{VERSION 4 0 "IBM INTEL LINUX22" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 
"" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }
{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 
1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 
1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 
0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 
0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 
12 12 0 0 0 0 0 0 19 0 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 19 "Relaxation Methods " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "with(LinearAlgebra):\nwith(plots):" }}{PARA 7 "" 1 "
" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "1D Poisson Equation" }}{PARA 0 "
" 0 "" {TEXT -1 42 " Number of unknowns: n+2 for n inner nodes" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n := 8;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "\nV
ector of unknowns:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "unkn := [seq(
u[i], i=1..n)];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%unknG7*&%\"uG6
#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%&F'6#\"\"&&F'6#\"\"'&F'6#\"\"(&F'
6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We can one linear equa
tion for each of the unknowns:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "s
le := [u[1]=0, seq( u[i]-(u[i-1]+u[i+1])/2= 0, i=2..n-1), u[n]=0];" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sleG7*/&%\"uG6#\"\"\"\"\"!/,(&F(6#
\"\"#F**&#F*F0F*F'F*!\"\"*&#F*F0F*&F(6#\"\"$F*F3F+/,(F6F**&#F*F0F*F.F*
F3*&#F*F0F*&F(6#\"\"%F*F3F+/,(F?F**&#F*F0F*F6F*F3*&#F*F0F*&F(6#\"\"&F*
F3F+/,(FHF**&#F*F0F*F?F*F3*&#F*F0F*&F(6#\"\"'F*F3F+/,(FQF**&#F*F0F*FHF
*F3*&#F*F0F*&F(6#\"\"(F*F3F+/,(FZF**&#F*F0F*FQF*F3*&#F*F0F*&F(6#\"\")F
*F3F+/F]oF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Using the function
 " }{TEXT 0 14 "GenerateMatrix" }{TEXT -1 17 " (defined in the " }
{TEXT 0 13 "LinearAlgebra" }{TEXT -1 97 " package) we can generate the
 system matrix and the right hand side from the system of equations:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A,b := GenerateMatrix(sle, unkn);
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"AG%\"bG6$-%'RTABLEG6$\"*O&
>n8-%'MATRIXG6#7*7*\"\"\"\"\"!F2F2F2F2F2F27*#!\"\"\"\"#F1F4F2F2F2F2F27
*F2F4F1F4F2F2F2F27*F2F2F4F1F4F2F2F27*F2F2F2F4F1F4F2F27*F2F2F2F2F4F1F4F
27*F2F2F2F2F2F4F1F47*F2F2F2F2F2F2F2F1-F)6$\"*?'Gn8-F-6#7*7#F2FCFCFCFCF
CFCFC" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Note that the zero funct
ion solves our problem:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "z := Vector(n,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"zG-%'RTABLEG6$\"*o(4p8-%'MATRIXG6#7*7#\"\"!F-F-F-F-F-F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Equal(A.z, b);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Jacobi-Relaxation" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Define the jacobi relaxation (line
wise computation of the new approximation):" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "jacobi := proc(A::Matrix, b::Vecto
r,x::Vector,n::posint)\n   local i,j,xnew;\n   xnew := Vector(x);\n   \+
for i from 1 to n do\n      xnew[i] := ( b[i] - add(A[i,j]*x[j], j=1..
i-1) \n                        - add(A[i,j]*x[j], j=i+1..n) )/A[i,i];
\n   end do;\nreturn xnew;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%'jacobiGR6&'%\"AG%'MatrixG'%\"bG%'VectorG'%\"xGF,'%\"nG%'posintG6
%%\"iG%\"jG%%xnewG6\"F6C%>8&-F,6#9&?(8$\"\"\"F?9'%%trueG>&F96#F>*&,(&9
%FDF?-%$addG6$*&&9$6$F>8%F?&F<6#FPF?/FP;F?,&F>F?F?!\"\"FV-FJ6$FL/FP;,&
F>F?F?F?F@FVF?&FN6$F>F>FVOF9F6F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 43 "Define a sine mode as the initial solution:" }{MPLTEXT 1 0 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "sinemode := proc(k::posint, n:
:posint)\n# returns a list x of initial solution\n# sine mode with wav
e number k is taken as the initial solution\n   return Vector( [evalf(
 seq( sin((i-1)*k*Pi/(n-1)), i=1..n ) )] );\nend proc;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%)sinemodeGR6$'%\"kG%'posintG'%\"nGF)6\"F,F,O-%'V
ectorG6#7#-%&evalfG6#-%$seqG6$-%$sinG6#*&*(,&%\"iG\"\"\"F?!\"\"F?9$F?%
#PiGF?F?,&9%F?F?F@F@/F>;F?FDF,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 120 "Prepare a plot ofthe error norms (note that the approximation \+
are equal to the error, beacause the exact solution is 0):" }{MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "jacobiplot := proc(A::M
atrix, b::Vector, x0::Vector,n::posint,iter::posint)\n   local norms,x
,i;\n   x := Vector(x0);\n   norms[0] := Norm(x);\n   for i from 1 to \+
iter do\n      x := jacobi(A,b,x,n);\n      norms[i] := Norm(x);\n   e
nd do;\n   return [seq(norms[i],i=0..iter)];\nend proc;" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%+jacobiplotGR6''%\"AG%'MatrixG'%\"bG%'VectorG'
%#x0GF,'%\"nG%'posintG'%%iterGF16%%&normsG%\"xG%\"iG6\"F8C&>8%-F,6#9&>
&8$6#\"\"!-_%.LinearAlgebraG%%NormG6#F;?(8&\"\"\"FK9(%%trueGC$>F;-%'ja
cobiG6&9$9%F;9'>&FA6#FJFDO7#-%$seqG6$FW/FJ;FCFLF8F8F8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "iter := 50;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%iterG\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "er
ror plot for sine mode k=1:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "x0 := sinemode(1,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#x0G-%'RTABLEG6$\"*'>Z*Q\"-%'MATRIXG6#7*7#$\"\"!F/7#$\"+$RP)QV!#5
7#$\"+D[J=yF37#$\"+B\"z#\\(*F3F7F4F0F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "errors := jacobiplot(A,b,x0,n,iter):\nlistplot(errors
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "error plots for sine modes \+
k=1, k=3, and k=5:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 256 "display( listplot( jacobiplot(A,b,sinemode(1,n),n,iter), color=
red, legend=\"k=1\" ),\n         listplot( jacobiplot(A,b,sinemode(3,n
),n,iter), color=green, legend=\"k=3\" ),\n         listplot( jacobipl
ot(A,b,sinemode(5,n),n,iter), color=blue, legend=\"k=5\" ) );" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Gauss-Seidel Relaxation" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Define the Gauss-Seidel relaxation
 (linewise computation of the new approximation):" }{MPLTEXT 1 0 0 "" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "gseidel := proc(A::Matrix, b::Ve
ctor,x::Vector,n::posint)\n   local i,j;\n   for i from 1 to n do\n   \+
   x[i] := ( b[i] - add(A[i,j]*x[j], j=1..i-1) \n                     \+
- add(A[i,j]*x[j], j=i+1..n) )/A[i,i];\n   end do;\nreturn x;\nend pro
c;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(gseidelGR6&'%\"AG%'MatrixG'%
\"bG%'VectorG'%\"xGF,'%\"nG%'posintG6$%\"iG%\"jG6\"F5C$?(8$\"\"\"F99'%
%trueG>&9&6#F8*&,(&9%F?F9-%$addG6$*&&9$6$F88%F9&F>6#FKF9/FK;F9,&F8F9F9
!\"\"FQ-FE6$FG/FK;,&F8F9F9F9F:FQF9&FI6$F8F8FQOF>F5F5F5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "Prepare a plot of the error norms  (depen
ding on number of iterations):" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 281 "gseidelplot := proc(A::Matrix, b::Vector, x0::Vect
or,n::posint,iter::posint)\n   local norms,x,i;\n   x := Vector(x0);\n
   norms[0] := Norm(x);\n   for i from 1 to iter do\n      x := gseide
l(A,b,x,n);\n      norms[i] := Norm(x);\n   end do;\n   return [seq(no
rms[i],i=0..iter)];\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,g
seidelplotGR6''%\"AG%'MatrixG'%\"bG%'VectorG'%#x0GF,'%\"nG%'posintG'%%
iterGF16%%&normsG%\"xG%\"iG6\"F8C&>8%-F,6#9&>&8$6#\"\"!-_%.LinearAlgeb
raG%%NormG6#F;?(8&\"\"\"FK9(%%trueGC$>F;-%(gseidelG6&9$9%F;9'>&FA6#FJF
DO7#-%$seqG6$FW/FJ;FCFLF8F8F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "iter := 50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%iterG\"#]" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "error plot for sine mode k=1:" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "errors := gseid
elplot(A,b,x0,n,iter):\nlistplot(errors);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "error plots for sine modes k=1, k=3, and k=5:" }{MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "display( listplot( gsei
delplot(A,b,sinemode(1,n),n,iter), color=red, legend=\"k=1\" ),\n     \+
    listplot( gseidelplot(A,b,sinemode(3,n),n,iter), color=green, lege
nd=\"k=3\" ),\n         listplot( gseidelplot(A,b,sinemode(5,n),n,iter
), color=blue, legend=\"k=5\" ) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{MARK "0 1 0" 8 }{VIEWOPTS 1 1 0 3 2 1804 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 136719536 136728620 
136909768 138947196 }{RTABLE 
M6R0
I6RTABLE_SAVE/136719536X,%)anythingG6"6"][[[[[p[o")")"""#!""""#""!F+F+F+F+F+F+F
'F(F+F+F+F+F+F+F(F'F(F+F+F+F+F+F+F(F'F(F+F+F+F+F+F+F(F'F(F+F+F+F+F+F+F(F'F(F+F+
F+F+F+F+F(F'F+F+F+F+F+F+F+F(F'F&
}
{RTABLE 
M6R0
I6RTABLE_SAVE/136728620X*%)anythingG6"6"\[[[[[t)")""!F'F'F'F'F'F'F'F&
}
{RTABLE 
M6R0
I6RTABLE_SAVE/136909768X*%)anythingG6"6"\[[[[[t)")""!F'F'F'F'F'F'F'F&
}
{RTABLE 
M6R0
I6RTABLE_SAVE/138947196X*%)anythingG6"6"\[[[[[t)")$""!F($"+$RP)QV!#5$"+D[J=yF+$
"+B"z#\(*F+F.F,F)F'F&
}