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ROOT FINDING

ROOT FINDING

For complex roots, the original polynomial:
(1) fn(x) = a0 + a1x + a2x2 + ... + anxn
becomes lower by two orders (n − 2) after deflating by the quadratic:
(2) fn − 2(x) = b2 + b3x + ... + bnxn − 2
with a remainder R = b1(x − d) + b0. The deflation scheme is similar to the real root scheme: 
b(n)=a(n)
b(n-1) = a(n-1) + r*b(n)
DO i=n-2, 0, -1
     b(i)=a(i)+r*b(i+1)+e*b(i+2)
ENDDO
While n >  = 3, the polynomial will be deflated. When the polynomial is a quadratic, then the two roots can be computed using the simple quadratic rule:
(3) x = (r±√(r2 + 4*e))/(2)
If the polynomial is first order, then x =  − e ⁄ r. The roots of the quadratic equation can be easily computed using 
SUBROUTINE quad(r,e,r1,i1,r2,i2)
REAL :: r, e, r1, i1, r2, i2, d
d = r2 + 4*e
IF (d > 0.0) THEN
         r1 = 0.5 * (r + SQRT(d))
         r2 = 0.5 * (r - SQRT(d))
         i1 = 0.0
         i2 = 0.0
ELSE
         r1 = r/2
         r2 = r1
         i1 = SQRT(ABS(d))*0.5
         i2 = -i1
ENDIF
END SUBROUTINE quad
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