chore: import upstream snapshot with attribution
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Function approximation
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----------------------
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Taylor series (``taylor``)
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..........................
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.. autofunction:: mpmath.taylor
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Pade approximation (``pade``)
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.............................
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.. autofunction:: mpmath.pade
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Chebyshev approximation (``chebyfit``)
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......................................
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.. autofunction:: mpmath.chebyfit
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Fourier series (``fourier``, ``fourierval``)
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............................................
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.. autofunction:: mpmath.fourier
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.. autofunction:: mpmath.fourierval
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Differentiation
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---------------
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Numerical derivatives (``diff``, ``diffs``)
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...........................................
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.. autofunction:: mpmath.diff
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.. autofunction:: mpmath.diffs
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Composition of derivatives (``diffs_prod``, ``diffs_exp``)
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..........................................................
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.. autofunction:: mpmath.diffs_prod
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.. autofunction:: mpmath.diffs_exp
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Fractional derivatives / differintegration (``differint``)
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............................................................
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.. autofunction:: mpmath.differint
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Numerical calculus
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==================
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.. toctree::
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:maxdepth: 2
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polynomials
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optimization
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sums_limits
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differentiation
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integration
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odes
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approximation
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inverselaplace
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Numerical integration (quadrature)
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----------------------------------
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Standard quadrature (``quad``)
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..............................
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.. autofunction:: mpmath.quad
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Quadrature with subdivision (``quadsubdiv``)
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............................................
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.. autofunction:: mpmath.quadsubdiv
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Oscillatory quadrature (``quadosc``)
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....................................
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.. autofunction:: mpmath.quadosc
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Quadrature rules
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................
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.. autoclass:: mpmath.calculus.quadrature.QuadratureRule
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:members:
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Tanh-sinh rule
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~~~~~~~~~~~~~~
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.. autoclass:: mpmath.calculus.quadrature.TanhSinh
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:members:
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Gauss-Legendre rule
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~~~~~~~~~~~~~~~~~~~
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.. autoclass:: mpmath.calculus.quadrature.GaussLegendre
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:members:
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Numerical inverse Laplace transform
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-----------------------------------
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One-step algorithm (``invertlaplace``)
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......................................
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.. autofunction:: mpmath.invertlaplace
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Specific algorithms
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...................
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Fixed Talbot algorithm
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~~~~~~~~~~~~~~~~~~~~~~
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.. autoclass:: mpmath.calculus.inverselaplace.FixedTalbot
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:members:
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Gaver-Stehfest algorithm
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~~~~~~~~~~~~~~~~~~~~~~~~
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.. autoclass:: mpmath.calculus.inverselaplace.Stehfest
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:members:
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de Hoog, Knight & Stokes algorithm
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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.. autoclass:: mpmath.calculus.inverselaplace.deHoog
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:members:
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Cohen acceleration algorithm
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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.. autoclass:: mpmath.calculus.inverselaplace.Cohen
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:members:
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Manual approach
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...............
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It is possible and sometimes beneficial to re-create some of the
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functionality in ``invertlaplace``. This could be used to compute the
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Laplace-space function evaluations in a different way. For example,
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the Laplace-space function evaluations could be the result of a
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quadrature or sum, solution to a system of ordinary differential
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equations, or possibly computed in parallel from some external library
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or function call.
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A trivial example showing the process (which could be implemented
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using the existing interface):
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>>> from mpmath import calculus, convert, exp, mp
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>>> myTalbot = calculus.inverselaplace.FixedTalbot(mp)
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>>> t = convert(0.25)
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>>> myTalbot.calc_laplace_parameter(t)
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>>> fp = lambda p: 1/(p + 1) - 1/(p + 1000)
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>>> ft = lambda t: exp(-t) - exp(-1000*t)
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>>> fpvec = [fp(p) for p in myTalbot.p]
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>>> ft(t)-myTalbot.calc_time_domain_solution(fpvec,t,manual_prec=True)
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mpf('1.92830017952889006175687218e-21')
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This manual approach is also useful to look at the Laplace parameter,
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order, or working precision which were computed.
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>>> myTalbot.degree
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34
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Credit
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......
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The numerical inverse Laplace transform functionality was contributed
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to mpmath by Kristopher L. Kuhlman in 2017. The Cohen method was contributed
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to mpmath by Guillermo Navas-Palencia in 2022.
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Ordinary differential equations
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-------------------------------
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Solving the ODE initial value problem (``odefun``)
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..................................................
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.. autofunction:: mpmath.odefun
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Root-finding and optimization
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-----------------------------
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Root-finding (``findroot``)
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...........................
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.. autofunction:: mpmath.findroot(f, x0, solver=Secant, tol=None, verbose=False, verify=True, **kwargs)
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Solvers
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^^^^^^^
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.. autoclass:: mpmath.calculus.optimization.Secant
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.. autoclass:: mpmath.calculus.optimization.Newton
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.. autoclass:: mpmath.calculus.optimization.MNewton
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.. autoclass:: mpmath.calculus.optimization.Halley
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.. autoclass:: mpmath.calculus.optimization.Muller
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.. autoclass:: mpmath.calculus.optimization.Bisection
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.. autoclass:: mpmath.calculus.optimization.Illinois
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.. autoclass:: mpmath.calculus.optimization.Pegasus
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.. autoclass:: mpmath.calculus.optimization.Anderson
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.. autoclass:: mpmath.calculus.optimization.Ridder
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.. autoclass:: mpmath.calculus.optimization.ANewton
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.. autoclass:: mpmath.calculus.optimization.MDNewton
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.. autoclass:: mpmath.calculus.optimization.ModAB
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.. autoclass:: mpmath.calculus.optimization.Brent
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Polynomials
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-----------
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See also :func:`~mpmath.taylor` and :func:`~mpmath.chebyfit` for
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approximation of functions by polynomials.
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Polynomial evaluation (``polyval``)
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...................................
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.. autofunction:: mpmath.polyval
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Polynomial roots (``polyroots``)
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................................
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.. autofunction:: mpmath.polyroots
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Sums, products, limits and extrapolation
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----------------------------------------
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The functions listed here permit approximation of infinite
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sums, products, and other sequence limits.
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Use :func:`mpmath.fsum` and :func:`mpmath.fprod`
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for summation and multiplication of finite sequences.
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Summation
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..........................................
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:func:`~mpmath.nsum`
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^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.nsum
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:func:`~mpmath.sumem`
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^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.sumem
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:func:`~mpmath.sumap`
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^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.sumap
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Products
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...............................
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:func:`~mpmath.nprod`
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^^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.nprod
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Limits (``limit``)
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..................
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:func:`~mpmath.limit`
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^^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.limit
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Extrapolation
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..........................................
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The following functions provide a direct interface to
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extrapolation algorithms. :func:`~mpmath.nsum` and :func:`~mpmath.limit`
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essentially work by calling the following functions with an increasing
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number of terms until the extrapolated limit is accurate enough.
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The following functions may be useful to call directly if the
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precise number of terms needed to achieve a desired accuracy is
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known in advance, or if one wishes to study the convergence
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properties of the algorithms.
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:func:`~mpmath.richardson`
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.richardson
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:func:`~mpmath.shanks`
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^^^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.shanks
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:func:`~mpmath.levin`
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^^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.levin
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:func:`~mpmath.cohen_alt`
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^^^^^^^^^^^^^^^^^^^^^^^^^^
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.. autofunction:: mpmath.cohen_alt
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