ(640320)
+CHUD_D = MPZ(12)
+
+def bs_chudnovsky(a, b, level, verbose):
+    """
+    Computes the sum from a to b of the series in the Chudnovsky
+    formula. Returns g, p, q where p/q is the sum as an exact
+    fraction and g is a temporary value used to save work
+    for recursive calls.
+    """
+    if b-a == 1:
+        g = MPZ((6*b-5)*(2*b-1)*(6*b-1))
+        p = b**3 * CHUD_C**3 // 24
+        q = (-1)**b * g * (CHUD_A+CHUD_B*b)
+    else:
+        if verbose and level < 4:
+            print("  binary splitting", a, b)
+        mid = (a+b)//2
+        g1, p1, q1 = bs_chudnovsky(a, mid, level+1, verbose)
+        g2, p2, q2 = bs_chudnovsky(mid, b, level+1, verbose)
+        p = p1*p2
+        g = g1*g2
+        q = q1*p2 + q2*g1
+    return g, p, q
+
+@constant_memo
+def pi_fixed(prec, verbose=False, verbose_base=None):
+    """
+    Compute floor(pi * 2**prec) as a big integer.
+
+    This is done using Chudnovsky's series (see comments in
+    libelefun.py for details).
+    """
+    # The Chudnovsky series gives 14.18 digits per term
+    N = int(prec/3.3219280948/14.181647462 + 2)
+    if verbose:
+        print("binary splitting with N =", N)
+    g, p, q = bs_chudnovsky(0, N, 0, verbose)
+    sqrtC = isqrt_fast(CHUD_C<<(2*prec))
+    v = p*CHUD_C*sqrtC//((q+CHUD_A*p)*CHUD_D)
+    return v
+
+def degree_fixed(prec):
+    return pi_fixed(prec)//180
+
+def bspe(a, b):
+    """
+    Sum series for exp(1)-1 between a, b, returning the result
+    as an exact fraction (p, q).
+    """
+    if b-a == 1:
+        return MPZ_ONE, MPZ(b)
+    m = (a+b)//2
+    p1, q1 = bspe(a, m)
+    p2, q2 = bspe(m, b)
+    return p1*q2+p2, q1*q2
+
+@constant_memo
+def e_fixed(prec):
+    """
+    Computes exp(1). This is done using the ordinary Taylor series for
+    exp, with binary splitting. For a description of the algorithm,
+    see:
+
+        http://numbers.computation.free.fr/Constants/
+            Algorithms/splitting.html
+    """
+    # Slight overestimate of N needed for 1/N! < 2**(-prec)
+    # This could be tightened for large N.
+    N = int(1.1*prec/math.log(prec) + 20)
+    p, q = bspe(0,N)
+    return ((p+q)<<prec)//q
+
+@constant_memo
+def phi_fixed(prec):
+    """
+    Computes the golden ratio, (1+sqrt(5))/2
+    """
+    prec += 10
+    a = isqrt_fast(MPZ_FIVE<<(2*prec)) + (MPZ_ONE << prec)
+    return a >> 11
+
+mpf_phi    = def_mpf_constant(phi_fixed)
+mpf_pi     = def_mpf_constant(pi_fixed)
+mpf_e      = def_mpf_constant(e_fixed)
+mpf_degree = def_mpf_constant(degree_fixed)
+mpf_ln2    = def_mpf_constant(ln2_fixed)
+mpf_ln10   = def_mpf_constant(ln10_fixed)
+
+
+@constant_memo
+def ln_sqrt2pi_fixed(prec):
+    wp = prec + 10
+    # ln(sqrt(2*pi)) = ln(2*pi)/2
+    return to_fixed(mpf_log(mpf_shift(mpf_pi(wp), 1), wp), prec-1)
+
+@constant_memo
+def sqrtpi_fixed(prec):
+    return sqrt_fixed(pi_fixed(prec), prec)
+
+mpf_sqrtpi   = def_mpf_constant(sqrtpi_fixed)
+mpf_ln_sqrt2pi   = def_mpf_constant(ln_sqrt2pi_fixed)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                                    Powers                                  #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+def mpf_pow(s, t, prec, rnd=round_fast):
+    """
+    Compute s**t. Raises ComplexResult if s is negative and t is
+    fractional.
+    """
+    ssign, sman, sexp, sbc = s
+    tsign, tman, texp, tbc = t
+    if ssign and texp < 0:
+        raise ComplexResult("negative number raised to a fractional power")
+    if texp >= 0:
+        return mpf_pow_int(s, (-1)**tsign * (tman<<texp), prec, rnd)
+    # s**(n/2) = sqrt(s)**n
+    if texp == -1:
+        if tman == 1:
+            if tsign:
+                return mpf_div(fone, mpf_sqrt(s, prec+10,
+                    reciprocal_rnd[rnd]), prec, rnd)
+            return mpf_sqrt(s, prec, rnd)
+        else:
+            if tsign:
+                return mpf_pow_int(mpf_sqrt(s, prec+10,
+                    reciprocal_rnd[rnd]), -tman, prec, rnd)
+            return mpf_pow_int(mpf_sqrt(s, prec+10, rnd), tman, prec, rnd)
+    # General formula: s**t = exp(t*log(s))
+    # TODO: handle rnd direction of the logarithm carefully
+    c = mpf_log(s, prec+10, rnd)
+    return mpf_exp(mpf_mul(t, c), prec, rnd)
+
+def int_pow_fixed(y, n, prec):
+    """n-th power of a fixed point number with precision prec
+
+       Returns the power in the form man, exp,
+       man * 2**exp ~= y**n
+    """
+    if n == 2:
+        return (y*y), 0
+    bc = bitcount(y)
+    exp = 0
+    workprec = 2 * (prec + 4*bitcount(n) + 4)
+    _, pm, pe, pbc = fone
+    while 1:
+        if n & 1:
+            pm = pm*y
+            pe = pe+exp
+            pbc += bc - 2
+            pbc = pbc + bctable[int(pm >> pbc)]
+            if pbc > workprec:
+                pm = pm >> (pbc-workprec)
+                pe += pbc - workprec
+                pbc = workprec
+            n -= 1
+            if not n:
+                break
+        y = y*y
+        exp = exp+exp
+        bc = bc + bc - 2
+        bc = bc + bctable[int(y >> bc)]
+        if bc > workprec:
+            y = y >> (bc-workprec)
+            exp += bc - workprec
+            bc = workprec
+        n = n // 2
+    return pm, pe
+
+# froot(s, n, prec, rnd) computes the real n-th root of a
+# positive mpf tuple s.
+# To compute the root we start from a 50-bit estimate for r
+# generated with ordinary floating-point arithmetic, and then refine
+# the value to full accuracy using the iteration
+
+#            1  /                     y       \
+#   r     = --- | (n-1)  * r   +  ----------  |
+#    n+1     n  \           n     r_n**(n-1)  /
+
+# which is simply Newton's method applied to the equation r**n = y.
+# With giant_steps(start, prec+extra) = [p0,...,pm, prec+extra]
+# and y = man * 2**-shift  one has
+# (man * 2**exp)**(1/n) =
+# y**(1/n) * 2**(start-prec/n) * 2**(p0-start) * ... * 2**(prec+extra-pm) *
+# 2**((exp+shift-(n-1)*prec)/n -extra))
+# The last factor is accounted for in the last line of froot.
+
+def nthroot_fixed(y, n, prec, exp1):
+    start = 50
+    try:
+        y1 = rshift(y, prec - n*start)
+        r = MPZ(int(y1**(1.0/n)))
+    except OverflowError:
+        y1 = from_int(y1, start)
+        fn = from_int(n)
+        fn = mpf_rdiv_int(1, fn, start)
+        r = mpf_pow(y1, fn, start)
+        r = to_int(r)
+    extra = 10
+    extra1 = n
+    prevp = start
+    for p in giant_steps(start, prec+extra):
+        pm, pe = int_pow_fixed(r, n-1, prevp)
+        r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
+        B = lshift(y, 2*p-prec+extra1)//r2
+        r = (B + (n-1) * lshift(r, p-prevp))//n
+        prevp = p
+    return r
+
+def mpf_nthroot(s, n, prec, rnd=round_fast):
+    """nth-root of a positive number
+
+    Use the Newton method when faster, otherwise use x**(1/n)
+    """
+    sign, man, exp, bc = s
+    if sign:
+        raise ComplexResult("nth root of a negative number")
+    if not man:
+        if s == fnan:
+            return fnan
+        if s == fzero:
+            if n > 0:
+                return fzero
+            if n == 0:
+                return fone
+            return finf
+        # Infinity
+        if not n:
+            return fnan
+        if n < 0:
+            return fzero
+        return finf
+    flag_inverse = False
+    if n < 2:
+        if n == 0:
+            return fone
+        if n == 1:
+            return mpf_pos(s, prec, rnd)
+        if n == -1:
+            return mpf_div(fone, s, prec, rnd)
+        # n < 0
+        rnd = reciprocal_rnd[rnd]
+        flag_inverse = True
+        extra_inverse = 5
+        prec += extra_inverse
+        n = -n
+    if n > 20 and (n >= 20000 or prec < int(233 + 28.3 * n**0.62)):
+        prec2 = prec + 10
+        fn = from_int(n)
+        nth = mpf_rdiv_int(1, fn, prec2)
+        r = mpf_pow(s, nth, prec2, rnd)
+        s = normalize(r[0], r[1], r[2], r[3], prec, rnd)
+        if flag_inverse:
+            return mpf_div(fone, s, prec-extra_inverse, rnd)
+        else:
+            return s
+    # Convert to a fixed-point number with prec2 bits.
+    prec2 = prec + 2*n - (prec%n)
+    # a few tests indicate that
+    # for 10 < n < 10**4 a bit more precision is needed
+    if n > 10:
+        prec2 += prec2//10
+        prec2 = prec2 - prec2%n
+    # Mantissa may have more bits than we need. Trim it down.
+    shift = bc - prec2
+    # Adjust exponents to make prec2 and exp+shift multiples of n.
+    sign1 = 0
+    es = exp+shift
+    if es < 0:
+        sign1 = 1
+        es = -es
+    if sign1:
+        shift += es%n
+    else:
+        shift -= es%n
+    man = rshift(man, shift)
+    extra = 10
+    exp1 = ((exp+shift-(n-1)*prec2)//n) - extra
+    rnd_shift = 0
+    if flag_inverse:
+        if rnd == 'u' or rnd == 'c':
+            rnd_shift = 1
+    else:
+        if rnd == 'd' or rnd == 'f':
+            rnd_shift = 1
+    man = nthroot_fixed(man+rnd_shift, n, prec2, exp1)
+    s = from_man_exp(man, exp1, prec, rnd)
+    if flag_inverse:
+        return mpf_div(fone, s, prec-extra_inverse, rnd)
+    else:
+        return s
+
+def mpf_cbrt(s, prec, rnd=round_fast):
+    """cubic root of a positive number"""
+    return mpf_nthroot(s, 3, prec, rnd)
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                                Logarithms                                  #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+def log_int_fixed(n, prec, ln2=None):
+    """
+    Fast computation of log(n), caching the value for small n,
+    intended for zeta sums.
+    """
+    if n in log_int_cache:
+        value, vprec = log_int_cache[n]
+        if vprec >= prec:
+            return value >> (vprec - prec)
+    wp = prec + 10
+    if wp <= LOG_TAYLOR_SHIFT:
+        if ln2 is None:
+            ln2 = ln2_fixed(wp)
+        r = bitcount(n)
+        x = n << (wp-r)
+        v = log_taylor_cached(x, wp) + r*ln2
+    else:
+        v = to_fixed(mpf_log(from_int(n), wp+5), wp)
+    if n < MAX_LOG_INT_CACHE:
+        log_int_cache[n] = (v, wp)
+    return v >> (wp-prec)
+
+def agm_fixed(a, b, prec):
+    """
+    Fixed-point computation of agm(a,b), assuming
+    a, b both close to unit magnitude.
+    """
+    i = 0
+    while 1:
+        anew = (a+b)>>1
+        if i > 4 and abs(a-anew) < 8:
+            return a
+        b = isqrt_fast(a*b)
+        a = anew
+        i += 1
+    return a
+
+def log_agm(x, prec):
+    """
+    Fixed-point computation of -log(x) = log(1/x), suitable
+    for large precision. It is required that 0 < x < 1. The
+    algorithm used is the Sasaki-Kanada formula
+
+        -log(x) = pi/agm(theta2(x)^2,theta3(x)^2). [1]
+
+    For faster convergence in the theta functions, x should
+    be chosen closer to 0.
+
+    Guard bits must be added by the caller.
+
+    HYPOTHESIS: if x = 2^(-n), n bits need to be added to
+    account for the truncation to a fixed-point number,
+    and this is the only significant cancellation error.
+
+    The number of bits lost to roundoff is small and can be
+    considered constant.
+
+    [1] Richard P. Brent, "Fast Algorithms for High-Precision
+        Computation of Elementary Functions (extended abstract)",
+        http://wwwmaths.anu.edu.au/~brent/pd/RNC7-Brent.pdf
+
+    """
+    x2 = (x*x) >> prec
+    # Compute jtheta2(x)**2
+    s = a = b = x2
+    while a:
+        b = (b*x2) >> prec
+        a = (a*b) >> prec
+        s += a
+    s += (MPZ_ONE<<prec)
+    s = (s*s)>>(prec-2)
+    s = (s*isqrt_fast(x<<prec))>>prec
+    # Compute jtheta3(x)**2
+    t = a = b = x
+    while a:
+        b = (b*x2) >> prec
+        a = (a*b) >> prec
+        t += a
+    t = (MPZ_ONE<<prec) + (t<<1)
+    t = (t*t)>>prec
+    # Final formula
+    p = agm_fixed(s, t, prec)
+    return (pi_fixed(prec) << prec) // p
+
+def log_taylor(x, prec, r=0):
+    """
+    Fixed-point calculation of log(x). It is assumed that x is close
+    enough to 1 for the Taylor series to converge quickly. Convergence
+    can be improved by specifying r > 0 to compute
+    log(x^(1/2^r))*2^r, at the cost of performing r square roots.
+
+    The caller must provide sufficient guard bits.
+    """
+    for i in xrange(r):
+        x = isqrt_fast(x<<prec)
+    one = MPZ_ONE << prec
+    v = ((x-one)<<prec)//(x+one)
+    sign = v < 0
+    if sign:
+        v = -v
+    v2 = (v*v) >> prec
+    v4 = (v2*v2) >> prec
+    s0 = v
+    s1 = v//3
+    v = (v*v4) >> prec
+    k = 5
+    while v:
+        s0 += v // k
+        k += 2
+        s1 += v // k
+        v = (v*v4) >> prec
+        k += 2
+    s1 = (s1*v2) >> prec
+    s = (s0+s1) << (1+r)
+    if sign:
+        return -s
+    return s
+
+def log_taylor_cached(x, prec):
+    """
+    Fixed-point computation of log(x), assuming x in (0.5, 2)
+    and prec <= LOG_TAYLOR_PREC.
+    """
+    n = x >> (prec-LOG_TAYLOR_SHIFT)
+    cached_prec = cache_prec_steps[prec]
+    dprec = cached_prec - prec
+    if (n, cached_prec) in log_taylor_cache:
+        a, log_a = log_taylor_cache[n, cached_prec]
+    else:
+        a = n << (cached_prec - LOG_TAYLOR_SHIFT)
+        log_a = log_taylor(a, cached_prec, 8)
+        log_taylor_cache[n, cached_prec] = (a, log_a)
+    a >>= dprec
+    log_a >>= dprec
+    u = ((x - a) << prec) // a
+    v = (u << prec) // ((MPZ_TWO << prec) + u)
+    v2 = (v*v) >> prec
+    v4 = (v2*v2) >> prec
+    s0 = v
+    s1 = v//3
+    v = (v*v4) >> prec
+    k = 5
+    while v:
+        s0 += v//k
+        k += 2
+        s1 += v//k
+        v = (v*v4) >> prec
+        k += 2
+    s1 = (s1*v2) >> prec
+    s = (s0+s1) << 1
+    return log_a + s
+
+def mpf_log(x, prec, rnd=round_fast):
+    """
+    Compute the natural logarithm of the mpf value x. If x is negative,
+    ComplexResult is raised.
+    """
+    sign, man, exp, bc = x
+    #------------------------------------------------------------------
+    # Handle special values
+    if not man:
+        if x == fzero: return fninf
+        if x == finf: return finf
+        if x == fnan: return fnan
+    if sign:
+        raise ComplexResult("logarithm of a negative number")
+    wp = prec + 20
+    #------------------------------------------------------------------
+    # Handle log(2^n) = log(n)*2.
+    # Here we catch the only possible exact value, log(1) = 0
+    if man == 1:
+        if not exp:
+            return fzero
+        return from_man_exp(exp*ln2_fixed(wp), -wp, prec, rnd)
+    mag = exp+bc
+    abs_mag = abs(mag)
+    #------------------------------------------------------------------
+    # Handle x = 1+eps, where log(x) ~ x. We need to check for
+    # cancellation when moving to fixed-point math and compensate
+    # by increasing the precision. Note that abs_mag in (0, 1) <=>
+    # 0.5 < x < 2 and x != 1
+    if abs_mag <= 1:
+        # Calculate t = x-1 to measure distance from 1 in bits
+        tsign = 1-abs_mag
+        if tsign:
+            tman = (MPZ_ONE<<bc) - man
+        else:
+            tman = man - (MPZ_ONE<<(bc-1))
+        tbc = bitcount(tman)
+        cancellation = bc - tbc
+        if cancellation > wp:
+            t = normalize(tsign, tman, abs_mag-bc, tbc, tbc, 'n')
+            return mpf_perturb(t, tsign, prec, rnd)
+        else:
+            wp += cancellation
+        # TODO: if close enough to 1, we could use Taylor series
+        # even in the AGM precision range, since the Taylor series
+        # converges rapidly
+    #------------------------------------------------------------------
+    # Another special case:
+    # n*log(2) is a good enough approximation
+    if abs_mag > 10000:
+        if bitcount(abs_mag) > wp:
+            return from_man_exp(exp*ln2_fixed(wp), -wp, prec, rnd)
+    #------------------------------------------------------------------
+    # General case.
+    # Perform argument reduction using log(x) = log(x*2^n) - n*log(2):
+    # If we are in the Taylor precision range, choose magnitude 0 or 1.
+    # If we are in the AGM precision range, choose magnitude -m for
+    # some large m; benchmarking on one machine showed m = prec/20 to be
+    # optimal between 1000 and 100,000 digits.
+    if wp <= LOG_TAYLOR_PREC:
+        m = log_taylor_cached(lshift(man, wp-bc), wp)
+        if mag:
+            m += mag*ln2_fixed(wp)
+    else:
+        optimal_mag = -wp//LOG_AGM_MAG_PREC_RATIO
+        n = optimal_mag - mag
+        x = mpf_shift(x, n)
+        wp += (-optimal_mag)
+        m = -log_agm(to_fixed(x, wp), wp)
+        m -= n*ln2_fixed(wp)
+    return from_man_exp(m, -wp, prec, rnd)
+
+def mpf_log_hypot(a, b, prec, rnd):
+    """
+    Computes log(sqrt(a^2+b^2)) accurately.
+    """
+    # If either a or b is inf/nan/0, assume it to be a
+    if not b[1]:
+        a, b = b, a
+    # a is inf/nan/0
+    if not a[1]:
+        # both are inf/nan/0
+        if not b[1]:
+            if a == b == fzero:
+                return fninf
+            if fnan in (a, b):
+                return fnan
+            # at least one term is (+/- inf)^2
+            return finf
+        # only a is inf/nan/0
+        if a == fzero:
+            # log(sqrt(0+b^2)) = log(|b|)
+            return mpf_log(mpf_abs(b), prec, rnd)
+        if a == fnan:
+            return fnan
+        return finf
+    # Exact
+    a2 = mpf_mul(a,a)
+    b2 = mpf_mul(b,b)
+    extra = 20
+    # Not exact
+    h2 = mpf_add(a2, b2, prec+extra)
+    cancelled = mpf_add(h2, fnone, 10)
+    mag_cancelled = cancelled[2]+cancelled[3]
+    # Just redo the sum exactly if necessary (could be smarter
+    # and avoid memory allocation when a or b is precisely 1
+    # and the other is tiny...)
+    if cancelled == fzero or mag_cancelled < -extra//2:
+        h2 = mpf_add(a2, b2, prec+extra-min(a2[2],b2[2]))
+    return mpf_shift(mpf_log(h2, prec, rnd), -1)
+
+
+#----------------------------------------------------------------------
+# Inverse tangent
+#
+
+def atan_newton(x, prec):
+    if prec >= 100:
+        r = math.atan(int((x>>(prec-53)))/2.0**53)
+    else:
+        r = math.atan(int(x)/2.0**prec)
+    prevp = 50
+    r = MPZ(int(r * 2.0**53) >> (53-prevp))
+    extra_p = 50
+    for wp in giant_steps(prevp, prec):
+        wp += extra_p
+        r = r << (wp-prevp)
+        cos, sin = cos_sin_fixed(r, wp)
+        tan = (sin << wp) // cos
+        a = ((tan-rshift(x, prec-wp)) << wp) // ((MPZ_ONE<<wp) + ((tan**2)>>wp))
+        r = r - a
+        prevp = wp
+    return rshift(r, prevp-prec)
+
+def atan_taylor_get_cached(n, prec):
+    # Taylor series with caching wins up to huge precisions
+    # To avoid unnecessary precomputation at low precision, we
+    # do it in steps
+    # Round to next power of 2
+    prec2 = (1<<(bitcount(prec-1))) + 20
+    dprec = prec2 - prec
+    if (n, prec2) in atan_taylor_cache:
+        a, atan_a = atan_taylor_cache[n, prec2]
+    else:
+        a = n << (prec2 - ATAN_TAYLOR_SHIFT)
+        atan_a = atan_newton(a, prec2)
+        atan_taylor_cache[n, prec2] = (a, atan_a)
+    return (a >> dprec), (atan_a >> dprec)
+
+def atan_taylor(x, prec):
+    n = (x >> (prec-ATAN_TAYLOR_SHIFT))
+    a, atan_a = atan_taylor_get_cached(n, prec)
+    d = x - a
+    s0 = v = (d << prec) // ((a**2 >> prec) + (a*d >> prec) + (MPZ_ONE << prec))
+    v2 = (v**2 >> prec)
+    v4 = (v2 * v2) >> prec
+    s1 = v//3
+    v = (v * v4) >> prec
+    k = 5
+    while v:
+        s0 += v // k
+        k += 2
+        s1 += v // k
+        v = (v * v4) >> prec
+        k += 2
+    s1 = (s1 * v2) >> prec
+    s = s0 - s1
+    return atan_a + s
+
+def atan_inf(sign, prec, rnd):
+    if not sign:
+        return mpf_shift(mpf_pi(prec, rnd), -1)
+    return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
+
+def mpf_atan(x, prec, rnd=round_fast):
+    sign, man, exp, bc = x
+    if not man:
+        if x == fzero: return fzero
+        if x == finf: return atan_inf(0, prec, rnd)
+        if x == fninf: return atan_inf(1, prec, rnd)
+        return fnan
+    mag = exp + bc
+    # Essentially infinity
+    if mag > prec+20:
+        return atan_inf(sign, prec, rnd)
+    # Essentially ~ x
+    if -mag > prec+20:
+        return mpf_perturb(x, 1-sign, prec, rnd)
+    wp = prec + 30 + abs(mag)
+    # For large x, use atan(x) = pi/2 - atan(1/x)
+    if mag >= 2:
+        x = mpf_rdiv_int(1, x, wp)
+        reciprocal = True
+    else:
+        reciprocal = False
+    t = to_fixed(x, wp)
+    if sign:
+        t = -t
+    if wp < ATAN_TAYLOR_PREC:
+        a = atan_taylor(t, wp)
+    else:
+        a = atan_newton(t, wp)
+    if reciprocal:
+        a = ((pi_fixed(wp)>>1)+1) - a
+    if sign:
+        a = -a
+    return from_man_exp(a, -wp, prec, rnd)
+
+# TODO: cleanup the special cases
+def mpf_atan2(y, x, prec, rnd=round_fast):
+    xsign, xman, xexp, xbc = x
+    ysign, yman, yexp, ybc = y
+    if not yman:
+        if y == fzero and x != fnan:
+            if mpf_sign(x) >= 0:
+                return fzero
+            return mpf_pi(prec, rnd)
+        if y in (finf, fninf):
+            if x in (finf, fninf):
+                return fnan
+            # pi/2
+            if y == finf:
+                return mpf_shift(mpf_pi(prec, rnd), -1)
+            # -pi/2
+            return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
+        return fnan
+    if ysign:
+        return mpf_neg(mpf_atan2(mpf_neg(y), x, prec, negative_rnd[rnd]))
+    if not xman:
+        if x == fnan:
+            return fnan
+        if x == finf:
+            return fzero
+        if x == fninf:
+            return mpf_pi(prec, rnd)
+        if y == fzero:
+            return fzero
+        return mpf_shift(mpf_pi(prec, rnd), -1)
+    tquo = mpf_atan(mpf_div(y, x, prec+4), prec+4)
+    if xsign:
+        return mpf_add(mpf_pi(prec+4), tquo, prec, rnd)
+    else:
+        return mpf_pos(tquo, prec, rnd)
+
+def mpf_asin(x, prec, rnd=round_fast):
+    sign, man, exp, bc = x
+    if bc+exp > 0 and x not in (fone, fnone):
+        raise ComplexResult("asin(x) is real only for -1 <= x <= 1")
+    # asin(x) = 2*atan(x/(1+sqrt(1-x**2)))
+    wp = prec + 15
+    a = mpf_mul(x, x)
+    b = mpf_add(fone, mpf_sqrt(mpf_sub(fone, a, wp), wp), wp)
+    c = mpf_div(x, b, wp)
+    return mpf_shift(mpf_atan(c, prec, rnd), 1)
+
+def mpf_acos(x, prec, rnd=round_fast):
+    # acos(x) = 2*atan(sqrt(1-x**2)/(1+x))
+    sign, man, exp, bc = x
+    if bc + exp > 0:
+        if x not in (fone, fnone):
+            raise ComplexResult("acos(x) is real only for -1 <= x <= 1")
+        if x == fnone:
+            return mpf_pi(prec, rnd)
+    wp = prec + 15
+    a = mpf_mul(x, x)
+    b = mpf_sqrt(mpf_sub(fone, a, wp), wp)
+    c = mpf_div(b, mpf_add(fone, x, wp), wp)
+    return mpf_shift(mpf_atan(c, prec, rnd), 1)
+
+def mpf_asinh(x, prec, rnd=round_fast):
+    wp = prec + 20
+    sign, man, exp, bc = x
+    mag = exp+bc
+    if mag < -8:
+        if mag < -wp:
+            return mpf_perturb(x, 1-sign, prec, rnd)
+        wp += (-mag)
+    # asinh(x) = log(x+sqrt(x**2+1))
+    # use reflection symmetry to avoid cancellation
+    q = mpf_sqrt(mpf_add(mpf_mul(x, x), fone, wp), wp)
+    q = mpf_add(mpf_abs(x), q, wp)
+    if sign:
+        return mpf_neg(mpf_log(q, prec, negative_rnd[rnd]))
+    else:
+        return mpf_log(q, prec, rnd)
+
+def mpf_acosh(x, prec, rnd=round_fast):
+    # acosh(x) = log(x+sqrt(x**2-1))
+    wp = prec + 15
+    if mpf_cmp(x, fone) == -1:
+        raise ComplexResult("acosh(x) is real only for x >= 1")
+    q = mpf_sqrt(mpf_add(mpf_mul(x,x), fnone, wp), wp)
+    return mpf_log(mpf_add(x, q, wp), prec, rnd)
+
+def mpf_atanh(x, prec, rnd=round_fast):
+    # atanh(x) = log((1+x)/(1-x))/2
+    sign, man, exp, bc = x
+    if (not man) and exp:
+        if x in (fzero, fnan):
+            return x
+        raise ComplexResult("atanh(x) is real only for -1 <= x <= 1")
+    mag = bc + exp
+    if mag > 0:
+        if mag == 1 and man == 1:
+            return [finf, fninf][sign]
+        raise ComplexResult("atanh(x) is real only for -1 <= x <= 1")
+    wp = prec + 15
+    if mag < -8:
+        if mag < -wp:
+            return mpf_perturb(x, sign, prec, rnd)
+        wp += (-mag)
+    a = mpf_add(x, fone, wp)
+    b = mpf_sub(fone, x, wp)
+    return mpf_shift(mpf_log(mpf_div(a, b, wp), prec, rnd), -1)
+
+def mpf_fibonacci(x, prec, rnd=round_fast):
+    sign, man, exp, bc = x
+    if not man:
+        if x == fninf:
+            return fnan
+        return x
+    # F(2^n) ~= 2^(2^n)
+    size = abs(exp+bc)
+    if exp >= 0:
+        # Exact
+        if size < 10 or size <= bitcount(prec):
+            return from_int(ifib(to_int(x)), prec, rnd)
+    # Use the modified Binet formula
+    wp = prec + size + 20
+    a = mpf_phi(wp)
+    b = mpf_add(mpf_shift(a, 1), fnone, wp)
+    u = mpf_pow(a, x, wp)
+    v = mpf_cos_pi(x, wp)
+    v = mpf_div(v, u, wp)
+    u = mpf_sub(u, v, wp)
+    u = mpf_div(u, b, prec, rnd)
+    return u
+
+
+#-------------------------------------------------------------------------------
+# Exponential-type functions
+#-------------------------------------------------------------------------------
+
+def exponential_series(x, prec, type=0):
+    """
+    Taylor series for cosh/sinh or cos/sin.
+
+    type = 0 -- returns exp(x)  (slightly faster than cosh+sinh)
+    type = 1 -- returns (cosh(x), sinh(x))
+    type = 2 -- returns (cos(x), sin(x))
+    """
+    if x < 0:
+        x = -x
+        sign = 1
+    else:
+        sign = 0
+    r = int(0.5*prec**0.5)
+    xmag = bitcount(x) - prec
+    r = max(0, xmag + r)
+    extra = 10 + 2*max(r,-xmag)
+    wp = prec + extra
+    x <<= (extra - r)
+    one = MPZ_ONE << wp
+    alt = (type == 2)
+    if prec < EXP_SERIES_U_CUTOFF:
+        x2 = a = (x*x) >> wp
+        x4 = (x2*x2) >> wp
+        s0 = s1 = MPZ_ZERO
+        k = 2
+        while a:
+            a //= (k-1)*k; s0 += a; k += 2
+            a //= (k-1)*k; s1 += a; k += 2
+            a = (a*x4) >> wp
+        s1 = (x2*s1) >> wp
+        if alt:
+            c = s1 - s0 + one
+        else:
+            c = s1 + s0 + one
+    else:
+        u = int(0.3*prec**0.35)
+        x2 = a = (x*x) >> wp
+        xpowers = [one, x2]
+        for i in xrange(1, u):
+            xpowers.append((xpowers[-1]*x2)>>wp)
+        sums = [MPZ_ZERO] * u
+        k = 2
+        while a:
+            for i in xrange(u):
+                a //= (k-1)*k
+                if alt and k & 2: sums[i] -= a
+                else:             sums[i] += a
+                k += 2
+            a = (a*xpowers[-1]) >> wp
+        for i in xrange(1, u):
+            sums[i] = (sums[i]*xpowers[i]) >> wp
+        c = sum(sums) + one
+    if type == 0:
+        s = isqrt_fast(c*c - (one<<wp))
+        if sign:
+            v = c - s
+        else:
+            v = c + s
+        for i in xrange(r):
+            v = (v*v) >> wp
+        return v >> extra
+    else:
+        # Repeatedly apply the double-angle formula
+        # cosh(2*x) = 2*cosh(x)^2 - 1
+        # cos(2*x) = 2*cos(x)^2 - 1
+        pshift = wp-1
+        for i in xrange(r):
+            c = ((c*c) >> pshift) - one
+        # With the abs, this is the same for sinh and sin
+        s = isqrt_fast(abs((one<<wp) - c*c))
+        if sign:
+            s = -s
+        return (c>>extra), (s>>extra)
+
+def exp_basecase(x, prec):
+    """
+    Compute exp(x) as a fixed-point number. Works for any x,
+    but for speed should have |x| < 1. For an arbitrary number,
+    use exp(x) = exp(x-m*log(2)) * 2^m where m = floor(x/log(2)).
+    """
+    if prec > EXP_COSH_CUTOFF:
+        return exponential_series(x, prec, 0)
+    r = int(prec**0.5)
+    prec += r
+    s0 = s1 = (MPZ_ONE << prec)
+    k = 2
+    a = x2 = (x*x) >> prec
+    while a:
+        a //= k; s0 += a; k += 1
+        a //= k; s1 += a; k += 1
+        a = (a*x2) >> prec
+    s1 = (s1*x) >> prec
+    s = s0 + s1
+    u = r
+    while r:
+        s = (s*s) >> prec
+        r -= 1
+    return s >> u
+
+def exp_expneg_basecase(x, prec):
+    """
+    Computation of exp(x), exp(-x)
+    """
+    if prec > EXP_COSH_CUTOFF:
+        cosh, sinh = exponential_series(x, prec, 1)
+        return cosh+sinh, cosh-sinh
+    a = exp_basecase(x, prec)
+    b = (MPZ_ONE << (prec+prec)) // a
+    return a, b
+
+def cos_sin_basecase(x, prec):
+    """
+    Compute cos(x), sin(x) as fixed-point numbers, assuming x
+    in [0, pi/2). For an arbitrary number, use x' = x - m*(pi/2)
+    where m = floor(x/(pi/2)) along with quarter-period symmetries.
+    """
+    if prec > COS_SIN_CACHE_PREC:
+        return exponential_series(x, prec, 2)
+    precs = prec - COS_SIN_CACHE_STEP
+    t = x >> precs
+    n = int(t)
+    if n not in cos_sin_cache:
+        w = t<<(10+COS_SIN_CACHE_PREC-COS_SIN_CACHE_STEP)
+        cos_t, sin_t = exponential_series(w, 10+COS_SIN_CACHE_PREC, 2)
+        cos_sin_cache[n] = (cos_t>>10), (sin_t>>10)
+    cos_t, sin_t = cos_sin_cache[n]
+    offset = COS_SIN_CACHE_PREC - prec
+    cos_t >>= offset
+    sin_t >>= offset
+    x -= t << precs
+    cos = MPZ_ONE << prec
+    sin = x
+    k = 2
+    a = -((x*x) >> prec)
+    while a:
+        a //= k; cos += a; k += 1; a = (a*x) >> prec
+        a //= k; sin += a; k += 1; a = -((a*x) >> prec)
+    return ((cos*cos_t-sin*sin_t) >> prec), ((sin*cos_t+cos*sin_t) >> prec)
+
+def mpf_exp(x, prec, rnd=round_fast):
+    sign, man, exp, bc = x
+    if man:
+        mag = bc + exp
+        wp = prec + 14
+        if sign:
+            man = -man
+        # TODO: the best cutoff depends on both x and the precision.
+        if prec > 600 and exp >= 0:
+            # Need about log2(exp(n)) ~= 1.45*mag extra precision
+            e = mpf_e(wp+int(1.45*mag))
+            return mpf_pow_int(e, man<<exp, prec, rnd)
+        if mag < -wp:
+            return mpf_perturb(fone, sign, prec, rnd)
+        # |x| >= 2
+        if mag > 1:
+            # For large arguments: exp(2^mag*(1+eps)) =
+            # exp(2^mag)*exp(2^mag*eps) = exp(2^mag)*(1 + 2^mag*eps + ...)
+            # so about mag extra bits is required.
+            wpmod = wp + mag
+            offset = exp + wpmod
+            if offset >= 0:
+                t = man << offset
+            else:
+                t = man >> (-offset)
+            lg2 = ln2_fixed(wpmod)
+            n, t = divmod(t, lg2)
+            n = int(n)
+            t >>= mag
+        else:
+            offset = exp + wp
+            if offset >= 0:
+                t = man << offset
+            else:
+                t = man >> (-offset)
+            n = 0
+        man = exp_basecase(t, wp)
+        return from_man_exp(man, n-wp, prec, rnd)
+    if not exp:
+        return fone
+    if x == fninf:
+        return fzero
+    return x
+
+
+def mpf_cosh_sinh(x, prec, rnd=round_fast, tanh=0):
+    """Simultaneously compute (cosh(x), sinh(x)) for real x"""
+    sign, man, exp, bc = x
+    if (not man) and exp:
+        if tanh:
+            if x == finf: return fone
+            if x == fninf: return fnone
+            return fnan
+        if x == finf: return (finf, finf)
+        if x == fninf: return (finf, fninf)
+        return fnan, fnan
+    mag = exp+bc
+    wp = prec+14
+    if mag < -4:
+        # Extremely close to 0, sinh(x) ~= x and cosh(x) ~= 1
+        if mag < -wp:
+            if tanh:
+                return mpf_perturb(x, 1-sign, prec, rnd)
+            cosh = mpf_perturb(fone, 0, prec, rnd)
+            sinh = mpf_perturb(x, sign, prec, rnd)
+            return cosh, sinh
+        # Fix for cancellation when computing sinh
+        wp += (-mag)
+    # Does exp(-2*x) vanish?
+    if mag > 10:
+        if 3*(1<<(mag-1)) > wp:
+            # XXX: rounding
+            if tanh:
+                return mpf_perturb([fone,fnone][sign], 1-sign, prec, rnd)
+            c = s = mpf_shift(mpf_exp(mpf_abs(x), prec, rnd), -1)
+            if sign:
+                s = mpf_neg(s)
+            return c, s
+    # |x| > 1
+    if mag > 1:
+        wpmod = wp + mag
+        offset = exp + wpmod
+        if offset >= 0:
+            t = man << offset
+        else:
+            t = man >> (-offset)
+        lg2 = ln2_fixed(wpmod)
+        n, t = divmod(t, lg2)
+        n = int(n)
+        t >>= mag
+    else:
+        offset = exp + wp
+        if offset >= 0:
+            t = man << offset
+        else:
+            t = man >> (-offset)
+        n = 0
+    a, b = exp_expneg_basecase(t, wp)
+    # TODO: optimize division precision
+    cosh = a + (b>>(2*n))
+    sinh = a - (b>>(2*n))
+    if sign:
+        sinh = -sinh
+    if tanh:
+        man = (sinh << wp) // cosh
+        return from_man_exp(man, -wp, prec, rnd)
+    else:
+        cosh = from_man_exp(cosh, n-wp-1, prec, rnd)
+        sinh = from_man_exp(sinh, n-wp-1, prec, rnd)
+        return cosh, sinh
+
+
+def mod_pi2(man, exp, mag, wp):
+    # Reduce to standard interval
+    if mag > 0:
+        i = 0
+        while 1:
+            cancellation_prec = 20 << i
+            wpmod = wp + mag + cancellation_prec
+            pi2 = pi_fixed(wpmod-1)
+            pi4 = pi2 >> 1
+            offset = wpmod + exp
+            if offset >= 0:
+                t = man << offset
+            else:
+                t = man >> (-offset)
+            n, y = divmod(t, pi2)
+            if y > pi4:
+                small = pi2 - y
+            else:
+                small = y
+            if small >> (wp+mag-10):
+                n = int(n)
+                t = y >> mag
+                wp = wpmod - mag
+                break
+            i += 1
+    else:
+        wp += (-mag)
+        offset = exp + wp
+        if offset >= 0:
+            t = man << offset
+        else:
+            t = man >> (-offset)
+        n = 0
+    return t, n, wp
+
+
+def mpf_cos_sin(x, prec, rnd=round_fast, which=0, pi=False):
+    """
+    which:
+    0 -- return cos(x), sin(x)
+    1 -- return cos(x)
+    2 -- return sin(x)
+    3 -- return tan(x)
+
+    if pi=True, compute for pi*x
+    """
+    sign, man, exp, bc = x
+    if not man:
+        if exp:
+            c, s = fnan, fnan
+        else:
+            c, s = fone, fzero
+        if which == 0: return c, s
+        if which == 1: return c
+        if which == 2: return s
+        if which == 3: return s
+
+    mag = bc + exp
+    wp = prec + 10
+
+    # Extremely small?
+    if mag < 0:
+        if mag < -wp:
+            if pi:
+                x = mpf_mul(x, mpf_pi(wp))
+            c = mpf_perturb(fone, 1, prec, rnd)
+            s = mpf_perturb(x, 1-sign, prec, rnd)
+            if which == 0: return c, s
+            if which == 1: return c
+            if which == 2: return s
+            if which == 3: return mpf_perturb(x, sign, prec, rnd)
+    if pi:
+        if exp >= -1:
+            if exp == -1:
+                c = fzero
+                s = (fone, fnone)[bool(man & 2) ^ sign]
+            elif exp == 0:
+                c, s = (fnone, fzero)
+            else:
+                c, s = (fone, fzero)
+            if which == 0: return c, s
+            if which == 1: return c
+            if which == 2: return s
+            if which == 3: return mpf_div(s, c, prec, rnd)
+        # Subtract nearest half-integer (= mod by pi/2)
+        n = ((man >> (-exp-2)) + 1) >> 1
+        man = man - (n << (-exp-1))
+        mag2 = bitcount(man) + exp
+        wp = prec + 10 - mag2
+        offset = exp + wp
+        if offset >= 0:
+            t = man << offset
+        else:
+            t = man >> (-offset)
+        t = (t*pi_fixed(wp)) >> wp
+    else:
+        t, n, wp = mod_pi2(man, exp, mag, wp)
+    c, s = cos_sin_basecase(t, wp)
+    m = n & 3
+    if   m == 1: c, s = -s, c
+    elif m == 2: c, s = -c, -s
+    elif m == 3: c, s = s, -c
+    if sign:
+        s = -s
+    if which == 0:
+        c = from_man_exp(c, -wp, prec, rnd)
+        s = from_man_exp(s, -wp, prec, rnd)
+        return c, s
+    if which == 1:
+        return from_man_exp(c, -wp, prec, rnd)
+    if which == 2:
+        return from_man_exp(s, -wp, prec, rnd)
+    if which == 3:
+        return from_rational(s, c, prec, rnd)
+
+def mpf_cos(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 1)
+def mpf_sin(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 2)
+def mpf_tan(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 3)
+def mpf_cos_sin_pi(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 0, 1)
+def mpf_cos_pi(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 1, 1)
+def mpf_sin_pi(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 2, 1)
+def mpf_cosh(x, prec, rnd=round_fast): return mpf_cosh_sinh(x, prec, rnd)[0]
+def mpf_sinh(x, prec, rnd=round_fast): return mpf_cosh_sinh(x, prec, rnd)[1]
+def mpf_tanh(x, prec, rnd=round_fast): return mpf_cosh_sinh(x, prec, rnd, tanh=1)
+
+
+# Low-overhead fixed-point versions
+
+def cos_sin_fixed(x, prec, pi2=None):
+    if pi2 is None:
+        pi2 = pi_fixed(prec-1)
+    n, t = divmod(x, pi2)
+    n = int(n)
+    c, s = cos_sin_basecase(t, prec)
+    m = n & 3
+    if m == 0: return c, s
+    if m == 1: return -s, c
+    if m == 2: return -c, -s
+    if m == 3: return s, -c
+
+def exp_fixed(x, prec, ln2=None):
+    if ln2 is None:
+        ln2 = ln2_fixed(prec)
+    n, t = divmod(x, ln2)
+    n = int(n)
+    v = exp_basecase(t, prec)
+    if n >= 0:
+        return v << n
+    else:
+        return v >> (-n)
+
+
+if BACKEND == 'sage':
+    try:
+        import sage.libs.mpmath.ext_libmp as _lbmp
+        mpf_sqrt = _lbmp.mpf_sqrt
+        mpf_exp = _lbmp.mpf_exp
+        mpf_log = _lbmp.mpf_log
+        mpf_cos = _lbmp.mpf_cos
+        mpf_sin = _lbmp.mpf_sin
+        mpf_pow = _lbmp.mpf_pow
+        exp_fixed = _lbmp.exp_fixed
+        cos_sin_fixed = _lbmp.cos_sin_fixed
+        log_int_fixed = _lbmp.log_int_fixed
+    except (ImportError, AttributeError):
+        print("Warning: Sage imports in libelefun failed")

.venv/lib/python3.11/site-packages/mpmath/tests/extratest_gamma.py

@@ -0,0 +1,215 @@
+from mpmath import *
+from mpmath.libmp import ifac
+
+import sys
+if "-dps" in sys.argv:
+    maxdps = int(sys.argv[sys.argv.index("-dps")+1])
+else:
+    maxdps = 1000
+
+raise_ = "-raise" in sys.argv
+
+errcount = 0
+
+def check(name, func, z, y):
+    global errcount
+    try:
+        x = func(z)
+    except:
+        errcount += 1
+        if raise_:
+            raise
+        print()
+        print(name)
+        print("EXCEPTION")
+        import traceback
+        traceback.print_tb(sys.exc_info()[2])
+        print()
+        return
+    xre = x.real
+    xim = x.imag
+    yre = y.real
+    yim = y.imag
+    tol = eps*8
+    err = 0
+    if abs(xre-yre) > abs(yre)*tol:
+        err = 1
+        print()
+        print("Error! %s (re = %s, wanted %s, err=%s)" % (name, nstr(xre,10), nstr(yre,10), nstr(abs(xre-yre))))
+        errcount += 1
+        if raise_:
+            raise SystemExit
+    if abs(xim-yim) > abs(yim)*tol:
+        err = 1
+        print()
+        print("Error! %s (im = %s, wanted %s, err=%s)" % (name, nstr(xim,10), nstr(yim,10), nstr(abs(xim-yim))))
+        errcount += 1
+        if raise_:
+            raise SystemExit
+    if not err:
+        sys.stdout.write("%s ok; " % name)
+
+def testcase(case):
+    z, result = case
+    print("Testing z =", z)
+    mp.dps = 1010
+    z = eval(z)
+    mp.dps = maxdps + 50
+    if result is None:
+        gamma_val = gamma(z)
+        loggamma_val = loggamma(z)
+        factorial_val = factorial(z)
+        rgamma_val = rgamma(z)
+    else:
+        loggamma_val = eval(result)
+        gamma_val = exp(loggamma_val)
+        factorial_val = z * gamma_val
+        rgamma_val = 1/gamma_val
+    for dps in [5, 10, 15, 25, 40, 60, 90, 120, 250, 600, 1000, 1800, 3600]:
+        if dps > maxdps:
+            break
+        mp.dps = dps
+        print("dps = %s" % dps)
+        check("gamma", gamma, z, gamma_val)
+        check("rgamma", rgamma, z, rgamma_val)
+        check("loggamma", loggamma, z, loggamma_val)
+        check("factorial", factorial, z, factorial_val)
+        print()
+        mp.dps = 15
+
+testcases = []
+
+# Basic values
+for n in list(range(1,200)) + list(range(201,2000,17)):
+    testcases.append(["%s" % n, None])
+for n in range(-200,200):
+    testcases.append(["%s+0.5" % n, None])
+    testcases.append(["%s+0.37" % n, None])
+
+testcases += [\
+["(0.1+1j)", None],
+["(-0.1+1j)", None],
+["(0.1-1j)", None],
+["(-0.1-1j)", None],
+["10j", None],
+["-10j", None],
+["100j", None],
+["10000j", None],
+["-10000000j", None],
+["(10**100)*j", None],
+["125+(10**100)*j", None],
+["-125+(10**100)*j", None],
+["(10**10)*(1+j)", None],
+["(10**10)*(-1+j)", None],
+["(10**100)*(1+j)", None],
+["(10**100)*(-1+j)", None],
+["(1.5-1j)", None],
+["(6+4j)", None],
+["(4+1j)", None],
+["(3.5+2j)", None],
+["(1.5-1j)", None],
+["(-6-4j)", None],
+["(-2-3j)", None],
+["(-2.5-2j)", None],
+["(4+1j)", None],
+["(3+3j)", None],
+["(2-2j)", None],
+["1", "0"],
+["2", "0"],
+["3", "log(2)"],
+["4", "log(6)"],
+["5", "log(24)"],
+["0.5", "log(pi)/2"],
+["1.5", "log(sqrt(pi)/2)"],
+["2.5", "log(3*sqrt(pi)/4)"],
+["mpf('0.37')", None],
+["0.25", "log(sqrt(2*sqrt(2*pi**3)/agm(1,sqrt(2))))"],
+["-0.4", None],
+["mpf('-1.9')", None],
+["mpf('12.8')", None],
+["mpf('33.7')", None],
+["mpf('95.2')", None],
+["mpf('160.3')", None],
+["mpf('2057.8')", None],
+["25", "log(ifac(24))"],
+["80", "log(ifac(79))"],
+["500", "log(ifac(500-1))"],
+["8000", "log(ifac(8000-1))"],
+["8000.5", None],
+["mpf('8000.1')", None],
+["mpf('1.37e10')", None],
+["mpf('1.37e10')*(1+j)", None],
+["mpf('1.37e10')*(-1+j)", None],
+["mpf('1.37e10')*(-1-j)", None],
+["mpf('1.37e10')*(-1+j)", None],
+["mpf('1.37e100')", None],
+["mpf('1.37e100')*(1+j)", None],
+["mpf('1.37e100')*(-1+j)", None],
+["mpf('1.37e100')*(-1-j)", None],
+["mpf('1.37e100')*(-1+j)", None],
+["3+4j",
+"mpc('"
+"-1.7566267846037841105306041816232757851567066070613445016197619371316057169"
+"4723618263960834804618463052988607348289672535780644470689771115236512106002"
+"5970873471563240537307638968509556191696167970488390423963867031934333890838"
+"8009531786948197210025029725361069435208930363494971027388382086721660805397"
+"9163230643216054580167976201709951509519218635460317367338612500626714783631"
+"7498317478048447525674016344322545858832610325861086336204591943822302971823"
+"5161814175530618223688296232894588415495615809337292518431903058265147109853"
+"1710568942184987827643886816200452860853873815413367529829631430146227470517"
+"6579967222200868632179482214312673161276976117132204633283806161971389519137"
+"1243359764435612951384238091232760634271570950240717650166551484551654327989"
+"9360285030081716934130446150245110557038117075172576825490035434069388648124"
+"6678152254554001586736120762641422590778766100376515737713938521275749049949"
+"1284143906816424244705094759339932733567910991920631339597278805393743140853"
+"391550313363278558195609260225928','"
+"4.74266443803465792819488940755002274088830335171164611359052405215840070271"
+"5906813009373171139767051863542508136875688550817670379002790304870822775498"
+"2809996675877564504192565392367259119610438951593128982646945990372179860613"
+"4294436498090428077839141927485901735557543641049637962003652638924845391650"
+"9546290137755550107224907606529385248390667634297183361902055842228798984200"
+"9591180450211798341715874477629099687609819466457990642030707080894518168924"
+"6805549314043258530272479246115112769957368212585759640878745385160943755234"
+"9398036774908108204370323896757543121853650025529763655312360354244898913463"
+"7115955702828838923393113618205074162812089732064414530813087483533203244056"
+"0546577484241423134079056537777170351934430586103623577814746004431994179990"
+"5318522939077992613855205801498201930221975721246498720895122345420698451980"
+"0051215797310305885845964334761831751370672996984756815410977750799748813563"
+"8784405288158432214886648743541773208808731479748217023665577802702269468013"
+"673719173759245720489020315779001')"],
+]
+
+for z in [4, 14, 34, 64]:
+    testcases.append(["(2+j)*%s/3" % z, None])
+    testcases.append(["(-2+j)*%s/3" % z, None])
+    testcases.append(["(1+2*j)*%s/3" % z, None])
+    testcases.append(["(2-j)*%s/3" % z, None])
+    testcases.append(["(20+j)*%s/3" % z, None])
+    testcases.append(["(-20+j)*%s/3" % z, None])
+    testcases.append(["(1+20*j)*%s/3" % z, None])
+    testcases.append(["(20-j)*%s/3" % z, None])
+    testcases.append(["(200+j)*%s/3" % z, None])
+    testcases.append(["(-200+j)*%s/3" % z, None])
+    testcases.append(["(1+200*j)*%s/3" % z, None])
+    testcases.append(["(200-j)*%s/3" % z, None])
+
+# Poles
+for n in [0,1,2,3,4,25,-1,-2,-3,-4,-20,-21,-50,-51,-200,-201,-20000,-20001]:
+    for t in ['1e-5', '1e-20', '1e-100', '1e-10000']:
+        testcases.append(["fadd(%s,'%s',exact=True)" % (n, t), None])
+        testcases.append(["fsub(%s,'%s',exact=True)" % (n, t), None])
+        testcases.append(["fadd(%s,'%sj',exact=True)" % (n, t), None])
+        testcases.append(["fsub(%s,'%sj',exact=True)" % (n, t), None])
+
+if __name__ == "__main__":
+    from timeit import default_timer as clock
+    tot_time = 0.0
+    for case in testcases:
+        t1 = clock()
+        testcase(case)
+        t2 = clock()
+        print("Test time:", t2-t1)
+        print()
+        tot_time += (t2-t1)
+    print("Total time:", tot_time)
+    print("Errors:", errcount)