<gh_stars>1-10
from math import isclose
from typing import Tuple, Union

from sympy import Symbol, diff, sympify
from sympy.core.function import Function
from sympy.parsing.sympy_parser import (convert_xor, function_exponentiation,
                                        implicit_multiplication_application,
                                        parse_expr, standard_transformations)

from ..error_handler import InfiniteIteration

x = Symbol('x')
transformations = standard_transformations + \
    (convert_xor, implicit_multiplication_application, function_exponentiation)


def fixed_point(f: Function, n: float, rational: bool = False,
                iterated_data: bool = False) -> Union[float, Tuple[float, list]]:
    """
    Using Fixed-Point method, returns float (or tupled with iterated data).
    - Known as the method of successive submission.

    ..Note:
        - Return iterated_data defaults to False
        - Always use 'x' as a symbol
        - Originally was to take list of formulae. Modularising thos method
        means the need to opt it out.

    ..Usage:
        * Find all the possible iterative formula of f(x),
        Transform them to x = f(x)
            Example:
                f(x) = x^2-8x+11 can be transformed to:
                    x = (x^2+11)/8;
                    x = sqrt(8x-11)
                    x = (8x-11)/x

    Algorithm
    =========
    1. Solve for the new value of n [f(n)] 
    2. If error % is converging to 0, return n as root
    3. if previous n is converging with current n, return n as root

    Examples
    ========
    ** Non-pythonic expressions will be parsed by SymPy module
    >>> fixed_point('(x^2+11)/8', 3)
    1.7639910281905442

    ** Using pythonic expressions is also accepted
    >>> fixed_point('(x**2+11)/8', 3)
    1.7639910281905442

    ** Turning rational True
    >>> fixed_point('(8x-11)/x', 3, rational=True)
    153301943/24583261

    ** Turning iterated_data True
    >>> fixed_point('(8x-11)/x', 3, iterated_data=True)
    (6.236029589402317, {0: {'Xn': 3, 'e%': None}, ...)

    Iterated data
    =============
    count / iteration :
        - The first key you see in the dictionary

    Xn :
        - Value of the iterated approximate root

    e% :
        - Percent error, how accurate the approximate root

    Exceptions
    ==========
    InfiniteIteration :
        - Raised when the function passed is indefinite and
        iteration reaches 99, assuming it is indefinite.

    Parameters
    ==========
    f :
        - Should be STRING
        - Mathematical expression
        - A mathematical function
        - Example: 
            '(x^2+11)/8'
            '(x**2+11)/8'
            '(8x-11)/x'

    n :
        - the 'x' value of the function

    rational :
        - Returns fraction/rational value
        - Defaults to False

    iterated_data:
        - Returns the iterated data in dictionary
        - Defaults to False
    """

    f = parse_expr(f, transformations=transformations)
    n = sympify(n)

    data = {}
    count = 0
    prev_n = 0

    while True:
        error = abs(((n-prev_n)/n)*100)
        data[count] = {'Xn': n, 'e%': error if count > 0 else None}

        prev_n = n
        n = f.subs(x, n)

        if isclose(error, 0.0000, rel_tol=1e-4):
            break
        elif isclose(prev_n, n, abs_tol=1e-4):
            break
        elif count >= 99:
            raise InfiniteIteration

        count += 1

    if not rational:
        n = float(n)

    if iterated_data:
        return n, data
    else:
        return n


def newton_raphson(f: Function, n: float, rational: bool = False,
                   iterated_data: bool = False) -> Union[float, Tuple[float, list]]:
    """
    Using Newton-Raphson method, returns float (or tupled with iterated data).
    - Utilizes the derivative of f(x)
    - is an open method that finds the root x of a function such that f(x)

    ..Note:
        - Return iterated_data defaults to False
        - Always use 'x' as a symbol

    Algorithm
    =========
    1. Solve f(x), f'(x), error %
    2. n = n - (f(x)/f'(x))
    3. if error % is converging to 0, return n as root
    4. if previous f'(x) is converging to next f'(x), return n as root

    Examples
    ========
    ** Non-pythonic expressions will be parsed by SymPy module
    >>> from newton_raphson import newton_raphson
    >>> newton_raphson('x^2-8x+11', 1)
    1.763932022500022

    ** Using pythonic expressions is also accepted
    >>> newton_raphson('x**2-8*x+11', 1)
    1.763932022500022

    ** Turning rational True
    >>> newton_raphson('x^3-3x+1', 0, rational=True)
    170999/492372

    ** Turning iterated_data True
    >>> newton_raphson('x^3-3x+1', 0, iterated_data=True)
    (0.347296353163868, {0: {'Xn': 0, 'fx': 1, 'fpx': -3, 'e%': None}, ...)

    Iterated data
    =============
    count / iteration :
        - The first key you see in the dictionary

    Xn :
        - Value of the iterated approximate root

    fx :
        - Value of the f(n)

    fpx :
        - Value of the f'(n) [computed derived f(n)]

    e% :
        - Percent error, how accurate the approximate root

    Exceptions
    ==========
    InfiniteIteration :
        - Raised when the function passed is indefinite and
        iteration reaches 99, assuming it is indefinite.

    Parameters
    ==========
    f :
        - Should be STRING
        - Mathematical expression
        - A mathematical function
        - Example: 
            'x^2-8x+11'
            'x**3-3*x+1'
            'x^3+10x^2-5'

    n :
        - the 'x' value of the function

    rational :
        - Returns fraction/rational value
        - Defaults to False

    iterated_data:
        - Returns the iterated data in dictionary
        - Defaults to False
    """
    f = parse_expr(f, transformations=transformations)
    n = sympify(n)

    data = {}
    count = 0
    prev_n = 0
    fp = diff(f, x)

    while True:
        fXn = f.subs(x, n)
        fpXn = fp.subs(x, n)
        error = abs(((n-prev_n)/n)*100)

        data[count] = {'Xn': n, 'fx': fXn, 'fpx': fpXn,
                       'e%': error if count > 0 else None}

        prev_n = n
        n = n - (fXn/fpXn)

        if isclose(round(error, 4), 0.0, rel_tol=1e-4):
            break
        elif isclose(fpXn, fp.subs(x, n), rel_tol=1e-4):
            break
        elif count >= 99:
            raise InfiniteIteration

        count += 1

    if not rational:
        n = float(n)

    if iterated_data:
        return n, data
    else:
        return n

from math import isclose
from typing import Tuple, Union

from sympy import Symbol, diff, sympify
from sympy.core.function import Function
from sympy.parsing.sympy_parser import (convert_xor, function_exponentiation,
                                        implicit_multiplication_application,
                                        parse_expr, standard_transformations)

from ..error_handler import InfiniteIteration

x = Symbol('x')
transformations = standard_transformations + \
    (convert_xor, implicit_multiplication_application, function_exponentiation)


def fixed_point(f: Function, n: float, rational: bool = False,
                iterated_data: bool = False) -> Union[float, Tuple[float, list]]:
    """
    Using Fixed-Point method, returns float (or tupled with iterated data).
    - Known as the method of successive submission.

    ..Note:
        - Return iterated_data defaults to False
        - Always use 'x' as a symbol
        - Originally was to take list of formulae. Modularising thos method
        means the need to opt it out.

    ..Usage:
        * Find all the possible iterative formula of f(x),
        Transform them to x = f(x)
            Example:
                f(x) = x^2-8x+11 can be transformed to:
                    x = (x^2+11)/8;
                    x = sqrt(8x-11)
                    x = (8x-11)/x

    Algorithm
    =========
    1. Solve for the new value of n [f(n)] 
    2. If error % is converging to 0, return n as root
    3. if previous n is converging with current n, return n as root

    Examples
    ========
    ** Non-pythonic expressions will be parsed by SymPy module
    >>> fixed_point('(x^2+11)/8', 3)
    1.7639910281905442

    ** Using pythonic expressions is also accepted
    >>> fixed_point('(x**2+11)/8', 3)
    1.7639910281905442

    ** Turning rational True
    >>> fixed_point('(8x-11)/x', 3, rational=True)
    153301943/24583261

    ** Turning iterated_data True
    >>> fixed_point('(8x-11)/x', 3, iterated_data=True)
    (6.236029589402317, {0: {'Xn': 3, 'e%': None}, ...)

    Iterated data
    =============
    count / iteration :
        - The first key you see in the dictionary

    Xn :
        - Value of the iterated approximate root

    e% :
        - Percent error, how accurate the approximate root

    Exceptions
    ==========
    InfiniteIteration :
        - Raised when the function passed is indefinite and
        iteration reaches 99, assuming it is indefinite.

    Parameters
    ==========
    f :
        - Should be STRING
        - Mathematical expression
        - A mathematical function
        - Example: 
            '(x^2+11)/8'
            '(x**2+11)/8'
            '(8x-11)/x'

    n :
        - the 'x' value of the function

    rational :
        - Returns fraction/rational value
        - Defaults to False

    iterated_data:
        - Returns the iterated data in dictionary
        - Defaults to False
    """

    f = parse_expr(f, transformations=transformations)
    n = sympify(n)

    data = {}
    count = 0
    prev_n = 0

    while True:
        error = abs(((n-prev_n)/n)*100)
        data[count] = {'Xn': n, 'e%': error if count > 0 else None}

        prev_n = n
        n = f.subs(x, n)

        if isclose(error, 0.0000, rel_tol=1e-4):
            break
        elif isclose(prev_n, n, abs_tol=1e-4):
            break
        elif count >= 99:
            raise InfiniteIteration

        count += 1

    if not rational:
        n = float(n)

    if iterated_data:
        return n, data
    else:
        return n


def newton_raphson(f: Function, n: float, rational: bool = False,
                   iterated_data: bool = False) -> Union[float, Tuple[float, list]]:
    """
    Using Newton-Raphson method, returns float (or tupled with iterated data).
    - Utilizes the derivative of f(x)
    - is an open method that finds the root x of a function such that f(x)

    ..Note:
        - Return iterated_data defaults to False
        - Always use 'x' as a symbol

    Algorithm
    =========
    1. Solve f(x), f'(x), error %
    2. n = n - (f(x)/f'(x))
    3. if error % is converging to 0, return n as root
    4. if previous f'(x) is converging to next f'(x), return n as root

    Examples
    ========
    ** Non-pythonic expressions will be parsed by SymPy module
    >>> from newton_raphson import newton_raphson
    >>> newton_raphson('x^2-8x+11', 1)
    1.763932022500022

    ** Using pythonic expressions is also accepted
    >>> newton_raphson('x**2-8*x+11', 1)
    1.763932022500022

    ** Turning rational True
    >>> newton_raphson('x^3-3x+1', 0, rational=True)
    170999/492372

    ** Turning iterated_data True
    >>> newton_raphson('x^3-3x+1', 0, iterated_data=True)
    (0.347296353163868, {0: {'Xn': 0, 'fx': 1, 'fpx': -3, 'e%': None}, ...)

    Iterated data
    =============
    count / iteration :
        - The first key you see in the dictionary

    Xn :
        - Value of the iterated approximate root

    fx :
        - Value of the f(n)

    fpx :
        - Value of the f'(n) [computed derived f(n)]

    e% :
        - Percent error, how accurate the approximate root

    Exceptions
    ==========
    InfiniteIteration :
        - Raised when the function passed is indefinite and
        iteration reaches 99, assuming it is indefinite.

    Parameters
    ==========
    f :
        - Should be STRING
        - Mathematical expression
        - A mathematical function
        - Example: 
            'x^2-8x+11'
            'x**3-3*x+1'
            'x^3+10x^2-5'

    n :
        - the 'x' value of the function

    rational :
        - Returns fraction/rational value
        - Defaults to False

    iterated_data:
        - Returns the iterated data in dictionary
        - Defaults to False
    """
    f = parse_expr(f, transformations=transformations)
    n = sympify(n)

    data = {}
    count = 0
    prev_n = 0
    fp = diff(f, x)

    while True:
        fXn = f.subs(x, n)
        fpXn = fp.subs(x, n)
        error = abs(((n-prev_n)/n)*100)

        data[count] = {'Xn': n, 'fx': fXn, 'fpx': fpXn,
                       'e%': error if count > 0 else None}

        prev_n = n
        n = n - (fXn/fpXn)

        if isclose(round(error, 4), 0.0, rel_tol=1e-4):
            break
        elif isclose(fpXn, fp.subs(x, n), rel_tol=1e-4):
            break
        elif count >= 99:
            raise InfiniteIteration

        count += 1

    if not rational:
        n = float(n)

    if iterated_data:
        return n, data
    else:
        return n
Using Fixed-Point method, returns float (or tupled with iterated data).     - Known as the method of successive submission.      ..Note:         - Return iterated_data defaults to False         - Always use 'x' as a symbol         - Originally was to take list of formulae. Modularising thos method         means the need to opt it out.      ..Usage:         * Find all the possible iterative formula of f(x),         Transform them to x = f(x)             Example:                 f(x) = x^2-8x+11 can be transformed to:                     x = (x^2+11)/8;                     x = sqrt(8x-11)                     x = (8x-11)/x      Algorithm     =========     1. Solve for the new value of n [f(n)]      2. If error % is converging to 0, return n as root     3. if previous n is converging with current n, return n as root      Examples     ========     ** Non-pythonic expressions will be parsed by SymPy module     >>> fixed_point('(x^2+11)/8', 3)     1.7639910281905442      ** Using pythonic expressions is also accepted     >>> fixed_point('(x**2+11)/8', 3)     1.7639910281905442      ** Turning rational True     >>> fixed_point('(8x-11)/x', 3, rational=True)     153301943/24583261      ** Turning iterated_data True     >>> fixed_point('(8x-11)/x', 3, iterated_data=True)     (6.236029589402317, {0: {'Xn': 3, 'e%': None}, ...)      Iterated data     =============     count / iteration :         - The first key you see in the dictionary      Xn :         - Value of the iterated approximate root      e% :         - Percent error, how accurate the approximate root      Exceptions     ==========     InfiniteIteration :         - Raised when the function passed is indefinite and         iteration reaches 99, assuming it is indefinite.      Parameters     ==========     f :         - Should be STRING         - Mathematical expression         - A mathematical function         - Example:              '(x^2+11)/8'             '(x**2+11)/8'             '(8x-11)/x'      n :         - the 'x' value of the function      rational :         - Returns fraction/rational value         - Defaults to False      iterated_data:         - Returns the iterated data in dictionary         - Defaults to False Using Newton-Raphson method, returns float (or tupled with iterated data).     - Utilizes the derivative of f(x)     - is an open method that finds the root x of a function such that f(x)      ..Note:         - Return iterated_data defaults to False         - Always use 'x' as a symbol      Algorithm     =========     1. Solve f(x), f'(x), error %     2. n = n - (f(x)/f'(x))     3. if error % is converging to 0, return n as root     4. if previous f'(x) is converging to next f'(x), return n as root      Examples     ========     ** Non-pythonic expressions will be parsed by SymPy module     >>> from newton_raphson import newton_raphson     >>> newton_raphson('x^2-8x+11', 1)     1.763932022500022      ** Using pythonic expressions is also accepted     >>> newton_raphson('x**2-8*x+11', 1)     1.763932022500022      ** Turning rational True     >>> newton_raphson('x^3-3x+1', 0, rational=True)     170999/492372      ** Turning iterated_data True     >>> newton_raphson('x^3-3x+1', 0, iterated_data=True)     (0.347296353163868, {0: {'Xn': 0, 'fx': 1, 'fpx': -3, 'e%': None}, ...)      Iterated data     =============     count / iteration :         - The first key you see in the dictionary      Xn :         - Value of the iterated approximate root      fx :         - Value of the f(n)      fpx :         - Value of the f'(n) [computed derived f(n)]      e% :         - Percent error, how accurate the approximate root      Exceptions     ==========     InfiniteIteration :         - Raised when the function passed is indefinite and         iteration reaches 99, assuming it is indefinite.      Parameters     ==========     f :         - Should be STRING         - Mathematical expression         - A mathematical function         - Example:              'x^2-8x+11'             'x**3-3*x+1'             'x^3+10x^2-5'      n :         - the 'x' value of the function      rational :         - Returns fraction/rational value         - Defaults to False      iterated_data:         - Returns the iterated data in dictionary         - Defaults to False