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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 52 "An Automated Linear Least Square
s Solution Generator" }}{PARA 19 "" 0 "" {TEXT -1 15 "Marc A. Murison
" }}{PARA 260 "" 0 "" {TEXT -1 74 "Astronomical Applications Departmen
t\nU.S. Naval Observatory\nWashington, DC" }}{PARA 261 "" 0 "" {TEXT 
-1 62 "murison@riemann.usno.navy.mil\nhttp://aa.usno.navy.mil/murison/
" }}{PARA 257 "" 0 "" {TEXT -1 12 "26 May, 1999" }}{SECT 0 {PARA 3 "" 
0 "1. Introduction" {TEXT -1 15 "1. Introduction" }}{EXCHG {PARA 259 "
" 0 "" {TEXT -1 766 "In day-to-day work, one often has need to perform
 a quick linear least squares calculation for some model which approxi
mates a physical process.  Since the model is likely to be something o
ther than simple linear regression, one winds up (often re-) deriving \+
the appropriate normal equations and solving for the model parameters.
  This is wasted effort, thanks to the availability of computer algebr
a systems, in which it is easy to automate this process.  In the secti
on which follows, I illustrate the basic \"quick and dirty\" linear le
ast squares process by means of a simple example.  In section 3, I pre
sent a Maple procedure that performs the algebra autmatically for any \+
model that is linear in the parameters, and in section 4 are a few exa
mples of its usage." }}}}{SECT 0 {PARA 3 "" 0 "2. A Linear Least Squar
es Example" {TEXT -1 33 "2. A Linear Least Squares Example" }}{EXCHG 
{PARA 259 "" 0 "" {TEXT -1 58 "Start with a simple second degree polyn
omial as our model:" }{TEXT 260 1 " " }{XPPEDIT 261 1 "model := A+B*t[
i]+C*t[i]^2:" "6#>%&modelG,(%\"AG\"\"\"*&%\"BGF'&%\"tG6#%\"iGF'F'*&%\"
CGF'*$&F+6#F-\"\"#F'F'" }{TEXT 262 2 ". " }{TEXT -1 10 " Then the " }
{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 14 " statistic is \+
" }{XPPEDIT 263 1 "chi^2 = Sum((y[i]-model)^2,i = 1 .. N);" "6#/*$%$ch
iG\"\"#-%$SumG6$*$,&&%\"yG6#%\"iG\"\"\"%&modelG!\"\"\"\"#/F/;\"\"\"%\"
NG" }{TEXT 264 3 ".  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$chiG\"
\"#\"\"\"-%$SumG6$*$),*&%\"yG6#%\"iG\"\"\"%\"AG!\"\"*&%\"BGF3&%\"tGF1F
3F5*&%\"CGF3)F8F'F(F5F'F(/F2;F3%\"NG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 22 "where the sum is over " }{TEXT 257 1 "N" }{TEXT -1 32 " availab
le data points, and the " }{XPPEDIT 18 0 "y[i];" "6#&%\"yG6#%\"iG" }
{TEXT -1 40 " are the data values, measured at times " }{XPPEDIT 18 0 
"t[i];" "6#&%\"tG6#%\"iG" }{TEXT -1 78 ".  In keeping with the \"quick
 and dirty\" premise, we do not take into account " }{TEXT 258 9 "a pr
iori " }{TEXT -1 167 "any weighting of the data.  It is a simple matte
r to add weight factors into the solutions afterwards; including them \+
now would serve only to clutter up the notation. " }}}{EXCHG {PARA 
259 "" 0 "" {TEXT -1 46 "Next, we calculate the partial derivatives of
 " }{XPPEDIT 18 0 "chi^2;" "6#*$%$chiG\"\"#" }{TEXT -1 111 " with resp
ect to the model parameters, setting these derivatives to zero and thu
s forming the normal equations." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "diff(rhs(%),A) = 0;" "6#/-%%diffG6$-%$rhsG6#%\"%G%\"AG\"\"!" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "diff(rhs(%%),B) = 0;" "6#/-%%diffG6$-%$
rhsG6#%#%%G%\"BG\"\"!" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "diff(rhs(%%%),
C) = 0;" "6#/-%%diffG6$-%$rhsG6#%$%%%G%\"CG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$,*&%\"yG6#%\"iG!\"#%\"AG\"\"#*&%\"BG\"\"\"&%
\"tGF*F1F.*&%\"CGF1)F2F.\"\"\"F./F+;F1%\"NG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$,$*&,*&%\"yG6#%\"iG\"\"\"%\"AG!\"\"*&%\"BGF.&
%\"tGF,F.F0*&%\"CGF.)F3\"\"#\"\"\"F0F.F3F9!\"#/F-;F.%\"NG\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,$*&,*&%\"yG6#%\"iG\"\"\"%\"
AG!\"\"*&%\"BGF.&%\"tGF,F.F0*&%\"CGF.)F3\"\"#\"\"\"F0F.F7F9!\"#/F-;F.%
\"NG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "NormalEqs := \{%, \+
%%, %%%\}:" "6#>%*NormalEqsG<%%\"%G%#%%G%$%%%G" }}}{EXCHG {PARA 259 "
" 0 "" {TEXT -1 117 "Finally, we solve the normal equations for the pa
rameter values.  These values are the linear least squares solution." 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "sols := sortsols(solve(value(
NormalEqs),\{A, B, C\}),[A, B, C]):" "6#>%%solsG-%)sortsolsG6$-%&solve
G6$-%&valueG6#%*NormalEqsG<%%\"AG%\"BG%\"CG7%F0F1F2" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "for p in sols do print(p) od;" "6#?&%\"pG%%so
lsG%%trueG-%&printG6#F$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"AG*&,.*
&-%$sumG6$&%\"yG6#%\"iG/F.;\"\"\"%\"NGF1)-F)6$*$)&%\"tGF-\"\"$\"\"\"F/
\"\"#F;F1*(F(F;-F)6$*$)F8\"\"%F;F/F1-F)6$*$)F8F<F;F/F1!\"\"*(-F)6$F8F/
F1-F)6$*&F8F1F+F1F/F1F>F;F1*(FIF;F4F1-F)6$*&FFF;F+F;F/F1FG*(FCF;F4F;FK
F;FG*&)FCF<F;FOF;F1F;,,*(FIF;FCF;F4F;!\"#*&F2F1F3F;F1*$)FCF:F;F1*&F>F;
)FIF<F;F1*(F>F;F2F;FCF;FG!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"
BG*&,.*(-%$sumG6$*$)&%\"tG6#%\"iG\"\"%\"\"\"/F0;\"\"\"%\"NGF5-F)6$&%\"
yGF/F3F5-F)6$F-F3F5F5*(F;F2-F)6$*&)F-\"\"#F2F9F5F3F5-F)6$*$FAF2F3F5!\"
\"*(-F)6$*&F-F5F9F2F3F5F6F5F(F2FF*(F7F2FCF2-F)6$*$)F-\"\"$F2F3F5FF*(F6
F2FLF2F>F2F5*&)FCFBF2FHF2F5F2,,*(F;F2FCF2FLF2!\"#*&F6F2)FLFBF2F5*$)FCF
PF2F5*&F(F2)F;FBF2F5*(F(F2F6F2FCF2FF!\"\"" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/%\"CG*&,.*(-%$sumG6$&%\"tG6#%\"iG/F.;\"\"\"%\"NGF1-F)6
$*$)F+\"\"#\"\"\"F/F1-F)6$*&F+F1&%\"yGF-F1F/F1!\"\"*(F2F1-F)6$*$)F+\"
\"$F8F/F1F9F8F1*(F@F8-F)6$F<F/F1F(F8F>*&-F)6$*&F6F8F<F8F/F1)F(F7F8F1*(
FIF8F2F8F3F8F>*&)F3F7F8FFF8F1F8,,*(F(F8F3F8F@F8!\"#*&F2F8)F@F7F8F1*$)F
3FDF8F1*&-F)6$*$)F+\"\"%F8F/F1FLF8F1*(FXF8F2F8F3F8F>!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 63 "This is a bit easier to look at if we mak
e a few substitutions." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(
seq(sum(t[i]^k,i = 1 .. N) = T[k],k = 1 .. 4),seq(sum(y[i]*t[i]^k,i = \+
1 .. N) = S[k],k = 0 .. 2),sols);" "6#-%%subsG6%-%$seqG6$/-%$sumG6$)&%
\"tG6#%\"iG%\"kG/F1;\"\"\"%\"NG&%\"TG6#F2/F2;\"\"\"\"\"%-F'6$/-F+6$*&&
%\"yG6#F1\"\"\")&F/6#F1F2FG/F1;\"\"\"F6&%\"SG6#F2/F2;\"\"!\"\"#%%solsG
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/%\"AG*&,.*&&%\"SG6#\"\"!\"\"\")
&%\"TG6#\"\"$\"\"#\"\"\"F-*(F)F4&F06#\"\"%F-&F06#F3F-!\"\"*(&F06#F-F-&
F*F>F-F6F4F-*(F=F4F/F-&F*F:F-F;*(F9F4F/F4F?F4F;*&)F9F3F4FAF4F-F4,,*(F=
F4F9F4F/F4!\"#*&%\"NGF-F.F4F-*$)F9F2F4F-*&F6F4)F=F3F4F-*(F6F4FIF4F9F4F
;!\"\"/%\"BG*&,.*(F6F4F)F4F=F4F-*(F=F4FAF4F9F4F;*(F?F4FIF4F6F4F;*(F)F4
F9F4F/F4F;*(FIF4F/F4FAF4F-*&FDF4F?F4F-F4FEFO/%\"CG*&,.*(F=F4F9F4F?F4F;
*(FIF4F/F4F?F4F-*(F/F4F)F4F=F4F;*&FAF4FMF4F-*(FAF4FIF4F9F4F;*&FDF4F)F4
F-F4FEFO" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Collecting on the mix
ed terms " }{XPPEDIT 18 0 "S[k];" "6#&%\"SG6#%\"kG" }{TEXT -1 20 ", we
 have the result" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(%,[
seq(S[k],k = 0 .. 2), N],factor):" "6#-%(collectG6%%\"%G7$-%$seqG6$&%
\"SG6#%\"kG/F.;\"\"!\"\"#%\"NG%'factorG" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "for p in % do print(p) od;" "6#?&%\"pG%\"%G%%trueG-%&pr
intG6#F$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"AG,(*&*&,&*$)&%\"TG6#
\"\"$\"\"#\"\"\"\"\"\"*&&F,6#\"\"%F1&F,6#F/F1!\"\"F0&%\"SG6#\"\"!F1F0,
**&F(F1%\"NGF1F1*(&F,6#F1F1F6F0F+F1!\"#*$)F6F.F0F1*&F3F0)FAF/F0F1!\"\"
F1*&*&,&*&F+F0F6F0F8*&FAF0F3F0F1F1&F:FBF1F0F=FHF1*&*&,&*$)F6F/F0F1*&FA
F0F+F0F8F1&F:F7F1F0F=FHF1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"BG,(*
&*&,&*&&%\"TG6#\"\"$\"\"\"&F+6#\"\"#F.!\"\"*&&F+6#F.F.&F+6#\"\"%F.F.F.
&%\"SG6#\"\"!F.\"\"\",**&,&*$)F*F1F=F.*&F6F=F/F=F2F.%\"NGF.F.*(F4F=F/F
=F*F=!\"#*$)F/F-F=F.*&F6F=)F4F1F=F.!\"\"F.*&*&,&*&FDF=F6F=F2*$)F/F1F=F
.F.&F:F5F.F=F>FKF.*&*&,&*&FDF=F*F=F.*&F4F=F/F=F2F.&F:F0F.F=F>FKF." }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"CG,(*&*&,&*$)&%\"TG6#\"\"#F.\"\"\"
\"\"\"*&&F,6#F0F0&F,6#\"\"$F0!\"\"F0&%\"SG6#\"\"!F0F/,**&,&*$)F4F.F/F0
*&&F,6#\"\"%F0F+F0F7F0%\"NGF0F0*(F2F/F+F/F4F/!\"#*$)F+F6F/F0*&FBF/)F2F
.F/F0!\"\"F0*&*&,&*&FEF/F4F/F0*&F2F/F+F/F7F0&F9F3F0F/F<FLF0*&*&,&*$FKF
/F0*&FEF/F+F/F7F0&F9F-F0F/F<FLF0" }}}}{SECT 0 {PARA 3 "" 0 "3. An Auto
mated Linear Least Squares Solution Generator" {TEXT -1 55 "3. An Auto
mated Linear Least Squares Solution Generator" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 95 "Here is a procedure that automates the process outlined
 in the example of the previous section." }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 6707 "#--------------------------------------------------
-------------------\n# model must be of the form Y = F(t[k],P), where \+
P are parameters\n# and t is the independent variable.  The sumindex s
ubscript [k] MUST \n# be attached to t in the model equation.  For exa
mple,\n#    leastsqrs( y=a+b*x[k], x, k, [a,b] );\n# Also valid is\n# \+
   leastsqrs( y[k]=a+b*x[k], x, k, [a,b] );\n#------------------------
---------------------------------------------\nleastsqrs := proc( mode
l::`=`, indepvar::name, \n                   sumindex::name, params::l
ist )\n\n  local chi2, y, i, t, partials, k, eqs, solset, T, Y, \n    \+
    replace_denominators, sums, YTsums, n, p, nmax, N, \n        Tcoun
t, Scount, ispoly, Delta, delta, tmp, goodsols;\n  global _subslist, t
ime0;\n\n  if not has(model,sumindex) then\n    ERROR(\"model must be \+
of the form Y=F(t[i],P)\");\n  fi;\n\n  time0 := time();\n\n  i := sum
index;\n  t := indepvar;\n  if not type(t,indexed) then\n    t := t[i]
;\n  fi;\n  y := lhs(model);\n  if not type(y,indexed) then\n    y := \+
y[i];\n  fi;\n  chi2 := Sum((y-rhs(model))^2,i=1..N);\n  if type(rhs(m
odel),polynom) then\n    ispoly := true;\n  else\n    ispoly := false;
\n  fi;\n  debug_print(procname,\"i,t,y,ispoly,chi2\",4,i,t,y,ispoly,c
hi2);\n\n  #-----------------------\n  # calculate the partials\n  #--
---------------------\n  debug_print(procname,\"Forming the normal equ
ations...\",3);\n  partials := [];\n  for k from 1 to nops(params) do
\n    partials := [ op(partials), diff(chi2,params[k])=0 ];\n  od;\n  \+
if printlevel >= 4 then\n    debug_print(procname,\"partials\",4);\n  \+
  for p in partials do\n      print(p);\n    od;\n  fi;\n\n  #--------
-----------------------------------\n  # form the summation substituti
ons variables\n  #-------------------------------------------\n  sums \+
:= convert( select( (x,y)->op(0,x)=y, \n                           ind
ets(value(expand(partials)),function), sum ), \n                   lis
t );\n  #degree() can't handle summation subscripts, so remove them\n \+
 YTsums    := subs( t=T, y=Y, sums );\n  _subslist := [];\n  nmax     \+
 := 0;\n  debug_print(procname,\"summation terms\",4,sums);\n  debug_p
rint(procname,\"YT terms\",5,YTsums);\n  Tcount := 0;\n  Scount := 0;
\n  for k from 1 to nops(YTsums) do\n    if ispoly then\n      select(
has,[op(YTsums[k])],T);\n      if %=[] then\n        n := 0;\n      el
se\n        n := degree( op(%), T );\n      fi;\n      nmax := max(n,n
max);\n    fi;\n    if has(YTsums[k],Y) then\n      if not ispoly then
 \n        n := Scount;\n        nmax := max(n,nmax);\n        Scount \+
:= Scount + 1;\n      fi;\n      _subslist := [op(_subslist),sums[k]='
S'[n]];\n    else\n      if not ispoly then \n        n := Tcount;\n  \+
      Tcount := Tcount + 1;\n      fi;\n      _subslist := [op(_subsli
st),sums[k]='T'[n]];\n    fi;\n  od;\n\n  #---------------------------
------------------------\n  # substitute for the summation terms in th
e partials\n  #---------------------------------------------------\n  \+
partials := collect( eval( value(expand(partials)), _subslist ), \n   \+
                    [seq(S[k],k=1..nmax),N], factor );\n\n  #put subst
itution list into form suitable for putting the sums back\n  _subslist
 := map( (x)->rhs(x)=lhs(x), _subslist );\n\n  #----------------------
----\n  # form the normal equations\n  #--------------------------\n  \+
eqs := collect( partials, [op(params),sum], factor );\n  if printlevel
 >= 1 then\n    debug_print(procname,\"normal equations\",1);\n    for
 p in eqs do\n      print(p);\n    od;\n  fi;\n\n  #------------------
---------\n  # solve the normal equations \n  #-----------------------
----\n  debug_print(procname,\"Solving the normal equations...\",1);\n
  _EnvAllSolutions := true;\n  map( allvalues, [solve( convert(eqs,set
), convert(params,set) )] );\n  map( sortsols, %, params );\n  if nops
(%)=1 then\n    solset := op(%);\n  else\n    solset := %;\n    debug_
print(procname,\"There are \".(nops(solset)).\" solutions\",1);\n  fi;
\n  debug_print(procname,\"raw solution(s)\",4,solset);\n\n  #--------
-------------------------------------------\n  # replace the denominat
ors with another substitution\n  #------------------------------------
---------------\n\n  #first define a workhorse procedure\n  replace_de
nominators := proc( expr, N, nmax, S, Delta )\n\n    local k, p, d;\n \+
   global _subslist;\n\n    #---------------------------------\n    # \+
determine the common denominator\n    #-------------------------------
--\n    #grab and filter denominators from solution parameter expressi
ons\n    map( denom, \{ seq(op(rhs(expr[k])),k=1..nops(expr)) \} );\n \+
   map( (x)->if type(x,\{integer,fraction\}) then 0 else x fi, % );\n \+
   d := % minus \{0\};\n    if nops(d) > 1 then\n      debug_print(pro
cname,\"denominator set\",0,d);\n      ERROR(\"Unable to determine a u
nique denominator!\");\n    elif d=\{\} then\n      debug_print(procna
me,\"no denominator for solution\",2,expr);\n      RETURN(expr);\n    \+
else\n      d := op(d);\n    fi;\n\n    #-----------------------------
---------\n    # make the substitution in the solution\n    #---------
-----------------------------\n    _subslist := [ Delta=eval(d,_subsli
st), op(_subslist) ];\n    collect( algsubs(d=Delta,expr), [Delta,seq(
S[k],k=0..nmax),N], factor );\n  end;\n\n  #now make the substitutions
\n  if type(solset,list(list)) then    #multiple solutions\n    for k \+
from 1 to nops(solset) do\n      subsop(k=replace_denominators(solset[
k],N,nmax,S,Delta),solset);\n      solset    := subs( Delta=delta[k], \+
% );\n      _subslist := subs( Delta=delta[k], _subslist );\n    od;\n
  else                               #only one solution\n    solset :=
 replace_denominators(solset,N,nmax,S,Delta);\n  fi;\n\n  debug_print(
procname,\"substitution list\",2,_subslist);\n\n  #-------------------
---\n  # check the solution(s)\n  #----------------------\n  if type(s
olset,list(list)) then  #multiple solutions\n    debug_print(procname,
\"Verifying the \".(nops(solset)).\" solutions...\",1);\n    goodsols \+
:= [];\n    for k from 1 to nops(solset) do\n      tmp := expand(subs(
_subslist,eval(eqs,solset[k])));\n      if not type(tmp,list(0=0)) the
n\n        debug_print(procname,\"Solution \".k.\" is invalid!\",0,sol
set[k]);\n        debug_print(procname,\"Solution \".k.\" substituted \+
into normal eqs\",0,eqs);\n        debug_print(procname,\"Throwing out
 solution \".k,0);\n      else\n        goodsols := [op(goodsols),sols
et[k]];\n      fi;\n    od;\n    solset := goodsols;\n    if nops(sols
et)=1 then \n      solset := op(solset); \n    fi;\n  else            \+
                 #single solution\n    debug_print(procname,\"Verifyin
g the solution...\",1);\n    eqs := factor(expand(subs(_subslist,eval(
eqs,solset))));\n    if not type(eqs,list(0=0)) then\n      debug_prin
t(procname,\"Solution substituted into normal eqs\",0,eqs);\n      ERR
OR(\"Invalid solution!\");\n    fi;\n  fi;\n\n  if time()-time0 > 10 t
hen\n    debug_print(procname,\"Done!\",1);\n  fi;\n  solset;\nend:" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}}{SECT 0 {PARA 
3 "" 0 "4. Examples" {TEXT -1 11 "4. Examples" }}{SECT 0 {PARA 4 "" 0 
"Polynomial of degree 1" {TEXT -1 22 "Polynomial of degree 1" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "We start with simple linear regre
ssion to check the program.  We set the procedure's \"chattiness\" to \+
maximum verbosity.  The solution appears as the last bracketed express
ion." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "printlevel := 4:" "6#>%
+printlevelG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsqrs(
y = A+B*t[i],t,i,[A, B]);" "6#-%*leastsqrsG6&/%\"yG,&%\"AG\"\"\"*&%\"B
GF*&%\"tG6#%\"iGF*F*F.F07$F)F," }}{PARA 6 "" 1 "" {TEXT -1 31 "leastsq
rs[0]: i,t,y,ispoly,chi2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%\"iG&%\"t
G6#F#&%\"yGF&%%trueG-%$SumG6$*$),(F'\"\"\"%\"AG!\"\"*&%\"BGF0F$F0F2\"
\"#\"\"\"/F#;F0%\"NG" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsqrs[0]: For
ming the normal equations..." }}{PARA 6 "" 1 "" {TEXT -1 22 "leastsqrs
[0]: partials" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,(&%\"yG6#%
\"iG!\"#%\"AG\"\"#*&%\"BG\"\"\"&%\"tGF*F1F./F+;F1%\"NG\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,$*&,(&%\"yG6#%\"iG\"\"\"%\"AG!\"
\"*&%\"BGF.&%\"tGF,F.F0F.F3\"\"\"!\"#/F-;F.%\"NG\"\"!" }}{PARA 6 "" 1 
"" {TEXT -1 29 "leastsqrs[0]: summation terms" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&-%$sumG6$&%\"yG6#%\"iG/F*;\"\"\"%\"NG-F%6$*&&%\"tGF)F
-F'F-F+-F%6$F2F+-F%6$*$)F2\"\"#\"\"\"F+" }}{PARA 6 "" 1 "" {TEXT -1 
30 "leastsqrs[0]: normal equations" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/,(&%\"SG6#\"\"!!\"#*&%\"AG\"\"\"%\"NGF,\"\"#*&%\"BGF,&%\"TG6#F,F,F.F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"SG6#\"\"\"!\"#*&%\"AGF(&%\"
TGF'F(\"\"#*&%\"BGF(&F-6#F.F(F.\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 45 "l
eastsqrs[0]: Solving the normal equations..." }}{PARA 6 "" 1 "" {TEXT 
-1 29 "leastsqrs[0]: raw solution(s)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#7$/%\"AG*&,&*&&%\"SG6#\"\"!\"\"\"&%\"TG6#\"\"#F-F-*&&F/6#F-F-&F*F4F-
!\"\"\"\"\",&*$)F3F1F7F6*&F.F7%\"NGF-F-!\"\"/%\"BG,$*&,&*&F5F7F<F7F6*&
F3F7F)F7F-F7F8F=F6" }}{PARA 6 "" 1 "" {TEXT -1 31 "leastsqrs[0]: subst
itution list" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/%&DeltaG,&*$)-%$sum
G6$&%\"tG6#%\"iG/F/;\"\"\"%\"NG\"\"#\"\"\"!\"\"*&-F*6$*$)F,F4F5F0F2F3F
2F2/&%\"SG6#\"\"!-F*6$&%\"yGF.F0/&F>6#F2-F*6$*&F,F2FCF2F0/&%\"TGFGF)/&
FM6#F4F8" }}{PARA 6 "" 1 "" {TEXT -1 39 "leastsqrs[0]: Verifying the s
olution..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"AG*&,&*&&%\"SG6#\"
\"!\"\"\"&%\"TG6#\"\"#F-F-*&&F/6#F-F-&F*F4F-!\"\"\"\"\"%&DeltaG!\"\"/%
\"BG*&,&*&F5F7%\"NGF-F-*&F3F7F)F7F6F7F8F9" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 52 "The substitutions are stored in the global variable " }
{TEXT 259 9 "_subslist" }{TEXT -1 30 ".  Hence, the full solution is" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "eval(%,_subslist);" "6#-%%eva
lG6$%\"%G%*_subslistG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"AG*&,&*
&-%$sumG6$&%\"yG6#%\"iG/F/;\"\"\"%\"NGF2-F*6$*$)&%\"tGF.\"\"#\"\"\"F0F
2F2*&-F*6$F8F0F2-F*6$*&F8F2F,F2F0F2!\"\"F;,&*$)F=F:F;FB*&F4F;F3F2F2!\"
\"/%\"BG*&,&*&F?F;F3F;F2*&F=F;F)F;FBF;FCFG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 229 "We can convert the solution into an optimized Maple proc
edure.  This procedure can then be used to evaluate the model with a s
et of data.  It is easily saved to disk for conversion into a C or for
tran procedure \227 also quite easy." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "tmp := array(map(proc (x) options operator, arrow; rhs(
x) end,%)):" "6#>%$tmpG-%&arrayG6#-%$mapG6$R6#%\"xG7\"6$%)operatorG%&a
rrowG6\"-%$rhsG6#F-F2F2F2%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "foo := optimize(makeproc(tmp,parameters = [N, y, t]));" "6#>%$fooG-
%)optimizeG6#-%)makeprocG6$%$tmpG/%+parametersG7%%\"NG%\"yG%\"tG" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$fooGR6%%\"NG%\"yG%\"tG6)%#t8G%$t11G
%#t4G%#t5G%#t1G%$tmpG%#t2G6\"F2C,>8)-%&arrayG6#;\"\"\"\"\"#>8(-%$sumG6
$&9%6#%\"iG/FD;F:%\"NG>8*-F?6$*$)&9&FCF;\"\"\"FE>8&-F?6$FNFE>8'-F?6$*&
FNF:FAF:FE>8$*$)FRF;FP>8%*&FPFP,&Fen!\"\"*&FIF:FGF:F:!\"\">&F56#F:*&,&
*&F=F:FIFPF:*&FRF:FVF:F\\oF:FinF:>&F56#F;*&,&*&FVFPFGFPF:*&FRFPF=FPF\\
oF:FinFPF5F2F2F2" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "writeto(\"d
:/Maple/test.p\");" "6#-%(writetoG6#Q0d:/Maple/test.p6\"" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "print(foo);" "6#-%&printG6#%$fooG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "writeto(terminal);" "6#-%(writet
oG6#%)terminalG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "This is simpl
e and straightforward in this example.  Later on, we will see that the
 solutions can get rather unwieldy, but that the optimized procedures \+
created from them are still relatively simple. " }}}}{SECT 0 {PARA 4 "
" 0 "Polynomial of degree 2" {TEXT -1 22 "Polynomial of degree 2" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "printlevel := 2:" "6#>%+printlev
elG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsqrs(Y = A+B*t
[i]+C*t[i]^2,t,i,[A, B, C]);" "6#-%*leastsqrsG6&/%\"YG,(%\"AG\"\"\"*&%
\"BGF*&%\"tG6#%\"iGF*F**&%\"CGF**$&F.6#F0\"\"#F*F*F.F07%F)F,F2" }}
{PARA 6 "" 1 "" {TEXT -1 30 "leastsqrs[0]: normal equations" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,*&%\"SG6#\"\"!!\"#*&%\"AG\"\"\"%\"NGF,\"\"
#*&%\"BGF,&%\"TG6#F,F,F.*&%\"CGF,&F26#F.F,F.F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,*&%\"SG6#\"\"\"!\"#*&%\"AGF(&%\"TGF'F(\"\"#*&%\"BGF(&
F-6#F.F(F.*&%\"CGF(&F-6#\"\"$F(F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,*&%\"SG6#\"\"#!\"#*&%\"AG\"\"\"&%\"TGF'F,F(*&%\"BGF,&F.6#\"\"$
F,F(*&%\"CGF,&F.6#\"\"%F,F(\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 45 "least
sqrs[0]: Solving the normal equations..." }}{PARA 6 "" 1 "" {TEXT -1 
31 "leastsqrs[0]: substitution list" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#7*/%&DeltaG,,*(-%$sumG6$&%\"tG6#%\"iG/F.;\"\"\"%\"NGF1-F)6$*$)F+\"\"#
\"\"\"F/F1-F)6$*$)F+\"\"$F8F/F1!\"#*&)F9F7F8F2F1F1*$)F3F=F8F1*(-F)6$*$
)F+\"\"%F8F/F1F2F8F3F8!\"\"*&FDF8)F(F7F8F1/&%\"SG6#\"\"!-F)6$&%\"YGF-F
//&%\"TG6#F1F(/&FW6#F7F3/&FNFX-F)6$*&F+F1FSF1F//&FW6#F=F9/&FNFen-F)6$*
&F6F8FSF8F//&FW6#FHFD" }}{PARA 6 "" 1 "" {TEXT -1 39 "leastsqrs[0]: Ve
rifying the solution..." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/%\"AG*&,
(*&,&*$)&%\"TG6#\"\"$\"\"#\"\"\"\"\"\"*&&F-6#\"\"%F2&F-6#F0F2!\"\"F2&%
\"SG6#\"\"!F2F2*&,&*&F7F1F,F2F9*&&F-6#F2F2F4F1F2F2&F;FCF2F2*&,&*&F,F1F
BF1F9*$)F7F0F1F2F2&F;F8F2F2F1%&DeltaG!\"\"/%\"BG*&,(*&F?F1F:F1F2*&,&FH
F2*&F4F1%\"NGF2F9F2FDF1F2*&,&*&FBF1F7F1F9*&F,F1FUF1F2F2FJF1F2F1FKFL/%
\"CG*&,(*&FFF1F:F1F2*&FWF1FDF1F2*&,&*&F7F1FUF1F9*$)FBF0F1F2F2FJF1F2F1F
KFL" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "tmp := array(map(proc (x
) options operator, arrow; rhs(x) end,eval(%,_subslist))):" "6#>%$tmpG
-%&arrayG6#-%$mapG6$R6#%\"xG7\"6$%)operatorG%&arrowG6\"-%$rhsG6#F-F2F2
F2-%%evalG6$%\"%G%*_subslistG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "optimize(makeproc(tmp,parameters = [N, Y, t]));" "6#-%)optimizeG6#-
%)makeprocG6$%$tmpG/%+parametersG7%%\"NG%\"YG%\"tG" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#R6%%\"NG%\"YG%\"tG63%#t7G%$t26G%$t31G%$t12G%$t16G%$t2
8G%$t17G%$t13G%$t21G%$t18G%$t36G%#t4G%$t10G%$tmpG%#t3G%#t1G%#t2G6\"F:C
7>81-%&arrayG6#;\"\"\"\"\"$>83-%$sumG6$*$)&9&6#%\"iGFC\"\"\"/FN;FB%\"N
G>84*$)FE\"\"#FO>82-FG6$*$)FK\"\"%FOFP>8/-FG6$*$)FKFWFOFP>8$-FG6$&9%FM
FP>80-FG6$FKFP>8',&*&FjnFBFEFB!\"\"*&FfoFBFYFBFB>8+-FG6$*&FKFBFcoFBFP>
8(*$)FjnFWFO>8*,&*&FEFOFfoFOF]pFepFB>8--FG6$*&F^oFOFcoFOFP>8,*&FfoFOFj
nFO>8%*&FYFOFRFB>8)*$)FfoFWFO>8&*&FOFO,,*&FbqFBFEFO!\"#*&FTFBFRFOFB*&F
epFBFjnFOFB*&FeqFBFjnFOF]p*&FYFOFhqFBFB!\"\">&F=6#FB*&,(*&,&FTFB*&FYFO
FjnFOF]pFBF`oFBFB*&FjoFBF`pFBFB*&FipFBF]qFBFBFBF\\rFB>8.,&FbqF]p*&FEFO
FRFOFB>&F=6#FW*&,(*&FjoFOF`oFOFB*&,&FepFBFeqF]pFBF`pFOFB*&FasFBF]qFOFB
FBF\\rFO>&F=6#FC*&,(*&FipFOF`oFOFB*&FasFOF`pFOFB*&,&*&FjnFOFRFOF]pFhqF
BFBF]qFOFBFBF\\rFOF=F:F:F:" }}}}{SECT 0 {PARA 4 "" 0 "Polynomial of de
gree 3" {TEXT -1 22 "Polynomial of degree 3" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 75 "In this example, we will not show the solution, since it \+
is a bit unwieldy." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsqrs
(Y = A+B*t[i]+C*t[i]^2+D*t[i]^3,t,i,[A, B, C, D]):" "6#-%*leastsqrsG6&
/%\"YG,*%\"AG\"\"\"*&%\"BGF*&%\"tG6#%\"iGF*F**&%\"CGF**$&F.6#F0\"\"#F*
F**&%\"DGF**$&F.6#F0\"\"$F*F*F.F07&F)F,F2F8" }}{PARA 6 "" 1 "" {TEXT 
-1 30 "leastsqrs[0]: normal equations" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/,,&%\"SG6#\"\"!!\"#*&%\"AG\"\"\"%\"NGF,\"\"#*&%\"BGF,&%\"TG6#F,F,F
.*&%\"CGF,&F26#F.F,F.*&%\"DGF,&F26#\"\"$F,F.F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,,&%\"SG6#\"\"\"!\"#*&%\"AGF(&%\"TGF'F(\"\"#*&%\"BGF(&
F-6#F.F(F.*&%\"CGF(&F-6#\"\"$F(F.*&%\"DGF(&F-6#\"\"%F(F.\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"SG6#\"\"#!\"#*&%\"AG\"\"\"&%\"T
GF'F,F(*&%\"BGF,&F.6#\"\"$F,F(*&%\"CGF,&F.6#\"\"%F,F(*&%\"DGF,&F.6#\"
\"&F,F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"SG6#\"\"$!\"#*&
%\"AG\"\"\"&%\"TGF'F,\"\"#*&%\"BGF,&F.6#\"\"%F,F/*&%\"CGF,&F.6#\"\"&F,
F/*&%\"DGF,&F.6#\"\"'F,F/\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsq
rs[0]: Solving the normal equations..." }}{PARA 6 "" 1 "" {TEXT -1 31 
"leastsqrs[1]: substitution list" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7-
/%&DeltaG,B*&-%$sumG6$*$)&%\"tG6#%\"iG\"\"'\"\"\"/F0;\"\"\"%\"NGF5)-F)
6$*$)F-\"\"#F2F3\"\"$F2F5*()F8F<F2-F)6$*$)F-F=F2F3F5-F)6$*$)F-\"\"&F2F
3F5!\"#*&F?F2)-F)6$*$)F-\"\"%F2F3F<F2!\"\"*(F8F5)FDF<F2F6F5F5**F8F2F(F
2FLF5F6F2FQ**F8F2-F)6$F-F3F5FLF2FDF2F<*(F8F2)F@F<F2FLF2F=**F8F2FVF2F(F
2F@F2FI*&)FLF=F2F6F2F5*(F(F2FYF2F6F2F5**FLF2F6F2F@F2FDF2FI*(FVF2F@F2FK
F2FI*(FVF2FYF2FDF2F<*()FVF<F2F(F2FLF2F5*&F\\oF2FSF2FQ*$)F@FPF2FQ/&%\"T
G6#F1F(/&Fbo6#FHFD/&%\"SG6#\"\"!-F)6$&%\"YGF/F3/&Fbo6#F<F8/&Fbo6#F5FV/
&Fbo6#F=F@/&FioFep-F)6$*&F-F5F^pF5F3/&FioFhp-F)6$*&FCF2F^pF2F3/&Fbo6#F
PFL/&FioFbp-F)6$*&F;F2F^pF2F3" }}{PARA 6 "" 1 "" {TEXT -1 39 "leastsqr
s[2]: Verifying the solution..." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "map(cost,%);" "6#-%$mapG6$%%costG%\"%G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7&,*%*additionsG\"#@%0multiplicationsG\"#]%*divisionsG
\"\"\"%,assignmentsGF*,*F%\"#AF'\"#[F)F*F+F*F,,*F%F&F'\"#ZF)F*F+F*" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The following commands produce an
 optimized procedure, which we will not show here." }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "tmp := array(map(proc (x) options operator, arr
ow; rhs(x) end,eval(%%,_subslist))):" "6#>%$tmpG-%&arrayG6#-%$mapG6$R6
#%\"xG7\"6$%)operatorG%&arrowG6\"-%$rhsG6#F-F2F2F2-%%evalG6$%#%%G%*_su
bslistG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "optimize(makeproc(tm
p,parameters = [N, Y, t])):" "6#-%)optimizeG6#-%)makeprocG6$%$tmpG/%+p
arametersG7%%\"NG%\"YG%\"tG" }}}}{SECT 0 {PARA 4 "" 0 "Sinusoid \227 R
epresentation 1" {TEXT -1 27 "Sinusoid \227 Representation 1" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "printlevel := 1:" "6#>%+printlev
elG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsqrs(Y = A*co
s(2*Pi*nu*t[i]+phi),t,i,[A, phi]);" "6#-%*leastsqrsG6&/%\"YG*&%\"AG\"
\"\"-%$cosG6#,&**\"\"#F*%#PiGF*%#nuGF*&%\"tG6#%\"iGF*F*%$phiGF*F*F4F67
$F)F7" }}{PARA 6 "" 1 "" {TEXT -1 30 "leastsqrs[0]: normal equations" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**&,.*&)-%$cosG6#%$phiG\"\"#\"\"\"
&%\"TG6#\"\"!\"\"\"\"\")*&F(F.&F06#F3F3!\")*(F)F3-%$sinGF+F3&F06#F-F3!
#;*&F(F.%\"NGF3F-*(F)F.F:F.&F06#\"\"$F3F4*&)F:F-F.&F06#\"\"%F3F4F3%\"A
GF3F3*&F)F.&%\"SGF1F3!\"%*&F)F.&FMF7F3F-*&F:F.&FMF=F3FIF2" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#/,&*&,2*(-%$cosG6#%$phiG\"\"\"-%$sinGF*F,%\"NGF
,!\"#*(F(\"\"\"F-F2&%\"TG6#\"\"!F,!\")*&)F(\"\"#F2&F46#\"\"$F,\"\"%*&F
9F2&F46#F:F,F7*(F(F2F-F2&F46#F>F,\"\")*(F(F2F-F2&F46#F,F,FE*&)F-F:F2F;
F2!\"%*&FJF2F@F2FEF,)%\"AGF:F2F,*&,(*&F-F2&%\"SGFHF,F0*&F(F2&FSFAF,F>*
&F-F2&FSF5F,F>F,FNF,F,F6" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsqrs[0]:
 Solving the normal equations..." }}{PARA 6 "" 1 "" {TEXT -1 35 "least
sqrs[2]: There are 3 solutions" }}{PARA 6 "" 1 "" {TEXT -1 42 "leastsq
rs[4]: Verifying the 3 solutions..." }}{PARA 6 "" 1 "" {TEXT -1 37 "le
astsqrs[18]: Solution 2 is invalid!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#7$/%\"AG,$*&*$-%%sqrtG6#,fo**&%\"TG6#\"\"#\"\"\"&%\"SGF0F2&F46#F2F2&F
/6#\"\"%F2\"#;**&F/6#\"\"$F2F3\"\"\"&F46#\"\"!F2F7F?F:**F<F?F3F?F5F?F7
F?!\")**F.F?F3F?F@F?F7F?!#K*(F.F?)F3F1F?F<F?!#;*(F@F?)F7F1F?F5F?FI*&)F
.F1F?FHF?F:*(F.F?)F5F1F?F<F?!\"%*()F<F1F?F@F?F5F?FP**F.F?F@F?F<F?F5F?F
:**F3F?&F/F6F2F<F?F5F?\"\")**F3F?&F/FAF2F<F?F5F?FD**F3F?FUF?F.F?F@F?\"
#K**F3F?FUF?F.F?F5F?FI**F3F?FUF?F<F?F@F?FI*(FMF?F@F?F5F?FI**F3F?%\"NGF
2F<F?F5F?!\"#**F3F?FXF?F.F?F@F?FF**F3F?FXF?F.F?F5F?F:**F3F?FXF?F<F?F@F
?F:**F3F?FinF?F.F?F5F?F9**F3F?FinF?F<F?F@F?F9*(F.F?)F@F1F?F<F?FI*(FHF?
FinF?FXF?FV*(FHF?FinF?FUF?FD*(FHF?FXF?FUF?FF*&FHF?)FinF1F?F2*&FHF?)FXF
1F?F:*&FHF?)FUF1F?F:*&FMF?FaoF?F:*&FMF?FOF?F9*&FRF?FaoF?F9*&FRF?FOF?F2
**F3F?FinF?F.F?F@F?FD*&FRF?FHF?F9*&FaoF?FKF?F:*&FOF?FKF?F9F?F?&%&delta
GF0!\"\"#F2F1/%$phiG,&-%'arctanG6$*&,0*&F3F?FinF?!\"\"*&F3F?FXF?FP*&F3
F?FUF?F9*&F.F?F@F?F9*&F.F?F5F?Fjn*&F<F?F@F?Fjn*&F<F?F5F?F2F?*$-F*6#F,F
?Fep,$*&,**&F.F?F3F?Fjn*&F<F?F3F?F2*&F@F?F7F?F1*&F5F?F7F?F`qF?*$-F*6#F
,F?FepF1F2*&%#PiGF2%$_Z|irGF2F1" }}{PARA 6 "" 1 "" {TEXT -1 53 "leasts
qrs[18]: Solution 2 substituted into normal eqs" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7$/,**&,.*&)-%$cosG6#%$phiG\"\"#\"\"\"&%\"TG6#\"\"!\"\"
\"\"\")*&F)F/&F16#F4F4!\")*(F*F4-%$sinGF,F4&F16#F.F4!#;*&F)F/%\"NGF4F.
*(F*F/F;F/&F16#\"\"$F4F5*&)F;F.F/&F16#\"\"%F4F5F4%\"AGF4F4*&F*F/&%\"SG
F2F4!\"%*&F*F/&FNF8F4F.*&F;F/&FNF>F4FJF3/,&*&,2*(F*F/F;F/FAF/!\"#*(F*F
/F;F/F0F/F9*&F)F/FCF/FJ*&F)F/F=F/F9*(F*F/F;F/FHF/F5*(F*F/F;F/F7F/F5*&F
GF/FCF/FO*&FGF/F=F/F5F4)FKF.F/F4*&,(*&F;F/FQF/FY*&F*F/FSF/FJ*&F;F/FMF/
FJF4FKF/F4F3" }}{PARA 6 "" 1 "" {TEXT -1 38 "leastsqrs[18]: Throwing o
ut solution 2" }}{PARA 6 "" 1 "" {TEXT -1 37 "leastsqrs[31]: Solution \+
3 is invalid!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$/%\"AG,$*&*$-%%sqrt
G6#,fo**&%\"TG6#\"\"#\"\"\"&%\"SGF0F2&F46#F2F2&F/6#\"\"%F2\"#;**&F/6#
\"\"$F2F3\"\"\"&F46#\"\"!F2F7F?F:**F<F?F3F?F5F?F7F?!\")**F.F?F3F?F@F?F
7F?!#K*(F.F?)F3F1F?F<F?!#;*(F@F?)F7F1F?F5F?FI*&)F.F1F?FHF?F:*(F.F?)F5F
1F?F<F?!\"%*()F<F1F?F@F?F5F?FP**F.F?F@F?F<F?F5F?F:**F3F?&F/F6F2F<F?F5F
?\"\")**F3F?&F/FAF2F<F?F5F?FD**F3F?FUF?F.F?F@F?\"#K**F3F?FUF?F.F?F5F?F
I**F3F?FUF?F<F?F@F?FI*(FMF?F@F?F5F?FI**F3F?%\"NGF2F<F?F5F?!\"#**F3F?FX
F?F.F?F@F?FF**F3F?FXF?F.F?F5F?F:**F3F?FXF?F<F?F@F?F:**F3F?FinF?F.F?F5F
?F9**F3F?FinF?F<F?F@F?F9*(F.F?)F@F1F?F<F?FI*(FHF?FinF?FXF?FV*(FHF?FinF
?FUF?FD*(FHF?FXF?FUF?FF*&FHF?)FinF1F?F2*&FHF?)FXF1F?F:*&FHF?)FUF1F?F:*
&FMF?FaoF?F:*&FMF?FOF?F9*&FRF?FaoF?F9*&FRF?FOF?F2**F3F?FinF?F.F?F@F?FD
*&FRF?FHF?F9*&FaoF?FKF?F:*&FOF?FKF?F9F?F?&%&deltaGF=!\"\"#!\"\"F1/%$ph
iG,&-%'arctanG6$,$*&,0*&F3F?FinF?Fgp*&F3F?FXF?FP*&F3F?FUF?F9*&F.F?F@F?
F9*&F.F?F5F?Fjn*&F<F?F@F?Fjn*&F<F?F5F?F2F?*$-F*6#F,F?FepFgp,$*&,**&F.F
?F3F?Fjn*&F<F?F3F?F2*&F@F?F7F?F1*&F5F?F7F?FgpF?*$-F*6#F,F?FepFjnF2*&%#
PiGF2%$_Z|irGF2F1" }}{PARA 6 "" 1 "" {TEXT -1 53 "leastsqrs[31]: Solut
ion 3 substituted into normal eqs" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7
$/,**&,.*&)-%$cosG6#%$phiG\"\"#\"\"\"&%\"TG6#\"\"!\"\"\"\"\")*&F)F/&F1
6#F4F4!\")*(F*F4-%$sinGF,F4&F16#F.F4!#;*&F)F/%\"NGF4F.*(F*F/F;F/&F16#
\"\"$F4F5*&)F;F.F/&F16#\"\"%F4F5F4%\"AGF4F4*&F*F/&%\"SGF2F4!\"%*&F*F/&
FNF8F4F.*&F;F/&FNF>F4FJF3/,&*&,2*(F*F/F;F/FAF/!\"#*(F*F/F;F/F0F/F9*&F)
F/FCF/FJ*&F)F/F=F/F9*(F*F/F;F/FHF/F5*(F*F/F;F/F7F/F5*&FGF/FCF/FO*&FGF/
F=F/F5F4)FKF.F/F4*&,(*&F;F/FQF/FY*&F*F/FSF/FJ*&F;F/FMF/FJF4FKF/F4F3" }
}{PARA 6 "" 1 "" {TEXT -1 38 "leastsqrs[31]: Throwing out solution 3" 
}}{PARA 6 "" 1 "" {TEXT -1 20 "leastsqrs[31]: Done!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$/%\"AG\"\"!/%$phiG,&-%'arctanG6#,$*&,&&%\"SG6#F&\"
\"#&F16#\"\"\"!\"\"\"\"\"&F16#F3!\"\"#F6F3F6*&%#PiGF6%$_Z|irGF6F6" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "As we can see, " }{XPPEDIT 18 0 "Y
 = A*cos(2*Pi*nu*t[i]+phi);" "6#/%\"YG*&%\"AG\"\"\"-%$cosG6#,&**\"\"#F
'%#PiGF'%#nuGF'&%\"tG6#%\"iGF'F'%$phiGF'F'" }{TEXT -1 143 " is not a p
articularly good choice of representation, since the solution involves
 inverse trigonometric functions and is particularly useless (" }
{XPPEDIT 18 0 "A = 0;" "6#/%\"AG\"\"!" }{TEXT -1 61 ").  The next, equ
ivalent, example leads to a useful solution." }}}}{SECT 0 {PARA 4 "" 
0 "Sinusoid \227 Representation 2" {TEXT -1 27 "Sinusoid \227 Represen
tation 2" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "printlevel := 2:" "6
#>%+printlevelG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsq
rs(Y = A*cos(2*Pi*nu*t[i])+B*sin(2*Pi*nu*t[i]),t,i,[A, B]);" "6#-%*lea
stsqrsG6&/%\"YG,&*&%\"AG\"\"\"-%$cosG6#**\"\"#F+%#PiGF+%#nuGF+&%\"tG6#
%\"iGF+F+F+*&%\"BGF+-%$sinG6#**\"\"#F+F1F+F2F+&F46#F6F+F+F+F4F67$F*F8
" }}{PARA 6 "" 1 "" {TEXT -1 30 "leastsqrs[0]: normal equations" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&,(&%\"TG6#\"\"\"!\")%\"NG\"\"#&F(
6#\"\"!\"\")F*%\"AGF*F**&,&&F(6#\"\"$!\"%&F(6#F-F1F*%\"BGF*F*&%\"SGF:F
8&F=F/F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&&%\"TG6#\"\"$!\"%
&F(6#\"\"#\"\")\"\"\"%\"AGF0F0&%\"SG6#F0F+*&%\"BGF0&F(6#\"\"%F0F/\"\"!
" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsqrs[0]: Solving the normal equa
tions..." }}{PARA 6 "" 1 "" {TEXT -1 31 "leastsqrs[1]: substitution li
st" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+/%&DeltaG,.*&-%$sumG6$*$)-%$co
sG6#*(%#PiG\"\"\"%#nuGF2&%\"tG6#%\"iGF2\"\"%\"\"\"/F7;F2%\"NGF2-F)6$*&
)-%$sinGF/\"\"#F9)F-FCF9F:F2F8*&-F)6$*$FDF9F:F2F=F9!\"%*&F<F2F=F9F2*$)
-F)6$*&)F-\"\"$F9FAF2F:FCF9FI*&-F)6$*&FAF9F-F2F:F2FMF2F8*$)FSFCF9!\"\"
/&%\"SG6#\"\"!-F)6$&%\"YGF6F:/&%\"TGFfnF(/&F^o6#F2FF/&F^o6#FCFM/&F^o6#
FQFS/&FenFao-F)6$*(FjnF2FAF9F-F9F:/&F^o6#F8F=/&FenFdo-F)6$*&FjnF9FDF9F
:" }}{PARA 6 "" 1 "" {TEXT -1 39 "leastsqrs[1]: Verifying the solution
..." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$/%\"AG*&,(*&&%\"SG6#\"\"!\"\"
\"&%\"TG6#\"\"%F-!\"\"*&,&&F/6#\"\"#!\"#&F/6#\"\"$F-F-&F*6#F-F-F-*&&F*
F6F-F.\"\"\"F7F@%&DeltaG!\"\"/%\"BG*&,(*&,&F5F-F9#F2F7F-F)F@F-*&,(&F/F
+F7&F/F=F8%\"NG#F-F7F-F<F@F-*&F4F@F?F@F-F@FAFB" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "tmp := array(map(proc (x) options operator, arrow;
 rhs(x) end,eval(%,_subslist))):" "6#>%$tmpG-%&arrayG6#-%$mapG6$R6#%\"
xG7\"6$%)operatorG%&arrowG6\"-%$rhsG6#F-F2F2F2-%%evalG6$%\"%G%*_subsli
stG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "optimize(makeproc(tmp,pa
rameters = [N, Y, t]));" "6#-%)optimizeG6#-%)makeprocG6$%$tmpG/%+param
etersG7%%\"NG%\"YG%\"tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6%%\"NG%\"
YG%\"tG6/%$tmpG%#t1G%#t2G%#t8G%$t14G%#t7G%#t6G%#t4G%$t10G%$t17G%$t21G%
$t27G%$t25G6\"F6C2>8$-%&arrayG6#;\"\"\"\"\"#>8%-%$sumG6$&9%6#%\"iG/FH;
F>%\"NG>8&-FC6$*&)-%$sinG6#*(%#PiGF>%#nuGF>&9&FGF>F?\"\"\")-%$cosGFTF?
FZFI>8+-FC6$*&)Ffn\"\"$FZFRF>FI>8*-FC6$*&FRFZFfnF>FI>8),&Fin!\"#F`oF>>
8'-FC6$*(FEF>FRFZFfnFZFI>8,-FC6$*&FEFZFenFZFI>8(-FC6$*$)Ffn\"\"%FZFI>8
--FC6$*$FenFZFI>8.*$)FinF?FZ>80*$)F`oF?FZ>8/*&FZFZ,.*&FMF>FcpF>Fhp*&Fj
pF>FMFZ!\"%*&FKF>FMFZF>F_qF\\r*&F`oF>FinF>FhpFcq!\"\"!\"\">&F96#F>*&,(
*&FAF>FMFZF_r*&FeoF>FioF>F>*&F^pF>FMFZF?F>FgqF>>&F96#F?*&,(*&,&FinF>F`
o#F_rF?F>FAFZF>*&,(FcpF?FjpFgoFK#F>F?F>FioFZF>*&FeoFZF^pFZF>F>FgqFZF9F
6F6F6" }}}}{SECT 0 {PARA 4 "" 0 "Logarithm" {TEXT -1 9 "Logarithm" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsqrs(Y = A+B*log(t[i]),t,i,
[A, B]);" "6#-%*leastsqrsG6&/%\"YG,&%\"AG\"\"\"*&%\"BGF*-%$logG6#&%\"t
G6#%\"iGF*F*F1F37$F)F," }}{PARA 6 "" 1 "" {TEXT -1 30 "leastsqrs[0]: n
ormal equations" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"SG6#\"\"!!\"
#*&%\"AG\"\"\"%\"NGF,\"\"#*&%\"BGF,&%\"TG6#F,F,F.F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/,(&%\"SG6#\"\"\"!\"#*&%\"AGF(&%\"TGF'F(\"\"#*&%\"BGF
(&F-6#\"\"!F(F.F3" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsqrs[0]: Solvin
g the normal equations..." }}{PARA 6 "" 1 "" {TEXT -1 31 "leastsqrs[0]
: substitution list" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'/%&DeltaG,&*$
)-%$sumG6$-%#lnG6#&%\"tG6#%\"iG/F2;\"\"\"%\"NG\"\"#\"\"\"!\"\"*&-F*6$*
$)F,F7F8F3F5F6F5F5/&%\"SG6#\"\"!-F*6$&%\"YGF1F3/&%\"TGFBF;/&FJ6#F5F)/&
FAFM-F*6$*&F,F5FFF5F3" }}{PARA 6 "" 1 "" {TEXT -1 39 "leastsqrs[0]: Ve
rifying the solution..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"AG*&,
&*&&%\"SG6#\"\"!\"\"\"&%\"TGF+F-F-*&&F/6#F-F-&F*F2F-!\"\"\"\"\"%&Delta
G!\"\"/%\"BG*&,&*&F3F5%\"NGF-F-*&F1F5F)F5F4F5F6F7" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "eval(%,_subslist);" "6#-%%evalG6$%\"%G%*_subslis
tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$/%\"AG*&,&*&-%$sumG6$&%\"YG6#%
\"iG/F/;\"\"\"%\"NGF2-F*6$*$)-%#lnG6#&%\"tGF.\"\"#\"\"\"F0F2F2*&-F*6$F
8F0F2-F*6$*&F8F2F,F2F0F2!\"\"F>,&*$)F@F=F>FE*&F4F>F3F2F2!\"\"/%\"BG*&,
&*&FBF>F3F>F2*&F@F>F)F>FEF>FFFJ" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "tmp := array(map(proc (x) options operator, arrow; rhs(x) end,ev
al(%%,_subslist))):" "6#>%$tmpG-%&arrayG6#-%$mapG6$R6#%\"xG7\"6$%)oper
atorG%&arrowG6\"-%$rhsG6#F-F2F2F2-%%evalG6$%#%%G%*_subslistG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "optimize(makeproc(tmp,parameters
 = [N, Y, t]));" "6#-%)optimizeG6#-%)makeprocG6$%$tmpG/%+parametersG7%
%\"NG%\"YG%\"tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6%%\"NG%\"YG%\"tG6
)%$tmpG%#t5G%$t11G%#t4G%#t8G%#t1G%#t2G6\"F0C,>8$-%&arrayG6#;\"\"\"\"\"
#>8)-%$sumG6$&9%6#%\"iG/FB;F8%\"NG>8*-F=6$*$)-%#lnG6#&9&FAF9\"\"\"FC>8
'-F=6$FLFC>8%-F=6$*&FLF8F?F8FC>8(*$)FSF9FQ>8&*&FQFQ,&Ffn!\"\"*&FGF8FEF
8F8!\"\">&F36#F8*&,&*&FGFQF;F8F8*&FSF8FWF8F]oF8FjnF8>&F36#F9*&,&*&FWFQ
FEFQF8*&F;FQFSFQF]oF8FjnFQF3F0F0F0" }}}}{SECT 0 {PARA 4 "" 0 "Exponent
ial" {TEXT -1 11 "Exponential" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 
"leastsqrs(Y = A+B*exp(t[i]),t,i,[A, B]);" "6#-%*leastsqrsG6&/%\"YG,&%
\"AG\"\"\"*&%\"BGF*-%$expG6#&%\"tG6#%\"iGF*F*F1F37$F)F," }}{PARA 6 "" 
1 "" {TEXT -1 30 "leastsqrs[0]: normal equations" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(&%\"SG6#\"\"!!\"#*&%\"AG\"\"\"%\"NGF,\"\"#*&%\"BGF,&
%\"TGF'F,F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"SG6#\"\"\"!\"#
*&%\"AGF(&%\"TG6#\"\"!F(\"\"#*&%\"BGF(&F-F'F(F0F/" }}{PARA 6 "" 1 "" 
{TEXT -1 45 "leastsqrs[0]: Solving the normal equations..." }}{PARA 6 
"" 1 "" {TEXT -1 31 "leastsqrs[0]: substitution list" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#7'/%&DeltaG,&*$)-%$sumG6$-%$expG6#&%\"tG6#%\"iG/F2;
\"\"\"%\"NG\"\"#\"\"\"!\"\"*&-F*6$*$)F,F7F8F3F5F6F5F5/&%\"SG6#\"\"!-F*
6$&%\"YGF1F3/&%\"TGFBF)/&FA6#F5-F*6$*&F,F5FFF5F3/&FJFMF;" }}{PARA 6 "
" 1 "" {TEXT -1 39 "leastsqrs[0]: Verifying the solution..." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$/%\"AG*&,&*&&%\"SG6#\"\"!\"\"\"&%\"TG6#F-F
-F-*&&F/F+F-&F*F0F-!\"\"\"\"\"%&DeltaG!\"\"/%\"BG*&,&*&F3F5%\"NGF-F-*&
F2F5F)F5F4F5F6F7" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "eval(%,_sub
slist);" "6#-%%evalG6$%\"%G%*_subslistG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$/%\"AG*&,&*&-%$sumG6$&%\"YG6#%\"iG/F/;\"\"\"%\"NGF2-F*6$*$)-%$
expG6#&%\"tGF.\"\"#\"\"\"F0F2F2*&-F*6$F8F0F2-F*6$*&F8F2F,F2F0F2!\"\"F>
,&*$)F@F=F>FE*&F4F>F3F2F2!\"\"/%\"BG*&,&*&FBF>F3F>F2*&F@F>F)F>FEF>FFFJ
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "tmp := array(map(proc (x) o
ptions operator, arrow; rhs(x) end,eval(%%,_subslist))):" "6#>%$tmpG-%
&arrayG6#-%$mapG6$R6#%\"xG7\"6$%)operatorG%&arrowG6\"-%$rhsG6#F-F2F2F2
-%%evalG6$%#%%G%*_subslistG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
optimize(makeproc(tmp,parameters = [N, Y, t]));" "6#-%)optimizeG6#-%)m
akeprocG6$%$tmpG/%+parametersG7%%\"NG%\"YG%\"tG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#R6%%\"NG%\"YG%\"tG6)%#t5G%$t11G%#t1G%#t2G%$tmpG%#t4G%#t
8G6\"F0C,>8(-%&arrayG6#;\"\"\"\"\"#>8&-%$sumG6$&9%6#%\"iG/FB;F8%\"NG>8
'-F=6$*$)-%$expG6#&9&FAF9\"\"\"FC>8)-F=6$FLFC>8$-F=6$*&FLF8F?F8FC>8**$
)FSF9FQ>8%*&FQFQ,&Ffn!\"\"*&FGF8FEF8F8!\"\">&F36#F8*&,&*&F;F8FGFQF8*&F
SF8FWF8F]oF8FjnF8>&F36#F9*&,&*&FWFQFEFQF8*&FSFQF;FQF]oF8FjnFQF3F0F0F0
" }}}}{SECT 0 {PARA 4 "" 0 "Exponentially Decaying Sinusoid" {TEXT -1 
31 "Exponentially Decaying Sinusoid" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 84 "The substitution list for this case is quite long, so its outpu
t will be suppressed." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "printl
evel := 1:" "6#>%+printlevelG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "leastsqrs(Y = A+exp(-t[i])*(B*cos(2*Pi*nu*t[i])+C*sin(2
*Pi*nu*t[i])),t,i,[A, B, C]):" "6#-%*leastsqrsG6&/%\"YG,&%\"AG\"\"\"*&
-%$expG6#,$&%\"tG6#%\"iG!\"\"F*,&*&%\"BGF*-%$cosG6#**\"\"#F*%#PiGF*%#n
uGF*&F16#F3F*F*F**&%\"CGF*-%$sinG6#**\"\"#F*F=F*F>F*&F16#F3F*F*F*F*F*F
1F37%F)F7FB" }}{PARA 6 "" 1 "" {TEXT -1 30 "leastsqrs[0]: normal equat
ions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%\"AG\"\"\"%\"NGF'\"\"#*&
,&&%\"TG6#\"\"!\"\"%&F-6#F)!\"#F'%\"BGF'F'&%\"SGF.F3*&%\"CGF'&F-6#\"\"
$F'F0F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&,&&%\"TG6#\"\"!\"\"%&F
(6#\"\"#!\"#\"\"\"%\"AGF0F0*&,(&F(6#F+\"\")&F(6#\"\"'!\")&F(6#\"\"&F.F
0%\"BGF0F0*&,&&F(6#F6!\"%&F(6#\"\"(F6F0%\"CGF0F0&%\"SG6#F0FC&FIF-F.F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%\"AG\"\"\"&%\"TG6#\"\"$F'\"
\"%*&,&&F)6#\"\")!\"%&F)6#\"\"(F1F'%\"BGF'F'&%\"SGF*F2*&%\"CGF'&F)6#F'
F'F1\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsqrs[0]: Solving the no
rmal equations..." }}{PARA 6 "" 1 "" {TEXT -1 39 "leastsqrs[2]: Verify
ing the solution..." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "soln := \+
%:" "6#>%%solnG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "for p i
n soln do print(p) od:" "6#?&%\"pG%%solnG%%trueG-%&printG6#F$" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"AG*&,**&,.*$)&%\"TG6#\"\"(\"\"#\"
\"\"\"\"%*$)&F,6#\"\")F/F0\"\"\"*&F+F7F4F7!\"%*&&F,6#\"\"&F7&F,6#F7F7!
\"\"*&&F,6#F1F7F>F0F9*&&F,6#\"\"'F7F>F0F1F7&%\"SG6#\"\"!F7F7*&,**&F+F0
&F,6#\"\"$F7F9*&&F,FJF7F>F0F1*&F4F0FOF0F/*&&F,6#F/F7F>F0!\"#F7&FIF?F7F
7*&,*FUF7FNF/FTF@FRFXF7&FIFWF7F7*&,0*&FOF0FEF0F9*&FOF0F;F0F7*&FOF0FBF0
F1*&FSF0F+F0F9*&F+F0FVF0F/*&FSF0F4F0F/*&FVF0F4F0F@F7&FIFPF7F7F0%&Delta
G!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"BG*&,**&,**&&%\"TG6#\"\"
(\"\"\"&F+6#\"\"$F.!\"#*&&F+6#\"\"!F.&F+6#F.F.\"\"#*&&F+6#\"\")F.F/\"
\"\"F.*&&F+6#F9F.F7F>!\"\"F.&%\"SGF5F.F.*&,&*&F7F>%\"NGF.F2*$)F/F9F>F9
F.&FDF8F.F.*&,&FGF.FIFBF.&FDFAF.F.*&,&*&,&F;FBF*F9F.FHF>F.*&F/F>,&F4F9
F@FBF.FBF.&FDF0F.F.F>%&DeltaG!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
/%\"CG*&,**&,0*&&%\"TG6#\"\"#\"\"\"&F+6#\"\")F.#!\"\"F-*&&F+6#\"\"!F.&
F+6#\"\"(F.!\"#*&F5\"\"\"F/F=F.*&F8F=F*F=F.*&&F+6#\"\"$F.&F+6#\"\"'F.F
;*&F@F=&F+6#\"\"&F.#F.F-*&F@F=&F+6#\"\"%F.F-F.&%\"SGF6F.F.*&,&*&,&F/F3
F8F-F.%\"NGF.F.*&F@F=,&F5F-F*F3F.F3F.&FP6#F.F.F.*&,&*&,&F/FJF8F3F.FUF=
F.FVFJF.&FPF,F.F.*&,&*&,(FCF-FGF2FLF;F.FUF=F.*$)FWF-F=FJF.&FPFAF.F.F=%
&DeltaG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The following com
mands produce an optimized procedure, which we will not show here." }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "tmp := array(map(proc (x) optio
ns operator, arrow; rhs(x) end,eval(soln,_subslist))):" "6#>%$tmpG-%&a
rrayG6#-%$mapG6$R6#%\"xG7\"6$%)operatorG%&arrowG6\"-%$rhsG6#F-F2F2F2-%
%evalG6$%%solnG%*_subslistG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
optimize(makeproc(tmp,parameters = [N, Y, t])):" "6#-%)optimizeG6#-%)m
akeprocG6$%$tmpG/%+parametersG7%%\"NG%\"YG%\"tG" }}}}{SECT 0 {PARA 4 "
" 0 "Sinusoid with Exponentially Decaying Component" {TEXT -1 46 "Sinu
soid with Exponentially Decaying Component" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "model := Y = A+B*cos(2*Pi*nu[1]*t[i])+C*sin(2*Pi*nu[1]*
t[i])+exp(-t[i])*(D*cos(2*Pi*nu[2]*t[i])+E*sin(2*Pi*nu[2]*t[i])):" "6#
>%&modelG/%\"YG,*%\"AG\"\"\"*&%\"BGF)-%$cosG6#**\"\"#F)%#PiGF)&%#nuG6#
\"\"\"F)&%\"tG6#%\"iGF)F)F)*&%\"CGF)-%$sinG6#**\"\"#F)F1F)&F36#\"\"\"F
)&F76#F9F)F)F)*&-%$expG6#,$&F76#F9!\"\"F),&*&%\"DGF)-F-6#**\"\"#F)F1F)
&F36#\"\"#F)&F76#F9F)F)F)*&%\"EGF)-F=6#**\"\"#F)F1F)&F36#\"\"#F)&F76#F
9F)F)F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "leastsqrs(model,
t,i,[A, B, C, D, E]):" "6#-%*leastsqrsG6&%&modelG%\"tG%\"iG7'%\"AG%\"B
G%\"CG%\"DG%\"EG" }}{PARA 6 "" 1 "" {TEXT -1 30 "leastsqrs[4]: normal \+
equations" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*&%\"NG\"\"\"%\"AGF'\"
\"#*&,&F&!\"#&%\"TG6#\"#6\"\"%F'%\"BGF'F'*&%\"CGF'&F.6#\"#7F'F1*&,&&F.
6#\"#8F1&F.6#\"#:F,F'%\"DGF'F'&%\"SG6#\"\"$F,*&%\"EGF'&F.6#\"#=F'F1\"
\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&,&%\"NG!\"#&%\"TG6#\"#6\"
\"%\"\"\"%\"AGF.F.*&,(F)!\")F'\"\"#&F*6#\"\"$\"\")F.%\"BGF.F.*&,&&F*6#
\"\"'F7&F*6#\"#7!\"%F.%\"CGF.F.*&,*&F*6#\"#8FA&F*6#\"\"*FA&F*6#\"#:F3&
F*6#\"\"(F7F.%\"DGF.F.*&,&&F*6#\"#5F7&F*6#\"#=FAF.%\"EGF.F.&%\"SGF5F3&
Ffn6#\"\"&FA\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*&%\"AG\"\"\"&
%\"TG6#\"#7F'\"\"%*&,&&F)6#\"\"'\"\")F(!\"%F'%\"BGF'F'*&%\"CGF'&F)6#\"
#;F'F2*&,&&F)6#\"#<F2&F)6#\"#>F3F'%\"DGF'F'&%\"SG6#F,F3*&%\"EGF'&F)6#F
2F'F2\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&,&&%\"TG6#\"#8\"\"%
&F(6#\"#:!\"#\"\"\"%\"AGF0F0*&,*F'!\"%&F(6#\"\"*F4F,\"\"#&F(6#\"\"(\"
\")F0%\"BGF0F0*&,&&F(6#\"#<F<&F(6#\"#>F4F0%\"CGF0F0*&,(&F(6#\"#9F<&F(6
#\"\"!!\")&F(6#F0F8F0%\"DGF0F0*&,&&F(6#F8F<&F(6#F+F4F0%\"EGF0F0&%\"SGF
MF4&FenFQF8FN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*&%\"AG\"\"\"&%\"T
G6#\"#=F'\"\"%*&,&&F)6#\"#5\"\")F(!\"%F'%\"BGF'F'*&%\"CGF'&F)6#F2F'F2*
&,&&F)6#\"\"#F2&F)6#F,F3F'%\"DGF'F'&%\"SGF<F3*&%\"EGF'&F)6#\"\"&F'F2\"
\"!" }}{PARA 6 "" 1 "" {TEXT -1 45 "leastsqrs[4]: Solving the normal e
quations..." }}{PARA 6 "" 1 "" {TEXT -1 40 "leastsqrs[46]: Verifying t
he solution..." }}{PARA 6 "" 1 "" {TEXT -1 21 "leastsqrs[569]: Done!" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "soln := %:" "6#>%%solnG%\"%G
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "As we can see, the solution f
or this model is quite complicated:" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "cost(soln);" "6#-%%costG6#%%solnG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*%*additionsG\"%)=#%0multiplicationsG\"%\\x%*divisions
G\"\"&%,assignmentsGF)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "map(c
ost,soln);" "6#-%$mapG6$%%costG%%solnG" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7',*%*additionsG\"$*y%0multiplicationsG\"%kG%*divisionsG\"\"\"%,
assignmentsGF*,*F%\"$%RF'\"%#Q\"F)F*F+F*F,,*F%\"$<#F'\"$R(F)F*F+F*F," 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(_subslist);" "6#-%%costG
6#%*_subslistG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*additionsG\"$a#%
0multiplicationsG\"%\\7%%sumsG\"%36%,assignmentsG\"#F" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "map(cost,eval(soln,_subslist));" "6#-%$
mapG6$%%costG-%%evalG6$%%solnG%*_subslistG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7',,%*additionsG\"%V5%0multiplicationsG\"%8T%%sumsG\"%o
R%*divisionsG\"\"\"%,assignmentsGF,,,F%\"$['F'\"%JEF)\"%([#F+F,F-F,F.,
,F%\"$r%F'\"%))>F)\"%U=F+F,F-F,F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
27 "However, it optimizes well:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "optsoln := optimize(soln,tryhard):" "6#>%(optsolnG-%)optimizeG6$
%%solnG%(tryhardG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(optso
ln);" "6#-%%costG6#%(optsolnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%*a
dditionsG\"$R(%0multiplicationsG\"%h5%*divisionsG\"\"\"%+subscriptsG\"
#E%,assignmentsG\"$Z%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The foll
owing command produces an optimized procedure, which we will not show \+
here." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "makeproc(eval([optsoln
],_subslist),parameters = [N, Y, t],globals = [A, B, C, D, E]):" "6#-%
)makeprocG6%-%%evalG6$7#%(optsolnG%*_subslistG/%+parametersG7%%\"NG%\"
YG%\"tG/%(globalsG7'%\"AG%\"BG%\"CG%\"DG%\"EG" }}}}}}{MARK "8 9 13 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }