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 then converges to p of order  with asymptotic error constant. 1. If  = 1, linear convergence. 2. If  = 2, quadratic convergence.

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 p 0 in [a,b], the sequence p n = g(p n-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 p 0  [p - , p + ], the sequence defined by p n = g(p n-1 ), when n  1, converges at least quadratically to p. Moreover, for sufficiently large values of n,

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  C 1 [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  C m [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.

Forward Difference (Def 2.12) For a given sequence, the forward difference p n, is defined by: p n = p n+1 – p n, for n  0. So: can be written:

“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 (p n – p)(p n+1 – p) > 0. Then the sequence converges to p faster than in the sense that

Steffensen’s Method Application of Aitken’s  2 Method To find the solution of p = g(p) with initial approximation p o. Find p 1 = g(p o ) & p 2 = g(p 1 ) Then form interation: Use successive values for p 0, p 1, p 2,.

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  C 3 [p- ,p+], then Steffenson’s method gives quadratic convergence for an p 0  [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 x 1, x 2,... x k, possibly complex, and unique positive integers m 1, m 2..., m k 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).

Rational Root Theorem: If P(x)= and all a i 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), a 0 = k·p and a n = c ·q with k and c elements of  (-the set of integers)

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