sympy/__init__.py

"""
SymPy is a Python library for symbolic mathematics. It aims to become a
full-featured computer algebra system (CAS) while keeping the code as simple
as possible in order to be comprehensible and easily extensible.  SymPy is
written entirely in Python. It depends on mpmath, and other external libraries
may be optionally for things like plotting support.

See the webpage for more information and documentation:

    https://sympy.org

"""


import sys
if sys.version_info < (3, 8):
    raise ImportError("Python version 3.8 or above is required for SymPy.")
del sys


try:
    import mpmath
except ImportError:
    raise ImportError("SymPy now depends on mpmath as an external library. "
    "See https://docs.sympy.org/latest/install.html#mpmath for more information.")

del mpmath

from sympy.release import __version__
from sympy.core.cache import lazy_function

if 'dev' in __version__:
    def enable_warnings():
        import warnings
        warnings.filterwarnings('default',   '.*',   DeprecationWarning, module='sympy.*')
        del warnings
    enable_warnings()
    del enable_warnings


def __sympy_debug():
    # helper function so we don't import os globally
    import os
    debug_str = os.getenv('SYMPY_DEBUG', 'False')
    if debug_str in ('True', 'False'):
        return eval(debug_str)
    else:
        raise RuntimeError("unrecognized value for SYMPY_DEBUG: %s" %
                           debug_str)
SYMPY_DEBUG = __sympy_debug()  # type: bool

from .core import (sympify, SympifyError, cacheit, Basic, Atom,
        preorder_traversal, S, Expr, AtomicExpr, UnevaluatedExpr, Symbol,
        Wild, Dummy, symbols, var, Number, Float, Rational, Integer,
        NumberSymbol, RealNumber, igcd, ilcm, seterr, E, I, nan, oo, pi, zoo,
        AlgebraicNumber, comp, mod_inverse, Pow, integer_nthroot, integer_log,
        Mul, prod, Add, Mod, Rel, Eq, Ne, Lt, Le, Gt, Ge, Equality,
        GreaterThan, LessThan, Unequality, StrictGreaterThan, StrictLessThan,
        vectorize, Lambda, WildFunction, Derivative, diff, FunctionClass,
        Function, Subs, expand, PoleError, count_ops, expand_mul, expand_log,
        expand_func, expand_trig, expand_complex, expand_multinomial, nfloat,
        expand_power_base, expand_power_exp, arity, PrecisionExhausted, N,
        evalf, Tuple, Dict, gcd_terms, factor_terms, factor_nc, evaluate,
        Catalan, EulerGamma, GoldenRatio, TribonacciConstant, bottom_up, use,
        postorder_traversal, default_sort_key, ordered)

from .logic import (to_cnf, to_dnf, to_nnf, And, Or, Not, Xor, Nand, Nor,
        Implies, Equivalent, ITE, POSform, SOPform, simplify_logic, bool_map,
        true, false, satisfiable)

from .assumptions import (AppliedPredicate, Predicate, AssumptionsContext,
        assuming, Q, ask, register_handler, remove_handler, refine)

from .polys import (Poly, PurePoly, poly_from_expr, parallel_poly_from_expr,
        degree, total_degree, degree_list, LC, LM, LT, pdiv, prem, pquo,
        pexquo, div, rem, quo, exquo, half_gcdex, gcdex, invert,
        subresultants, resultant, discriminant, cofactors, gcd_list, gcd,
        lcm_list, lcm, terms_gcd, trunc, monic, content, primitive, compose,
        decompose, sturm, gff_list, gff, sqf_norm, sqf_part, sqf_list, sqf,
        factor_list, factor, intervals, refine_root, count_roots, real_roots,
        nroots, ground_roots, nth_power_roots_poly, cancel, reduced, groebner,
        is_zero_dimensional, GroebnerBasis, poly, symmetrize, horner,
        interpolate, rational_interpolate, viete, together,
        BasePolynomialError, ExactQuotientFailed, PolynomialDivisionFailed,
        OperationNotSupported, HeuristicGCDFailed, HomomorphismFailed,
        IsomorphismFailed, ExtraneousFactors, EvaluationFailed,
        RefinementFailed, CoercionFailed, NotInvertible, NotReversible,
        NotAlgebraic, DomainError, PolynomialError, UnificationFailed,
        GeneratorsError, GeneratorsNeeded, ComputationFailed,
        UnivariatePolynomialError, MultivariatePolynomialError,
        PolificationFailed, OptionError, FlagError, minpoly,
        minimal_polynomial, primitive_element, field_isomorphism,
        to_number_field, isolate, round_two, prime_decomp, prime_valuation,
        itermonomials, Monomial, lex, grlex,
        grevlex, ilex, igrlex, igrevlex, CRootOf, rootof, RootOf,
        ComplexRootOf, RootSum, roots, Domain, FiniteField, IntegerRing,
        RationalField, RealField, ComplexField, PythonFiniteField,
        GMPYFiniteField, PythonIntegerRing, GMPYIntegerRing, PythonRational,
        GMPYRationalField, AlgebraicField, PolynomialRing, FractionField,
        ExpressionDomain, FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python,
        QQ_gmpy, GF, FF, ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, EXRAW,
        construct_domain, swinnerton_dyer_poly, cyclotomic_poly,
        symmetric_poly, random_poly, interpolating_poly, jacobi_poly,
        chebyshevt_poly, chebyshevu_poly, hermite_poly, hermite_prob_poly,
        legendre_poly, laguerre_poly, apart, apart_list, assemble_partfrac_list,
        Options, ring, xring, vring, sring, field, xfield, vfield, sfield)

from .series import (Order, O, limit, Limit, gruntz, series, approximants,
        residue, EmptySequence, SeqPer, SeqFormula, sequence, SeqAdd, SeqMul,
        fourier_series, fps, difference_delta, limit_seq)

from .functions import (factorial, factorial2, rf, ff, binomial,
        RisingFactorial, FallingFactorial, subfactorial, carmichael,
        fibonacci, lucas, motzkin, tribonacci, harmonic, bernoulli, bell, euler,
        catalan, genocchi, andre, partition, sqrt, root, Min, Max, Id,
        real_root, Rem, cbrt, re, im, sign, Abs, conjugate, arg, polar_lift,
        periodic_argument, unbranched_argument, principal_branch, transpose,
        adjoint, polarify, unpolarify, sin, cos, tan, sec, csc, cot, sinc,
        asin, acos, atan, asec, acsc, acot, atan2, exp_polar, exp, ln, log,
        LambertW, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh,
        acoth, asech, acsch, floor, ceiling, frac, Piecewise, piecewise_fold,
        piecewise_exclusive, erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv,
        Ei, expint, E1, li, Li, Si, Ci, Shi, Chi, fresnels, fresnelc, gamma,
        lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma,
        multigamma, dirichlet_eta, zeta, lerchphi, polylog, stieltjes, Eijk,
        LeviCivita, KroneckerDelta, SingularityFunction, DiracDelta, Heaviside,
        bspline_basis, bspline_basis_set, interpolating_spline, besselj,
        bessely, besseli, besselk, hankel1, hankel2, jn, yn, jn_zeros, hn1,
        hn2, airyai, airybi, airyaiprime, airybiprime, marcumq, hyper,
        meijerg, appellf1, legendre, assoc_legendre, hermite, hermite_prob,
        chebyshevt, chebyshevu, chebyshevu_root, chebyshevt_root, laguerre,
        assoc_laguerre, gegenbauer, jacobi, jacobi_normalized, Ynm, Ynm_c,
        Znm, elliptic_k, elliptic_f, elliptic_e, elliptic_pi, beta, mathieus,
        mathieuc, mathieusprime, mathieucprime, riemann_xi, betainc, betainc_regularized)

from .ntheory import (nextprime, prevprime, prime, primepi, primerange,
        randprime, Sieve, sieve, primorial, cycle_length, composite,
        compositepi, isprime, divisors, proper_divisors, factorint,
        multiplicity, perfect_power, pollard_pm1, pollard_rho, primefactors,
        totient, trailing, divisor_count, proper_divisor_count, divisor_sigma,
        factorrat, reduced_totient, primenu, primeomega,
        mersenne_prime_exponent, is_perfect, is_mersenne_prime, is_abundant,
        is_deficient, is_amicable, abundance, npartitions, is_primitive_root,
        is_quad_residue, legendre_symbol, jacobi_symbol, n_order, sqrt_mod,
        quadratic_residues, primitive_root, nthroot_mod, is_nthpow_residue,
        sqrt_mod_iter, mobius, discrete_log, quadratic_congruence,
        binomial_coefficients, binomial_coefficients_list,
        multinomial_coefficients, continued_fraction_periodic,
        continued_fraction_iterator, continued_fraction_reduce,
        continued_fraction_convergents, continued_fraction, egyptian_fraction)

from .concrete import product, Product, summation, Sum

from .discrete import (fft, ifft, ntt, intt, fwht, ifwht, mobius_transform,
        inverse_mobius_transform, convolution, covering_product,
        intersecting_product)

from .simplify import (simplify, hypersimp, hypersimilar, logcombine,
        separatevars, posify, besselsimp, kroneckersimp, signsimp,
        nsimplify, FU, fu, sqrtdenest, cse, epath, EPath, hyperexpand,
        collect, rcollect, radsimp, collect_const, fraction, numer, denom,
        trigsimp, exptrigsimp, powsimp, powdenest, combsimp, gammasimp,
        ratsimp, ratsimpmodprime)

from .sets import (Set, Interval, Union, EmptySet, FiniteSet, ProductSet,
        Intersection, DisjointUnion, imageset, Complement, SymmetricDifference, ImageSet,
        Range, ComplexRegion, Complexes, Reals, Contains, ConditionSet, Ordinal,
        OmegaPower, ord0, PowerSet, Naturals, Naturals0, UniversalSet,
        Integers, Rationals)

from .solvers import (solve, solve_linear_system, solve_linear_system_LU,
        solve_undetermined_coeffs, nsolve, solve_linear, checksol, det_quick,
        inv_quick, check_assumptions, failing_assumptions, diophantine,
        rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper, checkodesol,
        classify_ode, dsolve, homogeneous_order, solve_poly_system,
        solve_triangulated, pde_separate, pde_separate_add, pde_separate_mul,
        pdsolve, classify_pde, checkpdesol, ode_order, reduce_inequalities,
        reduce_abs_inequality, reduce_abs_inequalities, solve_poly_inequality,
        solve_rational_inequalities, solve_univariate_inequality, decompogen,
        solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution)

from .matrices import (ShapeError, NonSquareMatrixError, GramSchmidt,
        casoratian, diag, eye, hessian, jordan_cell, list2numpy, matrix2numpy,
        matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
        rot_axis3, symarray, wronskian, zeros, MutableDenseMatrix,
        DeferredVector, MatrixBase, Matrix, MutableMatrix,
        MutableSparseMatrix, banded, ImmutableDenseMatrix,
        ImmutableSparseMatrix, ImmutableMatrix, SparseMatrix, MatrixSlice,
        BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity, Inverse,
        MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace, Transpose,
        ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols,
        Adjoint, hadamard_product, HadamardProduct, HadamardPower,
        Determinant, det, diagonalize_vector, DiagMatrix, DiagonalMatrix,
        DiagonalOf, trace, DotProduct, kronecker_product, KroneckerProduct,
        PermutationMatrix, MatrixPermute, Permanent, per)

from .geometry import (Point, Point2D, Point3D, Line, Ray, Segment, Line2D,
        Segment2D, Ray2D, Line3D, Segment3D, Ray3D, Plane, Ellipse, Circle,
        Polygon, RegularPolygon, Triangle, rad, deg, are_similar, centroid,
        convex_hull, idiff, intersection, closest_points, farthest_points,
        GeometryError, Curve, Parabola)

from .utilities import (flatten, group, take, subsets, variations,
        numbered_symbols, cartes, capture, dict_merge, prefixes, postfixes,
        sift, topological_sort, unflatten, has_dups, has_variety, reshape,
        rotations, filldedent, lambdify, source,
        threaded, xthreaded, public, memoize_property, timed)

from .integrals import (integrate, Integral, line_integrate, mellin_transform,
        inverse_mellin_transform, MellinTransform, InverseMellinTransform,
        laplace_transform, inverse_laplace_transform, LaplaceTransform,
        InverseLaplaceTransform, fourier_transform, inverse_fourier_transform,
        FourierTransform, InverseFourierTransform, sine_transform,
        inverse_sine_transform, SineTransform, InverseSineTransform,
        cosine_transform, inverse_cosine_transform, CosineTransform,
        InverseCosineTransform, hankel_transform, inverse_hankel_transform,
        HankelTransform, InverseHankelTransform, singularityintegrate)

from .tensor import (IndexedBase, Idx, Indexed, get_contraction_structure,
        get_indices, shape, MutableDenseNDimArray, ImmutableDenseNDimArray,
        MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray,
        tensorproduct, tensorcontraction, tensordiagonal, derive_by_array,
        permutedims, Array, DenseNDimArray, SparseNDimArray)

from .parsing import parse_expr

from .calculus import (euler_equations, singularities, is_increasing,
        is_strictly_increasing, is_decreasing, is_strictly_decreasing,
        is_monotonic, finite_diff_weights, apply_finite_diff,
        differentiate_finite, periodicity, not_empty_in, AccumBounds,
        is_convex, stationary_points, minimum, maximum)

from .algebras import Quaternion

from .printing import (pager_print, pretty, pretty_print, pprint,
        pprint_use_unicode, pprint_try_use_unicode, latex, print_latex,
        multiline_latex, mathml, print_mathml, python, print_python, pycode,
        ccode, print_ccode, smtlib_code, glsl_code, print_glsl, cxxcode, fcode,
        print_fcode, rcode, print_rcode, jscode, print_jscode, julia_code,
        mathematica_code, octave_code, rust_code, print_gtk, preview, srepr,
        print_tree, StrPrinter, sstr, sstrrepr, TableForm, dotprint,
        maple_code, print_maple_code)

test = lazy_function('sympy.testing.runtests', 'test')
doctest = lazy_function('sympy.testing.runtests', 'doctest')

# This module causes conflicts with other modules:
# from .stats import *
# Adds about .04-.05 seconds of import time
# from combinatorics import *
# This module is slow to import:
#from physics import units
from .plotting import plot, textplot, plot_backends, plot_implicit, plot_parametric
from .interactive import init_session, init_printing, interactive_traversal

evalf._create_evalf_table()

__all__ = [
    '__version__',

    # sympy.core
    'sympify', 'SympifyError', 'cacheit', 'Basic', 'Atom',
    'preorder_traversal', 'S', 'Expr', 'AtomicExpr', 'UnevaluatedExpr',
    'Symbol', 'Wild', 'Dummy', 'symbols', 'var', 'Number', 'Float',
    'Rational', 'Integer', 'NumberSymbol', 'RealNumber', 'igcd', 'ilcm',
    'seterr', 'E', 'I', 'nan', 'oo', 'pi', 'zoo', 'AlgebraicNumber', 'comp',
    'mod_inverse', 'Pow', 'integer_nthroot', 'integer_log', 'Mul', 'prod',
    'Add', 'Mod', 'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge', 'Equality',
    'GreaterThan', 'LessThan', 'Unequality', 'StrictGreaterThan',
    'StrictLessThan', 'vectorize', 'Lambda', 'WildFunction', 'Derivative',
    'diff', 'FunctionClass', 'Function', 'Subs', 'expand', 'PoleError',
    'count_ops', 'expand_mul', 'expand_log', 'expand_func', 'expand_trig',
    'expand_complex', 'expand_multinomial', 'nfloat', 'expand_power_base',
    'expand_power_exp', 'arity', 'PrecisionExhausted', 'N', 'evalf', 'Tuple',
    'Dict', 'gcd_terms', 'factor_terms', 'factor_nc', 'evaluate', 'Catalan',
    'EulerGamma', 'GoldenRatio', 'TribonacciConstant', 'bottom_up', 'use',
    'postorder_traversal', 'default_sort_key', 'ordered',

    # sympy.logic
    'to_cnf', 'to_dnf', 'to_nnf', 'And', 'Or', 'Not', 'Xor', 'Nand', 'Nor',
    'Implies', 'Equivalent', 'ITE', 'POSform', 'SOPform', 'simplify_logic',
    'bool_map', 'true', 'false', 'satisfiable',

    # sympy.assumptions
    'AppliedPredicate', 'Predicate', 'AssumptionsContext', 'assuming', 'Q',
    'ask', 'register_handler', 'remove_handler', 'refine',

    # sympy.polys
    'Poly', 'PurePoly', 'poly_from_expr', 'parallel_poly_from_expr', 'degree',
    'total_degree', 'degree_list', 'LC', 'LM', 'LT', 'pdiv', 'prem', 'pquo',
    'pexquo', 'div', 'rem', 'quo', 'exquo', 'half_gcdex', 'gcdex', 'invert',
    'subresultants', 'resultant', 'discriminant', 'cofactors', 'gcd_list',
    'gcd', 'lcm_list', 'lcm', 'terms_gcd', 'trunc', 'monic', 'content',
    'primitive', 'compose', 'decompose', 'sturm', 'gff_list', 'gff',
    'sqf_norm', 'sqf_part', 'sqf_list', 'sqf', 'factor_list', 'factor',
    'intervals', 'refine_root', 'count_roots', 'real_roots', 'nroots',
    'ground_roots', 'nth_power_roots_poly', 'cancel', 'reduced', 'groebner',
    'is_zero_dimensional', 'GroebnerBasis', 'poly', 'symmetrize', 'horner',
    'interpolate', 'rational_interpolate', 'viete', 'together',
    'BasePolynomialError', 'ExactQuotientFailed', 'PolynomialDivisionFailed',
    'OperationNotSupported', 'HeuristicGCDFailed', 'HomomorphismFailed',
    'IsomorphismFailed', 'ExtraneousFactors', 'EvaluationFailed',
    'RefinementFailed', 'CoercionFailed', 'NotInvertible', 'NotReversible',
    'NotAlgebraic', 'DomainError', 'PolynomialError', 'UnificationFailed',
    'GeneratorsError', 'GeneratorsNeeded', 'ComputationFailed',
    'UnivariatePolynomialError', 'MultivariatePolynomialError',
    'PolificationFailed', 'OptionError', 'FlagError', 'minpoly',
    'minimal_polynomial', 'primitive_element', 'field_isomorphism',
    'to_number_field', 'isolate', 'round_two', 'prime_decomp',
    'prime_valuation', 'itermonomials', 'Monomial', 'lex', 'grlex',
    'grevlex', 'ilex', 'igrlex', 'igrevlex', 'CRootOf', 'rootof', 'RootOf',
    'ComplexRootOf', 'RootSum', 'roots', 'Domain', 'FiniteField',
    'IntegerRing', 'RationalField', 'RealField', 'ComplexField',
    'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing',
    'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField',
    'AlgebraicField', 'PolynomialRing', 'FractionField', 'ExpressionDomain',
    'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy',
    'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'EXRAW',
    'construct_domain', 'swinnerton_dyer_poly', 'cyclotomic_poly',
    'symmetric_poly', 'random_poly', 'interpolating_poly', 'jacobi_poly',
    'chebyshevt_poly', 'chebyshevu_poly', 'hermite_poly', 'hermite_prob_poly',
    'legendre_poly', 'laguerre_poly', 'apart', 'apart_list', 'assemble_partfrac_list',
    'Options', 'ring', 'xring', 'vring', 'sring', 'field', 'xfield', 'vfield',
    'sfield',

    # sympy.series
    'Order', 'O', 'limit', 'Limit', 'gruntz', 'series', 'approximants',
    'residue', 'EmptySequence', 'SeqPer', 'SeqFormula', 'sequence', 'SeqAdd',
    'SeqMul', 'fourier_series', 'fps', 'difference_delta', 'limit_seq',

    # sympy.functions
    'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial',
    'FallingFactorial', 'subfactorial', 'carmichael', 'fibonacci', 'lucas',
    'motzkin', 'tribonacci', 'harmonic', 'bernoulli', 'bell', 'euler', 'catalan',
    'genocchi', 'andre', 'partition', 'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root',
    'Rem', 'cbrt', 're', 'im', 'sign', 'Abs', 'conjugate', 'arg', 'polar_lift',
    'periodic_argument', 'unbranched_argument', 'principal_branch',
    'transpose', 'adjoint', 'polarify', 'unpolarify', 'sin', 'cos', 'tan',
    'sec', 'csc', 'cot', 'sinc', 'asin', 'acos', 'atan', 'asec', 'acsc',
    'acot', 'atan2', 'exp_polar', 'exp', 'ln', 'log', 'LambertW', 'sinh',
    'cosh', 'tanh', 'coth', 'sech', 'csch', 'asinh', 'acosh', 'atanh',
    'acoth', 'asech', 'acsch', 'floor', 'ceiling', 'frac', 'Piecewise',
    'piecewise_fold', 'piecewise_exclusive', 'erf', 'erfc', 'erfi', 'erf2',
    'erfinv', 'erfcinv', 'erf2inv', 'Ei', 'expint', 'E1', 'li', 'Li', 'Si',
    'Ci', 'Shi', 'Chi', 'fresnels', 'fresnelc', 'gamma', 'lowergamma',
    'uppergamma', 'polygamma', 'loggamma', 'digamma', 'trigamma', 'multigamma',
    'dirichlet_eta', 'zeta', 'lerchphi', 'polylog', 'stieltjes', 'Eijk', 'LeviCivita',
    'KroneckerDelta', 'SingularityFunction', 'DiracDelta', 'Heaviside',
    'bspline_basis', 'bspline_basis_set', 'interpolating_spline', 'besselj',
    'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn', 'yn',
    'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime',
    'airybiprime', 'marcumq', 'hyper', 'meijerg', 'appellf1', 'legendre',
    'assoc_legendre', 'hermite', 'hermite_prob', 'chebyshevt', 'chebyshevu',
    'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre',
    'gegenbauer', 'jacobi', 'jacobi_normalized', 'Ynm', 'Ynm_c', 'Znm',
    'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi', 'beta',
    'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime', 'riemann_xi','betainc',
    'betainc_regularized',

    # sympy.ntheory
    'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime',
    'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi',
    'isprime', 'divisors', 'proper_divisors', 'factorint', 'multiplicity',
    'perfect_power', 'pollard_pm1', 'pollard_rho', 'primefactors', 'totient',
    'trailing', 'divisor_count', 'proper_divisor_count', 'divisor_sigma',
    'factorrat', 'reduced_totient', 'primenu', 'primeomega',
    'mersenne_prime_exponent', 'is_perfect', 'is_mersenne_prime',
    'is_abundant', 'is_deficient', 'is_amicable', 'abundance', 'npartitions',
    'is_primitive_root', 'is_quad_residue', 'legendre_symbol',
    'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues',
    'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter',
    'mobius', 'discrete_log', 'quadratic_congruence', 'binomial_coefficients',
    'binomial_coefficients_list', 'multinomial_coefficients',
    'continued_fraction_periodic', 'continued_fraction_iterator',
    'continued_fraction_reduce', 'continued_fraction_convergents',
    'continued_fraction', 'egyptian_fraction',

    # sympy.concrete
    'product', 'Product', 'summation', 'Sum',

    # sympy.discrete
    'fft', 'ifft', 'ntt', 'intt', 'fwht', 'ifwht', 'mobius_transform',
    'inverse_mobius_transform', 'convolution', 'covering_product',
    'intersecting_product',

    # sympy.simplify
    'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars',
    'posify', 'besselsimp', 'kroneckersimp', 'signsimp',
    'nsimplify', 'FU', 'fu', 'sqrtdenest', 'cse', 'epath', 'EPath',
    'hyperexpand', 'collect', 'rcollect', 'radsimp', 'collect_const',
    'fraction', 'numer', 'denom', 'trigsimp', 'exptrigsimp', 'powsimp',
    'powdenest', 'combsimp', 'gammasimp', 'ratsimp', 'ratsimpmodprime',

    # sympy.sets
    'Set', 'Interval', 'Union', 'EmptySet', 'FiniteSet', 'ProductSet',
    'Intersection', 'imageset', 'DisjointUnion', 'Complement', 'SymmetricDifference',
    'ImageSet', 'Range', 'ComplexRegion', 'Reals', 'Contains', 'ConditionSet',
    'Ordinal', 'OmegaPower', 'ord0', 'PowerSet', 'Naturals',
    'Naturals0', 'UniversalSet', 'Integers', 'Rationals', 'Complexes',

    # sympy.solvers
    'solve', 'solve_linear_system', 'solve_linear_system_LU',
    'solve_undetermined_coeffs', 'nsolve', 'solve_linear', 'checksol',
    'det_quick', 'inv_quick', 'check_assumptions', 'failing_assumptions',
    'diophantine', 'rsolve', 'rsolve_poly', 'rsolve_ratio', 'rsolve_hyper',
    'checkodesol', 'classify_ode', 'dsolve', 'homogeneous_order',
    'solve_poly_system', 'solve_triangulated', 'pde_separate',
    'pde_separate_add', 'pde_separate_mul', 'pdsolve', 'classify_pde',
    'checkpdesol', 'ode_order', 'reduce_inequalities',
    'reduce_abs_inequality', 'reduce_abs_inequalities',
    'solve_poly_inequality', 'solve_rational_inequalities',
    'solve_univariate_inequality', 'decompogen', 'solveset', 'linsolve',
    'linear_eq_to_matrix', 'nonlinsolve', 'substitution',

    # sympy.matrices
    'ShapeError', 'NonSquareMatrixError', 'GramSchmidt', 'casoratian', 'diag',
    'eye', 'hessian', 'jordan_cell', 'list2numpy', 'matrix2numpy',
    'matrix_multiply_elementwise', 'ones', 'randMatrix', 'rot_axis1',
    'rot_axis2', 'rot_axis3', 'symarray', 'wronskian', 'zeros',
    'MutableDenseMatrix', 'DeferredVector', 'MatrixBase', 'Matrix',
    'MutableMatrix', 'MutableSparseMatrix', 'banded', 'ImmutableDenseMatrix',
    'ImmutableSparseMatrix', 'ImmutableMatrix', 'SparseMatrix', 'MatrixSlice',
    'BlockDiagMatrix', 'BlockMatrix', 'FunctionMatrix', 'Identity', 'Inverse',
    'MatAdd', 'MatMul', 'MatPow', 'MatrixExpr', 'MatrixSymbol', 'Trace',
    'Transpose', 'ZeroMatrix', 'OneMatrix', 'blockcut', 'block_collapse',
    'matrix_symbols', 'Adjoint', 'hadamard_product', 'HadamardProduct',
    'HadamardPower', 'Determinant', 'det', 'diagonalize_vector', 'DiagMatrix',
    'DiagonalMatrix', 'DiagonalOf', 'trace', 'DotProduct',
    'kronecker_product', 'KroneckerProduct', 'PermutationMatrix',
    'MatrixPermute', 'Permanent', 'per',

    # sympy.geometry
    'Point', 'Point2D', 'Point3D', 'Line', 'Ray', 'Segment', 'Line2D',
    'Segment2D', 'Ray2D', 'Line3D', 'Segment3D', 'Ray3D', 'Plane', 'Ellipse',
    'Circle', 'Polygon', 'RegularPolygon', 'Triangle', 'rad', 'deg',
    'are_similar', 'centroid', 'convex_hull', 'idiff', 'intersection',
    'closest_points', 'farthest_points', 'GeometryError', 'Curve', 'Parabola',

    # sympy.utilities
    'flatten', 'group', 'take', 'subsets', 'variations', 'numbered_symbols',
    'cartes', 'capture', 'dict_merge', 'prefixes', 'postfixes', 'sift',
    'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape',
    'rotations', 'filldedent', 'lambdify', 'source', 'threaded', 'xthreaded',
    'public', 'memoize_property', 'timed',

    # sympy.integrals
    'integrate', 'Integral', 'line_integrate', 'mellin_transform',
    'inverse_mellin_transform', 'MellinTransform', 'InverseMellinTransform',
    'laplace_transform', 'inverse_laplace_transform', 'LaplaceTransform',
    'InverseLaplaceTransform', 'fourier_transform',
    'inverse_fourier_transform', 'FourierTransform',
    'InverseFourierTransform', 'sine_transform', 'inverse_sine_transform',
    'SineTransform', 'InverseSineTransform', 'cosine_transform',
    'inverse_cosine_transform', 'CosineTransform', 'InverseCosineTransform',
    'hankel_transform', 'inverse_hankel_transform', 'HankelTransform',
    'InverseHankelTransform', 'singularityintegrate',

    # sympy.tensor
    'IndexedBase', 'Idx', 'Indexed', 'get_contraction_structure',
    'get_indices', 'shape', 'MutableDenseNDimArray', 'ImmutableDenseNDimArray',
    'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'NDimArray',
    'tensorproduct', 'tensorcontraction', 'tensordiagonal', 'derive_by_array',
    'permutedims', 'Array', 'DenseNDimArray', 'SparseNDimArray',

    # sympy.parsing
    'parse_expr',

    # sympy.calculus
    'euler_equations', 'singularities', 'is_increasing',
    'is_strictly_increasing', 'is_decreasing', 'is_strictly_decreasing',
    'is_monotonic', 'finite_diff_weights', 'apply_finite_diff',
    'differentiate_finite', 'periodicity', 'not_empty_in',
    'AccumBounds', 'is_convex', 'stationary_points', 'minimum', 'maximum',

    # sympy.algebras
    'Quaternion',

    # sympy.printing
    'pager_print', 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode',
    'pprint_try_use_unicode', 'latex', 'print_latex', 'multiline_latex',
    'mathml', 'print_mathml', 'python', 'print_python', 'pycode', 'ccode',
    'print_ccode', 'smtlib_code', 'glsl_code', 'print_glsl', 'cxxcode', 'fcode',
    'print_fcode', 'rcode', 'print_rcode', 'jscode', 'print_jscode',
    'julia_code', 'mathematica_code', 'octave_code', 'rust_code', 'print_gtk',
    'preview', 'srepr', 'print_tree', 'StrPrinter', 'sstr', 'sstrrepr',
    'TableForm', 'dotprint', 'maple_code', 'print_maple_code',

    # sympy.plotting
    'plot', 'textplot', 'plot_backends', 'plot_implicit', 'plot_parametric',

    # sympy.interactive
    'init_session', 'init_printing', 'interactive_traversal',

    # sympy.testing
    'test', 'doctest',
]


#===========================================================================#
#                                                                           #
# XXX: The names below were importable before SymPy 1.6 using               #
#                                                                           #
#          from sympy import *                                              #
#                                                                           #
# This happened implicitly because there was no __all__ defined in this     #
# __init__.py file. Not every package is imported. The list matches what    #
# would have been imported before. It is possible that these packages will  #
# not be imported by a star-import from sympy in future.                    #
#                                                                           #
#===========================================================================#


__all__.extend((
    'algebras',
    'assumptions',
    'calculus',
    'concrete',
    'discrete',
    'external',
    'functions',
    'geometry',
    'interactive',
    'multipledispatch',
    'ntheory',
    'parsing',
    'plotting',
    'polys',
    'printing',
    'release',
    'strategies',
    'tensor',
    'utilities',
))