(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 4.2'

                    Mathematica-Compatible Notebook

This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do
one of the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the
  application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info@wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
*******************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     37061,        956]*)
(*NotebookOutlinePosition[     37744,        979]*)
(*  CellTagsIndexPosition[     37700,        975]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[TextData[{
  "Here are some sample commands that should help you in implementing \
Newton's Method.  If you are not very familiar with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", you should start with the \"Intro\" notebook, which covers the basic \
commands."
}], "Text",
  FontSize->24],

Cell["\<\
I'm going to implement the bisection method on the function x - cos \
x = 0, so you can see the basics of defining functions and performing a \
simple iteration.\
\>", "Text",
  FontSize->24],

Cell["\<\
First I define f(x).  Note the particular notation f[x_] used to \
define a function which is waiting to receive an input value for x.  See the \
\"Intro\" notebook for more on this.\
\>", "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[x_] = \ x\  - \ Cos[x]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(x - Cos[x]\)], "Output",
  FontSize->24]
}, Open  ]],

Cell["\<\
Here's the mini-program that does bisection, with initially a=0, \
b=1.  Note that semicolons suppress output.  Carriage returns or semicolons \
separate different commands.   

The command Do[  stuff, {i,1,15} ] does \"stuff\" when i=1, then \"stuff\" \
when i=2, etc. through i=15.  Note that you can put multiple command in \
\"stuff\" using semicolons, as I do here (the c=(a+b)/2 and the If command).

The command If[ condition, stuff1, stuff2] says: If \"condition\", do \
\"stuff1\"; if \"condition\" is false; do \"stuff2\".  As above, you can put \
multiple commands in \"stuff1\" or \"stuff2\" using semicolons (but I don't \
do so here)

At the end, I print out the final values of a and b.\
\>", "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\(a = 0.0\ ;\)\ \[IndentingNewLine]
    \(b = 1.0;\)\ \[IndentingNewLine]
    \(Do[\[IndentingNewLine]\ \ \ \ \ \ c\  = \ \((a + b)\)/
            2; \[IndentingNewLine]\ \ \ \ \ \ If[\ \ f[c]*f[a]\  < \ 
            0.0, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ b = 
            c, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ a = c\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ ]\[IndentingNewLine]\ \ \ , \ \
{i, 1, 15}\[IndentingNewLine]];\)\[IndentingNewLine]
    \(\({a, b}\)\(\[IndentingNewLine]\)
    \)\)\(\ \ \ \)\)\)], "Input",
  FontSize->24],

Cell[BoxData[
    \({0.73907470703125`, 0.739105224609375`}\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[TextData[{
  "Now we can get a bit fancier.  Let's say we wanted to use bisection to \
solve x - p cos x =0 for various values of p.  We could keep changing p by \
hand over and over, but even nicer, we can let ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " do everything as a function of p.\n\nFirst we redefine f to depend on x \
and p."
}], "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[x_, p_] = \ x\  - \ p*Cos[x]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(x - p\ Cos[x]\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[TextData[{
  "Now we do bisection again, but this time, we tell ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " that this whole algorithm depends on p.  We assign the whole algorithm to \
be the function \"bisection[p_]\".  We use the colon-equals assignment to \
tell ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to not try to evaluate anything until it receives a value for p.  We wrap \
the algorithm  in the Block command; the syntax is Block[ {localvars}, \
stuff], where \"localvars\" are a list of variables used by the algorithm and \
\"stuff\" is a list of commands separated by semicolons.  \n\nI also made a \
few revisions to the algorithm to account for the presence of p:\n\nI write \
f[c,p] and f[a,p] instead of the old f[c] and f[a].\nI change the initial b \
from 1 to p, since f[1,p] may not be > 0, but f[p,p] is always >0.\nI set the \
number of iterations to n=Log[2,10^5(b-a)] (that's log base-2 of 10^5 (b-a)), \
to ensure that the width of the nth interval, (b-a)/2^n, is less than \
10^(-5)."
}], "Text",
  FontSize->24],

Cell[BoxData[{
    \(bisection[
        p_]\  := \[IndentingNewLine]Block[\ \ \ {a, b, c, n, 
          i}, \[IndentingNewLine]\ \ a = 0.0\ ; \ \[IndentingNewLine]\ \ b = 
          p; \ \[IndentingNewLine]\ \ n\  = 
          Log[2, 10^5*\((b - 
                  a)\)]; \[IndentingNewLine]\ \ Do[\[IndentingNewLine]\ \ \ \ \
\ \ \ \ c\  = \ \((a + b)\)/
              2; \[IndentingNewLine]\ \ \ \ \ \ \ \ If[\ \ f[c, p]*
                f[a, p]\  < \ 
              0.0, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ b = 
              c, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ a = c\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ ]\[IndentingNewLine]\ \ \ \
\ \ , \ {i, 1, 
            n}\[IndentingNewLine]\ \ ]; \[IndentingNewLine]\ \ \ \((a + b)\)/
          2\[IndentingNewLine]]\), "\[IndentingNewLine]", 
    \(\ \ \ \)}], "Input",
  FontSize->24],

Cell["Now we can run bisection for various values of p:", "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(bisection[1]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(0.7390823364257812`\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(bisection[2]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(1.0298690795898438`\)], "Output",
  FontSize->24]
}, Open  ]],

Cell["\<\
We can even plot the solution as a function of p (this amount to \
solving numerically the implicit equation x - p cos x = 0 for p in terms of \
x.\
\>", "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(Plot[bisection[p], {p, 0, 4}]\)], "Input",
  FontSize->24],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.238095 0.0147151 0.469997 [
[.2619 .00222 -3 -9 ]
[.2619 .00222 3 0 ]
[.5 .00222 -3 -9 ]
[.5 .00222 3 0 ]
[.7381 .00222 -3 -9 ]
[.7381 .00222 3 0 ]
[.97619 .00222 -3 -9 ]
[.97619 .00222 3 0 ]
[.01131 .10871 -18 -4.5 ]
[.01131 .10871 0 4.5 ]
[.01131 .20271 -18 -4.5 ]
[.01131 .20271 0 4.5 ]
[.01131 .29671 -18 -4.5 ]
[.01131 .29671 0 4.5 ]
[.01131 .39071 -18 -4.5 ]
[.01131 .39071 0 4.5 ]
[.01131 .48471 -6 -4.5 ]
[.01131 .48471 0 4.5 ]
[.01131 .57871 -18 -4.5 ]
[.01131 .57871 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .10871 m
.03006 .10871 L
s
[(0.2)] .01131 .10871 1 0 Mshowa
.02381 .20271 m
.03006 .20271 L
s
[(0.4)] .01131 .20271 1 0 Mshowa
.02381 .29671 m
.03006 .29671 L
s
[(0.6)] .01131 .29671 1 0 Mshowa
.02381 .39071 m
.03006 .39071 L
s
[(0.8)] .01131 .39071 1 0 Mshowa
.02381 .48471 m
.03006 .48471 L
s
[(1)] .01131 .48471 1 0 Mshowa
.02381 .57871 m
.03006 .57871 L
s
[(1.2)] .01131 .57871 1 0 Mshowa
.125 Mabswid
.02381 .03821 m
.02756 .03821 L
s
.02381 .06171 m
.02756 .06171 L
s
.02381 .08521 m
.02756 .08521 L
s
.02381 .13221 m
.02756 .13221 L
s
.02381 .15571 m
.02756 .15571 L
s
.02381 .17921 m
.02756 .17921 L
s
.02381 .22621 m
.02756 .22621 L
s
.02381 .24971 m
.02756 .24971 L
s
.02381 .27321 m
.02756 .27321 L
s
.02381 .32021 m
.02756 .32021 L
s
.02381 .34371 m
.02756 .34371 L
s
.02381 .36721 m
.02756 .36721 L
s
.02381 .41421 m
.02756 .41421 L
s
.02381 .43771 m
.02756 .43771 L
s
.02381 .46121 m
.02756 .46121 L
s
.02381 .50821 m
.02756 .50821 L
s
.02381 .53171 m
.02756 .53171 L
s
.02381 .55521 m
.02756 .55521 L
s
.02381 .60221 m
.02756 .60221 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .01472 m
.06244 .09001 L
.10458 .16597 L
.14415 .22818 L
.18221 .27918 L
.22272 .3249 L
.26171 .36191 L
.30316 .39514 L
.34309 .42238 L
.3815 .44499 L
.42237 .46592 L
.46172 .48355 L
.49955 .49858 L
.53984 .51284 L
.57861 .52512 L
.61984 .53686 L
.65954 .54708 L
.69774 .55603 L
.73838 .56473 L
.77751 .5724 L
.81909 .57989 L
.85916 .58654 L
.89771 .59247 L
.93871 .59833 L
.97619 .60332 L
s
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{444, 274.313},
  ImageMargins->{{43, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  FontSize->72,
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006l00014R000`400?l00000o`00003oo`3Vc82m
0>K<P000NP3Vc8001P3Vc600J@00000000000000000S08fdP640i/b000H0iYTj0000000000000000
0000>P2/c81Q0>K<P0040<Ym8`0000000000001MH6<0i/b000@0bWdS0000000002<0SKB02@3Vc800
07`0i/b000<0iYTj041mL03Vc800H`3Vc8000`2=>@00000j0:c<P01R0>K<P0060>JdCP10>Dh0i/b0
0>K<H01Y02<0SKB0H`3Vc8000`3VVCX0@7e`0>K<P0090>K<P000O03Vc8000`3VVCX0@7e`0>K<P01T
0>K<P0030>JI>P000000JIV006<0i/b000<0[5d00000001YVH00H@3Vc8001P3Vc600J@0000000000
0000000S08fdP0T0i/b0001l0>K<P0030>JI>P10OG00i/b006D0i/b000<0i/aP0:bIP03Vc800H`3V
c8000`3Vc600JCU>0>K<P01Q0>K<P0040:aM001YVH00iYTj041mL0X0i/b0001k0>K<P00308di>P2/
VCX0@7e`06<0i/b000H0i[A>040iCP3Vc800i/aP06T0>P2/c81P0>K<P0060>K<H01Y02<0SKB00>K<
H01Y03X0[<b0H@3Vc800103Vc600J@0S08em>P10OG0:0>K<P000N`3Vc8000`3VVCX00000041mL01T
0>K<P0040<Ym8`0000000000041mL680i/b000@0i[A>040000000000JIV0H`3Vc8000`3V]4h0@000
041mL00:0>K<P000o`3Vc82m0>K<P0005`3Vc8000`000000i/b00>K<P03o0>K<P:<0i/b0000G0>K<
P0030000003Vc800i/b00?l0i/b0X`3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc82S0>K<P000
5`3Vc8000`000000i/b00>K<P03o0>K<P:<0i/b0000G0>K<P0030000003Vc800i/b00?l0i/b0X`3V
c80001L0i/b000<000000>K<P03Vc800o`3Vc82S0>K<P0005`3Vc8000`000000i/b00>K<P03o0>K<
P:<0i/b0000=0>K<P?l00000[`0000010>K<P0005`3Vc802000001<0i/b000<000000>K<P03Vc800
4@3Vc8000`000000i/b00>K<P00B0>K<P0030000003Vc800i/b00140i/b000<000000>K<P03Vc800
4@3Vc8000`000000i/b00>K<P00B0>K<P0030000003Vc800i/b00140i/b000<000000>K<P03Vc800
4P3Vc8000`000000i/b00>K<P00A0>K<P0030000003Vc800i/b00180i/b000<000000>K<P03Vc800
4@3Vc8000`000000i/b00>K<P00A0>K<P0030000003Vc800i/b00180i/b000<000000>K<P03Vc800
4@3Vc8000`000000i/b00>K<P00B0>K<P0030000003Vc800i/b00140i/b000<000000>K<P03Vc800
4P3Vc8000`000000i/b00>K<P00A0>K<P0030000003Vc800i/b00180i/b000<000000>K<P03Vc800
4@3Vc8000`000000i/b00>K<P0090>K<P0005`3Vc802000006@0i/b000<000000>K<P03Vc800I03V
c8000`000000i/b00>K<P01S0>K<P0030000003Vc800i/b006<0i/b000<000000>K<P03Vc8002@3V
c80001L0i/b000<000000>K<P0000000o`3Vc82S0>K<P0005`3Vc8000`000000i/b00000003o0>K<
P:<0i/b0000G0>K<P0040000003Vc800i/b000000?l0i/b0XP3Vc80001L0i/b000@000000>K<P03V
c8000000o`3Vc82R0>K<P0005`3Vc8001@000000i/b00>K<P03Vc80000000?l0i/b0X@3Vc80001L0
i/b000D000000>K<P03Vc800i/b00000003o0>K<P:40i/b0000G0>K<P0030000003Vc800i/b00080
i/b000<000000>K<P03Vc800o`3Vc82N0>K<P0005`3Vc80300000080i/b000<000000>K<P03Vc800
o`3Vc82N0>K<P0005`3Vc8000`000000i/b00>K<P0030>K<P0030000003Vc800i/b00?l0i/b0W@3V
c80001L0i/b000<000000>K<P03Vc8000`3Vc8000`000000i/b00>K<P03o0>K<P9d0i/b0000G0>K<
P0030000003Vc800i/b000@0i/b000<000000>K<P03Vc800o`3Vc82L0>K<P0005`3Vc8000`000000
i/b00>K<P0040>K<P0030000003Vc800i/b00?l0i/b0W03Vc80001L0i/b000<000000>K<P03Vc800
1@3Vc8000`000000i/b00>K<P03o0>K<P9/0i/b0000G0>K<P0030000003Vc800i/b000H0i/b000<0
00000>K<P03Vc800o`3Vc82J0>K<P0005`3Vc8000`000000i/b00>K<P0060>K<P0030000003Vc800
i/b00?l0i/b0VP3Vc80001L0i/b000<000000>K<P03Vc8001`3Vc8000`000000i/b00>K<P03o0>K<
P9T0i/b0000G0>K<P0030000003Vc800i/b000L0i/b000<000000>K<P03Vc800o`3Vc82I0>K<P000
5`3Vc803000000P0i/b000<000000>K<P03Vc800o`3Vc82H0>K<P0005`3Vc8000`000000i/b00>K<
P0080>K<P0030000003Vc800i/b00?l0i/b0V03Vc80001L0i/b000<000000>K<P03Vc8002@3Vc800
0`000000i/b00>K<P03o0>K<P9L0i/b0000G0>K<P0030000003Vc800i/b000T0i/b000<000000>K<
P03Vc800o`3Vc82G0>K<P0005`3Vc8000`000000i/b00>K<P00:0>K<P0030000003Vc800i/b00?l0
i/b0UP3Vc80001L0i/b000<000000>K<P03Vc8002P3Vc8000`000000i/b00>K<P03o0>K<P9H0i/b0
000G0>K<P0030000003Vc800i/b000/0i/b000<000000>K<P03Vc800o`3Vc82E0>K<P0005`3Vc800
0`000000i/b00>K<P00;0>K<P0030000003Vc800i/b00?l0i/b0U@3Vc80001L0i/b000<000000>K<
P03Vc800303Vc8000`000000i/b00>K<P03o0>K<P9@0i/b0000G0>K<P0030000003Vc800i/b000`0
i/b000<000000>K<P03Vc800o`3Vc82D0>K<P0005`3Vc803000000d0i/b000<000000>K<P03Vc800
o`3Vc82C0>K<P0005`3Vc8000`000000i/b00>K<P00=0>K<P0030000003Vc800i/b00?l0i/b0T`3V
c80001L0i/b000<000000>K<P03Vc8003P3Vc8000`000000i/b00>K<P03o0>K<P980i/b0000G0>K<
P0030000003Vc800i/b000l0i/b000<000000>K<P03Vc800o`3Vc82A0>K<P0005`3Vc8000`000000
i/b00>K<P00?0>K<P0030000003Vc800i/b00?l0i/b0T@3Vc80001L0i/b000<000000>K<P03Vc800
403Vc8000`000000i/b00>K<P03o0>K<P900i/b0000G0>K<P0030000003Vc800i/b00100i/b000<0
00000>K<P03Vc800o`3Vc82@0>K<P0005`3Vc8000`000000i/b00>K<P00A0>K<P0030000003Vc800
i/b00?l0i/b0S`3Vc80000050>K<P03:OB<0000000000010OG000P3Vc8000`3Vc600J@00000iCP02
0>K<P0060>JI>P00000000000000000003X0[<b01@3Vc8000`000000i/b00>K<P00A0>K<P0030000
003Vc800i/b00?l0i/b0S`3Vc80000060>K<P02=GDh0i/b00>K<P02=>CX0[<b01`3Vc8000`2=>@00
000j0:c<P0070>K<P0030000003Vc800i/b00180i/b000<000000>K<P03Vc800o`3Vc82>0>K<P000
00H0i/aP06TiCP3Vc800i/b00:aM8`2=]8080>K<P0030>JI>P000000JIV000H0i/b01000000B0>K<
P0030000003Vc800i/b00?l0i/b0S@3Vc80000060>K<H01Y>Dh0i/b00>K<P02/GB<0SKB02@3Vc800
0`3Vc600[9V00>K<P0050>K<P0030000003Vc800i/b001<0i/b000<000000>K<P03Vc800o`3Vc82=
0>K<P00000H0i/b008di8`2=]800i/aP06T0>P2/c8060>K<P0060>JdCP10>Dh0i/b00>K<H01Y03X0
[<b01@3Vc8000`000000i/b00>K<P00D0>K<P0030000003Vc800i/b00?l0i/b0S03Vc80000050>K<
P03V]4h0@0000000001YVH00203Vc800103:OB<0000000000010OG060>K<P0030000003Vc800i/b0
01@0i/b000<000000>K<P03Vc800o`3Vc82<0>K<P0005`3Vc8000`000000i/b00>K<P00E0>K<P003
0000003Vc800i/b00?l0i/b0R`3Vc80001L0i/b000<000000>K<P03Vc8005@3Vc8000`000000i/b0
0>K<P03o0>K<P8/0i/b0000G0>K<P0030000003Vc800i/b001H0i/b000<000000>K<P03Vc800o`3V
c82:0>K<P0005`3Vc8000`000000i/b00>K<P00F0>K<P0030000003Vc800i/b00?l0i/b0RP3Vc800
01L0i/b000<000000>K<P03Vc8005`3Vc8000`000000i/b00>K<P03o0>K<P8T0i/b0000G0>K<P003
0000003Vc800i/b001P0i/b000<000000>K<P03Vc800o`3Vc8280>K<P0005`3Vc803000001P0i/b0
00<000000>K<P03Vc800o`3Vc8280>K<P0005`3Vc8000`000000i/b00>K<P00I0>K<P0030000003V
c800i/b00?l0i/b0Q`3Vc80001L0i/b000<000000>K<P03Vc8006@3Vc8000`000000i/b00>K<P03o
0>K<P8L0i/b0000G0>K<P0030000003Vc800i/b001X0i/b000<000000>K<P03Vc800o`3Vc8260>K<
P0005`3Vc8000`000000i/b00>K<P00J0>K<P0030000003Vc800i/b00?l0i/b0QP3Vc80001L0i/b0
00<000000>K<P03Vc8006`3Vc8000`000000i/b00>K<P03o0>K<P8D0i/b0000G0>K<P0030000003V
c800i/b001`0i/b000<000000>K<P03Vc800o`3Vc8240>K<P0005`3Vc8000`000000i/b00>K<P00L
0>K<P0030000003Vc800i/b00?l0i/b0Q03Vc80001L0i/b000<000000>K<P03Vc8007@3Vc8000`00
0000i/b00>K<P03o0>K<P8<0i/b0000G0>K<P0030000003Vc800i/b001d0i/b000<000000>K<P03V
c800o`3Vc8230>K<P0005`3Vc803000001h0i/b000<000000>K<P03Vc800o`3Vc8220>K<P0005`3V
c8000`000000i/b00>K<P00N0>K<P0030000003Vc800i/b00?l0i/b0PP3Vc80001L0i/b000<00000
0>K<P03Vc8007`3Vc8000`000000i/b00>K<P03o0>K<P840i/b0000G0>K<P0030000003Vc800i/b0
01l0i/b000<000000>K<P03Vc800o`3Vc8210>K<P0005`3Vc8000`000000i/b00>K<P00P0>K<P003
0000003Vc800i/b00?l0i/b0P03Vc80001L0i/b000<000000>K<P03Vc8008@3Vc8000`000000i/b0
0>K<P03o0>K<P7l0i/b0000G0>K<P0030000003Vc800i/b00240i/b000<000000>K<P03Vc800o`3V
c81o0>K<P0005`3Vc8000`000000i/b00>K<P00R0>K<P0030000003Vc800i/b00?l0i/b0OP3Vc800
01L0i/b000<000000>K<P03Vc8008`3Vc8000`000000i/b00>K<P03o0>K<P7d0i/b0000G0>K<P003
0000003Vc800i/b002<0i/b000<000000>K<P03Vc800o`3Vc81m0>K<P0005`3Vc803000002@0i/b0
00<000000>K<P03Vc800o`3Vc81l0>K<P0005`3Vc8000`000000i/b00>K<P00T0>K<P0030000003V
c800i/b00?l0i/b0O03Vc80001L0i/b000<000000>K<P03Vc8009@3Vc8000`000000i/b00>K<P03o
0>K<P7/0i/b0000G0>K<P0030000003Vc800i/b002H0i/b000<000000>K<P03Vc800o`3Vc81j0>K<
P0005`3Vc8000`000000i/b00>K<P00V0>K<P0030000003Vc800i/b00?l0i/b0NP3Vc80001L0i/b0
00<000000>K<P03Vc8009`3Vc8000`000000i/b00>K<P03o0>K<P7T0i/b0000G0>K<P0030000003V
c800i/b002P0i/b000<000000>K<P03Vc800o`3Vc81h0>K<P0005`3Vc8000`000000i/b00>K<P00X
0>K<P0030000003Vc800i/b00?l0i/b0N03Vc80000050>K<P03:OB<0000000000010OG000P3Vc800
0`3Vc600J@00000iCP040>K<P0040<Ym8`000000000S08fdP0D0i/b000<000000>K<P03Vc800:@3V
c8000`000000i/b00>K<P03o0>K<P7L0i/b000001P3Vc800SEe>0>K<P03Vc800SCTj0:c<P0T0i/b0
00<0iYTj041mL03Vc8001@3Vc8000`000000i/b00>K<P00Y0>K<P0030000003Vc800i/b00?l0i/b0
M`3Vc80000060>K<H01Y>Dh0i/b00>K<P02/GB<0SKB01P3Vc8001P3Vc600J@00000000000000000S
08fdP0D0i/b01000000Y0>K<P0030000003Vc800i/b00?l0i/b0MP3Vc80000060>K<H01Y>Dh0i/b0
0>K<P02/GB<0SKB01`3Vc800102/G@00JIV00>JI>P10OG060>K<P0030000003Vc800i/b002/0i/b0
00<000000>K<P03Vc800o`3Vc81e0>K<P00000H0i/b008di8`2=]800i/aP06T0>P2/c8070>K<P004
0>K<H01Y02<0SGdj041mL0H0i/b000<000000>K<P03Vc800:`3Vc8000`000000i/b00>K<P03o0>K<
P7D0i/b000001@3Vc800i[A>040000000000JIV000T0i/b000<0i[A>04000010OG001P3Vc8000`00
0000i/b00>K<P00/0>K<P0030000003Vc800i/b00?l0i/b0M03Vc80001L0i/b000<000000>K<P03V
c800;@3Vc8000`000000i/b00>K<P03o0>K<P7<0i/b0000G0>K<P0030000003Vc800i/b002d0i/b0
00<000000>K<P03Vc800o`3Vc81c0>K<P0005`3Vc8000`000000i/b00>K<P00^0>K<P0030000003V
c800i/b00?l0i/b0LP3Vc80001L0i/b000<000000>K<P03Vc800;P3Vc8000`000000i/b00>K<P03o
0>K<P780i/b0000G0>K<P0030000003Vc800i/b002l0i/b000<000000>K<P03Vc800o`3Vc81a0>K<
P0005`3Vc8000`000000i/b00>K<P00`0>K<P0030000003Vc800i/b00?l0i/b0L03Vc80001L0i/b0
0`00000`0>K<P0030000003Vc800i/b00?l0i/b0L03Vc80001L0i/b000<000000>K<P03Vc800<@3V
c8000`000000i/b00>K<P03o0>K<P6l0i/b0000G0>K<P0030000003Vc800i/b00380i/b000<00000
0>K<P03Vc800o`3Vc81^0>K<P0005`3Vc8000`000000i/b00>K<P00b0>K<P0030000003Vc800i/b0
0?l0i/b0KP3Vc80001L0i/b000<000000>K<P03Vc800<`3Vc8000`000000i/b00>K<P03o0>K<P6d0
i/b0000G0>K<P0030000003Vc800i/b003@0i/b000<000000>K<P03Vc800o`3Vc81/0>K<P0005`3V
c8000`000000i/b00>K<P00e0>K<P0030000003Vc800i/b00?l0i/b0J`3Vc80001L0i/b000<00000
0>K<P03Vc800=@3Vc8000`000000i/b00>K<P03o0>K<P6/0i/b0000G0>K<P0030000003Vc800i/b0
03H0i/b000<000000>K<P03Vc800o`3Vc81Z0>K<P0005`3Vc8000`000000i/b00>K<P00g0>K<P003
0000003Vc800i/b00?l0i/b0J@3Vc80001L0i/b00`00000h0>K<P0030000003Vc800i/b00?l0i/b0
J03Vc80001L0i/b000<000000>K<P03Vc800>03Vc8000`000000i/b00>K<P03o0>K<P6P0i/b0000G
0>K<P0030000003Vc800i/b003T0i/b000<000000>K<P03Vc800o`3Vc81W0>K<P0005`3Vc8000`00
0000i/b00>K<P00j0>K<P0030000003Vc800i/b00?l0i/b0IP3Vc80001L0i/b000<000000>K<P03V
c800>P3Vc8000`000000i/b00>K<P03o0>K<P6H0i/b0000G0>K<P0030000003Vc800i/b003/0i/b0
00<000000>K<P03Vc800o`3Vc81U0>K<P0005`3Vc8000`000000i/b00>K<P00l0>K<P0030000003V
c800i/b00?l0i/b0I03Vc80001L0i/b000<000000>K<P03Vc800?@3Vc8000`000000i/b00>K<P03o
0>K<P6<0i/b0000G0>K<P0030000003Vc800i/b003d0i/b000<000000>K<P03Vc800o`3Vc81S0>K<
P0005`3Vc8000`000000i/b00>K<P00n0>K<P0030000003Vc800i/b00?l0i/b0HP3Vc80001L0i/b0
0`00000o0>K<P0030000003Vc800i/b00?l0i/b0H@3Vc80001L0i/b000<000000>K<P03Vc800@03V
c8000`000000i/b00>K<P03o0>K<P600i/b0000G0>K<P0030000003Vc800i/b00400i/b000<00000
0>K<P03Vc800o`3Vc81P0>K<P0005`3Vc8000`000000i/b00>K<P0110>K<P0030000003Vc800i/b0
0?l0i/b0G`3Vc80001L0i/b000<000000>K<P03Vc800@P3Vc8000`000000i/b00>K<P03o0>K<P5h0
i/b0000G0>K<P0030000003Vc800i/b004<0i/b000<000000>K<P03Vc800o`3Vc81M0>K<P0005`3V
c8000`000000i/b00>K<P0140>K<P0030000003Vc800i/b00?l0i/b0G03Vc80001L0i/b000<00000
0>K<P03Vc800A@3Vc8000`000000i/b00>K<P03o0>K<P5/0i/b0000G0>K<P0030000003Vc800i/b0
04H0i/b000<000000>K<P03Vc800o`3Vc81J0>K<P00000D0i/b00<Ym8`0000000000041mL0020>K<
P0030>K<H01Y000003U>00<0i/b000D0i/aP06T000000000000S08fdP0050>K<P0030000003Vc800
i/b004H0i/b000<000000>K<P03Vc800o`3Vc81J0>K<P00000H0i/b008eMCP3Vc800i/b008di>P2/
c8070>K<P0050:aM001YVH00i/b00>K<H02/VH001@3Vc8000`000000i/b00>K<P0170>K<P0030000
003Vc800i/b00?l0i/b0F@3Vc80000060>K<H01Y>Dh0i/b00>K<P02/GB<0SKB01P3Vc8001P3Vc600
J@00041mL03Vc800bWdS06VIP0D0i/b0100000170>K<P0030000003Vc800i/b00?l0i/b0F03Vc800
00060>K<H01Y>Dh0i/b00>K<P02/GB<0SKB01`3Vc800102=>CX0@3T000000000>Dh60>K<P0030000
003Vc800i/b004T0i/b000<000000>K<P03Vc800o`3Vc81G0>K<P00000H0i/b008di8`2=]800i/aP
06T0>P2/c8070>K<P0030>JI>P00>Dh0i/b000L0i/b000<000000>K<P03Vc800BP3Vc8000`000000
i/b00>K<P03o0>K<P5H0i/b000001@3Vc800i[A>040000000000JIV000T0i/b000@0bWdS00000000
0000JIV01@3Vc8000`000000i/b00>K<P01;0>K<P0030000003Vc800i/b00?l0i/b0E@3Vc80001L0
i/b000<000000>K<P03Vc800C03Vc8000`000000i/b00>K<P03o0>K<P5@0i/b0000G0>K<P0030000
003Vc800i/b004d0i/b000<000000>K<P03Vc800o`3Vc81C0>K<P0005`3Vc8000`000000i/b00>K<
P01>0>K<P0030000003Vc800i/b00?l0i/b0DP3Vc80001L0i/b000<000000>K<P03Vc800C`3Vc800
0`000000i/b00>K<P03o0>K<P540i/b0000G0>K<P0030000003Vc800i/b004l0i/b000<000000>K<
P03Vc800o`3Vc81A0>K<P0005`3Vc8000`000000i/b00>K<P01@0>K<P0030000003Vc800i/b00?l0
i/b0D03Vc80001L0i/b00`00001A0>K<P0030000003Vc800i/b00?l0i/b0C`3Vc80001L0i/b000<0
00000>K<P03Vc800DP3Vc8000`000000i/b00>K<P03o0>K<P4h0i/b0000G0>K<P0030000003Vc800
i/b005<0i/b000<000000>K<P03Vc800o`3Vc81=0>K<P0005`3Vc8000`000000i/b00>K<P01D0>K<
P0030000003Vc800i/b00?l0i/b0C03Vc80001L0i/b000<000000>K<P03Vc800E@3Vc8000`000000
i/b00>K<P03o0>K<P4/0i/b0000G0>K<P0030000003Vc800i/b005H0i/b000<000000>K<P03Vc800
o`3Vc81:0>K<P0005`3Vc8000`000000i/b00>K<P01G0>K<P0030000003Vc800i/b00?l0i/b0B@3V
c80001L0i/b000<000000>K<P03Vc800F03Vc8000`000000i/b00>K<P03o0>K<P4P0i/b0000G0>K<
P0030000003Vc800i/b005T0i/b000<000000>K<P03Vc800o`3Vc8170>K<P0005`3Vc8000`000000
i/b00>K<P01J0>K<P0030000003Vc800i/b00?l0i/b0AP3Vc80001L0i/b00`00001K0>K<P0030000
003Vc800i/b00?l0i/b0A@3Vc80001L0i/b000<000000>K<P03Vc800G03Vc8000`000000i/b00>K<
P03o0>K<P4@0i/b0000G0>K<P0030000003Vc800i/b005d0i/b000<000000>K<P03Vc800o`3Vc813
0>K<P0005`3Vc8000`000000i/b00>K<P01N0>K<P0030000003Vc800i/b00?l0i/b0@P3Vc80001L0
i/b000<000000>K<P03Vc800G`3Vc8000`000000i/b00>K<P03o0>K<P440i/b0000G0>K<P0030000
003Vc800i/b00600i/b000<000000>K<P03Vc800o`3Vc8100>K<P0005`3Vc8000`000000i/b00>K<
P01Q0>K<P0030000003Vc800i/b00?l0i/b0?`3Vc80001L0i/b000<000000>K<P03Vc800HP3Vc800
0`000000i/b00>K<P03o0>K<P3h0i/b0000G0>K<P0030000003Vc800i/b006<0i/b000<000000>K<
P03Vc800o`3Vc80m0>K<P0005`3Vc8000`000000i/b00>K<P01T0>K<P0030000003Vc800i/b00?l0
i/b0?03Vc80001L0i/b00`00001U0>K<P0800000o`3Vc80l0>K<P0005`3Vc8000`000000i/b00>K<
P01W0>K<P0030000003Vc800i/b00?l0i/b0>@3Vc80001L0i/b000<000000>K<P03Vc800J03Vc800
0`000000i/b00>K<P03o0>K<P3P0i/b0000G0>K<P0030000003Vc800i/b006T0i/b00P00003o0>K<
P3P0i/b0000G0>K<P0030000003Vc800i/b006/0i/b000<000000>K<P03Vc800o`3Vc80e0>K<P000
5`3Vc8000`000000i/b00>K<P01/0>K<P0030000003Vc800i/b00?l0i/b0=03Vc80001L0i/b000<0
00000>K<P03Vc800K@3Vc8000`000000i/b00>K<P03o0>K<P3<0i/b0000G0>K<P0030000003Vc800
i/b006h0i/b00P00003o0>K<P3<0i/b000001@3Vc800bWdS000000000000@7e`0080i/b000<0i/aP
06T00000>Dh00`3Vc800102/G@00000000000010OG060>K<P0030000003Vc800i/b00700i/b000<0
00000>K<P03Vc800o`3Vc80`0>K<P00000H0i/b008eMCP3Vc800i/b008di>P2/c8060>K<P0050>K<
H01YGF00i/b00>K<P02=GDh01P3Vc8000`000000i/b00>K<P01a0>K<P0030000003Vc800i/b00?l0
i/b0;`3Vc80000060>K<H01Y>Dh0i/b00>K<P02/GB<0SKB01`3Vc800102=GDh0i/b00>K<H01YGF06
0>K<P0@00000L@3Vc80200000?l0i/b0;`3Vc80000060>K<H01Y>Dh0i/b00>K<P02/GB<0SKB01`3V
c800102/G@00000000000010OG060>K<P0030000003Vc800i/b007@0i/b000<000000>K<P03Vc800
o`3Vc80/0>K<P00000H0i/b008di8`2=]800i/aP06T0>P2/c8060>K<P0050>K<H01Y>Dh0i/b00>K<
H01Y>Dh01P3Vc8000`000000i/b00>K<P01e0>K<P0030000003Vc800i/b00?l0i/b0:`3Vc8000005
0>K<P03V]4h0@0000000001YVH00203Vc800103:OB<000000000001YVH060>K<P0030000003Vc800
i/b007H0i/b00P00003o0>K<P2/0i/b0000G0>K<P0030000003Vc800i/b007P0i/b000<000000>K<
P03Vc800o`3Vc80X0>K<P0005`3Vc8000`000000i/b00>K<P01i0>K<P0030000003Vc800i/b00?l0
i/b09`3Vc80001L0i/b000<000000>K<P03Vc800NP3Vc80200000?l0i/b09`3Vc80001L0i/b000<0
00000>K<P03Vc800O03Vc8000`000000i/b00>K<P03o0>K<P2@0i/b0000G0>K<P0030000003Vc800
i/b007d0i/b00P00003o0>K<P2@0i/b0000G0>K<P0030000003Vc800i/b007l0i/b000<000000>K<
P03Vc800o`3Vc80Q0>K<P0005`3Vc80300000800i/b00P00003o0>K<P240i/b0000G0>K<P0030000
003Vc800i/b00880i/b000<000000>K<P03Vc800o`3Vc80N0>K<P0005`3Vc8000`000000i/b00>K<
P0230>K<P0030000003Vc800i/b00?l0i/b07@3Vc80001L0i/b000<000000>K<P03Vc800Q03Vc802
00000?l0i/b07@3Vc80001L0i/b000<000000>K<P03Vc800QP3Vc8000`000000i/b00>K<P03o0>K<
P1X0i/b0000G0>K<P0030000003Vc800i/b008L0i/b00P00003o0>K<P1X0i/b0000G0>K<P0030000
003Vc800i/b008T0i/b00P00003o0>K<P1P0i/b0000G0>K<P0030000003Vc800i/b008/0i/b00P00
003o0>K<P1H0i/b0000G0>K<P0030000003Vc800i/b008d0i/b00P00003o0>K<P1@0i/b0000G0>K<
P0030000003Vc800i/b008l0i/b00P00003o0>K<P180i/b0000G0>K<P0<00000T@3Vc80200000?l0
i/b0403Vc80001L0i/b000<000000>K<P03Vc800T`3Vc80200000?l0i/b03P3Vc80001L0i/b000<0
00000>K<P03Vc800U@3Vc80200000?l0i/b0303Vc80001L0i/b000<000000>K<P03Vc800U`3Vc800
0`000000i/b00>K<P03o0>K<P0T0i/b0000G0>K<P0030000003Vc800i/b009P0i/b00P00003o0>K<
P0T0i/b0000G0>K<P0030000003Vc800i/b009X0i/b00P00003o0>K<P0L0i/b0000G0>K<P0030000
003Vc800i/b009`0i/b00P00003o0>K<P0D0i/b0000G0>K<P0030000003Vc800i/b009h0i/b00P00
003o0>K<P0<0i/b0000G0>K<P0030000003Vc800i/b00:00i/b00P00003o0>K<P040i/b0000G0>K<
P0030000003Vc800i/b00:80i/b00P00003n0>K<P0005`3Vc80300000:@0i/b00P00003l0>K<P000
5`3Vc8000`000000i/b00>K<P02V0>K<P0800000nP3Vc80001L0i/b000<000000>K<P03Vc800Z03V
c80200000?P0i/b0000G0>K<P0030000003Vc800i/b00:X0i/b00P00003f0>K<P0005`3Vc8000`00
0000i/b00>K<P02/0>K<P0800000m03Vc80001L0i/b000<000000>K<P03Vc800[P3Vc80200000?80
i/b0000G0>K<P0030000003Vc800i/b00;00i/b00P00003`0>K<P0005`3Vc8000`000000i/b00>K<
P02b0>K<P0800000kP3Vc80000`0i/b000H0i/aP06T000000000000000008`2=]8050>K<P0030000
003Vc800i/b00;@0i/b00P00003/0>K<P0003P3Vc8000`3VVCX0@7e`0>K<P0060>K<P0030000003V
c800i/b00;H0i/b00P00003Z0>K<P0003P3Vc8000`3VVCX0@7e`0>K<P0060>K<P0@00000]`3Vc803
00000>L0i/b0000>0>K<P0030>JI>P10OG00i/b000H0i/b000<000000>K<P03Vc800^`3Vc8020000
0>D0i/b0000=0>K<P00308di>P2/VCX0@7e`00L0i/b000<000000>K<P03Vc800_@3Vc80300000>80
i/b0000=0>K<P0030>JI>P000000@7e`00L0i/b000<000000>K<P03Vc800`03Vc80300000=l0i/b0
000G0>K<P0030000003Vc800i/b00<<0i/b00P00003M0>K<P0005`3Vc8000`000000i/b00>K<P035
0>K<P0<00000fP3Vc80001L0i/b000<000000>K<P03Vc800b03Vc80300000=L0i/b0000G0>K<P003
0000003Vc800i/b00</0i/b00`00003D0>K<P0005`3Vc8000`000000i/b00>K<P03>0>K<P0<00000
d@3Vc80001L0i/b000<000000>K<P03Vc800d@3Vc80300000<h0i/b0000G0>K<P0<00000e03Vc803
00000</0i/b0000G0>K<P0030000003Vc800i/b00=L0i/b00`0000380>K<P0005`3Vc8000`000000
i/b00>K<P03J0>K<P0<00000a@3Vc80001L0i/b000<000000>K<P03Vc800g@3Vc80200000<<0i/b0
000G0>K<P0030000003Vc800i/b00=l0i/b00`0000300>K<P0005`3Vc8000`000000i/b00>K<P03R
0>K<P0<00000_@3Vc80001L0i/b000<000000>K<P03Vc800i@3Vc80200000;/0i/b0000G0>K<P003
0000003Vc800i/b00>L0i/b00`00002h0>K<P0005`3Vc8000`000000i/b00>K<P03Z0>K<P0<00000
]@3Vc80001L0i/b000<000000>K<P03Vc800k@3Vc80400000;40i/b0000G0>K<P0<00000l@3Vc803
00000:h0i/b0000G0>K<P0030000003Vc800i/b00?@0i/b01000002Z0>K<P0005`3Vc8000`000000
i/b00>K<P03h0>K<P0@00000YP3Vc80001L0i/b000<000000>K<P03Vc800o03Vc80400000:80i/b0
000G0>K<P0030000003Vc800i/b00?l0i/b00@3Vc804000009h0i/b0000G0>K<P0030000003Vc800
i/b00?l0i/b01@3Vc804000009X0i/b0000G0>K<P0030000003Vc800i/b00?l0i/b02@3Vc8040000
09H0i/b0000G0>K<P0030000003Vc800i/b00?l0i/b03@3Vc80400000980i/b0000G0>K<P0030000
003Vc800i/b00?l0i/b04@3Vc804000008h0i/b0000G0>K<P0030000003Vc800i/b00?l0i/b05@3V
c804000008X0i/b0000G0>K<P0030000003Vc800i/b00?l0i/b06@3Vc804000008H0i/b0000G0>K<
P0<00000o`3Vc80M0>K<P0D00000P@3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc80R0>K<P0@0
0000O@3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc80V0>K<P0D00000N03Vc80001L0i/b000<0
00000>K<P03Vc800o`3Vc80[0>K<P0@00000M03Vc80001L0i/b000<000000>K<P03Vc800o`3Vc80_
0>K<P0D00000K`3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc80d0>K<P0H00000J@3Vc80001L0
i/b000<000000>K<P03Vc800o`3Vc80j0>K<P0H00000H`3Vc80001L0i/b000<000000>K<P03Vc800
o`3Vc8100>K<P0D00000GP3Vc800000:0>K<H01Y000000000000000002<0SKB00>K<P03Vc600J@00
000iCP80i/b000H0iYTj00000000000000000000>P2/c8050>K<P0030000003Vc800i/b00?l0i/b0
A@3Vc806000005P0i/b000020>K<P0030>JI>P10OG00i/b000P0i/b000<0SCT00000>P2/c8001`3V
c8000`000000i/b00>K<P03o0>K<P4/0i/b01P00001B0>K<P0000P3Vc8000`3VVCX0@7e`0>K<P009
0>K<P0030>JI>P000000JIV000H0i/b01000003o0>K<P500i/b01P00001<0>K<P0000P3Vc8000`3V
VCX0@7e`0>K<P00:0>K<P0030>K<H02/VH00i/b000D0i/b000<000000>K<P03Vc800o`3Vc81G0>K<
P0H00000AP3Vc80000040>K<P02=>CX0[9Tj041mL0P0i/b000H0i[A>040iCP3Vc800i/aP06T0>P2/
c8050>K<P0030000003Vc800i/b00?l0i/b0G@3Vc80600000400i/b00000103Vc800iYTj00000010
OG090>K<P0040<Ym8`0000000000041mL0H0i/b000<000000>K<P03Vc800o`3Vc81S0>K<P0D00000
>`3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc81X0>K<P0D00000=P3Vc80001L0i/b000<00000
0>K<P03Vc800o`3Vc81]0>K<P0H00000<03Vc80001L0i/b000<000000>K<P03Vc800o`3Vc81c0>K<
P0L00000:@3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc81j0>K<P0T00000803Vc80001L0i/b0
00<000000>K<P03Vc800o`3Vc8230>K<P0P00000603Vc80001L0i/b000<000000>K<P03Vc800o`3V
c82;0>K<P0P00000403Vc80001L0i/b00`00003o0>K<P9<0i/b01@00000;0>K<P0005`3Vc8000`00
0000i/b00>K<P03o0>K<P:<0i/b0000G0>K<P0030000003Vc800i/b00?l0i/b0X`3Vc80001L0i/b0
00<000000>K<P03Vc800o`3Vc82S0>K<P0005`3Vc8000`000000i/b00>K<P03o0>K<P:<0i/b0000G
0>K<P0030000003Vc800i/b00?l0i/b0X`3Vc80001L0i/b000<000000>K<P03Vc800o`3Vc82S0>K<
P000\
\>"],
  ImageRangeCache->{{{0, 443}, {273.313, 0}} -> {-0.231805, -0.071207, \
0.0097855, 0.00495722}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output",
  FontSize->24]
}, Open  ]],

Cell[TextData[{
  "When implementing Newton's method, you will need derivatives of functions. \
 You can, of course, do this by hand, or you can let ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " do it using the \"D\" command:"
}], "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[x_, p_] = \ x\  - p*\ Cos[x]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(x - p\ Cos[x]\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dfdx[x_, p_]\  = \ D[f[x, p], x]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(1 + p\ Sin[x]\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dfdp[x_, p_]\  = \ D[f[x, p], p]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(\(-Cos[x]\)\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[TextData[{
  "When implementing Newton's method, you will have to program the step ",
  Cell[BoxData[
      \(TraditionalForm\`x\_\(k + 1\)\)]],
  " = ",
  Cell[BoxData[
      \(TraditionalForm\`x\_k\)]],
  " - f(",
  Cell[BoxData[
      \(TraditionalForm\`x\_k\)]],
  ")/f'(",
  Cell[BoxData[
      \(TraditionalForm\`x\_k\)]],
  ").  You can actually use the same variable name for ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(x\_\(k + 1\)\ as\ you\ use\ for\ x\_k\)\(,\)\(\ \
\)\(since\)\(\ \)\)\)]],
  "you don't need to reuse ",
  Cell[BoxData[
      \(TraditionalForm\`x\_k\)]],
  ".  So, assuming you have defined functions f[x] and dfdx[x], you can just \
do x = x - f[x]/dfdx[x]:"
}], "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[x_]\  = \ x^2\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(x\^2\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dfdx[x_]\  = \ D[f[x], x]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(2\ x\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\  = \ 2\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(2\)], "Output",
  FontSize->24]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\  = \ x\  - \ f[x]/dfdx[x]\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(1\)], "Output",
  FontSize->24]
}, Open  ]],

Cell["\<\
When implementing Newton's method in n dimensions, you can use \
almost the same code if you set things up as vector functions.  Here's the \
example I did in class:  \
\>", "Text",
  FontSize->24],

Cell["\<\
Here are our two functions.  We want to solve f1=0,f2=0 \
simultaneously for x and y.\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(\(f1[x_, y_]\  = \ 
        x + y + Cos[x\^2 + y\^2];\)\), "\[IndentingNewLine]", 
    \(\(f2[x_, y_]\  = \ 2*x*y - Sin[x\^2 - y\^2];\)\)}], "Input",
  FontSize->24],

Cell["\<\
Here's the vector function we're looking to set equal to zero.  \
\
\>", "Text",
  FontSize->24],

Cell[BoxData[
    \(f[x_, y_]\  = \ {f1[x, y], f2[x, y]}\)], "Input",
  FontSize->24],

Cell["Here's the Jacobian.  ", "Text",
  FontSize->24],

Cell[BoxData[
    RowBox[{\(J[x_, y_]\), " ", "=", " ", 
      RowBox[{"(", GridBox[{
            {\(D[f1[x, y], x]\), \(D[f1[x, y], y]\)},
            {\(D[f2[x, y], x]\), \(D[f2[x, y], y]\)}
            }], ")"}]}]], "Input",
  FontSize->24],

Cell[TextData[{
  "Here's a version of f that accepts as input a vector v.  That will make \
the Newton's method code nicer (\"vector version\").  Notice \"Block\" to run \
a set of commands at once.  Notice the := (\"delayed evaluation\") .  That \
tells ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " not to try to compute anything until a specific v is plugged in.  Right \
now it doesn't know what to make of v[[1]] and v[[2]], the first and second \
slots of the vector v."
}], "Text",
  FontSize->24],

Cell[BoxData[
    \(fvec[v_]\  := \ f[\ v[\([1]\)], \ v[\([2]\)]\ ]\)], "Input",
  FontSize->24],

Cell["\<\
Similarly, here's a version of J that accepts as input a vector v.  \
\
\>", "Text",
  FontSize->24],

Cell[BoxData[
    \(Jvec[v_]\  := \ J[\ v[\([1]\)], \ v[\([2]\)]\ ]\)], "Input",
  FontSize->24],

Cell[TextData[{
  "Now let's run Newton's method.  I use initial guess {x,y}={-1/2,0}.  My \
stopping condition is that the length of f is less than ",
  Cell[BoxData[
      \(TraditionalForm\`10\^\(-12\)\)]],
  ".  Notice the use of LinearSolve to compute ",
  Cell[BoxData[
      \(TraditionalForm\`J\^\(-1\)\)]],
  "f. "
}], "Text",
  FontSize->24],

Cell[BoxData[{
    \(\(v\  = \ {\(-0.5\), 0.0};\)\), "\[IndentingNewLine]", 
    \(\(Print[{"\<{x,y}\>", "\<f(x,y)\>"}];\)\), "\[IndentingNewLine]", 
    \(\(Print[v, "\<,\>", fvec[v]];\)\), "\[IndentingNewLine]", 
    \(While[\@\(fvec[v] . fvec[v]\) > \ 
        10\^\(-12\), \[IndentingNewLine]v\  = \ 
        v\  - \ LinearSolve[Jvec[v], fvec[v]]; \[IndentingNewLine]Print[
        v, "\<,\>", fvec[v]]\[IndentingNewLine]]\), "\[IndentingNewLine]", 
    \(\)}], "Input",
  FontSize->24],

Cell["\<\
Where did I get that initial guess from?  Well, a contour plot of \
the two functions, with only the zero contour plotted, gives useful \
information: \
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(plot1 = 
      ContourPlot[f1[x, y], {x, \(-3\), 3}, {y, \(-3\), 3}, 
        Contours \[Rule] {0}, PlotPoints \[Rule] 50, 
        ContourShading \[Rule] False]\), "\[IndentingNewLine]", 
    \(plot2 = 
      ContourPlot[f2[x, y], {x, \(-3\), 3}, {y, \(-3\), 3}, 
        Contours \[Rule] {0}, PlotPoints \[Rule] 50, 
        ContourShading \[Rule] False, 
        ContourStyle \[Rule] RGBColor[1, 0, 0]]\), "\[IndentingNewLine]", 
    \(Show[plot1, plot2]\)}], "Input",
  FontSize->24]
},
FrontEndVersion->"4.2 for Microsoft Windows",
ScreenRectangle->{{0, 1400}, {0, 967}},
WindowSize->{962, 737},
WindowMargins->{{42, Automatic}, {Automatic, 15}},
StyleDefinitions -> "PastelColor.nb"
]

(*******************************************************************
Cached data follows.  If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of  the file.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{
Cell[1754, 51, 309, 8, 107, "Text"],
Cell[2066, 61, 201, 5, 74, "Text"],
Cell[2270, 68, 222, 5, 74, "Text"],

Cell[CellGroupData[{
Cell[2517, 77, 73, 2, 49, "Input"],
Cell[2593, 81, 60, 2, 49, "Output"]
}, Open  ]],
Cell[2668, 86, 741, 16, 404, "Text"],

Cell[CellGroupData[{
Cell[3434, 106, 620, 12, 423, "Input"],
Cell[4057, 120, 89, 2, 49, "Output"]
}, Open  ]],
Cell[4161, 125, 384, 9, 173, "Text"],

Cell[CellGroupData[{
Cell[4570, 138, 79, 2, 49, "Input"],
Cell[4652, 142, 63, 2, 49, "Output"]
}, Open  ]],
Cell[4730, 147, 1076, 20, 437, "Text"],
Cell[5809, 169, 903, 18, 559, "Input"],
Cell[6715, 189, 81, 1, 41, "Text"],

Cell[CellGroupData[{
Cell[6821, 194, 61, 2, 49, "Input"],
Cell[6885, 198, 69, 2, 49, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[6991, 205, 61, 2, 49, "Input"],
Cell[7055, 209, 69, 2, 49, "Output"]
}, Open  ]],
Cell[7139, 214, 187, 5, 74, "Text"],

Cell[CellGroupData[{
Cell[7351, 223, 78, 2, 49, "Input"],
Cell[7432, 227, 23864, 499, 291, 3430, 241, "GraphicsData", "PostScript", \
"Graphics"],
Cell[31299, 728, 146, 4, 49, "Output"]
}, Open  ]],
Cell[31460, 735, 267, 7, 74, "Text"],

Cell[CellGroupData[{
Cell[31752, 746, 79, 2, 49, "Input"],
Cell[31834, 750, 63, 2, 49, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[31934, 757, 81, 2, 49, "Input"],
Cell[32018, 761, 63, 2, 49, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[32118, 768, 81, 2, 49, "Input"],
Cell[32202, 772, 61, 2, 49, "Output"]
}, Open  ]],
Cell[32278, 777, 725, 23, 140, "Text"],

Cell[CellGroupData[{
Cell[33028, 804, 64, 2, 49, "Input"],
Cell[33095, 808, 54, 2, 50, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[33186, 815, 74, 2, 49, "Input"],
Cell[33263, 819, 54, 2, 49, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[33354, 826, 58, 2, 49, "Input"],
Cell[33415, 830, 51, 2, 49, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[33503, 837, 77, 2, 49, "Input"],
Cell[33583, 841, 51, 2, 49, "Output"]
}, Open  ]],
Cell[33649, 846, 207, 5, 74, "Text"],
Cell[33859, 853, 125, 4, 41, "Text"],
Cell[33987, 859, 186, 4, 85, "Input"],
Cell[34176, 865, 106, 4, 41, "Text"],
Cell[34285, 871, 85, 2, 49, "Input"],
Cell[34373, 875, 54, 1, 41, "Text"],
Cell[34430, 878, 243, 6, 74, "Input"],
Cell[34676, 886, 518, 11, 173, "Text"],
Cell[35197, 899, 96, 2, 49, "Input"],
Cell[35296, 903, 110, 4, 41, "Text"],
Cell[35409, 909, 96, 2, 49, "Input"],
Cell[35508, 913, 351, 10, 74, "Text"],
Cell[35862, 925, 490, 9, 301, "Input"],
Cell[36355, 936, 191, 5, 74, "Text"],
Cell[36549, 943, 508, 11, 219, "Input"]
}
]
*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)