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* Python mirror_formulae.py is added to the repository * Changes done after reading readme.md * Changes for running doctest on all platforms * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Change 2 for Doctests * Changes for doctest 2 * updating DIRECTORY.md * Doctest whitespace error rectification to mirror_formulae.py * updating DIRECTORY.md * Adding Thermodynamic Work Done Formulae * Work done on/by body in a thermodynamic setting * updating DIRECTORY.md * updating DIRECTORY.md * Doctest adiition to binary_exponentiation_3.py * Change 1 * updating DIRECTORY.md * Rename binary_exponentiation_3.py to binary_exponentiation_2.py * updating DIRECTORY.md * updating DIRECTORY.md * Formatting --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com> Co-authored-by: Christian Clauss <cclauss@me.com> Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>
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Vipin Karthic
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DIRECTORY.md
26
DIRECTORY.md
@ -54,13 +54,12 @@
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* [Largest Pow Of Two Le Num](bit_manipulation/largest_pow_of_two_le_num.py)
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* [Missing Number](bit_manipulation/missing_number.py)
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* [Numbers Different Signs](bit_manipulation/numbers_different_signs.py)
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* [Power Of 4](bit_manipulation/power_of_4.py)
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* [Reverse Bits](bit_manipulation/reverse_bits.py)
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* [Single Bit Manipulation Operations](bit_manipulation/single_bit_manipulation_operations.py)
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## Blockchain
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* [Chinese Remainder Theorem](blockchain/chinese_remainder_theorem.py)
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* [Diophantine Equation](blockchain/diophantine_equation.py)
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* [Modular Division](blockchain/modular_division.py)
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## Boolean Algebra
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* [And Gate](boolean_algebra/and_gate.py)
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@ -101,11 +100,13 @@
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* [Diffie Hellman](ciphers/diffie_hellman.py)
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* [Elgamal Key Generator](ciphers/elgamal_key_generator.py)
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* [Enigma Machine2](ciphers/enigma_machine2.py)
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* [Fractionated Morse Cipher](ciphers/fractionated_morse_cipher.py)
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* [Hill Cipher](ciphers/hill_cipher.py)
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* [Mixed Keyword Cypher](ciphers/mixed_keyword_cypher.py)
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* [Mono Alphabetic Ciphers](ciphers/mono_alphabetic_ciphers.py)
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* [Morse Code](ciphers/morse_code.py)
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* [Onepad Cipher](ciphers/onepad_cipher.py)
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* [Permutation Cipher](ciphers/permutation_cipher.py)
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* [Playfair Cipher](ciphers/playfair_cipher.py)
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* [Polybius](ciphers/polybius.py)
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* [Porta Cipher](ciphers/porta_cipher.py)
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@ -172,6 +173,7 @@
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## Data Structures
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* Arrays
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* [Equilibrium Index In Array](data_structures/arrays/equilibrium_index_in_array.py)
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* [Median Two Array](data_structures/arrays/median_two_array.py)
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* [Permutations](data_structures/arrays/permutations.py)
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* [Prefix Sum](data_structures/arrays/prefix_sum.py)
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@ -352,6 +354,7 @@
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* [Smith Waterman](dynamic_programming/smith_waterman.py)
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* [Subset Generation](dynamic_programming/subset_generation.py)
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* [Sum Of Subset](dynamic_programming/sum_of_subset.py)
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* [Trapped Water](dynamic_programming/trapped_water.py)
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* [Tribonacci](dynamic_programming/tribonacci.py)
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* [Viterbi](dynamic_programming/viterbi.py)
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* [Word Break](dynamic_programming/word_break.py)
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@ -360,6 +363,7 @@
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* [Apparent Power](electronics/apparent_power.py)
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* [Builtin Voltage](electronics/builtin_voltage.py)
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* [Carrier Concentration](electronics/carrier_concentration.py)
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* [Charging Capacitor](electronics/charging_capacitor.py)
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* [Circular Convolution](electronics/circular_convolution.py)
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* [Coulombs Law](electronics/coulombs_law.py)
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* [Electric Conductivity](electronics/electric_conductivity.py)
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@ -466,6 +470,8 @@
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* [Test Min Spanning Tree Prim](graphs/tests/test_min_spanning_tree_prim.py)
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## Greedy Methods
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* [Best Time To Buy And Sell Stock](greedy_methods/best_time_to_buy_and_sell_stock.py)
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* [Fractional Cover Problem](greedy_methods/fractional_cover_problem.py)
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* [Fractional Knapsack](greedy_methods/fractional_knapsack.py)
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* [Fractional Knapsack 2](greedy_methods/fractional_knapsack_2.py)
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* [Gas Station](greedy_methods/gas_station.py)
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@ -524,6 +530,10 @@
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* Local Weighted Learning
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* [Local Weighted Learning](machine_learning/local_weighted_learning/local_weighted_learning.py)
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* [Logistic Regression](machine_learning/logistic_regression.py)
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* Loss Functions
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* [Binary Cross Entropy](machine_learning/loss_functions/binary_cross_entropy.py)
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* [Huber Loss](machine_learning/loss_functions/huber_loss.py)
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* [Mean Squared Error](machine_learning/loss_functions/mean_squared_error.py)
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* [Mfcc](machine_learning/mfcc.py)
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* [Multilayer Perceptron Classifier](machine_learning/multilayer_perceptron_classifier.py)
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* [Polynomial Regression](machine_learning/polynomial_regression.py)
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@ -564,7 +574,9 @@
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* [Carmichael Number](maths/carmichael_number.py)
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* [Catalan Number](maths/catalan_number.py)
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* [Ceil](maths/ceil.py)
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* [Chebyshev Distance](maths/chebyshev_distance.py)
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* [Check Polygon](maths/check_polygon.py)
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* [Chinese Remainder Theorem](maths/chinese_remainder_theorem.py)
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* [Chudnovsky Algorithm](maths/chudnovsky_algorithm.py)
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* [Collatz Sequence](maths/collatz_sequence.py)
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* [Combinations](maths/combinations.py)
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@ -591,6 +603,7 @@
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* [Gaussian](maths/gaussian.py)
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* [Gaussian Error Linear Unit](maths/gaussian_error_linear_unit.py)
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* [Gcd Of N Numbers](maths/gcd_of_n_numbers.py)
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* [Germain Primes](maths/germain_primes.py)
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* [Greatest Common Divisor](maths/greatest_common_divisor.py)
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* [Greedy Coin Change](maths/greedy_coin_change.py)
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* [Hamming Numbers](maths/hamming_numbers.py)
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@ -618,7 +631,9 @@
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* [Matrix Exponentiation](maths/matrix_exponentiation.py)
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* [Max Sum Sliding Window](maths/max_sum_sliding_window.py)
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* [Median Of Two Arrays](maths/median_of_two_arrays.py)
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* [Minkowski Distance](maths/minkowski_distance.py)
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* [Mobius Function](maths/mobius_function.py)
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* [Modular Division](maths/modular_division.py)
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* [Modular Exponential](maths/modular_exponential.py)
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* [Monte Carlo](maths/monte_carlo.py)
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* [Monte Carlo Dice](maths/monte_carlo_dice.py)
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@ -720,12 +735,16 @@
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## Neural Network
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* [2 Hidden Layers Neural Network](neural_network/2_hidden_layers_neural_network.py)
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* Activation Functions
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* [Binary Step](neural_network/activation_functions/binary_step.py)
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* [Exponential Linear Unit](neural_network/activation_functions/exponential_linear_unit.py)
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* [Leaky Rectified Linear Unit](neural_network/activation_functions/leaky_rectified_linear_unit.py)
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* [Mish](neural_network/activation_functions/mish.py)
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* [Rectified Linear Unit](neural_network/activation_functions/rectified_linear_unit.py)
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* [Scaled Exponential Linear Unit](neural_network/activation_functions/scaled_exponential_linear_unit.py)
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* [Sigmoid Linear Unit](neural_network/activation_functions/sigmoid_linear_unit.py)
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* [Soboleva Modified Hyperbolic Tangent](neural_network/activation_functions/soboleva_modified_hyperbolic_tangent.py)
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* [Softplus](neural_network/activation_functions/softplus.py)
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* [Squareplus](neural_network/activation_functions/squareplus.py)
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* [Back Propagation Neural Network](neural_network/back_propagation_neural_network.py)
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* [Convolution Neural Network](neural_network/convolution_neural_network.py)
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* [Perceptron](neural_network/perceptron.py)
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* [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
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* [Photoelectric Effect](physics/photoelectric_effect.py)
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* [Potential Energy](physics/potential_energy.py)
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* [Reynolds Number](physics/reynolds_number.py)
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* [Rms Speed Of Molecule](physics/rms_speed_of_molecule.py)
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* [Shear Stress](physics/shear_stress.py)
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* [Speed Of Sound](physics/speed_of_sound.py)
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* [Interpolation Search](searches/interpolation_search.py)
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* [Jump Search](searches/jump_search.py)
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* [Linear Search](searches/linear_search.py)
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* [Median Of Medians](searches/median_of_medians.py)
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* [Quick Select](searches/quick_select.py)
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* [Sentinel Linear Search](searches/sentinel_linear_search.py)
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* [Simple Binary Search](searches/simple_binary_search.py)
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* [Snake Case To Camel Pascal Case](strings/snake_case_to_camel_pascal_case.py)
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* [Split](strings/split.py)
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* [String Switch Case](strings/string_switch_case.py)
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* [Strip](strings/strip.py)
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* [Text Justification](strings/text_justification.py)
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* [Top K Frequent Words](strings/top_k_frequent_words.py)
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* [Upper](strings/upper.py)
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"""
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* Binary Exponentiation for Powers
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* This is a method to find a^b in a time complexity of O(log b)
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* This is one of the most commonly used methods of finding powers.
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* Also useful in cases where solution to (a^b)%c is required,
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* where a,b,c can be numbers over the computers calculation limits.
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* Done using iteration, can also be done using recursion
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Binary Exponentiation
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This is a method to find a^b in O(log b) time complexity
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This is one of the most commonly used methods of exponentiation
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It's also useful when the solution to (a^b) % c is required because a, b, c may be
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over the computer's calculation limits
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* @author chinmoy159
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* @version 1.0 dated 10/08/2017
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Let's say you need to calculate a ^ b
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- RULE 1 : a ^ b = (a*a) ^ (b/2) ---- example : 4 ^ 4 = (4*4) ^ (4/2) = 16 ^ 2
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- RULE 2 : IF b is odd, then a ^ b = a * (a ^ (b - 1)), where b - 1 is even
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Once b is even, repeat the process until b = 1 or b = 0, because a^1 = a and a^0 = 1
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For modular exponentiation, we use the fact that (a*b) % c = ((a%c) * (b%c)) % c
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Now apply RULE 1 or 2 as required
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@author chinmoy159
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"""
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def b_expo(a: int, b: int) -> int:
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"""
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>>> b_expo(2, 10)
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1024
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>>> b_expo(9, 0)
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1
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>>> b_expo(0, 12)
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0
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>>> b_expo(4, 12)
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16777216
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"""
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res = 1
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while b > 0:
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if b & 1:
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@ -24,6 +40,16 @@ def b_expo(a: int, b: int) -> int:
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def b_expo_mod(a: int, b: int, c: int) -> int:
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"""
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>>> b_expo_mod(2, 10, 1000000007)
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1024
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>>> b_expo_mod(11, 13, 19)
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11
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>>> b_expo_mod(0, 19, 20)
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0
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>>> b_expo_mod(15, 5, 4)
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3
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"""
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res = 1
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while b > 0:
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if b & 1:
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b >>= 1
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return res
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"""
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* Wondering how this method works !
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* It's pretty simple.
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* Let's say you need to calculate a ^ b
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* RULE 1 : a ^ b = (a*a) ^ (b/2) ---- example : 4 ^ 4 = (4*4) ^ (4/2) = 16 ^ 2
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* RULE 2 : IF b is ODD, then ---- a ^ b = a * (a ^ (b - 1)) :: where (b - 1) is even.
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* Once b is even, repeat the process to get a ^ b
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* Repeat the process till b = 1 OR b = 0, because a^1 = a AND a^0 = 1
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*
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* As far as the modulo is concerned,
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* the fact : (a*b) % c = ((a%c) * (b%c)) % c
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* Now apply RULE 1 OR 2 whichever is required.
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"""
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