N-th Order Linear Differential Equations 197- Use equations (1) and (2) to compute R = p(tn) and S = p'(tn) for
n=l,2,3, .... - Provided S-=/=-0, use Newton-Raphson's method to compute
- If for some n, R is as near zero as desired, then take a root of p(x) to be
tn. If n becomes large, but R is not as near zero as desired, start over
with a different value for t 1.
ICo=ents on Computer Software! The software accompanying this text
includes a program named POLYRTS. This program computes all the roots
of a polynomial with complex coefficients of degree less than or equal to 10. It
uses the Newton-Raphson method, Homer's method and deflation in conjunc-
tion with other techniques to approximate the roots of a polynomial. Com-
plete instructions for running this program appear in Appendix A. The next
two examples illustrate the typical output of POLYRTS. You should compare
the results you obtain using your software with these results. The following
two MAPLE statements also solve numerically the polynomial equation (3)
appearing in example 1 below.
polyeqn:= x /\ 7 + x /\ 6 + 12 * x /\ 5 - 28 * x /\ 4 - 733 * x /\ 3+1011 * x /\ 2 - 1784 * x - 38480 = 0:
factor(polyeqn,complex);EXAMPLE 1 Calculation of the Roots of a PolynomialFind the roots of the polynomial equation(3) x^7 + x^6 + 12x^5 - 28x^4 - 733 x^3 + 10llx^2 - l 784x - 38480 = 0SOLUTION
We entered the value 7 into POLYRTS for the degree of the polynomial
equation to be solved. Then we input the coefficients of the polynomial-
namely, a1 = 1, a5 = 1, as = 12, a4 = -28, a3 = -733, a2 = 1011,
a 1 = -1784, and a 0 = -38480. The roots of the polynomial as calculated
by POLYRTS appear in Figure 4.2. One zero is computed to be 2.000000 +
3.000000i. The actual zero of the polynomial corresponding to this com-
puted zero is 2 + 3i. So, in this case, the approximation is excellent. The
second and third zeros computed were -4.000000 - 1.153824 x 10-^11 i and
-4.000000 + 1.153809 x 10-^11 i. The actu al zeros of the polynomial corre-
sponding to these roots are -4 and -4. This is the worst approximation of
a zero for this polynomial, but it is not unacceptable. One thing you must