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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "Digits:=12; Levels:=8; " }{MPLTEXT 1 0 56 "
BSplineDegree:=7; # Experiment with different parameters" }}{PARA 11 "
" 1 "" {XPPMATH 20 "\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "uniform:=proc(r::constant..constant)\n" }{MPLTEXT 1 
0 10 "   proc()\n" }{MPLTEXT 1 0 24 "      local intrange,f;\n" }
{MPLTEXT 1 0 42 "      intrange:=map(x->round(x*10^10),r);\n" }
{MPLTEXT 1 0 25 "      f:=rand(intrange);\n" }{MPLTEXT 1 0 17 "      f
()/10^10;\n" }{MPLTEXT 1 0 13 "   end proc;\n" }{MPLTEXT 1 0 9 "end pr
oc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Generate_Spherical_D
ata:=proc(N,r)\n" }{MPLTEXT 1 0 28 "local phi,theta,data,i,U,V;\n" }
{MPLTEXT 1 0 10 "data:=[];\n" }{MPLTEXT 1 0 18 "U:=uniform(0..1):\n" }
{MPLTEXT 1 0 18 "V:=uniform(0..2):\n" }{MPLTEXT 1 0 14 "for i to N do
\n" }{MPLTEXT 1 0 32 "    phi:=U()*Pi; theta:=V()*Pi;\n" }{MPLTEXT 1 
0 83 "    data:=[op(data),r*Vector([sin(phi)*cos(theta),sin(phi)*sin(t
heta),cos(phi)])];\n" }{MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 13 "re
turn data;\n" }{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "BSWeights:=proc(degree)::Matrix;\n" }{MPLTEXT 1 0 16 
"local i,mask,n;\n" }{MPLTEXT 1 0 96 "if type(degree,odd) then mask:=V
ector[row](degree+3); else mask:=Vector[row](degree+2); end if;\n" }
{MPLTEXT 1 0 30 "for i from 0 to (degree+1) do\n" }{MPLTEXT 1 0 46 "  \+
  mask[i+1]:=binomial(degree+1,i)/2^degree;\n" }{MPLTEXT 1 0 8 "end do
;\n" }{MPLTEXT 1 0 20 "n:=Dimension(mask);\n" }{MPLTEXT 1 0 110 "retur
n Matrix([[Vector[row]([seq(mask[n+1-2*i],i=1..n/2)])],[Vector[row]([s
eq(mask[n-2*i],i=0..(n/2-1))])]]);\n" }{MPLTEXT 1 0 9 "end proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Log_Sphere:=proc(v0,v1)\n" }
{MPLTEXT 1 0 25 "local cos_theta,i,theta;\n" }{MPLTEXT 1 0 68 "cos_the
ta:=1-sum('(v0[i]-v1[i])^2',i=1..3)/2/sum('v0[i]^2',i=1..3);\n" }
{MPLTEXT 1 0 26 "theta:=arccos(cos_theta);\n" }{MPLTEXT 1 0 50 "return
 evalf((v1-v0*cos_theta)*theta/sin(theta));\n" }{MPLTEXT 1 0 9 "end pr
oc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Exp_Sphere:=proc(v0,
A)\n" }{MPLTEXT 1 0 15 "local i,theta;\n" }{MPLTEXT 1 0 63 "theta:=sqr
t(sum('A[i]^2',i=1..3))/sqrt(sum('v0[i]^2',i=1..3));\n" }{MPLTEXT 1 0 
48 "return evalf(A*sin(theta)/theta+v0*cos(theta));\n" }{MPLTEXT 1 0 
9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Geodesic_Mi
dpt:=proc(v0,v1)\n" }{MPLTEXT 1 0 11 "local i,v;\n" }{MPLTEXT 1 0 14 "
v:=(v0+v1)/2;\n" }{MPLTEXT 1 0 72 "return evalf(v*sqrt(sum('v0[i]^2',i
=1..3))/sqrt(sum('v[i]^2',i=1..3)));\n" }{MPLTEXT 1 0 9 "end proc:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "SphereSubdivide_LogExp_NonI
nterp_Periodic:=proc(data0,weights)\n" }{MPLTEXT 1 0 42 "local data,i,
s,L,m,temp1,temp2,j,newdata;\n" }{MPLTEXT 1 0 13 "newdata:=[];\n" }
{MPLTEXT 1 0 16 "m:=nops(data0);\n" }{MPLTEXT 1 0 33 "s:=Dimension(wei
ghts); s:=s[2]; \n" }{MPLTEXT 1 0 13 "data:=data0;\n" }{MPLTEXT 1 0 
20 "if type(s,odd) then\n" }{MPLTEXT 1 0 17 "    L:=(s-1)/2; \n" }
{MPLTEXT 1 0 65 "    while nops(data)<L+1 do data:=[op(data), op(data)
]; end do; \n" }{MPLTEXT 1 0 53 "    data:=[op(-L..-1,data),op(data),o
p(1..L,data)]; \n" }{MPLTEXT 1 0 25 "    for i from 1 to m do\n" }
{MPLTEXT 1 0 60 "        temp1:=map2(Log_Sphere,data[i+L],data[i..(i+L
-1)]);\n" }{MPLTEXT 1 0 66 "        temp2:=map2(Log_Sphere,data[i+L],d
ata[(i+L+1)..(i+2*L)]);\n" }{MPLTEXT 1 0 236 "       newdata:=[op(newd
ata),op(map2(Exp_Sphere,data[i+L],[eval(sum('temp1[j]*weights[1,j]',j=
1..L))+eval(sum('temp2[-j]*weights[1,-j]',j=1..L)),eval(sum('temp1[j]*
weights[2,j]',j=1..L))+eval(sum('temp2[-j]*weights[2,-j]',j=1..L))]))]
;\n" }{MPLTEXT 1 0 12 "    end do;\n" }{MPLTEXT 1 0 5 "else\n" }
{MPLTEXT 1 0 12 "    L:=s/2;\n" }{MPLTEXT 1 0 69 "    while nops(data)
<max(L,3) do data:=[op(data), op(data)]; end do;\n" }{MPLTEXT 1 0 55 "
    data:=[op((1-L)..-1,data),op(data),op(1..L,data)];\n" }{MPLTEXT 1 
0 25 "    for i from 1 to m do\n" }{MPLTEXT 1 0 55 "         temp1:=Ge
odesic_Midpt(data[i+L-1],data[i+L]);\n" }{MPLTEXT 1 0 57 "         tem
p2:=map2(Log_Sphere,temp1,data[i..(i+s-1)]);\n" }{MPLTEXT 1 0 146 "   \+
      newdata:=[op(newdata),op(map2(Exp_Sphere,temp1,[eval(sum('temp2[
j]*weights[1,j]',j=1..s)),eval(sum('temp2[j]*weights[2,j]',j=1..s))]))
];\n" }{MPLTEXT 1 0 12 "    end do;\n" }{MPLTEXT 1 0 8 "end if;\n" }
{MPLTEXT 1 0 16 "return newdata;\n" }{MPLTEXT 1 0 9 "end proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Difference:=proc(data,order)
\n" }{MPLTEXT 1 0 11 "local c,j;\n" }{MPLTEXT 1 0 55 "c:=[seq((-1)^(or
der-j)*binomial(order,j),j=0..order)]:\n" }{MPLTEXT 1 0 27 "# c:=Vecto
r[row](order+1):\n" }{MPLTEXT 1 0 77 "# for j from 0 to order do c[j+1
]:=(-1)^(order-j)*binomial(order,j): end do:\n" }{MPLTEXT 1 0 26 "if t
ype(data,'list') then\n" }{MPLTEXT 1 0 80 "return [seq(eval(sum('c[j+1
]*data[i+j]',j=0..order)),i=1..(nops(data)-order))];\n" }{MPLTEXT 1 0 
90 "else return [seq(eval(sum('c[j+1]*data[i+j]',j=0..order)),i=1..(Di
mension(data)-order))];\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 
9 "end proc:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "InfinityNorm:=proc(
data)\n" }{MPLTEXT 1 0 9 "local n;\n" }{MPLTEXT 1 0 32 "if type(data[1
],'Vector') then \n" }{MPLTEXT 1 0 23 "n:=Dimension(data[1]);\n" }
{MPLTEXT 1 0 67 "return max(seq(max(seq(abs(data[i][j]),j=1..n)),i=1..
nops(data)));\n" }{MPLTEXT 1 0 52 "else return max(seq(abs(data[i]),i=
1..nops(data)));\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 9 "end p
roc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "AIWeights:=proc(deg
ree)::Matrix;\n" }{MPLTEXT 1 0 24 "local i,L,temp,x,a,m,w;\n" }
{MPLTEXT 1 0 15 "L:=degree/2+1;\n" }{MPLTEXT 1 0 18 "m:=array(1..2*L);
\n" }{MPLTEXT 1 0 33 "temp:=product(1/2-x,x=(1-L)..L);\n" }{MPLTEXT 1 
0 21 "for i from 1 to L do\n" }{MPLTEXT 1 0 67 "   a:=product(i-L-x,x=
(1-L)..(i-1-L))*product(i-L-x,x=(i+1-L)..L);\n" }{MPLTEXT 1 0 45 "   m
[i]:=temp/a/(1/2-i+L); m[2*L+1-i]:=m[i];\n" }{MPLTEXT 1 0 8 "end do;\n
" }{MPLTEXT 1 0 18 "w:=array(1..2*L);\n" }{MPLTEXT 1 0 12 "w[1]:=m[1];
\n" }{MPLTEXT 1 0 23 "for i from 2 to 2*L do\n" }{MPLTEXT 1 0 22 "   w
[i]:=w[i-1]+m[i];\n" }{MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 217 "Ma
trix([[2*(Vector[row]([seq(w[degree+2-i],i=1..(degree+1))])-Vector[row
]([seq(1,i=1..(L-1)),seq(0,i=1..L)]))],[2*(Vector[row]([seq(1,i=1..L),
seq(0,i=1..(L-1))])-Vector[row]([seq(w[degree+2-i],i=1..(degree+1))]))
]]);\n" }{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "DDWeights:=proc(degree)::Vector;\n" }{MPLTEXT 1 0 22 
"local i,L,temp,x,a,m;\n" }{MPLTEXT 1 0 17 "L:=(1+degree)/2;\n" }
{MPLTEXT 1 0 21 "m:=Vector[row](2*L);\n" }{MPLTEXT 1 0 33 "temp:=produ
ct(1/2-x,x=(1-L)..L);\n" }{MPLTEXT 1 0 21 "for i from 1 to L do\n" }
{MPLTEXT 1 0 67 "   a:=product(i-L-x,x=(1-L)..(i-1-L))*product(i-L-x,x
=(i+1-L)..L);\n" }{MPLTEXT 1 0 45 "   m[i]:=temp/a/(1/2-i+L); m[2*L+1-
i]:=m[i];\n" }{MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 10 "return m;\n
" }{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "Subdivide:=proc(data::Vector,weights::Matrix)::Vector;\n" }
{MPLTEXT 1 0 37 "local i,s,n,weight1,weight2,newdata;\n" }{MPLTEXT 1 
0 25 "s:=[Dimension(weights)];\n" }{MPLTEXT 1 0 20 "n:=Dimension(data)
;\n" }{MPLTEXT 1 0 29 "weight1:=weights[1,1..s[2]];\n" }{MPLTEXT 1 0 
29 "weight2:=weights[2,1..s[2]];\n" }{MPLTEXT 1 0 36 "newdata:=Vector[
row](2*(n+1-s[2]));\n" }{MPLTEXT 1 0 28 "for i from 1 to n+1-s[2] do\n
" }{MPLTEXT 1 0 49 "    newdata[2*i-1]:=weight1.data[i..(i+s[2]-1)];\n
" }{MPLTEXT 1 0 47 "    newdata[2*i]:=weight2.data[i..(i+s[2]-1)];\n" 
}{MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 16 "return newdata;\n" }
{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "SphereSubdivide_LogExp_NonInterp:=proc(data,weights)\n" }{MPLTEXT 
1 0 37 "local i,s,L,m,temp1,temp2,j,newdata;\n" }{MPLTEXT 1 0 13 "newd
ata:=[];\n" }{MPLTEXT 1 0 15 "m:=nops(data);\n" }{MPLTEXT 1 0 33 "s:=D
imension(weights); s:=s[2]; \n" }{MPLTEXT 1 0 20 "if type(s,odd) then
\n" }{MPLTEXT 1 0 17 "    L:=(s-1)/2; \n" }{MPLTEXT 1 0 24 "    for i \+
to (m-s+1) do\n" }{MPLTEXT 1 0 60 "        temp1:=map2(Log_Sphere,data
[i+L],data[i..(i+L-1)]);\n" }{MPLTEXT 1 0 66 "        temp2:=map2(Log_
Sphere,data[i+L],data[(i+L+1)..(i+2*L)]);\n" }{MPLTEXT 1 0 236 "      \+
 newdata:=[op(newdata),op(map2(Exp_Sphere,data[i+L],[eval(sum('temp1[j
]*weights[1,j]',j=1..L))+eval(sum('temp2[-j]*weights[1,-j]',j=1..L)),e
val(sum('temp1[j]*weights[2,j]',j=1..L))+eval(sum('temp2[-j]*weights[2
,-j]',j=1..L))]))];\n" }{MPLTEXT 1 0 12 "    end do;\n" }{MPLTEXT 1 0 
5 "else\n" }{MPLTEXT 1 0 12 "    L:=s/2;\n" }{MPLTEXT 1 0 24 "    for \+
i to (m-s+1) do\n" }{MPLTEXT 1 0 55 "         temp1:=Geodesic_Midpt(da
ta[i+L-1],data[i+L]);\n" }{MPLTEXT 1 0 57 "         temp2:=map2(Log_Sp
here,temp1,data[i..(i+s-1)]);\n" }{MPLTEXT 1 0 146 "         newdata:=
[op(newdata),op(map2(Exp_Sphere,temp1,[eval(sum('temp2[j]*weights[1,j]
',j=1..s)),eval(sum('temp2[j]*weights[2,j]',j=1..s))]))];\n" }{MPLTEXT
 1 0 12 "    end do;\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 16 "
return newdata;\n" }{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 71 "SphereSubdivide_LogExpModified_NonInterp_Periodi
c:=proc(data0,weights)\n" }{MPLTEXT 1 0 72 " local data,i,s,L,m,temp1,
temp2,j,newdata,ddweight,y,newdata1,newdata2;\n" }{MPLTEXT 1 0 14 " ne
wdata:=[];\n" }{MPLTEXT 1 0 17 " m:=nops(data0);\n" }{MPLTEXT 1 0 33 "
 s:=Dimension(weights); s:=s[2];\n" }{MPLTEXT 1 0 30 " ddweight:=DDWei
ghts(2*s-1); \n" }{MPLTEXT 1 0 14 " data:=data0;\n" }{MPLTEXT 1 0 16 "
 L:=floor(s/2);\n" }{MPLTEXT 1 0 60 " while nops(data)<s do data:=[op(
data), op(data)]; end do; \n" }{MPLTEXT 1 0 53 " data:=[op((1-s)..-1,d
ata),op(data),op(1..s,data)]; \n" }{MPLTEXT 1 0 26 " for i from s to m
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i],data[(i+1)..(i+L)]);\n" }{MPLTEXT 1 0 126 "   newdata1:=Exp_Sphere(
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Sphere,y,data[(i-s+L+1)..(i+L)]);\n" }{MPLTEXT 1 0 70 "   newdata2:=Ex
p_Sphere(y,eval(sum('temp2[j]*weights[2,j]',j=1..s)));\n" }{MPLTEXT 1 
0 45 "   newdata:=[op(newdata),newdata1,newdata2];\n" }{MPLTEXT 1 0 9 
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