Simultaneous determination of quadratic factors by optimization methods

Mathematics and Computers @North-Holland Publishing

SIMULTANEOUS

in Simulation Company

XIX (1977)

DETERMINATION

57-59

OF QUADRATIC

FACTORS BY OPTIMIZATION

METHODS

I. TANG Department

of Applied Mathematics,

The N.S. W. Institute of Technology,

P.O. Box 123, Broadway, N.S. W. 2007, Austrak

A method developed for the analog computer using minimization to find a quadratic tended to include a deflation process, so that all the quadratic factors can be determined

1. Introduction

p(x) =xn +a,_1x ‘-l

F =F(U,V?,

of the form:

+ . . . + QlX

+ Qg

= u, + U,_~Un_~ + .. . + LzlUl)

equations

For (1) we assume the coefficients ai are so scaled that max lai 1< 1. For n > 2, P(x) will have exactly [n/2] quadratic factors of the form (2)

for which, depending on the coefficients ai, Vmay be positive or negative. To find a quadratic factor, define the recursion relationship uj = uu.I-1 - vuj-2, and u1= 1.

and G = G(U, V), =u,+1

+ a,_lun_l

j=2,3

) ...) n + 1

(3)

+ . . . +a129 +a+1,

then it can be shown that the solution

(4)

of the set

F(U, u) = 0, GW 0 = 0,

(9

gives the values U, V for the quadratic factor Q of (2).

3. The optimization

Q(x) = x2 -- Ux + I’,

(4)

(1)

in which all the coefficients are real. It is well known that P(x) may be factorized into [n/2] quadratic factors, each having real coefficients. There are many methods of finding such quadratic factors on a digital computer. An earlier article by this author [l] describes a method of finding individual quadratic factors using an analog/hybrid computer. This article extends the method by including a deflation process to find all the quadratic factors simultaneously.

uo=o,

is ex-

If we let

Consider a polynomial

2. The remainder

factor of a polynomial simultaneously.

process

Many of the numerical schemes of finding quadratic factors are developed basing on (5). To devise a method for use on an analog computer, we attempt a solution of (5) by minimizing a non-negative function H: H(U,v=F2

+G2.

(6)

We start the process by choosing arbitrarily a pair of values (I/(‘), V(O)) and proceed to find successively in a systematic manner values (I+‘! V”$ (lb*! Vc2Q, ...~e~~&c~)) such that H(O)> H(l)> H(*)> . .. > Hem) = H( u<‘); V(‘)). For arbitrarily small drn), (UC”), Vcm)) may be taken as an approximate solution of (5).

58

I. Tang/Simultaneous determination of quadratic factors

This process may be realized by any of the wellknown optimization schemes. Here we consider H to vary with time so that dH

aH c+aH

dt

aU dt

-_=-

dV aV

(7)

dt’

Q(x)= x2 _ uWx + v(m),

(12)

is a quadratic factor of (1). In the event that U, V each converges but H does not converge to zero, the integration of (9) must be repeated with different initial values (U(O), V(O)).

If we require that 4. Deflating the polynomial Assume that at each t, U, and V, are coefficients of a quadratic factor such that (1) may be factorized thus:

and

gc

_k,

g,

kl,k2>0

P(x) = (x2 - U,x + V,)(x”-2

then dH/dt < 0, and H will be strictly decreasing as t increases. Setting k, = k2 = i in (8) and differentiating (6), we get

t ... + b,x tbo),

b n-3 =a,_1

dt=

bn-4 = an-2 +b

and

b n-5 =an-3

. (9)

where aF ,,=g,

+a,-1

g=

-En-1

gn-l

+...

- ~n-~tn_2

+ a3t3

-

+ a&,

. . . - qt3

- a3.52,

and ac

aF

(10)

av=-Z with

aqau =-aui+llav, = Ulj_1 - V&$2,

li =

Uj_1

+

i= 2, .. . . n

(11)

subject to go = .$I = 0. Suppose (9) were now integrated subject to U = 17~‘) and V = V(O) at t = 0. If as t increases, H + 0 and (U, V) converge toward their terminal values, then for small values of Hem), (U”), Vcm)) may be taken as the solution of (5). In this case

(13)

where

dU

dV _=dt

+ bn_3xn-3

.

.

+

U,, n-3”l

-

vl,

+b n_4 U1 -- b,_,

.

V, ,

.

bo=a2tblUl-b2V1.

The (n-2)th order polynomial with coefficients bi on the right of (13) can be factorized in the same manner to yield a quadratic factor with coefficients U2 and VI, each now a function of U, and VI, as well as a polynomial or order (n - 4) with coefficients Cn_5, c n-4, . . . . Co, each a function of U, and V2. The (n-4)th order polynomial can further be deflated until, depending on in (1) whether n is even or odd, either a quadratic or a linear factor remains. After complete deflation, we will have obtained a set of differential equations in the form of (9) involving dU,/dt, dV,/dt, dU2/dt, . .. . dUNldt and d I/,/dt where N = [(n - 1)/2]. Applying to these 2N equations the minimization procedure discussed, we will find for each variable a terminal value which is the value of the appropriate coefficient in the N quadratic factors, and the factorization P(x) = (x2 - u,x + V,)(x2 - u,x + V,) . . . (x2 - u,x

+ V&(x>

(14)

where P(X) =x2 - u,,,x or

+ v,,,,

n even,

I. TangfSimultaneous determination of quadratic factors

59

P(x) =X6 t 0.9X5 + 0.7x4

1.4 -

UZ

+ 0.4x3 + 0.2X2 + 0.1x f 1, Vl

(15)

for which quadratic factors are to be found. We proceed by finding according to (4) F1 = F(U, >V, ) = ti,j + 0.9u5 •t-0.7#, + 0.4u3 + 0.2u, + O.lu, and G, = G(U,, V,) = u7 t 0.9u6 t 0.7u5 t 0.4u4 t 0.2u3 + 0.1u2 t ul. (16)

\

0

-0.4

t

t Fig. 1

=x+cq

n odd,

is now complete if we note further that the quantities u n,2, Vni2 or (Yare given by the coefficients calculated from the final deflation.

5. An example As an example, consider

Then dU,/dt, d V,/dt are obtained according to (9). Next, F2, G,, are found using the coefficients bi given in (13). Eq. (9) is applied once again to obtain dU2/dt, d V2/dt. These four differential equations must now be solved subject to starting values. For U,(u) = Vi@ = @u) = Vi’) = 1, a simulated run was made on a digital computer. The results are shown in fig. 1, and the following coefficients are obtained U, = -0.280, V, = 1.112 and U, = 1.308, V2 = 0.743.

Reference 1l] I. Tang, Finding quadratic factors by an analog minimisation process, Simulation 26, No. 4 (April 1976), 128129.