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%MODULE 1: Optimization
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%BUILD  TITLE PAGE 
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\begin_layout Title
Newton's Method
\end_layout

\begin_layout Subtitle
So far we have explored two different methods for finding the root of a
 function: one based on the Intermediate Value Theorem, and one based on
 Linear Approximation.
 Both of those approaches are quite inefficient.
 In this module, we study a method developed by Isaac Newton that is much
 more efficient.
 
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{
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Sample problem
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}
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\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "page:modf-problem"

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Approximate a positive zero of 
\begin_inset Formula $\ec x^{3}+0.4x^{2}-0.51x-0.16$
\end_inset

 to the ten thousandths place.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
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%1
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\end_layout

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{
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SAGE worksheets
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}
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\begin_layout Standard
For this module you will need the SAGE worksheet 
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{
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\end_inset


\begin_inset ERT
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\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 Newton's Method
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
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{
\end_layout

\end_inset

Newton's Method: the idea
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Newton's Method is based on the fact that the line tangent to a function
 follows the same general direction as the function.
 If you start 
\begin_inset Quotes eld
\end_inset

close enough
\begin_inset Quotes erd
\end_inset

 to a root of the function, the root of the tangent line will be close to
 the root of the function.
\end_layout

\begin_layout Standard
\noindent
The illustration below gives an example.
 The root of 
\begin_inset Formula $\ec f$
\end_inset

 (the black curve) is approximately 
\begin_inset Formula $\ec0.70$
\end_inset

.
 A line tangent to the graph of 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=1$
\end_inset

 has a root at approximately 
\begin_inset Formula $\ec0.78$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/motivating_ex.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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{
\end_layout

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Newton's Method: the idea
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
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status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
By itself, 
\begin_inset Formula $\ec0.78$
\end_inset

 is not that good an approximation.
 Estimating from the graph, it looks as if 
\begin_inset Formula $\ec f(0.78)\approx0.15$
\end_inset

, which is not very close to zero.
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's build a new line tangent to the graph of 
\begin_inset Formula $\ec f$
\end_inset

\SpecialChar \ldots{}
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/motivating_ex.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let's build a new line tangent to the graph of 
\begin_inset Formula $\ec f$
\end_inset

\SpecialChar \ldots{}
 at 
\begin_inset Formula $\ec x=0.78$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/motivating_ex_2step.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
The root of the second line is approximately 
\begin_inset Formula $\ec0.70$
\end_inset

.
 The graph shows that it is closer to the zero of 
\begin_inset Formula $\ec f$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Newton's Method: the algorithm
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
This experiment suggests an algorithm to find zeroes of a function.
\end_layout

\begin_layout Standard
\noindent

\emph on
Given
\emph default
 
\begin_inset Formula $\ec f$
\end_inset

 and a point 
\begin_inset Formula $\ec a$
\end_inset

 that you think is 
\begin_inset Quotes eld
\end_inset

close
\begin_inset Quotes erd
\end_inset

 to a root of 
\begin_inset Formula $\ec f$
\end_inset

, you can approximate a root of 
\begin_inset Formula $\ec f$
\end_inset

 by repeating these steps:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Compute the line tangent to 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Find the root of this equation, and call it 
\begin_inset Formula $\ec b$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Is 
\begin_inset Formula $\ec b$
\end_inset

 
\begin_inset Quotes eld
\end_inset

close enough
\begin_inset Quotes erd
\end_inset

 to the actual root? 
\end_layout

\begin_deeper
\begin_layout Itemize
If not, repeat the process using 
\begin_inset Formula $\ec b$
\end_inset

 in place of 
\begin_inset Formula $\ec a$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If so, you are done! 
\end_layout

\end_deeper
\begin_layout Standard
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\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
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{
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\end_inset


\begin_inset Quotes eld
\end_inset

Close enough
\begin_inset Quotes erd
\end_inset

?!?
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
How do we decide that our approximation is 
\begin_inset Quotes eld
\end_inset

close enough
\begin_inset Quotes erd
\end_inset

? We will base our decision on whether the decimal digits of the current
 approximation (
\begin_inset Formula $\ec b$
\end_inset

) are the same as those of the previous approximation (
\begin_inset Formula $\ec a$
\end_inset

) up to the desired accuracy.
\end_layout

\begin_layout BeginFrame
\noindent
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{
\end_layout

\end_inset

Newton's Method: example
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Let's try an example.
 Fire up the Newton's Method SAGElet.
 It should start automatically, showing an approximation of the root of
 
\begin_inset Formula $\ec x^{3}+4x^{2}-.51x-.16$
\end_inset

 using an initial guess of 
\begin_inset Formula $\ec a=2$
\end_inset

, and two tangent lines.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/sagelet.png
	lyxscale 40
	scale 15

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
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{
\end_layout

\end_inset

Newton's Method: example
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The approximation given (
\begin_inset Formula $\ec0.9661$
\end_inset

) is clearly insufficient.
 Move the slider labeled 
\begin_inset Quotes eld
\end_inset

Number of approximations
\begin_inset Quotes erd
\end_inset

 to the midway point, and the applet will try three approximations.
 You should obtain 
\begin_inset Formula $\ec0.7629$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Increase the number of approximations repeatedly, one step each time, seeing
 if the first four digits will remain equal.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Four approximations gives 
\begin_inset Formula $\ec0.6937$
\end_inset

, and five gives 
\begin_inset Formula $\ec0.6853$
\end_inset

.
 We've managed to get the 
\emph on
first
\emph default
 digit to repeat, but not the rest.
 To obtain more accuracy, we will have to change the starting point.
 Move the slider all the way to the left, then change the starting point
 
\begin_inset Formula $\ec x_{0}$
\end_inset

 to our last approximation, 
\begin_inset Formula $\ec0.6853.$
\end_inset

 It will be hard to see what is going on, so change the window (
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 xmin
\family default
, 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 xmax
\family default
) the details grow clear.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Newton's Method: example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Putting 
\begin_inset Formula $\ec\mathtt{\ttcolor xmin}=0.685$
\end_inset

 and 
\begin_inset Formula $\ec\mathtt{\ttcolor xmax}=0.6854$
\end_inset

, we obtained the following diagram.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ex1_rezoom.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
The SAGElet reports that running Newton's method one step from 
\begin_inset Formula $\ec0.6853$
\end_inset

 gives the approximation 
\begin_inset Formula $\ec0.6852$
\end_inset

.
 We've now managed to keep 
\emph on
three
\emph default
 digits the same.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Newton's Method: exampe
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Move the slider right one spot, and you should obtain the new approximation
 
\begin_inset Formula $\ec0.6852$
\end_inset

.
 We've managed to keep four digits the same, so we will decide that this
 approximation is 
\emph on
close enough
\emph default
, or at least accurate to the ten thousandths place.
\end_layout

\begin_layout Standard
\noindent
(In fact, we've managed to keep six digits the same.)
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Newton's Method: Dangers
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
There are several cautions to observe with Newton's Method.
 
\end_layout

\begin_layout Itemize
If we encounter a point whose tangent line is horizontal, the method fails
 because the line has no root.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/bad_start.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
If the method leads us into a 
\begin_inset Quotes eld
\end_inset

bowl
\begin_inset Quotes erd
\end_inset

, there may be no way out and towards the zero.
 The algorithm may 
\begin_inset Quotes eld
\end_inset

bounce around
\begin_inset Quotes erd
\end_inset

 in bad estimates.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ex_bounce_around.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
A bad starting point can lead to a root different than the one desired,
 or even to no root at all.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/wrong_root.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Newton's Method: Simple example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We will conclude with a simple example that you should work by hand, on
 paper.
 In this example, we will estimate a root of 
\begin_inset Formula $\ec\cos x-x$
\end_inset

, correct to the hundredths place, starting at 
\begin_inset Formula $\ec x=0$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the first tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3-4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the first tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-x+1.
\]

\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
(Why? The slope is 
\begin_inset Formula $\ec f'(0)=-1$
\end_inset

.
 Substitute into 
\begin_inset Formula $\ec y-y_{0}=m(x-x_{0})$
\end_inset

, where 
\begin_inset Formula $\ec x_{0}=0$
\end_inset

 and 
\begin_inset Formula $\ec y_{0}=f(x_{0})=1$
\end_inset

.
 Solve for 
\begin_inset Formula $\ec y$
\end_inset

.)
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Now solve for the root of the tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Now solve for the root of the tangent line.
 You should get
\begin_inset Formula 
\[
\ec x=1.
\]

\end_inset

 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/easy_step1.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the second tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<7>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the second tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-(\sin1-1)(x-1)+(\cos1-1)\approx-1.841x+1.382.
\]

\end_inset

 Now solve for the root of the tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<8>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the second tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-(\sin1-1)(x-1)+(\cos1-1)\approx-1.841x+1.382.
\]

\end_inset

 Now solve for the root of the tangent line.
 You should get
\begin_inset Formula 
\[
\ec x\approx0.751.
\]

\end_inset

 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/easy_step2.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<9>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the third tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<10>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the third tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-(\sin0.75-1)(x-0.75)+(\cos0.75-0.75)\approx-1.682x+1.243.
\]

\end_inset

 Now solve for the root of the tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<11>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the third tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-(\sin0.75-1)(x-0.75)+(\cos0.75-0.75)\approx-1.682x+1.243.
\]

\end_inset

 Now solve for the root of the tangent line.
 You should get
\begin_inset Formula 
\[
\ec x\approx0.739.
\]

\end_inset

 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/easy_step3.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<12>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the fourth tangent line.
 (Don't fret; we're almost there!)
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<13>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the fourth tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-(\sin0.739-1)(x-0.739)+(\cos0.739-0.739)\approx-1.674x+1.237.
\]

\end_inset

 Now solve for the root of the tangent line.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<14->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Compute the equation of the fourth tangent line.
 You should get
\begin_inset Formula 
\[
\ec y=-(\sin0.739-1)(x-0.739)+(\cos0.739-0.739)\approx-1.674x+1.237.
\]

\end_inset

 Now solve for the root of the tangent line.
 You should get
\begin_inset Formula 
\[
\ec x\approx0.739.
\]

\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<14>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/easy_step4.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
The third and fourth step agree in the hundredths place; the root is approximate
ly 
\begin_inset Formula $\ec0.74.$
\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Newton's method provides a simple method of approximating the roots of functions.
 It is based on the observation that, in the neighborhood of the root of
 a function, a line tangent to the function leads one to the root.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Compute the line tangent to 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Find the root of this equation, and call it 
\begin_inset Formula $\ec b$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Is 
\begin_inset Formula $\ec b$
\end_inset

 
\begin_inset Quotes eld
\end_inset

close enough
\begin_inset Quotes erd
\end_inset

 to the actual root? 
\end_layout

\begin_deeper
\begin_layout Itemize
If not, repeat the process using 
\begin_inset Formula $\ec b$
\end_inset

 in place of 
\begin_inset Formula $\ec a$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If so, you are done! 
\end_layout

\end_deeper
\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
