Applied Numerical Analysis

1. Applied Numerical Analysis Chapter 2 Notes (continued)

2. 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.

3. 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].

4. 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,

5. 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.

6. 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.

7. 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 .

8. 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:

9. “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

10. 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,.

11. 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+].

12. 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

13. 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).

14. 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)

15. 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).

16. 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