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Applied Numerical Analysis . Chapter 2 Notes (continued). Order of Convergence of a Sequence Asymptotic Error Constant (Defs). Suppose is a sequence that converges to p, with p n  p for all n. If positive constants  and  exist with

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applied numerical analysis

Applied Numerical Analysis

Chapter 2 Notes (continued)

order of convergence of a sequence asymptotic error constant defs
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 .

    • If  =1, linear convergence.
    • If  = 2, quadratic convergence.
test for linear convergence thm 2 7
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
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,

solution of multiplicity zero def 2 9
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
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
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 .

forward difference def 2 12
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:

converges more rapidly thm 2 13
“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

steffensen s method
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,.
steffenson s theorem thm2 14
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
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 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).
rational root theorem
Rational Root Theorem:
  • If P(x)=

and all ai Q, i=0n, (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)

descartes rule of signs
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
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

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