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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

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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 .

    • If  =1, linear convergence.

    • 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 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,


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 .


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)

  • 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

  • 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)

  • 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).


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:

  • 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


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