Applied Numerical Analysis

Chapter 2 Notes (continued)

Order of Convergence of a SequenceAsymptotic Error Constant (Defs)

Suppose is a sequence that converges to p, with pn p for all n. If positive constants and exist with

then converges to p of order with asymptotic error constant .

1. If = 1, linear convergence.2. If = 2, quadratic convergence.

1nnp

1nnp

||

|| 1limpp

pp

n

n

n

Test for Linear Convergence (Thm 2.7)

Let g C[a,b] be such that g(x) [a,b], for all x [a,b]. Suppose in addition that g’ is continuous on (a,b) and a positive constant k <1 exists with

|g’(x)| < k, for all x (a,b). If g’(p) 0, then for any number p0 in

[a,b], the sequence pn = g(pn-1), for n 1, converges only linearly to the unique fixed point p in [a,b].

Test for Quadratic Convergence (Thm 2.8)

Let p be a solution for the equation x = g(x). Suppose that g’(p) = 0 and g” is contin-uous and strictly bounded by M on an open interval I containing p. Then there exists a > 0 such that, for p0 [p - , p + ], the sequence defined by pn = g(pn-1), when

n 1, converges at least quadratically to p. Moreover, for sufficiently large values of n,

.||2

|| 21 pp

Mpp nn

Solution of multiplicity zero. (Def 2.9) A solution p of f(x) = 0 is a zero of

multiplicity m of f if for x p, we can write:

f(x) = (x – p)m q(x), where q(x) 0.

Functions with zeros of multiplicity m (Thms2.10,11)

f C1[a,b]has a simple zero at p in (a,b) if and only if f(p) = 0 but f’(p) 0.

The function f Cm[a,b] has a zero of multiplicity m at p in (a,b) if and only if

0 = f(p) = f’(p) =f”(p) = ... = f(m-1)(p) but f(m)(p) 0.

Aitken’s 2 Method Assumption

Suppose is a linearly covergent sequence with limit p. If we can assume for n “suf-

ficiently large” then by algebra:

and the sequence converges “more rapidly” than does .

1nnp

pp

pp

pp

pp

n

n

n

n

1

21

nnn

nnnn ppp

pppp

12

21

^

2

)(

1

^

nnp

1nnp

Forward Difference (Def 2.12)

For a given sequence , the forward difference pn, is defined by:

pn = pn+1 – pn, for n 0. So:

can be written:

1nnp

nnn

nnnn ppp

pppp

12

21

^

2

)(

n

nnn

p

ppp

2

2^ )(

“Converges more rapidly” (Thm 2.13)

Suppose that is a sequence that converges linearly to the limit p and that for all sufficiently large values of n we have (pn– p)(pn+1– p) > 0. Then

the sequence converges to p

faster than in the sense that

1nnp

1nnp

1

^

nnp

.0lim

^

pp

pp

n

n

n

Steffensen’s Method Application of Aitken’s 2 Method To find the solution of p = g(p) with

initial approximation po.

Find p1 = g(po) & p2 = g(p1) Then form interation:

Use successive values for p0, p1, p2,.

012

201

01

^

2

)(

ppp

pppp

^p

Steffenson’s Theorem (Thm2.14) Suppose tht x = g(x) has the solution p

with g’(p) 1. If there exists a > 0 such that g C3[p-,p+], then Steffenson’s method gives quadratic convergence for an p0 [p-,p+].

Fundamental Theorem of Algebra (Thm:2.15)

If P(x) is a polynomial of degree n 1 with real or complex coefficients, then P(x) = 0 has a least one (possibly complex) root.

If P(x) is a polynomial of degree n 1 with real or complex coefficients, then there exist unique constants x1, x2, ... xk, possibly complex, and unique positive integers m1, m2..., mk such that

Remainder and Factor Theorems:

Remainder Theorem: If P(x) is divided by x-a, then the

remainder upon dividing is P(a).

Factor Theorem: If R(a) = 0, the x-a is a factor of P(x).

)())(()()(

)()(

xRaxxQxPorax

xRxQ

ax

xP

)(0)()())(()( aRaPoraRaaaQaP

00))(()()())(()( aaaQaPaRaaaQaP

Rational Root Theorem: If P(x)= and all ai Q, i=0n, (Q-the set of

rational numbers) then if P(x) has rational roots of the

form p/q (in lowest terms), a0 = k·p and an = c ·q with k and c

elements of (-the set of integers)

nn

nn

n

k

kk xaxaxaaxa

1

11

100

Descartes’ Rule of Signs:

The number of positive real roots of P(x) = 0, where P(x) is a polynomial with real coefficients, is eual to the number of variations in sign occurring in P(x), or else is less than this number by a positive even integer.

Then number of negative real roots can by found by using the same rule on P(-x).

Horner’s Method (Synthetic Division)

Example:

2|1 0 –9 4 12 |_ +2(1) +2(2) +2(-5) +2(-6) 1 2 -5 -6 | 0 = R

6522

1249 2324

xxx

x

xxx