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{SECT 0 {PARA 261 "" 0 "" {TEXT -1 24 "Polynomial Approximation" }}
{PARA 257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 "" {TEXT -1
12 "P O Box 1038" }}{PARA 259 "" 0 "" {TEXT -1 25 "Grove Hill, Alabama
36451" }}{PARA 260 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 56 " If f is a function with N derivatives at t
he point " }{TEXT 277 1 "a" }{TEXT -1 82 ", then it is not so hard to \+
give a polynomial which has the same N derivatives at " }{TEXT 265 1 "
a" }{TEXT -1 93 " that f has. We ask: Suppose we select an interval \+
on which f is defined. How do we choose " }{TEXT 259 1 "a" }{TEXT -1
150 " so that the polynomial described above approximates f on the spe
cified interval best? We suggest an answer for this question with a sp
ecific example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 ""
{TEXT 262 14 "THE QUESTION: " }{TEXT -1 8 " Choose " }{TEXT 260 1 "a"
}{TEXT -1 136 " in the interval [0, 4 ] so that the polynomial with de
gree six and with the same first six derivatives that the cosine funct
ion has at " }{TEXT 261 1 "a" }{TEXT -1 10 " will be a" }{TEXT 282 19
" best approximation" }{TEXT -1 42 " for the cosine function on that i
nterval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
100 " There are several things that must be decided: Given a funct
ion f (such as the cosine function)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 15 "" 0 "" {TEXT -1 69 "What is a polynomial with the same N (=
6) derivatives as f at x = a?" }}{PARA 15 "" 0 "" {TEXT -1 42 "What i
s a criteria for deciding what is a " }{TEXT 257 18 "best approximatio
n" }{TEXT -1 28 " in the process of choosing " }{TEXT 263 1 "a" }
{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 49 " We formulate an answer to the two questions." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT
1 {PARA 3 "" 0 "" {TEXT -1 57 "Defining a polynomial with prescribed d
erivatives at x = " }{TEXT 264 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "Define a polynomial as
follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 39 "P:=x->b[0]+sum(b[p]/p!*(x-a)^p,p=1..6);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We determine the v
alue and first six derivatives of P(x)." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P(a),seq(subs(x=a,diff(
P(x),x$i)),i=1..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 132 " The result of this application is that if we want
a function with the first six derivatives specified, we use the coeff
icients " }{XPPEDIT 18 0 "b[p]" "6#&%\"bG6#%\"pG" }{TEXT -1 48 ". Here
is an example: we make a polynomial with " }{XPPEDIT 18 0 "n^th" "6#)
%\"nG%#thG" }{TEXT -1 37 " derivative equal to1/(n+1) at x = 3." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ps:=x->1+sum(1/(p+1)/p!*(x-3
)^p,p=1..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 51 " As a check, we find the first six derivatives." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ps(a);seq(subs(x=3,diff(ps(x
),x$i)),i=1..6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 217 "It should be the case that if you wish t
o make a polynomial with specified derivatives, you now know how to do
this. As a test of this, you might make a cubic polynomial with the f
irst three derivatives all 7. Try it!" }}}{PARA 0 "" 0 "" {TEXT -1 1 "
" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Giving a meaning for " }
{TEXT 258 18 "best approximation" }{TEXT -1 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " We measure " }{TEXT
256 9 "closeness" }{TEXT -1 200 " using the integral of the square of \+
the difference between the two functions. Thus, if f and g are the pre
scribed functions, we measure the distance between f and g on the inte
rval [c, d] as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 68 " distance between f an
d g = d(f, g) = " }{XPPEDIT 18 0 "sqrt(int((f(x)-g(x))^2,x=c..d))" "6#
-%%sqrtG6#-%$intG6$*$,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F.!\"\"\"\"#/F.;%\"c
G%\"dG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT 266 9 "Examples:" }{TEXT -1 79 " The function f(x) = x and \+
g(x) = sin(x) look close together on the interval [-" }{XPPEDIT 18 0 "
Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT -1 3 "/8," }{XPPEDIT 18 0 "Pi;" "
6#%#PiG" }{TEXT -1 0 "" }{TEXT -1 6 " /8]. " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "plot([x,sin(x)],x=-Pi/8..Pi/8);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Compare the graph on [-
" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT -1 3 "/4," }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT -1 5 " /4]." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([x,sin(x)],x=-Pi/4..Pi/
4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 96 " We measure how far \+
these two are apart on both intervals, with the measure described abov
e." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "sqrt(Int((x-sin(x))^2
,x=-Pi/8..Pi/8))\n=sqrt(evalf(int((x-sin(x))^2,x=-Pi/8..Pi/8)));\nsqrt
(Int((x-sin(x))^2,x=-Pi/4..Pi/4))\n=sqrt(evalf(int((x-sin(x))^2,x=-Pi/
4..Pi/4)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0
"" 0 "" {TEXT -1 184 " You should now be able to use this measure \+
to understand the distance between functions on an interval. As a test
, find how far apart sin(x) and cos(x) are on the interval [0, 2]." }}
}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "The remaining two jobs" }}{PARA
0 "" 0 "" {TEXT -1 33 "There remain two jobs to be done." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "(1) Give the pol
ynomial with the same first six derivatives as the cosine function at \+
x = " }{TEXT 267 1 "a" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 69 "P:=x->cos(a)+sum(subs(x=a,diff(cos(x),x$p))/p!*(x-a)^
p,p=1..6):\nP(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 87 " That function is a polynomial in x with degree 6. Bu
t, it really is a function of " }{TEXT 278 1 "a" }{TEXT -1 5 " and " }
{TEXT 279 1 "x" }{TEXT -1 29 ". Why not write it that way? " }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "F:=unapply(P(x),(a,x));" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(2) Choose " }
{TEXT 270 1 "a" }{TEXT -1 51 " so that the approximation is as the bes
t possible." }}{PARA 0 "" 0 "" {TEXT -1 132 "To illustrate that not al
l these approximate the cosine function equally well, we make several \+
plots by choosing various values for " }{TEXT 268 1 "a" }{TEXT -1 1 ".
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot([cos(x),F(Pi/4,x),F
(3*Pi,x)],x=0..4*Pi,y=-2..2,color=[black,blue,green]);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 69 "Ne
ither graph seems to fit the cosine function. We ask how to choose " }
{TEXT 269 1 "a" }{TEXT -1 85 " so that the distance between the graph \+
of cos(x) and F(a,x) is as small as possible?" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Choosing " }{TEXT
271 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 179 " Surely the output of the next computation is
so long that we learn nothing from reading it. Recall that the output
measures the closeness of F(a,x) to cos(x). We call this " }{TEXT
272 3 "mcF" }{TEXT -1 39 " and note that since x integrates out, " }
{TEXT 273 3 "mcF" }{TEXT -1 18 " is a function of " }{TEXT 274 1 "a" }
{TEXT -1 6 " only." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int((c
os(x)-F(a,x))^2,x=0..4*Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
18 "mcF:=unapply(%,a):" }}}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0
"" 0 "" {TEXT -1 47 " We would like to know where this function "
}{TEXT 275 3 "mcF" }{TEXT -1 95 " takes on its minimum value. The usua
l idea is to take the derivative and set it equal to zero." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(mcF(a),a):" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 " It surely will no
t be easy to see where this derivative is zero. We draw a graph and de
termine a numerical approximation to the solution." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "plot(mcF(a),a=0..4*Pi);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "plot(mcF(a),a=5..6,y=0..1000);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fsolve(diff(mcF(a),a)=0,a,5.4..5.6)
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " \+
This last number may be rational multiple of Pi." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "evalf(%/Pi);\nconvert(%,fraction);" }}}
{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 120 " That \+
seems to be about 7/4. Using that value, we draw the graph of F( 7 / \+
4, x) to see how close this best fit looks.." }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 37 "plot(\{cos(x),F(7*Pi/4,x)\},x=0..4*Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 12 "There is no " }{TEXT 280 1 "a" }{TEXT -1 87 " in [0, 4] for whi
ch the graph of the polynomial F(a ,x) will fit better than this one.
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1
24 "Exercise for the student" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }
{TEXT 281 1 "a" }{TEXT -1 95 " in [0, 10] so that the polynomial with \+
degree six which has the same first six derivatives as " }{XPPEDIT 18
0 "x^2*exp(-x)" "6#*&%\"xG\"\"#-%$expG6#,$F$!\"\"\"\"\"" }{TEXT -1 6 "
is a " }{TEXT 276 18 "best approximation" }{TEXT -1 1 "." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(x^2*exp(-x),x=0..10);" }}}
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