# Roots for polynomianl

## Roots for quadratic equation

In this problem we try to solve a quadractic equation of the form of
\begin{equation}
ax^2 + bx + c = 0.
\tag{1}
\end{equation}
We can factorize (1) to
\begin{equation}
(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}.
\tag{2}
\end{equation}
So the solution to x is
\begin{equation}
x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}
\tag{3}
\end{equation}
Fisrtly, you enter three coefficient (a, b, c) which are floating point number. Then I tranfrom data type REAL to COMPLEX by using intrinsic function *cmplx*. Lastly, I calculate the roots of this quadractic equation by eq (3).
To present the result I seperate real part and imaginary part by using another intrinsic function *real* and *aimag*.

### Code

!   Purpose: Finding roots for quadratic equation
!
!   Methods: Using quadratic root formula 
!   
!   Input: d, e, f (a real number floating point coef)
!
!   Output: x1, x2 (complex solution)
!       
!       Date        Programer       Description of change
!       ====        =========       =====================
!      9/27/20        Morris        Original Code
!========================================================
PROGRAM quadratic
    IMPLICIT NONE
    REAL :: d, e, f
    COMPLEX :: a, b, c
    COMPLEX :: x1, x2
    CHARACTER(len=15) :: str1, str2, str3

    write(6, *) 'Enter the coef of a quafractic equation (ax^2+bx+c=0)'
    read(5, *) d, e, f
    write(6, *) 'The coef of the equation is:'
    write(str1, '(F6.2)') d
    write(str2, '(F6.2)') e
    write(str3, '(F6.2)') f
    write(6, *) 'a = '//str1
    write(6, *) 'b = '//str2
    write(6, *) 'c = '//str3
    a = cmplx(d, 0.0)
    b = cmplx(e, 0.0)
    c = cmplx(f, 0.0)

    x1 = (-b + sqrt(b**2 - 4*a*c)) / (2*a)
    x2 = (-b - sqrt(b**2 - 4*a*c)) / (2*a)
    write(6, *) 'The solution to this equation is'
    write(6, *) 'x1 = (', real(x1), ')   +   (', aimag(x1), ')i'
    write(6, *) 'x2 = (', real(x2), ')   +   (', aimag(x2), ')i'
END PROGRAM quadratic

### Result
Output information are presented below in which I calculate the roots of
\begin{equation}
5x^2 + 6x + 3 = 0.
\tag{4}
\end{equation}
And it gives the solution of
\begin{equation}
x = \frac{-6 \pm \sqrt{-24}}{10} = -0.6 \pm 2\sqrt{6}i
\tag{5}
\end{equation}

## Roots for cubic equation
In this problem we try to solve a cubic equation of the form of
\begin{equation}
ax^3 + bx^2 + cx + d = 0.
\tag{6}
\end{equation}
First we define following relation
\begin{equation}
Q = \frac{3(\frac{c}{a})-(\frac{b}{a})^2}{9}
\tag{7}
\end{equation}
\begin{equation}
R = \frac{9(\frac{b}{a})(\frac{c}{a})-27(\frac{d}{a})-2(\frac{b}{a})^2}{54}
\tag{8}
\end{equation}
\begin{equation}
S = \sqrt[3]{R + \sqrt{Q^3 + R^2}}
\tag{9}
\end{equation}
\begin{equation}
T = \sqrt[3]{R - \sqrt{Q^3 + R^2}}
\tag{10}
\end{equation}
Then the solution to x is
\begin{align*}
x_1 &= S + T - \frac{1}{3}(\frac{b}{a})\\
x_2 &= -\frac{1}{2}(S + T) - \frac{1}{3}(\frac{b}{a}) + \frac{1}{2}i\sqrt{3}(S-T)\\
x_3 &= -\frac{1}{2}(S + T) - \frac{1}{3}(\frac{b}{a}) - \frac{1}{2}i\sqrt{3}(S-T)\\
\tag{11}
\end{align*}
Fisrtly, you enter four coefficient (a, b, c, c) which are floating point number. Then I calculate these parameter(7) to (10). Lastly, I calculate the roots of this cubic equation by using above equation.
To present the result I seperate real part and imaginary part by using another intrinsic function *real* and *aimag*.
I futher check the validity using below relation.
\begin{align*}
&x_1 + x_2 + x_3 = -b/a \\
&x_1x_2+x_2x_3+x_3x_1=c/a\\
&x_1x_2x_3=-d/a
\tag{12}
\end{align*}

### Code

!========================================================
!   Purpose: Finding roots for cubic equation
!
!   Methods: Cubic equation formula
!
!   Input: a, b, c, d(coef. of the equation)
!
!       Date        Programer       Description of change
!       ====        =========       =====================
!      9/27/20        Morris        Original Code
!========================================================
PROGRAM cubic
    IMPLICIT NONE
    COMPLEX, PARAMETER :: i=(0., 1.)
    REAL :: a, b, c, d
    REAL :: a1, a2, a3
    REAL :: q, r, s, t
    COMPLEX :: x1, x2, x3
    CHARACTER(len=15) :: str1, str2, str3, str4

    write(6, *) 'Enter the coef of a cubic equation (ax^3+bx^2+cx+d=0)'
    read(5, *) a, b, c, d
    write(6, *) 'The coef of the equation is:'
    write(str1, '(F6.2)') a
    write(str2, '(F6.2)') b
    write(str3, '(F6.2)') c
    write(str4, '(F6.2)') d
    write(6, *) 'a = '//str1
    write(6, *) 'b = '//str2
    write(6, *) 'c = '//str3
    write(6, *) 'd = '//str4
    a1 = b / a
    a2 = c / a
    a3 = d / a

    q = (3. * a2 - a1**2) / 9.
    r = (9. * a1 * a2 - 27 * a3 - 2 * a1**3) / 54
    s = (r + (q**3 + r**2)**0.5)**(1./3.)
    t = (r - (q**3 + r**2)**0.5)**(1./3.)

    x1 = s + t - a1 / 3.
    x2 = -0.5 * (s + t) - a1 / 3. + 0.5 * i * sqrt(3.) * &
         (s - t)
    x3 = -0.5 * (s + t) - a1 / 3. - 0.5 * i * sqrt(3.) * &
         (s - t)

    write(6, *) '================================================='
    write(6, *) 'The solution to this equation is'
    write(6, *) 'x1 = (', real(x1), ')   +   (', aimag(x1), ')i'
    write(6, *) 'x2 = (', real(x2), ')   +   (', aimag(x2), ')i'
    write(6, *) 'x3 = (', real(x3), ')   +   (', aimag(x3), ')i'
    write(6, *) '================================================='

    write(6, *) '================================================='
    write(6, *) 'below check the if it is a right solution(zero means right)'
    write(6, *) 'x1 + x2 + x3 + b/a : '
    write(6, '(F6.3)') abs(x1+x2+x3+a1)
    write(6, *) 'x1x2 + x2x3 + x3x1 - c/a : ' 
    write(6, '(F6.3)') abs(x1*x2 + x2*x3 + x3*x1 - a2)
    write(6, *) 'x1x2x3 + d/a : ' 
    write(6, '(F6.3)') abs(x1*x2*x3 + a3)
    write(6, *) '================================================='
END PROGRAM cubic

### Result
Output information are presented below in which I calculate the roots of
\begin{equation}
5.5x^3 + x^2 - 2x + 3 = 0.\tag{13}
\end{equation}