Çözüldü Üstel Fonksiyonlu Denklem-Türev-Ara Değer (Intermediate Value) Teoremi-Programlama

University of Oregon'dan çözümlü bir problemin fen lisesi için hesap makinesi kullanımının da serbest olduğu klasik sınav uyarlaması:

e^x = 1000x denklemini sağlayan kaç kök olduğunu cebirsel olarak açıklayınız, grafiğini çiziniz ve bildiğiniz bir programlama dilinde çözümü veren kodu yazınız.


https://i.ibb.co/N2m789q2/UOregon.png
https://pages.uoregon.edu/math/syllabi/syllabi2012-2013/Math 251/251Final2-Fall2012.pdf
(Sayfa 1 - 2, Soru 2)

Grafik:

https://i.ibb.co/WWVLfFpF/UOregon-Graph.png
https://www.wolframalpha.com/input?i=plot y=e^x, y=1000x

"Google Gemini Yapay Zekâ"nın Cebirsel Açıklaması ve Fortran Çözümü:
(Anlaşılamayan yerler varsa lütfen haberimiz olsun.)

https://i.ibb.co/YF4YVrPb/UOregon-Gemini.png


https://i.ibb.co/XfWLW9LV/UOregon-Gemini-Fortran.png

Program:

Kod:

! e^x = 1000x
! Program written by Google Gemini AI

program newton_raphson_all_roots
    implicit none

    ! Parameters for numerical precision and limits
    real(8), parameter :: TOLERANCE = 1.0d-12
    integer, parameter :: MAX_ITERATIONS = 100

    ! Array of initial guesses to find both roots
    real(8), dimension(2) :: guesses
    real(8) :: x_curr, x_next, fx, dfx, error
    integer :: i, iter
    logical :: converged

    ! Define the two initial guesses based on mathematical analysis
    guesses(1) = 0.0d0   ! Targets the smaller root near x ~ 0.001
    guesses(2) = 10.0d0  ! Targets the larger root near x ~ 9.118

    print *, "========================================================="
    print *, "   Newton-Raphson Root Finder for: exp(x) - 1000x = 0    "
    print *, "========================================================="

    ! Loop through each initial guess to locate all roots
    do i = 1, 2
        x_curr = guesses(i)
        converged = .false.
      
        print *
        print '(A, I1, A, F5.1)', " Searching for Root ", i, " using initial guess: ", x_curr
        print *, "---------------------------------------------------------"
        print '(A10, A20, A20)', "Iter", "x_value", "f(x)"
        print *, "---------------------------------------------------------"

        do iter = 1, MAX_ITERATIONS
            fx = f(x_curr)
            dfx = df(x_curr)

            ! Print progress tracking for transparency
            print '(I10, F20.12, E20.4E2)', iter, x_curr, fx

            ! Safety check: Prevent division by zero if derivative vanishes
            if (abs(dfx) < 1.0d-30) then
                print *, " Error: Derivative is too close to zero. Aborting."
                exit
            end if

            ! Core Newton-Raphson update equation
            x_next = x_curr - (fx / dfx)
            error = abs(x_next - x_curr)

            ! Check for convergence criteria
            if (error < TOLERANCE) then
                x_curr = x_next
                converged = .true.
                exit
            end if

            ! Update the current approximation for the next loop iteration
            x_curr = x_next
        end do

        ! Output the results for the targeted root
        print *, "---------------------------------------------------------"
        if (converged) then
            print '(A, F20.12)', " SUCCESS: Found root at x = ", x_curr
            print '(A, E20.4E2)', " Final residual f(x)     = ", f(x_curr)
        else
            print *, " FAILURE: Method did not converge within maximum iterations."
        end if
        print *, "========================================================="
    end do

contains

    ! Evaluation of the objective function f(x)
    real(8) function f(x)
        real(8), intent(in) :: x
        f = exp(x) - 1000.0d0 * x
    end function f

    ! Evaluation of the derivative function f'(x)
    real(8) function df(x)
        real(8), intent(in) :: x
        df = exp(x) - 1000.0d0
    end function df

end program newton_raphson_all_roots