6. Roots of Equations Flashcards
Roots of Equations
Outline
-for a non-linear function f(x,ε) we seek roots f(x,ε)=0
Regular vs Singular Algebraic Equations
Definition
- an algebraic equation is regular if the highest power of x does not vanish when ε=0
- otherwise the equation is singular
Regular Algebraic Equations
Simple Roots - Description
-for a complicated algebraic equations that we otherwise can’t solve the idea is to rewrite it as a perturbation of an equation that we can solve easily and then develop a perturbation expansion for the latter system
Regular Algebraic Equations
Simple Roots - Steps
-consider placing an ε in front of each of the terms of the equation in turn in order to create a perturbation problem
-want to choose ε placement so that the new leading order solution is close to the root we are searching for
-when ε=0, find x=xo then seek an expansion of the form:
x = xo + εx1 + ε²x2 + …
-sub this in and consider balancing terms at O(1), O(ε) etc.
Regular Algebraic Equations
Repeated Roots - Steps
-seek solutions of the form:
x = xo + εx1 + ε²x2 + …
-this may give a root that works, however the expansion may also break down, remember we assume x0>x1>x2>…
-if this happens then we can seek more roots of the form:
x = xo + δ(ε)
-using xo from the previous expansion
-sub this new expansion in to find δ
Regular Algebraic Equations
Repeated Roots - General Case
-suppose that xo is a root of f(x,ε)=0 when ε=0
-then f(xo,0) = 0
=>
x = xo - ε fε(xo,0) / fx(xo,0)
-provided that fε(xo,0)≠0 & fx(xo,0)≠0
What is the generic scaling for a double root?
-square root scaling:
x = xo + ε^1/2 x1/2 + ε x1 + ε^3/2 x3/2 + ε² x2 + …
Regular Algebraic Equations
Repeated Roots - General Case
-suppose that f(x,ε)=0 has a double root at x=xo when ε=0, then:
f(xo,0) = 0
& fx(xo,0)
-where fx indicates the partial derivative of f with respect to x
-can derive square root scaling:
x + xo ± √[-2ε fε(xo,0) / fxx(xo,0)]
-provided fε(xo,0)≠0 & fxx(xo,0)≠0
Singular Algebraic Equations
Roots
-a singular algebraic equation has fewer roots when ε=0 then when ε≠0
Singular Algebraic Equations
Regular Expansion
-suppose that:
x = xo + εx1 + ε²x2 + …
-consider O(1), O(ε), etc. terms, this will give expansions for some roots
-but there may not be enough roots for the order of the equation
-other techniques will be required to find the rest
Singular Algebraic Equations
Dominant Balance
-searching for roots where x becomes large as ε->0
-let:
x = δ(ε) X where X=Os(1) as ε->0
-sub into the equation and seek δ(ε) that gives a permissable balance
-consider the new equation in X, if it is non-singular can now take a regular expansion in X:
X = Xo + ε X1 + ε² X2 + …
-if it is another singular equation we can take only the real roots with expansion:
X = Xo + ε^1/2 X1/2 + ε X1 + …
-finally convert back to x from X
Transcendental Equations
Definition
- as opposed to cases where f(x,ε) is polynomial in x, transcendental equations involve transcendental functions such as trigonometric, exponential or logarithmic functions
- there are no general rules about how roots may be found, equations must be dealt with on a case by case basis
Transcendental Equations
Regular Expansion
-find root of the unperturbed (ε=0) problem x=xo
-suppose that this is perturbed according to:
x = xo + x1ε + x2ε² + …
-sub in to the perturbed problem and consider O(1), O(ε) etc. terms
Transcendental Equations
Dominant Balance
-suppose that: x = δ0(ε) + δ1(ε) + δ2(ε) + ... -where 1 >> δ0 >> δ1 >> ... as ε->0 -sub in -balance the largest terms on each side -then the second largest -etc.
Transcendental Equations
Exponentials
-make the exponential the dominant term by taking natural log of the equation
-then make expansion:
x = δ0(ε) + δ1(ε) + δ2(ε) + …
-where δ0»_space; δ1»_space; … as ε->0 AND δ0»1 as ε->0
-expand out the log argument so that a Taylor expansion can be applied
-balance the largest terms
-then the second largest terms
-etc.
