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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT 256 244 "Taylor-Newton-Gauss program to\nfind the un
known parameters of a NonLINEAR or LINEAR\nfunction from a series of o
bservations \{X,Y\}.\nWe assume that reasonable initial approximations
 \{Guess values\} of\nthe desired unknowns \{a[1] ... a[n]\} are known
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2510 "restart;\nwith(linal
g):\nwith(stats):\nwith(plots):\nTNG:=proc()\nglobal Guess,n;\nlocal E
quation,guess_dim,Taylor1,Taylor,z,A1,A,ma1,ma,AA,BB,abc,answer,answer
1,new_y,correl,correlation,correlation1,rms_error,rms_error1,avg_new_y
,RMS_error,SSQ,RMS_percent_avg_response,eq_function,Yvalues_predicted,
Residuals,C,q,p,P,P1,D1,D2,j,m,nn;\nEquation:=y-Fit:\nguess_dim:=nops(
Guess):\n     #Find the partial differentiation of each parameter usin
g the trucated Taylor series.\nTaylor1:=seq(diff(Fit,a[n])*q[n],n=1..g
uess_dim):\nTaylor:=-(Equation+sum(Taylor1[n],n=1..guess_dim)):\nfor z
 from 1 to iterations do\n     #substitute the Guess values, X and Y o
bservations into the Taylor function.\nA1:=subs(seq(a[t]=Guess[t],t=1.
.guess_dim),Taylor);\nA:=[evalf(seq(subs(x=X[s],y=Y[s],A1),s=1..N))];
\n    #The nonlinear error equations are now linearized with the error
 q[n]=a[n]-Guess[n].\n    #Form a matrix of the coefficients of the q[
n] equations and find the leastsquares solution.\nma1:=seq([seq(coeff(
A[nn],q[p]),nn=1..N)],p=1..guess_dim);\nma:=array([ma1]);\nAA:=transpo
se(ma);\nBB := array( [seq(tcoeff(A[n]),n=1..N)] );\nabc:=leastsqrs(AA
, BB);\nGuess:=[seq(abc[j]+Guess[j],j=1..guess_dim)] od:\n     #Calcul
ate and print: answer, correlation, etc\nanswer:=evalf(subs([seq(a[k]=
Guess[k],k=1..guess_dim)],Fit),6);\nanswer1:=subs([seq(a[k]=Guess[k],k
=1..guess_dim)],Fit):\nprint(`Answer=`,answer1);\nprint(`coefficients=
`,Guess);\nnew_y:=[seq(subs(x=X[m],answer),m=1..N)]:\ncorrel:=evalf(de
scribe[linearcorrelation](Y,new_y)):\ncorrelation1:=evalf(correl):\nco
rrelation:=evalf(correlation1,6);\nprint(`correlation=`,correlation);
\nrms_error:=sqrt(1/N*sum((Y[i]-new_y[i])^2,i=1..N)):\nrms_error1:=eva
lf(rms_error):\navg_new_y:=sum(new_y[i],i=1..N)/N:\nRMS_error:=evalf(r
ms_error1,6);\nprint(`RMS error=`,RMS_error);\nSSQ:=evalf(sum((Y[i]-ne
w_y[i])^2,i=1..N),6);\nprint(`Sum of the squares=`,SSQ);\nRMS_percent_
avg_response:=evalf(rms_error/avg_new_y*100,6);\nprint(`RMS percent av
erage response=`,RMS_percent_avg_response);\neq_function:=unapply(answ
er1,x):\nYvalues_predicted:=evalf(transform[apply[eq_function]](X),6):
\nprint(`new Y values=`,Yvalues_predicted);\nResiduals:=transform[mult
iapply[(x,y)-> x-y]]([Y, Yvalues_predicted]):\nprint(`old Y and new Y \+
residuals=`,Residuals);\nC:=evalf([describe[variance](Residuals)]*N/(N
-1),6):\nprint(`Curve fitting variance=`,C);\nfor q from 1 to N do p[q
]:=[X[q],Y[q]] od:\nP:=seq(p[m],m=1..N):\nunassign('n'):\nA1:=pointplo
t(\{P\},color=red):\nP1:=plot(answer,x=X[1]..X[N],color=blue):\nD1:=di
splay(A1,P1);\nprint(D1);\nend:" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warnin
g, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, ne
w definition for trace" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, `n` in
 call to `seq` is not local" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
165 "Fit:=a[1]*x/(a[2]+x);\nX:=[0.3, 0.5, 1, 2, 3, 4, 5, 10];\nY := [0
.17, 0.27, 0.43, 0.65, 0.73, 0.78, 0.79, 0.81];\nGuess:=[1,1];\niterat
ions:=10;\nN:=nops(X);\nNy:=nops(Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$FitG*&*&&%\"aG6#\"\"\"F*%\"xGF*\"\"\",&&F(6#\"\"#F*F+F*!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG7*$\"\"$!\"\"$\"\"&F(\"\"\"\"\"#
F'\"\"%F*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7*$\"#<!\"#$\"#
FF($\"#VF($\"#lF($\"#tF($\"#yF($\"#zF($\"#\")F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&GuessG7$\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%+iterationsG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NyG\"\")" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 6 "TNG();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(Answer=
G,$*&%\"xG\"\"\",&$\"+LEsP6!\"*\"\"\"F&F,!\"\"$\"+fX:N'*!#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%.coefficients=G7$$\"+fX:N'*!#5$\"+LEsP6!\"*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-correlation=G$\"'`<**!\"'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%+RMS~error=G$\"'(=?$!\"(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%4Sum~of~the~squares=G$\"'k,#)!\")" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%>RMS~percent~average~response=G$\"';)\\&!\"&" 
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F'$\"'@2XF'$\"']ThF'$\"'%e)pF'$\"'\\,vF'$\"'8\\yF'$\"'$4l)F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%;old~Y~and~new~Y~residuals=G7*$!&^5$!\"'$!&
jT#F'$!&@2#F'$\"&]e$F'$\"&;9$F'$\"&^)HF'$\"%(3&F'$!&$4bF'" }}{PARA 11 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "Fit:=1./(1.+exp(-a[1]*x-a[2
]));\nX:=[10, 20, 30, 40, 50];\nY := [0.05, 0.1, 0.20, 0.40, 0.50];\nG
uess:=[0.05, -2];\niterations:=10;\nN:=nops(X);\nNy:=nops(Y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FitG,$*&\"\"\"F',&$\"\"\"\"\"!F*-%$
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"" 1 "" {XPPMATH 20 "6#>%+iterationsG\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NyG\"
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