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# Edit detail for SandBoxMLE revision 3 of 3

 1 2 3 Editor: Mark Clements Time: 2009/12/08 20:06:44 GMT-8 Note: Update for recent version of Fricas

changed:
-		(Bvec(2)-Bvec(1)**2)**2*100.0+(1.0-Bvec(1))**2
(Bvec(2)-Bvec(1)^2)^2*100.0+(1.0-Bvec(1))^2

changed:
-		eps : Float := (10.0**(-digits()$Float+1))$Float
eps : Float := (10.0^(-digits()$Float+1))$Float

changed:
-	erf(x:Float):Float == if x<0.0 then -erf(-x) else pgamma(x**2,0.5)
erf(x:Float):Float == if x<0.0 then -erf(-x) else pgamma(x^2,0.5)

changed:
- 		reduce(_+,[exp(-lambda)*lambda**yi/factorial(yi) for yi in 0..y])$List(Float) reduce(_+,[exp(-lambda)*lambda^yi/factorial(yi) for yi in 0..y])$List(Float)

changed:
-			cumP := cumP + exp(-lambda)*lambda**i/factorial(i)
cumP := cumP + exp(-lambda)*lambda^i/factorial(i)

changed:
- 		reduce(_+,[(x(i)-meanx)**2 for i in 1..#x])$List(Float)/(#x-1) reduce(_+,[(x(i)-meanx)^2 for i in 1..#x])$List(Float)/(#x-1)

changed:
-			x**a*(1.0-x)**b/Beta(a,b)
x^a*(1.0-x)^b/Beta(a,b)

changed:
-		eps : Float := (10.0**(-digits()$Float+1))$Float
eps : Float := (10.0^(-digits()$Float+1))$Float

changed:
-	pt(x:Float,df:Float):Float == 1-betai(df/2.0,0.5,df/(df+x**2))/2.0
pt(x:Float,df:Float):Float == 1-betai(df/2.0,0.5,df/(df+x^2))/2.0

changed:
-	nmmin(vals, (x +-> eval(e, vars, x)), -1)$OPTIM nmmin(vals, ((x:Vector Float):Float +-> eval(e, vars, x)), -1)$OPTIM

changed:
-optimFun((b-a**2)**2*100.0+(1.0-a)**2, [a=11.0,b=1.0])
optimFun((b-a^2)^2*100.0+(1.0-a)^2, [a=11.0,b=1.0])


I have recently been experimenting with using Axiom for maximum likelihood estimation,
borrowing code (with permission) from "Compact Numerical Methods" by Nash.
)abbrev package OPTIM OptimNash
++ Author: M. Clements
++ Date Created: 24 August 2008
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classification:
++ Keywords:
++ References:
++ Description:
++ Numerical optimisation using methods from Nash
++ Permission granted by Nash to translate from his Pascal code to Axiom
axisResults ==> Record(X: Vector Float, Fmin:Float, Lowerfun: Boolean)
nmminResults ==> Record(X:Vector Float, Fmin:Float, Fail:Boolean)
OptimNash: Exports == Implementation where
Exports == with
Rosen: Vector Float -> Float
++ Rosen(Bvec) returns the Rosen function
nmmin: (Vector Float, (Vector Float -> Float), Float) -> nmminResults
++ nmmin(Bvec, fminfn, intol) returns values for the optimised function
axissrch :(Vector Float, Vector Float -> Float) -> axisResults
++ axissrch(Bvec, fminfn) returns the axis search method
demo: () -> nmminResults
++ demo() returns the rsults of a simple demo using the Rosen function
demo2: () -> nmminResults
++ demo2() returns the results of a more complex demo
--optimFun: (Expression Float,List Equation Expression Float) -> nmminResults
--writeln(Classical starting point (-1.2,1))
Rosen(Bvec: Vector Float):Float ==
(Bvec(2)-Bvec(1)^2)^2*100.0+(1.0-Bvec(1))^2        --nmmin(vector([11.0, 1.0]), Rosen$OPTIM, -1.0)$OPTIM
nmmin(Bvec: Vector Float,
fminfn: Vector Float -> Float,
intol: Float):nmminResults ==
Pcol ==> 27
Prow ==> 26
alpha ==> 1.0
beta ==> 0.5
gamma ==> 2.0
Calceps ==> 1.0e-35
big ==> 1.0e35 -- as per R/src/main/optim.c
n : Integer := #Bvec
action    : String
C         : Integer
calcvert  : Boolean
convtol   : Float
f         : Float
funcount  : Integer
H         : Integer
i         : Integer
j         : Integer
L         : Integer
notcomp   : Boolean
n1        : Integer
oldsize   : Float
P         : Matrix Float := new(Prow,Pcol,0.0)--[1..Prow,1..Pcol]
shrinkfail: Boolean
size      : Float
step      : Float
temp      : Float
trystep   : Float
VH,VL,VN  : Float
VR        : Float
x         : Vector Float := Bvec -- ugly declaration
--writeln(Nash Algorithm 19 version 2 1988-03-17)
--writeln(  Nelder Mead polytope direct search function minimiser)
--writeln(confile,Nash Algorithm 19 version 2 1988-03-17)
--writeln(confile,  Nelder Mead polytope direct search function minimiser)
fail := false
(f,notcomp) := (fminfn(Bvec),false)
if notcomp then
--writeln(**** Function cannot be evaluated at initial parameters ****)
--writeln(confile,
--        **** Function cannot be evaluated at initial parameters ****)
fail := true
else
--writeln(Function value for initial parameters = ,f)
--writeln(confile,Function value for initial parameters = ,f)
if intol<0.0 then intol := Calceps
funcount := 1
convtol := intol*(abs(f)+intol)
--writeln(  Scaled convergence tolerance is ,convtol)
--writeln(confile,  Scaled convergence tolerance is ,convtol)
n1 := n+1
C := n+2
P(n1,1) := f
for i in 1..n repeat P(i,1) := Bvec(i)
L := 1
size := 0.0
step := 0.0
for i in 1..n repeat if 0.1abs(Bvec(i))>step then step := 0.1abs(Bvec(i))
--writeln(Stepsize computed as ,step)
--writeln(confile,Stepsize computed as ,step)
for j in 2..n1 repeat
action := "BUILD          "
for i in 1..n repeat P(i,j) := Bvec(i)
trystep := step
while P(j-1,j)=Bvec(j-1) repeat                                        P(j-1,j) := Bvec(j-1)+trystep
trystep := trystep*10.0
size := size+trystep
oldsize := size
calcvert := true
shrinkfail := false
repeat
if calcvert then
for j in 1..n1 repeat
if j~=L then
for i in 1..n repeat Bvec(i) := P(i,j)
(f,notcomp) := (fminfn(Bvec),false)
if notcomp or f>=big then f := big
funcount := funcount+1
P(n1,j) := f
calcvert := false
VL := P(n1,L)
VH := VL
H := L
for j in 1..n1 repeat
if j~=L then
f := P(n1,j)
if fVH then
H := j
VH := f
if VH>VL+convtol then
--str(funcount:5,tstr)
--writeln(action,tstr,' ,VH, ,VL)
--writeln(confile,action,tstr, ,VH, ',VL)
VN := betaVL+(1.0-beta)VH
for i in 1..n repeat
temp := -P(i,H)
for j in 1..n1 repeat temp := temp+P(i,j)
P(i,C) := temp/n
for i in 1..n repeat
Bvec(i) := (1.0+alpha)P(i,C)-alphaP(i,H)
(f,notcomp) := (fminfn(Bvec),false)
if notcomp or f>=big then f := big
funcount := funcount+1
action := "REFLECTION     "
VR := f
if VRBvec(i)+(1-gamma)P(i,C)
P(i,C) := Bvec(i)
Bvec(i) := f
(f,notcomp) := (fminfn(Bvec),false)
if notcomp or f>=big then f := big
funcount := funcount+1
if fP(i,H)+betaP(i,C)
(f,notcomp) := (fminfn(Bvec),false)
if notcomp or f>=big then f := big
funcount := funcount+1
if f=VH then
action := "SHRINK         "
calcvert := true
size := 0.0
for j in 1..n1 repeat
if j~=L then
for i in 1..n repeat
P(i,j) := beta*(P(i,j)-P(i,L))+P(i,L)
size := size+abs(P(i,j)-P(i,L))
if sizePolytope size measure not decreased in shrink)
--writeln(confile,
--        Polytope size measure not decreased in shrink)
shrinkfail := true
if ((VH<=VL+convtol) or shrinkfail ) then break -- repeat loop
--writeln(Exiting from Alg19.pas Nelder Mead polytope minimiser)
--writeln('    ,funcount, function evaluations used')
--writeln(confile,Exiting from Alg19.pas Nelder Mead polytope minimiser)
--writeln(confile,'    ,funcount, function evaluations used')
fmin := P(n1,L)
for i in 1..n repeat x(i) := P(i,L)
if shrinkfail then fail := true
return [x, fmin, fail]

axissrch(Bvec: Vector Float, fminfn: Vector Float -> Float): axisResults ==
cradius, eps, f, fplus, step, temp, tilt, fmin: Float
i : Integer
--notcomp : Boolean
--writeln(alg20.pas -- axial search)
--writeln(confile,alg20.pas -- axial search)
Calceps   : Float := 1.0e-35
big       : Float := 1.0e35 -- as per R/src/main/optim.c
n :Integer := #(Bvec)::NonNegativeInteger::Integer
fmin := fminfn(Bvec)
eps := Calceps
eps := sqrt(eps)
--writeln('  Axis':6,' Stepsize  ':14,'function +  ':14,
--    'function -  :14, rad. of curv.:14,   tilt')
--writeln(confile,'  Axis':6,' Stepsize  ':14,'function +  ':14,
--    'function -  :14, rad. of curv.:14,   tilt')
lowerfn := false
for i in 1..n repeat
if (not lowerfn) then
temp := Bvec(i)
step := eps*(abs(temp)+eps)
Bvec(i) := temp+step
f := fminfn(Bvec)
if (f>=big) then f := big
--write(i:5,' ,step:12,  ,f:12,  )
--write(confile,i:5, ,step:12,  ,f:12,  )
if f<fmin then lowerfn := true
if (not lowerfn) then
fplus := f
Bvec(i) := temp-step
f := fminfn(Bvec)
if f>=big then f := big
--write(f:12,  )  write(confile,f:12,  ')
if f                            temp := 0.5(fplus-f)/step
fplus := 0.5(fplus+f-2.0fmin)/(stepstep)
if fplus~=0.0 then
else
tilt := 45.0*atan(temp)/atan(1.0)
--write(cradius:12,'  ,tilt:12)
--write(confile,cradius:12,  ',tilt:12)
--writeln writeln(confile)
[Bvec, f, lowerfn]

demo(): nmminResults ==
nmmin([-1.2, 1.0], Rosen, -1.0)
demo2(): nmminResults ==
--banner:=dr1920.pas -- driver for Nelder-Mead minimisation
--startup
--fminset(n,B,Workdata) {sets up problem and defines starting
--               values of B}
lowerfn:Boolean := false --{safety setting}
B:Vector Float := [-1.2,1.0] --initial values
mynmminResults :nmminResults
myaxisResults : axisResults
--n :Integer := #(B)::NonNegativeInteger::Integer
repeat
mytol:=-1.0 --Note: set the tolerance negative to indicate that
--procedure must obtain an appropriate value.
mynmminResults := nmmin(B,Rosen,mytol) --{minimise the function}
--writeln
--writeln(confile)
--writeln( Minimum function value found =,Fmin)
--writeln( At parameters)
--writeln(confile, Minimum function value found =,Fmin)
--writeln(confile, At parameters)
--for i in 1..n repeat
--{
--writeln(' B[',i,]=,X[i])
--writeln(confile,' B[',i,]=,X[i])
--} --{loop to write out parameters}
B := mynmminResults.X
myaxisResults := axissrch(B, Rosen)  --{alg20.pas}
lowerfn := myaxisResults.Lowerfun
if lowerfn then
B := myaxisResults.X
--writeln(Lower function value found)
--writeln(confile,Lower function value found)
if (not lowerfn) then break
return mynmminResults
--flush(confile) close(confile)
--if infname<>con then close(infile)
--{dr1920.pas -- Nelder Mead minimisation with axial search}
--      optimFun(e:Expression Float,initial:List Equation Expression Float):nmminResults ==
--              vars := [lhs(x) for x in initial]
--              vals := [rhs(x) for x in initial]::Vector Float
--              nmmin(vals, (x +-> eval(e, vars, x)), -1)

)abbrev package SOMESTAT SomeStatisticsPackage
SomeStatisticsPackage: Exports == Implementation where
DF  ==> DoubleFloat
Exports ==> with
pgamma:(Float,Float) -> Float
++ pgamma(y,p) returns the incomplete gamma function (Alg AS 147)
erf:(Float) -> Float
++ erf(x) returns the error function
erfc:(Float) -> Float
++ erfc(x) returns the complementary error function
pnorm:(Float) -> Float
++ pnorm(x) returns the CDF for the normal(0,1) function
pnorm:(Float,Float,Float) -> Float
++ pnorm(x,mean,sd) returns the CDF for a normal(mean,sd) function
pchisq:(Float,Float) -> Float
ppois:(Integer,Float) -> Float
ppois2:(Integer,Float) -> Float
rpois:(Float) -> Integer
++ rpois(lambda) returns a random Poisson variable with mean lambda
rbinom:(Integer,Float) -> Integer
++ rbinom(n,p) returns a random binomial variable from n trials
++ and probability p
mean:(Vector Float) -> Float
mean:(List Float) -> Float
mean:(Vector DF) -> DF
mean:(List DF) -> DF
var:(Vector Float) -> Float
Beta:(Float,Float) -> Float
betai:(Float,Float,Float) -> Float
betacf:(Float,Float,Float) -> Float
pbeta:(Float,Float,Float) -> Float
++ pbeta(x,a,b) returns the incomplete beta function betai(a,b,x)
pt:(Float,Float) -> Float
++ pt(x,df) returns the CDF for a t distribution
pbinom:(Float,Float,Float) -> Float
++ pbinom(k,n,p) returns the CDF for 0..k events given n trials with probability p
pf:(Float,Float,Float) -> Float
++ pf(f,df1,df2) returns the CDF of the F distribution with df1 and df1 degrees of freedom
import FloatSpecialFunctions
import RandomFloatDistributions
pgamma(y,p) ==
eps : Float := (10.0^(-digits()$Float+1))$Float
g : Float := 0.0
if (y=0.0 or p=0.0) then return g
if (y < 0.0 or p < 0.0) then error "Invalid arguments"
a : Float := p + 1.0
f : Float := exp(p  log(y) - logGamma(a) - y)
if (f < eps) then
return g
c : Float := 1.0
g : Float := 1.0
a : Float := p
while (c > eps  g) repeat
a := a+1.0
c := c(y / a)
g := g+c
g := gf
return g
erf(x:Float):Float == if x<0.0 then -erf(-x) else pgamma(x^2,0.5)
erfc(x:Float):Float == 1-erf(x)
pnorm(x:Float):Float == 0.5(1+erf(x/sqrt(2.0)))
pnorm(x:Float,mu:Float,sigma:Float):Float == 0.5(1+erf((x-mu)/sigma/sqrt(2.0)))
pchisq(x:Float,df:Float):Float == pgamma(x/2,df/2)
ppois(y:Integer,lambda:Float):Float ==
if y<0 then return 0.0
reduce(_+,[exp(-lambda)lambda^yi/factorial(yi) for yi in 0..y])$List(Float) ppois2(y:Integer,lambda:Float):Float == 1-pgamma(lambda,y::Float+1) rpois(lambda:Float):Integer == cumP : Float := 0 x : Float := uniform01() for i in 0.. repeat cumP := cumP + exp(-lambda)lambda^i/factorial(i) if x1.0 then error "bad argument x in betai" bt := if x=0.0 or x=1.0 then 0.0 else x^a(1.0-x)^b/Beta(a,b) -- exp(logGamma(a+b)$FloatSpecialFunctions-
--                          logGamma(a)$FloatSpecialFunctions- -- logGamma(b)$FloatSpecialFunctions+
--                          alog(x)+blog(1.0-x))
if x<(a+1.0)/(a+b+2.0) then return bt  betacf(a,b,x)/a else
return 1.0-btbetacf(b,a,1.0-x)/b
betacf(a:Float,b:Float,x:Float):Float ==
itmax : Integer := 1000
eps : Float := (10.0^(-digits()$Float+1))$Float
qab : Float := a+b
qap : Float := a+1.0
qam : Float := a-1.0
c : Float := 1.0
d : Float := 1.0-qabx/qap
d := 1.0/d
h : Float := d
for m in 1..itmax repeat
em : Float := m::Float
m2 := 2.0em
aa : Float := m(b-em)x/((qam+m2)(a+m2))
c := 1.0+aa/c
d := 1.0/d
h := hdc
aa := -(a+em)(qab+em)x/((a+m2)(qap+m2))
c := 1.0+aa/c
d := 1.0/d
del : Float := dc
h := hdel
if abs(del-1.0)f))
Now, let's try this.
\begin{axiom}
nmminResults ==> Record(X:Vector Float, Fmin:Float, Fail:Boolean)
optimFun(e:Expression Float,initial:List Equation Expression Float):nmminResults ==
vars := [lhs(x) for x in initial]
vals := [rhs(x) for x in initial]::Vector Float
nmmin(vals, ((x:Vector Float):Float +-> eval(e, vars, x)), -1)$OPTIM -- some examples dpois := exp(-lambda)*lambda^y/Gamma(y+1) negll1 := -log(eval(dpois, y=10.0)) optimFun(negll1, [lambda=5.0]) optimFun((b-a^2)^2*100.0+(1.0-a)^2, [a=11.0,b=1.0]) x := [xi::Float for xi in 1..5] Y := [10, 15, 20, 25, 31] negll1 := reduce(+, [eval(-log(dpois),[y=Y(i),lambda=exp(alpha+beta*x(i))]) for i in 1..#Y]); out1 := optimFun(negll1, [alpha=2, beta=0]) -- R: summary(glm(c(10, 15, 20, 25, 31) ~ I(1:5), family=poisson)) mleResult ==> Record(coef:Matrix Float,vars:List Symbol, AIC:Float) mle(negll:Expression Float, initial:List Equation Expression Float):mleResult == vars := [lhs(x) for x in initial]::List Symbol vals := [rhs(x) for x in initial]::Vector Float out : nmminResults := nmmin(vals, (x +-> eval(negll, vars, x)), -1)$OPTIM
estimate := out.X
covMatrix := inverse(matrix([[eval(D(D(negll,x),y),vars,estimate) for y in vars] for x in vars]))
se := [sqrt(covMatrix(i,i)) for i in 1..nrows(covMatrix)]
zscore := [estimate(i)/se(i) for i in 1..nrows(covMatrix)]
pvalues := [2(1-pnorm(zscore(i)::Float)$SOMESTAT) for i in 1..nrows(covMatrix)] AIC := out.Fmin2+2*#(out.X) coef := transpose(matrix([estimate,se,zscore,pvalues])) return [coef,vars,AIC] mle(negll1, [alpha=2.0, beta=0.0]) negll2 := -log(eval(dpois, y=10.0)) mle(negll2, [lambda=5.0]) \end{axiom} As an aside: can we calculate an elasticity from logistic regression? \begin{axiom} Y:=exp(alpha+betakPk)/(1+exp(alpha+betakPk)) D(Y,Pk)/Y D(Y,Pk)/Y/(1-Y) \end{axiom} Some or all expressions may not have rendered properly, because Axiom returned the following error:Error: export FRICAS=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-unknown-linux/lib; LANG=en_US.UTF-8$FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7245897146845666469-25px.axm


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l.153               (Bvec(2)-Bvec(1)^
2)^2*100.0+(1.0-Bvec(1))^2
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l.154  You can't use macro parameter character #' in horizontal mode.
l.166               n : Integer := #
Bvec

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[]\T1/cmr/m/n/12 --nmmin(vector([11.0, 1.0]), Rosen$\OML/cmm/m/it/12 OPTIM; \OM S/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 :\OT1/cmr/m/n/12 0)$\T1/cmr/
m/n/12 OPTIM nm-min(Bvec: Vec-tor Float, fminfn: Vec-tor Float -> Float, in-tol
: Float):nmminResults
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l.333           n :Integer := #
(Bvec)::NonNegativeInteger::Integer
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l.388           --n :Integer := #
(B)::NonNegativeInteger::Integer
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l.464               eps : Float := (10.0^
(-digits()$Float+1))$Float
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l.494                       cumP := cumP + exp(-lambda)*lambda^
i/factorial(i)
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l.504       mean(x:Vector Float):Float == reduce(_
+,x)/#x
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l.504 ...n(x:Vector Float):Float == reduce(_+,x)/#
x
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l.505       mean(x:List Float):Float == reduce(_+,x)/#
x
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l.506       mean(x:Vector DF):DF == reduce(_+,x)/#
x
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l.507       mean(x:List DF):DF == reduce(_+,x)/#
x
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l.510               reduce(_+,[(x(i)-meanx)^2 for i in 1..#
x])$List(Float)/(#x-1) You can't use macro parameter character #' in horizontal mode. l.510 ...-meanx)^2 for i in 1..#x])$List(Float)/(#
x-1)
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l.527               eps : Float := (10.0^
(-digits()$Float+1))$Float
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\OML/cmm/m/it/12 Float \OT1/cmr/m/n/12 == \OML/cmm/m/it/12 ify < \OT1/cmr/m/n/1
2 0\OML/cmm/m/it/12 thenreturn\OT1/cmr/m/n/12 0\OML/cmm/m/it/12 :\OT1/cmr/m/n/1
2 0\OML/cmm/m/it/12 reduce\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12
[\OML/cmm/m/it/12 exp\OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OML/cmm/m/it/12 lam
bda\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 ^^C \OML/cmm/m/it/12 lambda[]i=factorial\
OT1/cmr/m/n/12 (\OML/cmm/m/it/12 yi\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 foryiin\OT
1/cmr/m/n/12 0\OML/cmm/m/it/12 ::y\OT1/cmr/m/n/12 ])$\T1/cmr/m/n/12 List(Float) ppois2(y:Integer,lambda:Float):Float Overfull \hbox (65.58339pt too wide) in paragraph at lines 422--557 \OML/cmm/m/it/12 meanx \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 Float \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 mean\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\O ML/cmm/m/it/12 reduce\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 [(\O ML/cmm/m/it/12 x\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 i\OT1/cmr/m/n/12 ) \OMS/cmsy/ m/n/12 ^^@ \OML/cmm/m/it/12 meanx\OT1/cmr/m/n/12 )[]\OML/cmm/m/it/12 foriin\OT1 /cmr/m/n/12 1\OML/cmm/m/it/12 ::x\OT1/cmr/m/n/12 ])$\T1/cmr/m/n/12 List(Float)/
(x-1) --Gamma(x:Float):Float == Gamma(x)\$\OML/cmm/m/it/12 FloatSpecialFunctions
Beta\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 a \OT1/cmr/m/n/12 :
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(see the transcript file for additional information)
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