Çözüldü Kinematik - Yaklaşık İntegrasyonda Serilerin Kullanımı - Sayısal Çözümleme - Programlama

Princeton University'den çözümlü bir problemin fen lisesi için klasik sınav uyarlaması:

(n = 0'dan sonsuza), Σ (-1)^n·x^(2n) / [ (2n)! ] = cos(x) olduğuna göre yatay düzlemde duran ve hız denklemi
v(t) = cos(√t) olan bir cismin ilk 0,1 birim zamanda aldığı yolun 1 / 1000'den daha az hatalı değerini cebirsel yoldan bulunuz ve bildiğiniz bir programlama dilinde de bu sonucu veren kodlamayı yapınız.


https://i72.servimg.com/u/f72/19/97/10/39/prince30.png
https://web.math.princeton.edu/~nelson/104/fall06finalsolutions.pdf
(Sayfa 5, Soru 4)

Fortran Çözümü:

https://i72.servimg.com/u/f72/19/97/10/39/simpso10.png

Program:

Kod:

! Program written by Google Gemini AI and slightly changed by me
program simpson_integration
    implicit none
  
    ! Define variables using double precision (8 bytes)
    double precision :: a, b, h, total_sum, integral
    integer :: n, i
    double precision :: x
  
    ! 1. Inputs: Define boundaries and intervals
    a = 0.0d0          ! Lower limit of integration
    b = 0.1d0          ! Upper limit of integration
    n = 100            ! Number of sub-intervals (must be even)
  
    ! Ensure n is even; if odd, add 1 to satisfy Simpson's rule requirements
    if (mod(n, 2) /= 0) then
        n = n + 1
    end if
  
    ! 2. Calculate the step size (h)
    h = (b - a) / dble(n)
  
    ! Initialize the sum with the boundary elements: f(a) + f(b)
    total_sum = f(a) + f(b)
  
    ! 3. Loop through internal points and apply coefficients
    ! Alternate coefficients: 4 for odd indexes, 2 for even indexes
    do i = 1, n - 1
        x = a + dble(i) * h
        if (mod(i, 2) /= 0) then
            total_sum = total_sum + 4.0d0 * f(x) ! Odd terms multiplied by 4
        else
            total_sum = total_sum + 2.0d0 * f(x) ! Even terms multiplied by 2
        end if
    end do
  
    ! 4. Compute final composite integration result
    integral = (h / 3.0d0) * total_sum
  
    ! 5. Print out the formatted result
    print '(A, F6.4)', ' The approximated integral value is 39 / 400 = ', integral

contains

    ! Define the target function to integrate f(x)
    ! Example here integrates f(x) = cos(√x) (Expected analytical answer over [0,.1] is 0.0975)
    double precision function f(x)
        implicit none
        double precision, intent(in) :: x
      
        f = cos(sqrt(x))
    end function f

end program simpson_integration