diff --git a/d0/da4/namespacemonte__carlo.html b/d0/da4/namespacemonte__carlo.html new file mode 100644 index 000000000..a1893e43b --- /dev/null +++ b/d0/da4/namespacemonte__carlo.html @@ -0,0 +1,111 @@ + + + + + + + +Algorithms_in_C++: monte_carlo Namespace Reference + + + + + + + + + + + + + + + +
+
+ + + + + + +
+
Algorithms_in_C++ 1.0.0 +
+
Set of algorithms implemented in C++.
+
+
+ + + + + + + +
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+
monte_carlo Namespace Reference
+
+
+ +

Functions for the Monte Carlo Integration implementation. +More...

+

Detailed Description

+

Functions for the Monte Carlo Integration implementation.

+
+
+ + + + diff --git a/d5/d5c/integral__approximation2_8cpp__incl.map b/d5/d5c/integral__approximation2_8cpp__incl.map new file mode 100644 index 000000000..37250ea3f --- /dev/null +++ b/d5/d5c/integral__approximation2_8cpp__incl.map @@ -0,0 +1,10 @@ + + + + + + + + + + diff --git a/d5/d5c/integral__approximation2_8cpp__incl.md5 b/d5/d5c/integral__approximation2_8cpp__incl.md5 new file mode 100644 index 000000000..f37a84e97 --- /dev/null +++ b/d5/d5c/integral__approximation2_8cpp__incl.md5 @@ -0,0 +1 @@ +f1e84904021d3d8fadd7b878793f49fd \ No newline at end of file diff --git a/d5/d5c/integral__approximation2_8cpp__incl.svg b/d5/d5c/integral__approximation2_8cpp__incl.svg new file mode 100644 index 000000000..3780276b5 --- /dev/null +++ b/d5/d5c/integral__approximation2_8cpp__incl.svg @@ -0,0 +1,127 @@ + + + + + + +math/integral_approximation2.cpp + + + +Node1 + + +math/integral_approximation2.cpp + + + + + +Node2 + + +cmath + + + + + +Node1->Node2 + + + + + +Node3 + + +cstdint + + + + + +Node1->Node3 + + + + + +Node4 + + +ctime + + + + + +Node1->Node4 + + + + + +Node5 + + +functional + + + + + +Node1->Node5 + + + + + +Node6 + + +iostream + + + + + +Node1->Node6 + + + + + +Node7 + + +random + + + + + +Node1->Node7 + + + + + +Node8 + + +vector + + + + + +Node1->Node8 + + + + + diff --git a/d5/d88/md__d_i_r_e_c_t_o_r_y.html b/d5/d88/md__d_i_r_e_c_t_o_r_y.html index c5271859c..928d90381 100644 --- a/d5/d88/md__d_i_r_e_c_t_o_r_y.html +++ b/d5/d88/md__d_i_r_e_c_t_o_r_y.html @@ -306,6 +306,7 @@ Math
  • Gcd Of N Numbers
  • Gcd Recursive Euclidean
  • Integral Approximation
    • +
  • Integral Approximation2
  • Inv Sqrt
  • Large Factorial
  • Large Number
    • diff --git a/db/d40/integral__approximation2_8cpp.html b/db/d40/integral__approximation2_8cpp.html new file mode 100644 index 000000000..7fe973040 --- /dev/null +++ b/db/d40/integral__approximation2_8cpp.html @@ -0,0 +1,478 @@ + + + + + + + +Algorithms_in_C++: math/integral_approximation2.cpp File Reference + + + + + + + + + + + + + + + +
    +
    + + + + + + +
    +
    Algorithms_in_C++ 1.0.0 +
    +
    Set of algorithms implemented in C++.
    +
    +
    + + + + + + + +
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    +Namespaces | +Macros | +Typedefs | +Functions
    +
    integral_approximation2.cpp File Reference
    +
    +
    + +

    Monte Carlo Integration +More...

    +
    #include <cmath>
    +#include <cstdint>
    +#include <ctime>
    +#include <functional>
    +#include <iostream>
    +#include <random>
    +#include <vector>
    +
    +Include dependency graph for integral_approximation2.cpp:
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    +
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    +
    +
    + + + + + + + +

    +Namespaces

    namespace  math
     for IO operations
     
    namespace  monte_carlo
     Functions for the Monte Carlo Integration implementation.
     
    + + + +

    +Typedefs

    +using math::monte_carlo::Function = std::function< double(double &)>
     
    + + + + + + + + + + + + + +

    +Functions

    std::vector< double > math::monte_carlo::generate_samples (const double &start_point, const Function &pdf, const uint32_t &num_samples, const uint32_t &discard=100000)
     short-hand for std::functions used in this implementation More...
     
    double math::monte_carlo::integral_monte_carlo (const double &start_point, const Function &function, const Function &pdf, const uint32_t &num_samples=1000000)
     Compute an approximation of an integral using Monte Carlo integration. More...
     
    static void test ()
     Self-test implementations. More...
     
    int main ()
     Main function. More...
     
    +

    Detailed Description

    +

    Monte Carlo Integration

    +

    In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.

    +

    This implementation supports arbitrary pdfs. These pdfs are sampled using the Metropolis-Hastings algorithm. This can be swapped out by every other sampling techniques for example the inverse method. Metropolis-Hastings was chosen because it is the most general and can also be extended for a higher dimensional sampling space.

    +
    Author
    Domenic Zingsheim
    +

    Function Documentation

    + +

    ◆ generate_samples()

    + +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    std::vector< double > math::monte_carlo::generate_samples (const double & start_point,
    const Function & pdf,
    const uint32_t & num_samples,
    const uint32_t & discard = 100000 
    )
    +
    + +

    short-hand for std::functions used in this implementation

    +

    Generate samples according to some pdf

    +

    This function uses Metropolis-Hastings to generate random numbers. It generates a sequence of random numbers by using a markov chain. Therefore, we need to define a start_point and the number of samples we want to generate. Because the first samples generated by the markov chain may not be distributed according to the given pdf, one can specify how many samples should be discarded before storing samples.

    Parameters
    + + + + + +
    start_pointThe starting point of the markov chain
    pdfThe pdf to sample
    num_samplesThe number of samples to generate
    discardHow many samples should be discarded at the start
    +
    +
    +
    Returns
    A vector of size num_samples with samples distributed according to the pdf
    +
    53 {
    +
    54 std::vector<double> samples;
    +
    55 samples.reserve(num_samples);
    +
    56
    +
    57 double x_t = start_point;
    +
    58
    +
    59 std::default_random_engine generator;
    +
    60 std::uniform_real_distribution<double> uniform(0.0, 1.0);
    +
    61 std::normal_distribution<double> normal(0.0, 1.0);
    +
    62 generator.seed(time(nullptr));
    +
    63
    +
    64 for(uint32_t t = 0; t < num_samples + discard; ++t) {
    +
    65 // Generate a new proposal according to some mutation strategy.
    +
    66 // This is arbitrary and can be swapped.
    +
    67 double x_dash = normal(generator) + x_t;
    +
    68 double acceptance_probability = std::min(pdf(x_dash)/pdf(x_t), 1.0);
    +
    69 double u = uniform(generator);
    +
    70
    +
    71 // Accept "new state" according to the acceptance_probability
    +
    72 if(u <= acceptance_probability) {
    +
    73 x_t = x_dash;
    +
    74 }
    +
    75
    +
    76 if(t >= discard) {
    +
    77 samples.push_back(x_t);
    +
    78 }
    +
    79 }
    +
    80
    +
    81 return samples;
    +
    82}
    +
    std::min
    T min(T... args)
    + +
    std::vector::push_back
    T push_back(T... args)
    + +
    std::vector::reserve
    T reserve(T... args)
    +
    std::time
    T time(T... args)
    + + +
    +Here is the call graph for this function:
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    +
    +
    + +
    +
    + +

    ◆ integral_monte_carlo()

    + +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    double math::monte_carlo::integral_monte_carlo (const double & start_point,
    const Function & function,
    const Function & pdf,
    const uint32_t & num_samples = 1000000 
    )
    +
    + +

    Compute an approximation of an integral using Monte Carlo integration.

    +

    The integration domain [a,b] is given by the pdf. The pdf has to fulfill the following conditions: 1) for all x \in [a,b] : p(x) > 0 2) for all x \not\in [a,b] : p(x) = 0 3) \int_a^b p(x) dx = 1

    Parameters
    + + + + + +
    start_pointThe start point of the Markov Chain (see generate_samples)
    functionThe function to integrate
    pdfThe pdf to sample
    num_samplesThe number of samples used to approximate the integral
    +
    +
    +
    Returns
    The approximation of the integral according to 1/N \sum_{i}^N f(x_i) / p(x_i)
    +
    97 {
    +
    98 double integral = 0.0;
    +
    99 std::vector<double> samples = generate_samples(start_point, pdf, num_samples);
    +
    100
    +
    101 for(double sample : samples) {
    +
    102 integral += function(sample) / pdf(sample);
    +
    103 }
    +
    104
    +
    105 return integral / static_cast<double>(samples.size());
    +
    106}
    +
    math::monte_carlo::generate_samples
    std::vector< double > generate_samples(const double &start_point, const Function &pdf, const uint32_t &num_samples, const uint32_t &discard=100000)
    short-hand for std::functions used in this implementation
    Definition: integral_approximation2.cpp:53
    +
    +Here is the call graph for this function:
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    + +
    +
    + +

    ◆ main()

    + +
    +
    + + + + + + + + +
    int main (void )
    +
    + +

    Main function.

    +
    Returns
    0 on exit
    +
    193 {
    +
    194 test(); // run self-test implementations
    +
    195 return 0;
    +
    196}
    +
    test
    static void test()
    Self-test implementations.
    Definition: integral_approximation2.cpp:115
    +
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    +
    +
    + +
    +
    + +

    ◆ test()

    + +
    +
    + + + + + +
    + + + + + + + +
    static void test ()
    +     		+static
    +
    + +

    Self-test implementations.

    +
    Returns
    void
    +
    115 {
    +
    116 std::cout << "Disclaimer: Because this is a randomized algorithm," << std::endl;
    +
    117 std::cout << "it may happen that singular samples deviate from the true result." << std::endl << std::endl;;
    +
    118
    +
    119 math::monte_carlo::Function f;
    +
    120 math::monte_carlo::Function pdf;
    +
    121 double integral = 0;
    +
    122 double lower_bound = 0, upper_bound = 0;
    +
    123
    +
    124 /* \int_{-2}^{2} -x^2 + 4 dx */
    +
    125 f = [&](double& x) {
    +
    126 return -x*x + 4.0;
    +
    127 };
    +
    128
    +
    129 lower_bound = -2.0;
    +
    130 upper_bound = 2.0;
    +
    131 pdf = [&](double& x) {
    +
    132 if(x >= lower_bound && x <= -1.0) {
    +
    133 return 0.1;
    +
    134 }
    +
    135 if(x <= upper_bound && x >= 1.0) {
    +
    136 return 0.1;
    +
    137 }
    +
    138 if(x > -1.0 && x < 1.0) {
    +
    139 return 0.4;
    +
    140 }
    +
    141 return 0.0;
    +
    142 };
    +
    143
    +
    144 integral = math::monte_carlo::integral_monte_carlo((upper_bound - lower_bound) / 2.0, f, pdf);
    +
    145
    +
    146 std::cout << "This number should be close to 10.666666: " << integral << std::endl;
    +
    147
    +
    148 /* \int_{0}^{1} e^x dx */
    +
    149 f = [&](double& x) {
    +
    150 return std::exp(x);
    +
    151 };
    +
    152
    +
    153 lower_bound = 0.0;
    +
    154 upper_bound = 1.0;
    +
    155 pdf = [&](double& x) {
    +
    156 if(x >= lower_bound && x <= 0.2) {
    +
    157 return 0.1;
    +
    158 }
    +
    159 if(x > 0.2 && x <= 0.4) {
    +
    160 return 0.4;
    +
    161 }
    +
    162 if(x > 0.4 && x < upper_bound) {
    +
    163 return 1.5;
    +
    164 }
    +
    165 return 0.0;
    +
    166 };
    +
    167
    +
    168 integral = math::monte_carlo::integral_monte_carlo((upper_bound - lower_bound) / 2.0, f, pdf);
    +
    169
    +
    170 std::cout << "This number should be close to 1.7182818: " << integral << std::endl;
    +
    171
    +
    172 /* \int_{-\infty}^{\infty} sinc(x) dx, sinc(x) = sin(pi * x) / (pi * x)
    +
    173 This is a difficult integral because of its infinite domain.
    +
    174 Therefore, it may deviate largely from the expected result.
    +
    175 */
    +
    176 f = [&](double& x) {
    +
    177 return std::sin(M_PI * x) / (M_PI * x);
    +
    178 };
    +
    179
    +
    180 pdf = [&](double& x) {
    +
    181 return 1.0 / std::sqrt(2.0 * M_PI) * std::exp(-x * x / 2.0);
    +
    182 };
    +
    183
    +
    184 integral = math::monte_carlo::integral_monte_carlo(0.0, f, pdf, 10000000);
    +
    185
    +
    186 std::cout << "This number should be close to 1.0: " << integral << std::endl;
    +
    187}
    + +
    std::endl
    T endl(T... args)
    +
    std::exp
    T exp(T... args)
    + +
    math::monte_carlo::integral_monte_carlo
    double integral_monte_carlo(const double &start_point, const Function &function, const Function &pdf, const uint32_t &num_samples=1000000)
    Compute an approximation of an integral using Monte Carlo integration.
    Definition: integral_approximation2.cpp:97
    +
    std::lower_bound
    T lower_bound(T... args)
    +
    std::sin
    T sin(T... args)
    +
    std::sqrt
    T sqrt(T... args)
    +
    std::upper_bound
    T upper_bound(T... args)
    +
    +Here is the call graph for this function:
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    + + + + diff --git a/db/d40/integral__approximation2_8cpp.js b/db/d40/integral__approximation2_8cpp.js new file mode 100644 index 000000000..36791c0a5 --- /dev/null +++ b/db/d40/integral__approximation2_8cpp.js @@ -0,0 +1,7 @@ +var integral__approximation2_8cpp = +[ + [ "generate_samples", "db/d40/integral__approximation2_8cpp.html#a71249ee535f16f8ed2e9cc8f0199a2cf", null ], + [ "integral_monte_carlo", "db/d40/integral__approximation2_8cpp.html#af7da9ba8932f1f48b9bbc2d80471af51", null ], + [ "main", "db/d40/integral__approximation2_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4", null ], + [ "test", "db/d40/integral__approximation2_8cpp.html#aa8dca7b867074164d5f45b0f3851269d", null ] +]; \ No newline at end of file diff --git a/db/d40/integral__approximation2_8cpp_a71249ee535f16f8ed2e9cc8f0199a2cf_cgraph.map b/db/d40/integral__approximation2_8cpp_a71249ee535f16f8ed2e9cc8f0199a2cf_cgraph.map new file mode 100644 index 000000000..c6259ce41 --- /dev/null +++ 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    Detailed Description

    for IO operations

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    Math algorithms.

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    for std::cout

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    Evaluate recurrence relation using matrix exponentiation.

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    Mathematical algorithms

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    Mathematical functions

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      +file  integral_approximation2.cpp + Monte Carlo Integration
    +  file  inv_sqrt.cpp  Implementation of the inverse square root Root.
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(factorial) using Legendre's Formula - lcm_sum.cppAn algorithm to calculate the sum of LCM: \(\mathrm{LCM}(1,n) + \mathrm{LCM}(2,n) + \ldots + \mathrm{LCM}(n,n)\) - least_common_multiple.cpp - magic_number.cppA simple program to check if the given number is a magic number or not. A number is said to be a magic number, if the sum of its digits are calculated till a single digit recursively by adding the sum of the digits after every addition. If the single digit comes out to be 1,then the number is a magic number - miller_rabin.cpp - modular_division.cppAn algorithm to divide two numbers under modulo p Modular Division - modular_exponentiation.cppC++ Program for Modular Exponentiation Iteratively - modular_inverse_fermat_little_theorem.cppC++ Program to find the modular inverse using Fermat's Little Theorem - n_bonacci.cppImplementation of the N-bonacci series - n_choose_r.cppCombinations n choose r function implementation - ncr_modulo_p.cppThis program aims at calculating nCr modulo p - number_of_positive_divisors.cppC++ Program to calculate the number of positive divisors - power_for_huge_numbers.cppCompute powers of large numbers - power_of_two.cppImplementation to check whether a number is a power of 2 or not - prime_factorization.cppPrime factorization of positive integers - prime_numbers.cppGet list of prime numbers - primes_up_to_billion.cppCompute prime numbers upto 1 billion - realtime_stats.cppCompute statistics for data entered in rreal-time - sieve_of_eratosthenes.cppGet list of prime numbers using Sieve of Eratosthenes - sqrt_double.cppCalculate the square root of any positive real number in \(O(\log N)\) time, with precision fixed using bisection method of root-finding - string_fibonacci.cppThis Programme returns the Nth fibonacci as a string - sum_of_binomial_coefficient.cppAlgorithm to find sum of binomial coefficients of a given positive integer - sum_of_digits.cppA C++ Program to find the Sum of Digits of input integer - vector_cross_product.cppCalculates the Cross Product and the magnitude of two mathematical 3D vectors + integral_approximation2.cppMonte Carlo Integration + inv_sqrt.cppImplementation of the inverse square root Root + large_factorial.cppCompute factorial of any arbitratily large number/ + large_number.hLibrary to perform arithmatic operations on arbitrarily large numbers + largest_power.cppAlgorithm to find largest x such that p^x divides n! (factorial) using Legendre's Formula + lcm_sum.cppAn algorithm to calculate the sum of LCM: \(\mathrm{LCM}(1,n) + \mathrm{LCM}(2,n) + \ldots + \mathrm{LCM}(n,n)\) + least_common_multiple.cpp + magic_number.cppA simple program to check if the given number is a magic number or not. A number is said to be a magic number, if the sum of its digits are calculated till a single digit recursively by adding the sum of the digits after every addition. If the single digit comes out to be 1,then the number is a magic number + miller_rabin.cpp + modular_division.cppAn algorithm to divide two numbers under modulo p Modular Division + modular_exponentiation.cppC++ Program for Modular Exponentiation Iteratively + modular_inverse_fermat_little_theorem.cppC++ Program to find the modular inverse using Fermat's Little Theorem + n_bonacci.cppImplementation of the N-bonacci series + n_choose_r.cppCombinations n choose r function implementation + ncr_modulo_p.cppThis program aims at calculating nCr modulo p + number_of_positive_divisors.cppC++ Program to calculate the number of positive divisors + power_for_huge_numbers.cppCompute powers of large numbers + power_of_two.cppImplementation to check whether a number is a power of 2 or not + prime_factorization.cppPrime factorization of positive integers + prime_numbers.cppGet list of prime numbers + primes_up_to_billion.cppCompute prime numbers upto 1 billion + realtime_stats.cppCompute statistics for data entered in rreal-time + sieve_of_eratosthenes.cppGet list of prime numbers using Sieve of Eratosthenes + sqrt_double.cppCalculate the square root of any positive real number in \(O(\log N)\) time, with precision fixed using bisection method of root-finding + string_fibonacci.cppThis Programme returns the Nth fibonacci as a string + sum_of_binomial_coefficient.cppAlgorithm to find sum of binomial coefficients of a given positive integer + sum_of_digits.cppA C++ Program to find the Sum of Digits of input integer + vector_cross_product.cppCalculates the Cross Product and the magnitude of two mathematical 3D vectors  ► numerical_methods  bisection_method.cppSolve the equation \(f(x)=0\) using bisection method  brent_method_extrema.cppFind real extrema of a univariate real function in a given interval using Brent's method diff --git a/globals_func_m.html b/globals_func_m.html index 875a674bd..fdc700c45 100644 --- a/globals_func_m.html +++ b/globals_func_m.html @@ -93,7 +93,7 @@ $(document).ready(function(){initNavTree('globals_func_m.html',''); initResizabl  

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