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 ? Ca,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 lt1
  exists with
• g(x) lt 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 ? gt 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 ? C1a,bhas a simple zero at p in (a,b) if and
  only if f(p) 0 but f(p) ? 0.
• The function f ? Cma,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
Aitkens ?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 pn1 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)(pn1 p) gt 0. Then
• the sequence converges to p
• faster than in the sense that

10
Steffensens Method

• Application of Aitkens ?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
Steffensons Theorem (Thm2.14)

• Suppose tht x g(x) has the solution p with
  g(p) ? 1. If there exists a ? gt 0 such that g ?
  C3p-?,p?, then Steffensons method gives
  quadratic convergence for an p0 ? p-?,p?.

12
Fundamental Theorem of Algebra (Thm2.15)

• 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, i0?n, (Q-the set of rational
  numbers)
• then if P(x) has rational roots of the form p/q
  (in lowest terms),
• a0 kp 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
Horners Method (Synthetic Division)

• Example
• 21 0 9 4 12
• _ 2(1) 2(2) 2(-5) 2(-6)
• 1 2 -5 -6 0 R