3. Asymptotic Expansions Flashcards
Function in terms of General Truncated Sum
f(ε) = Sn(ε) + Rn(ε)
-where:
Sn = Σ an*ε^n
-sum from n=0 to n=N
-here Sn is a partial sum involving expansion coefficients a0,a1,…,aN and Rn(ε) is a remainder term
-note that this is an exact expression for f(ε) so if Sn starts to diverge then Rn must diverge (in the opposite direction) so that the equality holds
Convergent Series
Definition
-convergence asks about the expression: f(ε) = Sn(ε) + Rn(ε) -as N->∞ at fixed ε we say that Sn->f if: lim Rn(ε) = 0 at fixed ε -limit as N->∞ -then: f(ε) = Σ an*ε^n -sum from n=0 to n=∞
Factorial vs Power Growth
-factorial growth always beats power growth
Asymptotic Series
Definition
-we can also ask what happens if Rn decreases rapidly to 0 as ε->0
-in particular, we demand:
|Rn| «_space;ε^N as ε->0
<=>
lim |Rn|/ε^n -> 0 as ε->0
-limit as ε->0
-since the final term of Sn is aNε^N this means that the remainder will be much smaller than Sn as ε->0
-if this limit applies for all N, then:
f(ε) ~ Σ anε^n as ε->0
-sum from n=0 to n=∞
-we call this the asymptotic expansion of f (as ε-> 0)
Asymptotic vs. Convergent Series
-in practice the conditions for asymptotic and convergent series mean very different things for approximating f(ε)
Convergent Series
Tolerance
-given a tolerance δ, a convergent series can achieve this tolerance with sufficiently large N, for some range of ε, i.e. |R| < radius of convergence
Asymptotic Series
Tolerance
-for tolerance δ, an asymptotic series when truncated to any N can meet the tolerance but only for sufficiently small ε
Evaluating Asymptotic Series
- for divergent series, there comes a point where adding more terms to the sequence makes the estimate worse and not better
- and even if a series is convergent, simply summing many terms form the infinite series may note be efficient, especially if convergence is slow
Severe Truncation
-just take the first few terms of a series to make an estimate, e.g.
S1 = a0 + a1ε
S2 = a0 + a1ε + a2*ε²
-provided the series is asymptotic, Rn «_space;ε^n as ε->0, so |R1|<
Optimal Truncation
- here we seek the specific value of N=N* where the terms of the series are minimised in magnitude, i.e. minimise aN*ε^N
- for a divergent series, as a good working rule N* is found by taking N* to be the point where the terms of the series stop decreasing, i.e. when |aN*ε^N| is minimised
- for a convergent series there may well never be such a point since the terms typically keep decreasing
- this value, N*, depends on the chosen value of ε
Pade Approximant
Outline
-a superior technique for accelerating the convergence of a series given only its first few terms
-the idea is to rewrite the asymptotic expression:
f(ε) ~ Σ ak*ε^k
-sum from k=0 to k=∞
-as a rational function in which the numerator and denominator are polynomials of degree M and N respectively
Pade Approximant
Equation
-the [M/N] Pade approximant of f(ε):
PM/N = Σnkε^k / Σdkε^k
= [n0+n1ε+n2ε²+…+nMε^M] / [d0+d1ε+d2ε²+…+dNε^N]
-without loss of generality we can set d0=1 with M+N+1 remaining unknown constants
Pade Approximant
Determining Coefficients
- the M+N+1 unknown constants are determined from the first M+N+1 coefficients in the asymptotic expansion for f(ε)
- specifically, we require that the Pade approximant is the same as f(ε) ~ Σ ak*ε^k for the first M+N+1 terms (i.e. up to and including O(ε^M+N)
The Shanks Transform
Outline
- we consider a series f(ε) ~ Σ akε^k (sum k=0 to k=∞) along with its partial sums SN(ε)=Σ akε^k (sum k=0 to k=N)
- the Shanks transform is a technique for estimating f(ε) given a sequence of three consecutive partial sums
- can be used on a divergent series or to accelerate convergence of a convergent series
The Shanks Transform
Equation
-given a sequence of three consecutive partial sums SN-1, SN, SN+1
-the prediction for f(ε) is:
σ(SN) = [SN-1*SN+1 - SN²] / [SN-1 + SN+1 - 2SN]
The Shanks Transform
Reapplication
-if the Shanks transform is applied three times to produce σ(SN-1), σ(SN), σ(SN+1), the Shanks transform can then be reapplied on these values:
σ(σ(SN)) = [σ(SN-1)*σ(SN+1) - σ(SN)²] / [σ(SN-1) + σ(SN+1) - 2σ(SN)]
The Shanks Transform
Basis
-the basis for the Shanks transform is to suppose that the partial sums are in a geometric progression:
SN-1 = b + cr^(N-1)
SN = b + cr^N
SN-1 = b + r^(N-1)
-we can then eliminate c and r to find b:
b = [SN-1SN+1 - SN²] / [SN-1 + SN+1 - 2SN]
The Shanks Transform
ε
-we can also use the Shanks transform with general ε
e.g.
f(ε) = 1 + ε + ε² + …
=>
S0 = 1
S1 = 1+ε
S2 = 1+ε+ε²
-then sub these into the Shanks transform equation
General Asymptotic Expansion
Form
f(ε) = Σ an δn(ε) + RN(ε)
-sum from n=0 to n=N
-where {δn(ε)} is an asymptotic sequence of functions with:
δ0(ε) >> δ1(ε) >> δ2(ε) >> ...
-as ε->0
-more precisely:
lim δn+1(ε)/δn(ε) = 0
-limit as ε->0 for all n≥0General Asymptotic Expansion
Asymptotic Expansion
-if, for all N: lim RN/δN(ε) = 0 -limit as ε->0 -i.e. RN(ε) << δN(ε) -we say that f(ε) has an asymptotic expansion, write: f(ε) ~ Σ an δn(ε) as ε->0
o Notation
-if lim f(ε)/g(ε) = 0
-limit as ε->0
-then we say that:
f(ε) = o(g(ε)) as ε->0
-i.e. f(ε) «_space;g(ε)
-can actually be any limit, doesn’t have to be ε->0
Os Notation
-if lim f(ε)/g(ε) = C for C∈R{0}
-limit taken as ε->0
-if C is finite and non-zero we can write:
f(ε) = Os(g(ε)) as ε->0
-and we say that f(ε) is strictly of order g(ε) as ε->0
-can actually be any limit, doesn’t have to be ε->0
O Notation
-if lim f(ε)/g(ε) = C for C∈R then:
f(ε) = O(g(ε)) as ε->0
-f(ε) is no larger than g(ε)
-can actually be any limit, doesn’t have to be ε->0
Expansion Coefficients
Formula for aN
aN = lim [f(ε) - Σan*δn(ε)]/δN(ε)
- for N≥1
- limit as ε->0 and sum from n=0 to n=N-1
Expansion Coefficients
Generating δn(ε)
- the formula for aN can also be used to generate δn(ε)
- sub into the formula as normal leaving δN(ε) as a general expression
- test possible function for δN(ε) searching for a function which gives a finite, non-zero limit
Uniqueness of Asymptotic Expansions
- for a given function has different asymptotic expansions in terms of different asymptotic sequences {δn(ε)} and {δn^(ε)}
- for a given sequence {δn(ε)}, f(ε) has a unique asymptotic expansion
- for a given sequence {δn(ε)}, different functions may have the same asymptotic expansion
Exponentially/Transcendentally Small Terms and Uniqueness of Asymptotic Expansions
-any two functions which differ by an exponentially small term, e.g. e^(-1/ε²), have the same asymptotic expansion in δn(ε)=ε^n
Operations on Asymptotic Expansions
Addition / Subtraction / Multiplication / Division
-asymptotic expansions can be added, subtracted multiplied and divided in the obvious way
Operations on Asymptotic Expansions
Integration
-asymptotic expansions can also be integrated term by term, although you may need to be careful with terms in logε or inverse powers of ε as ε->0
Operations on Asymptotic Expansions
Differentiation
-asymptotic expansions cannot IN GENERAL be differentiated term by term
Uniform Asymptotic Expansion
Definition
-consider f(x;ε) where ε«1 with {δn(ε)}:
f(x;ε) ~ Σ an(x) δn(ε), ε->0
-we then need an+1(x)δn+1(ε) «_space;an(x)δn(ε) as ε->0 to have an asymptotic expansion
-if the above holds for all x for which f is defined then the expansion is a uniform asymptotic expansion
Non-Uniform Asymptotic Expansion
Definition
ε«1 with {δn(ε)}:
f(x;ε) ~ Σ an(x) δn(ε), ε->0
-we then need an+1(x)δn+1(ε) «_space;an(x)δn(ε) as ε->0 to have an asymptotic expansion
-if the inequality does not hold then the expansion is non-uniform
