Merge pull request #25 from Ramy-Badr-Ahmed/feature/numerical_integra…

…tion

Implemented Several Numerical Integration Algorithms in `maths/`

DIRECTORY.md

@@ -1,5 +1,11 @@
 # The Algorithms - Directory of Fortran
 ## Maths
+* numerical_integration
+  * [trapezoidal_rule](/modules/maths/numerical_integration/trapezoid.f90)
+  * [simpson_rule](/modules/maths/numerical_integration/simpson.f90)
+  * [midpoint_rule](/modules/maths/numerical_integration/midpoint.f90)
+  * [monte_carlo](/modules/maths/numerical_integration/monte_carlo.f90)
+  * [gauss_legendre](/modules/maths/numerical_integration/gaussian_legendre.f90)
 * [euclid_gcd](/modules/maths/euclid_gcd.f90)
 * [factorial](/modules/maths/factorial.f90)
 * [fibonacci](/modules/maths/fibonacci.f90)

examples/maths/numerical_integration/gaussian_legendre.f90

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+!> Example Program for Gaussian-Legendre Quadrature Module
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #25
+!!  https://github.com/TheAlgorithms/Fortran/pull/25
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
+!! This program demonstrates the use of Gaussian-Legendre Quadrature Module for numerical integration.
+!!
+!! It sets the integration limits and the number of quadrature points (n), and calls the
+!! gauss_legendre_quadrature subroutine to compute the approximate value of the definite integral
+!! of the specified function.
+!!
+!! Example function: f(x) = exp(-x^2) * cos(2.0_dp * x)
+
+program example_gaussian_quadrature
+    use gaussian_legendre_quadrature
+    implicit none
+
+    real(dp) :: lower_bound, upper_bound, integral_result
+    integer :: quadrature_points_number
+
+    ! Set the integration limits and number of quadrature points
+    lower_bound = -1.0_dp
+    upper_bound = 1.0_dp
+    quadrature_points_number = 5       !! Number of quadrature points (order of accuracy) up to 5
+
+    ! Call Gaussian quadrature to compute the integral with the function passed as an argument
+    call gauss_legendre_quadrature(integral_result, lower_bound, upper_bound, quadrature_points_number, function)
+
+    write (*, '(A, F12.6)') "Gaussian Quadrature result: ", integral_result          !! ≈ 0.858574
+
+contains
+
+    function function(x) result(fx)
+        implicit none
+        real(dp), intent(in) :: x
+        real(dp) :: fx
+
+        fx = exp(-x**2)*cos(2.0_dp*x)       !! Example function to integrate
+    end function function
+
+end program example_gaussian_quadrature

examples/maths/numerical_integration/midpoint.f90

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+!> Example Program for Midpoint Rule
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #25
+!!  https://github.com/TheAlgorithms/Fortran/pull/25
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
+!! This program demonstrates the use of Midpoint Rule for numerical integration.
+!!
+!! It sets the integration limits and number of subintervals (panels), and calls the
+!! midpoint subroutine to compute the approximate value of the definite integral
+!! of the specified function.
+!!
+!! Example function: f(x) = exp(-x^2) * cos(2.0_dp * x)
+
+program example_midpoint
+    use midpoint_rule
+    implicit none
+
+    real(dp) :: lower_bound, upper_bound, integral_result
+    integer :: panels_number
+
+    ! Set the integration limits and number of panels
+    lower_bound = -1.0_dp
+    upper_bound = 1.0_dp
+    panels_number = 400     !! Number of subdivisions
+
+    ! Call the midpoint rule subroutine with the function passed as an argument
+    call midpoint(integral_result, lower_bound, upper_bound, panels_number, function)
+
+    write (*, '(A, F12.6)') "Midpoint rule yields: ", integral_result  !! ≈ 0.858196
+
+contains
+
+    function function(x) result(fx)
+        implicit none
+        real(dp), intent(in) :: x
+        real(dp) :: fx
+
+        fx = exp(-x**2)*cos(2.0_dp*x)       !! Example function to integrate
+    end function function
+
+end program example_midpoint

examples/maths/numerical_integration/monte_carlo.f90

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+!> Example Program for Monte Carlo Integration
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #25
+!!  https://github.com/TheAlgorithms/Fortran/pull/25
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
+!! This program demonstrates the use of Monte Carlo module for numerical integration.
+!!
+!! It sets the integration limits and number of random samples, and calls the
+!! monte_carlo subroutine to compute the approximate value of the definite integral
+!! of the specified function.
+!!
+!! Example function: f(x) = exp(-x^2) * cos(2.0_dp * x)
+
+program example_monte_carlo
+    use monte_carlo_integration
+    implicit none
+
+    real(dp) :: lower_bound, upper_bound, integral_result, error_estimate
+    integer :: random_samples_number
+
+    ! Set the integration limits and number of random samples
+    lower_bound = -1.0_dp
+    upper_bound = 1.0_dp
+    random_samples_number = 1000000     !! 1E6 Number of random samples
+
+    ! Call Monte Carlo integration with the function passed as an argument
+    call monte_carlo(integral_result, error_estimate, lower_bound, upper_bound, random_samples_number, function)
+
+    write (*, '(A, F12.6, A, F12.6)') "Monte Carlo result: ", integral_result, " +- ", error_estimate     !! ≈ 0.858421
+
+contains
+
+    function function(x) result(fx)
+        implicit none
+        real(dp), intent(in) :: x
+        real(dp) :: fx
+
+        fx = exp(-x**2)*cos(2.0_dp*x)       !! Example function to integrate
+    end function function
+
+end program example_monte_carlo

examples/maths/numerical_integration/simpson.f90

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+!> Example Program for Simpson's Rule
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #25
+!!  https://github.com/TheAlgorithms/Fortran/pull/25
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
+!! This program demonstrates the use of Simpson's rule for numerical integration.
+!!
+!! It sets the integration limits and number of panels, and calls the
+!! simpson subroutine to compute the approximate value of the definite integral
+!! of the specified function.
+!!
+!! Example function: f(x) = exp(-x^2) * cos(2.0_dp * x)
+
+program example_simpson
+    use simpson_rule
+    implicit none
+
+    real(dp) :: lower_bound, upper_bound, integral_result
+    integer :: panels_number
+
+    ! Set the integration limits and number of panels
+    lower_bound = -1.0_dp
+    upper_bound = 1.0_dp
+    panels_number = 100     !! Number of subdivisions (must be even)
+
+    ! Call Simpson's rule with the function passed as an argument
+    call simpson(integral_result, lower_bound, upper_bound, panels_number, function)
+
+    write (*, '(A, F12.8)') "Simpson's rule yields: ", integral_result       !! ≈ 0.85819555
+
+contains
+
+    function function(x) result(fx)
+        implicit none
+        real(dp), intent(in) :: x
+        real(dp) :: fx
+
+        fx = exp(-x**2)*cos(2.0_dp*x)       !! Example function to integrate
+    end function function
+
+end program example_simpson

examples/maths/numerical_integration/trapezoid.f90

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+!> Example Program for Trapezoidal Rule
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #25
+!!  https://github.com/TheAlgorithms/Fortran/pull/25
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
+!! This program demonstrates the use of the Trapezoidal rule for numerical integration.
+!!
+!! It sets the integration limits and number of panels, and calls the
+!! trapezoid subroutine to compute the approximate value of the definite integral
+!! of the specified function.
+!!
+!! Example function: f(x) = exp(-x^2) * cos(2.0_dp * x)
+
+program example_tapezoid
+    use trapezoidal_rule
+    implicit none
+
+    real(dp) :: lower_bound, upper_bound, integral_result
+    integer :: panels_number
+
+    ! Set the integration limits and number of panels
+    lower_bound = -1.0_dp
+    upper_bound = 1.0_dp
+    panels_number = 1000000     !! 1E6 Number of subdivisions
+
+    ! Call the trapezoidal rule with the function passed as an argument
+    call trapezoid(integral_result, lower_bound, upper_bound, panels_number, function)
+
+    write (*, '(A, F12.6)') 'Trapezoidal rule yields: ', integral_result     !! ≈ 0.858195
+
+contains
+
+    function function(x) result(fx)
+        implicit none
+        real(dp), intent(in) :: x
+        real(dp) :: fx
+
+        fx = exp(-x**2)*cos(2.0_dp*x)       !! Example function to integrate
+    end function function
+
+end program example_tapezoid

examples/sorts/example_usage_gnome_sort.f90

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 !> Test program for the Gnome Sort algorithm
-
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #09
+!!  https://github.com/TheAlgorithms/Fortran/pull/9
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
 !! This program demonstrates the use of the gnome_sort_module by sorting an array of integers.
 
 program test_gnome_sort

examples/sorts/example_usage_heap_sort.f90

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 !> Example program for the Heap Sort algorithm
 !!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #8
+!!  https://github.com/TheAlgorithms/Fortran/pull/8
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
 !! This program demonstrates the use of the heap_sort_module by sorting an array of integers.
 
 program test_heap_sort

examples/sorts/example_usage_merge_sort.f90

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 !> Test program for the Merge Sort algorithm
-
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #7
+!!  https://github.com/TheAlgorithms/Fortran/pull/7
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
 !! This program demonstrates the use of the merge_sort_module by sorting an array of integers.
 
 program test_merge_sort

examples/sorts/example_usage_quick_sort.f90

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 !> Example program for the Quick Sort algorithm
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #10
+!!  https://github.com/TheAlgorithms/Fortran/pull/10
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
 !! This program demonstrates the use of the quick_sort_module by sorting an array of integers.
 
 program test_quick_sort

examples/sorts/example_usage_radix_sort.f90

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 !> Test program for the Radix Sort algorithm
-
+!!
+!!  Created by: Ramy-Badr-Ahmed (https://github.com/Ramy-Badr-Ahmed)
+!!  in Pull Request: #11
+!!  https://github.com/TheAlgorithms/Fortran/pull/11
+!!
+!!  Please mention me (@Ramy-Badr-Ahmed) in any issue or pull request
+!!  addressing bugs/corrections to this file. Thank you!
+!!
 !! This program demonstrates the use of the radix_sort_module by sorting an array of integers.
 !! The base parameter affects the internal digit processing but does not change the final sorted order
 !! of decimal integers. The output is always in decimal form.