HORNER'S METHOD OF APPROXIMATION.
626. Let it be required to solve the equation
x^3-3x^2-2x+5 = 0 (1).
By Sturm's Theorem there are 3 real roots and one of them lies between 1 and 2 ; we will find its value to four places of decimals, which will sufficiently illustrate the method.
Diminishing the roots of the equation by 1 [Arts. 583, 584] , we have
1 -3 -2 +5 |1 1 -2 -4 -2 -4 1 1 -1 -1 -5 1 0
The transformed equation is
y^4-5y+1 = 0 (2).
Equation (1) has a root between 1 and 2. The roots of equation (2) are each less by 1 than those of equation (1) ; hence equation (2) has a root between 0 and 1 . This root being less than unity the higher powers of y are each less than y. Neglecting them, we obtain an approximate value
of y from - 5y +1 =0, or y = .2.
Diminishing the roots of (2), the first transformed equation, by .2, we have
1 ±0 .2 -5 .04 + 1 .992 .2 .2 -4.96 .08 .008 .4 .2 -4.88 .6 .2
