Added Chakravala method to solve pell's equation

DIRECTORY.md

@@ -9,6 +9,7 @@
   * [Newton Forward Interpolation](arithmetic_analysis/newton_forward_interpolation.py)
   * [Newton Method](arithmetic_analysis/newton_method.py)
   * [Newton Raphson](arithmetic_analysis/newton_raphson.py)
+  * [Newton Raphson New](arithmetic_analysis/newton_raphson_new.py)
   * [Secant Method](arithmetic_analysis/secant_method.py)
 
 ## Audio Filters
@@ -107,6 +108,7 @@
   * [Lempel Ziv](compression/lempel_ziv.py)
   * [Lempel Ziv Decompress](compression/lempel_ziv_decompress.py)
   * [Peak Signal To Noise Ratio](compression/peak_signal_to_noise_ratio.py)
+  * [Run Length Encoding](compression/run_length_encoding.py)
 
 ## Computer Vision
   * [Cnn Classification](computer_vision/cnn_classification.py)
@@ -474,6 +476,7 @@
   * [Binomial Distribution](maths/binomial_distribution.py)
   * [Bisection](maths/bisection.py)
   * [Ceil](maths/ceil.py)
+  * [Chakravala](maths/chakravala.py)
   * [Check Polygon](maths/check_polygon.py)
   * [Chudnovsky Algorithm](maths/chudnovsky_algorithm.py)
   * [Collatz Sequence](maths/collatz_sequence.py)
@@ -621,6 +624,7 @@
   * [Linear Congruential Generator](other/linear_congruential_generator.py)
   * [Lru Cache](other/lru_cache.py)
   * [Magicdiamondpattern](other/magicdiamondpattern.py)
+  * [Maximum Subarray](other/maximum_subarray.py)
   * [Nested Brackets](other/nested_brackets.py)
   * [Password Generator](other/password_generator.py)
   * [Scoring Algorithm](other/scoring_algorithm.py)
@@ -1053,6 +1057,7 @@
   * [Fetch Bbc News](web_programming/fetch_bbc_news.py)
   * [Fetch Github Info](web_programming/fetch_github_info.py)
   * [Fetch Jobs](web_programming/fetch_jobs.py)
+  * [Fetch Quotes](web_programming/fetch_quotes.py)
   * [Fetch Well Rx Price](web_programming/fetch_well_rx_price.py)
   * [Get Imdb Top 250 Movies Csv](web_programming/get_imdb_top_250_movies_csv.py)
   * [Get Imdbtop](web_programming/get_imdbtop.py)

maths/chakravala.py

@@ -0,0 +1,120 @@
+"""
+Implementing Chakravala method using python
+
+https://en.wikipedia.org/wiki/Chakravala_method
+https://kappadath-gopal.blogspot.com/2013/04/ancient-medieval-indian-mathematics.html
+
+The chakravala method is a cyclic algorithm to solve indeterminate quadratic equations,
+including Pell's equation.
+
+"""
+
+import math
+
+
+def is_perfect_square(num: int) -> bool:
+    """
+    Check if a number is perfect square number or not
+    :param num: the number to be checked
+    :return: True if number is square number, otherwise False
+
+    >>> is_perfect_square(9)
+    True
+    >>> is_perfect_square(16)
+    True
+    >>> is_perfect_square(1)
+    True
+    >>> is_perfect_square(0)
+    True
+    >>> is_perfect_square(10)
+    False
+    """
+
+    sr = int(math.sqrt(num))
+    return sr * sr == num
+
+
+def chakravala_method(num: int) -> (tuple[int, int] | tuple):
+    """
+    This method takes in the value of N in the equation
+
+    x^2 = N*y^2 + 1
+    :param num: the number N equals to
+    :return: empty tuple if N is perfect square else tuple(x,y)
+
+    >>> chakravala_method(1)
+    ()
+    >>> chakravala_method(2)
+    (3, 2)
+    >>> chakravala_method(4)
+    ()
+    >>> chakravala_method(5)
+    (9, 4)
+    >>> chakravala_method(7)
+    (8, 3)
+
+    """
+
+    if is_perfect_square(num):
+        return ()
+
+    # Takes b = 1 and finds a and k accordingly, refer to algorithm link
+    # variable naming is used as same as algorithm, (a,b,k,m) except N = num
+
+    b = 1
+
+    min_diff = num
+    a = 0
+    while True:
+        diff = abs(num - (a + 1) ** 2)
+        if min_diff > diff:
+            min_diff = diff
+            a += 1
+            continue
+        break
+
+    k = a**2 - num
+
+    while True:
+
+        kabs = abs(k)
+
+        if k == 1:
+            return (a, b)
+
+        if k == -1 or kabs == 2 or (kabs == 4 and (a % 2 == 0 or b % 2 == 0)):
+            return (abs((a**2 + num * b**2) // k), abs(2 * a * b // k))
+
+        min_diff = num
+        n = 1  # loop variable
+        n2 = n  # stores the correct value of n
+        while True:
+            if kabs * n <= a:
+                n += 1
+                continue
+            if (kabs * n - a) % b == 0:
+                m = (kabs * n - a) // b
+            else:
+                n += 1
+                continue
+
+            diff = abs(m**2 - num)
+            if min_diff > diff:
+                min_diff = diff
+                n2 = n
+                n += 1
+                continue
+            break
+        m = (kabs * n2 - a) // b
+
+        a, b = abs((a * m + num * b) // k), abs((a + b * m) // k)
+        k = (m**2 - num) // k
+
+
+if __name__ == "__main__":
+
+    import doctest
+
+    doctest.testmod()
+
+    print("X and Y for the equation X^2 - 13Y^2 = 1 is: ", chakravala_method(13))