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Mathematics for computer algebra - Mignotte M.

Mignotte M. Mathematics for computer algebra - New York, 1992. - 92 p.
ISBN 0-387-97675-2
Download (direct link): mathforcomputer1992.djvu
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Proof

Prove the corollary as เ๋ exercise. []

Corollary 5.2. -— Let f be a polynomial with real coefficients, and let a and b be two real numbers, with a < b. Then there exists at least one real number c, with a < ๑ <b, that satisfies the relation

f(b) — f (a) — (b — a)f'(c).

Proof

Apply the theorem to the polynomial

u(X) = (6 - a) {/(X) - /(a)} - (X - a) {f(b) - /(a)}.

The result is immediate . []

Corollary 3. — Let f be a polynomial with real coefficients, and let a and b be two real numbers, with a < b. Suppose that the derivative of f takes a positive value (and respectively a negative value) at each point x of the interval]a, b[. Then the function f(x) is strictly increasing (or respectively strictly decreasing) on the interval ]o, b[.

Proof

The result is an immediate consequence of Corollary 2. []

3. Estimates of real roots

Most of the results of this section are variants of the estimates of complex roots that were proved in Section 4.2; when it is the case, the new demonstrations will be quite short.

I. The rule of Newton

Proposition 5.2. — Let f be a polynomial of degree n with real coefficients. Let L be a real number such that we have f^(L) > 0 for ณ = 0, I, ..., n. Then any real root x of the polynomial f satisfies x < L.
3. ESTIMATES OF REAL ROOTS

191

Proof

Let x = L+y be a real number, where ๓ is positive. By Taylor’s formula, we have

/<*>-??>น>.

i=0

hence f(x) is strictly positive for X > L. Q

2. The rule of Lagrange and MacLaurin

Proposition 5.3. — Let f be a polynomial with real coefficients, f(X) = Xn + U1Xn-1 + ... + a*,

with ai > 0 for ณ = I, ..., m — I, and let A = max {—am,..., — an, 0}. Then any real root x of the polynomial f satisfies

x <1 +A1^m.

Proof

For x > I + A1Zm, we have the relation

fix) > xn - ห(1 + x + • • • + xn~m) > 0.

Hence, we have the result. []

3. A special case of the rule of Descartes

Proposition 5.4. — Let f be a polynomial with real coefficients like f(X) =Xn + O1Xn-1 + • • • + amXn~m - am+iXn_m_1-------------------an,

with ai > 0 for ใ = I, ..., n. If ๑ is a nonnegative real number for which f(c) is also nonnegative, then for any real number x > ๑ we have the inequality f(x) > 0. Thus, in such a case, any real root x of the polynomial f satisfies x < c.
JtjTXfOj

This is another formulation of Lemma 4.1. []

4ฆ The rule of Cauchy

Proposition 5.5. — Let am, am>, ... with m > m' > ••• be the

Let M be the value of the maximum considered in the statement of the proposition. For x > M, we have

f(x) >xn- (Om xn m + am< xn m' ห-------------------) > O.

5. An example of an estimation of the real roots of a polynomial Let us consider the polynomial

f{x) =Xe- 12x4 - 2x3 + 37x2 + IOx - 10.

The successive derivatives of / are

f'{x) = 2(3x5 - 24x3 - 3x2 + 37x + 5), f"{x) = 2(15x4 - 72x2 - 6x + 37), /(3Hx) = 12(10x3 — 24x — I),

/(4)(*) = 72(5x2-4),

/(5) (x) = 720x, and /(6) (x) = 720.

strictly negative coefficients o/ a polynomial / with real coefficients,

and let ๊ be number o/ these negative coefficients. Then any real root x o/ the polynomial / satisfies

Let us apply the various rules that we have seen to polynomial /.
IVUic Ul new IUil . TVC UiUOU nuu ivui muuxuv* W uซv** -J —

positive for ณ = I, 2, ..., 6. It is easy to verify that we have /W(c) > O for * = 2, ..., 6 and ๑ > %/7. The inequality f(\/8) > O shows that we can take the value L = \/8 = 2.828... as an upper bound of the roots.

Rule of Lagrange and MacLaurin : We find at once the upper bound I + \/ฏ2 = 4.464-----

Rule of Cauchy : With the notations of Proposition 5.5, we have here ๊ = 3 and we find the bound

max {(3 x 12)1/2, (3 x 2)1/3, (3 x 10)1/6} = 6.

Rule of Laguerre (see Exercise 5.12) : It gives the bound ๋/ฏว = 3.605...

4. The number of zeros of a polynomial in a real interval

I. The rule of Sturm

Before stating the theorem of Sturm, we need two new definitions.

Let / be a polynomial and let a and b be two real numbers, where a < b. We say that a sequence of polynomials /o = /, /i, •••>/ซ is a sequence of Sturm for / on the interval [a,b], when the following conditions are satisfied :

(i) f(a) • f(b) ิ 0.

(ii) The polynomial function fs has no zero on the closed real interval [a, 6].

(iii) If ๑ is a real number such that a < ๑ < b and fj (c) = 0 for an index j, with 0 < j < s, then we have fj-i(c)fj+1(c) < 0.

(iv) If ๑ is a real number such that a < ๑ < b and /(c) = 0, then the product polynomial /(x) ฆ fi(x) has the same sign as x — ๑ in the neighborhood of the point c.

To such a sequence and to a point x of the interval ] a, b [, we associate the number of variations of sign of this sequence at the point x; this number is written V (/o, ฆ • •, fs ; x), or in a simpler way V(x), which means that V (x) is defined by the formula

V(x) = Card{(i,j);/i(x)/j(x) < 0 and ฤ(๕) = 0 if i<k<j}, where the indices ณ and j satisfy 0 < ณ < j < n.
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