Page 81 - Standard Handbook Of Petroleum & Natural Gas Engineering

70 Mathematics

Because of round off errors, the Regula Falsa method should include a check
for excessive iterations. A modified Regula FaLsa method is based on the use of a
relaxationfactor, Le., a number used to alter the results of one iteration before
inserting into the next. (See the section on relaxation methods and "Solution
of Sets of Simultaneous Linear Equations.")
By iteration, the general expression for the Newton-Raphson method may be
written (if f' can be evaluated and is continuous near the root):

where (n) denotes values obtained on the nth iteration and (n + 1) those obtained
on the (n + l)lh iteration. The iterations are terminated when the magnitude of
16("+') - 6(")1 < E, being the predetermined error factor and E s 0.1 of the
permissible error in the root.
The modified Newton method [ 121 offers one way of dealing with multiple roots.
If a new function is defined

since u(x) = 0 when f(x) = 0 and if f(x) has a multiple root at x = c of multiplicity
r, then Newton's method can be applied and

where

u'(x) = 1 - f (x)f"( x)
[f'(x)I2

If multiple or closely spaced roots exist, both f and f' may vanish near a root
and therefore methods that depend on tangents will not work. Deflation of the
polynomial P(x) produces, by factoring,

where Q(x) is a polynomial of one degree lower than P(x) and the roots of Q
are the remaining roots of P after factorization by synthetic division. Deflation
avoids convergence to the same root more than one time. Although the cal-
culated roots become progressively more inaccurate, errors may be minimized
by using the results as initial guesses to iterate for the actual roots in P.
Methods such as Graeffe's root-squaring method, Muller's method, Laguerre's
method, and others exist for finding all roots of polynomials with real coeffi-
cients. [12 and others]