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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT 256 27 "NUMEERISEN ANALYYSIN DEMO
T " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 51 "Differenssimenetelm\344 r
euna-arvoteht\344v\344n ratkaisussa" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart:with(linalg):" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 39 " Tridiagonaalisen yht\344l\366ryhm\344n ratkaisu" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "tridiag:=proc(A,B,C,v,n)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "   local omega,alpha,i,u,y;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "   omega[1]:=B[1]: alpha[1]:=C[1]/o
mega[1]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "   for i from 2 to n-1 \+
do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "         omega[i]:=B[i]-alpha
[i-1]*A[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "         alpha[i]:=C
[i]/omega[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "   omega[n]:=B[n]-alpha[n-1]*A[n]:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "   u[1]:=v[1]/omega[1]: " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "   for i from 2 to n do    " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "      u[i]:=(v[i]-A[i]*u[i-1])/omega[i];  " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "   y:=vector(n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "   y[n]:=u[n]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "   for i fro
m n-1 by -1 to 1 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "       y[i]
:=u[i]-alpha[i]*y[i+1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   y; " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end;" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 " p(x) y'' + q
(x) y' + r(x) y = s(x), y(a)=ya, y(b)=yb" }{TEXT 258 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "`y''` - 'y' = 'x', 'y(a)'=5, 'y(b)'=-1;a:=0.;b:=5.;ya
:=5.:yb:=-1.: n:=5;  h:=(b-a)/n:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "x:=[seq(i*h,i=1..n-1)];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "p:=([seq(1.,i=1..n-1)]);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "q:=[seq(0.,i=1..n-1)];" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "r:=[seq(-1,i=1..n-1)];" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "s:=x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
pperhh:=p/h^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "qper2h:=0
.5*q/h;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d:=r-2*pperhh;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "u:=pperhh+qper2h;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "l:=pperhh-qper2h;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "s[1]:=s[1]-ya*l[1];s[n-1]:=s
[n-1]-yb*u[n-1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 36 "Muodostetaan
 tridiagonaalinen ryhm\344:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with
(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "K:=diag(op(d)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for i to n-2 do K[i,i+1]:=u[i];K[i+
1,i]:=l[i+1];od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "K:=evalm(K);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s:=s;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 265 51 "Ratkaistaan tridiagonaalinen yht\344l\366ryhm\344 K y = \+
s:  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Digits:=10:y:=tridiag(l,d,u
,s,n-1):Digits:=3:evalm(y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
137 "with(plots):Tarkka_ratkaisu:='c1'*exp('x')+'c2'*exp(-'x')-'x';dif
f_p:=pointplot([[a,ya],seq([x[i],y[i]],i=1..n-1),[b,yb]],symbol=circle
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits:=10:c2:=(4.-5*exp(5.)
)/(exp(-5.)-exp(5.));c1:=5.-c2;t_p:=plot(c1*exp(t)+c2*exp(-t)-t,t=0..5
,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{t_p,diff
_p\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 68 " Toinen esimerkki : x^2  y'' - x y' + (1 \+
- x) y = 0, y(1)=0, y(5)=3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
111 "'x'^2*`y''`-'x'*`y'`+(1-'x')*y = 0, 'y'('a')=0, 'y'('b')=3.;a:=1.
;b:=5.;ya:=0.:yb:=3.: n:=8;  h:=0.5;Digits:=5:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "x:=[seq(a+i*h,i=1..n-1)];  " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "p:=[seq(x[i]^2,i=1..n-1)]; " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "q:=-x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r:=[seq(1
-x[i],i=1..n-1)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "s:=[seq(0.,i=1
..n-1)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pperhh:=p/h^2;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "qper2h:=0.5*q/h; " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "d:=r-2*pperhh;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "u:=pperhh+qper2h;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "l:=pperh
h-qper2h; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s[1]:=s[1]-ya*l[1];" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s[n-1]:=s[n-1]-yb*u[n-1];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 261 36 "Muodostetaan tridiagonaalinen ryh
m\344:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "K:=diag(op(d)): for i to \+
n-2 do K[i,i+1]:=u[i];K[i+1,i]:=l[i+1];od:K:=evalm(K);s:=s;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 260 51 "Ratkaistaan tridiagonaalinen yht\344l\366ryhm\344 K y = \+
s:  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Digits:=10:y:=tridiag(l,d,u
,s,n-1):Digits:=4:y=evalm(y);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 
15 "Tarkka ratkaisu" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "dsolve(\{t^2
*diff(z(t),t$2)-t*diff(z(t),t)+(1-t)*z(t)=0,z(1)=0,z(5)=3\},z(t)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "tr:=rhs(%);" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "diff_p:=poi
ntplot([[a,ya],seq([x[i],y[i]],i=1..n-1),[b,yb]],symbol=circle):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "TR:=plot(tr,t=1..5,color=blue):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(\{diff_p,TR\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 23 " Ep\344lineaarinen yht\344l\366:" }{TEXT 262 39 "  y'' = 3 y^2 \+
/ 2,  y(0) = 2, y(1) = 3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 
"`y''` = 3*'y'^2 / 2 -'x'*`y'`;  'y'(a) = 2; 'y'(b) = 3;a:=0;b:=1;ya:=
2.:yb:=3.: N:=8; h:=(b-a)/N;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 21 "
Alkuarvauksena suora:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x':" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "for i from 0 to N do x[i]:=i*h; v[i]:=2.+
x[i];od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Digits:=4;print(x=[seq(
evalf(x[i]),i=0..N)]);print(y=[seq(evalf(v[i]),i=0..N)]);Digits:=10;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ero:=100:kl:=0:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 264 20 
"Kiintopisteiterointi" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "whi
le ero>0.001 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  ero:=0:kl:=kl+
1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "  for i from 1 to N-1 do " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "     u[i]:=(v[i-1]+v[i+1]-h^2*(3*v[
i]^2/2-x[i]*(v[i+1]-v[i-1])/(2*h)))/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "     ero:=max(ero,abs(u[i]-v[i]));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "  od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  for i fr
om 1 to N-1 do v[i]:=u[i]; od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " \+
 Digits:=4;print('kl'=kl,y=[seq(evalf(v[i]),i=0..N)]);Digits:=10;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{PARA 11 "" 1 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "pointplot([seq([a+i*h,v
[i]],i=0..N)],style=point,symbol=circle,color=blue);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "dj:=plot([seq([a+i*h,v[i]],i=0..N)],style=point,
symbol=circle,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 5 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Newtonin menetelm\344" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:='y':" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "l:=[seq(1-h*x[i]/2.,i=1..N-1)];" }{TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "u:=[seq(1+h*x[i]/2.,i=1..N-1)];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "d:=[seq(-2-3*h^2*y[i],i=1..N-1)];" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "b:=[seq(y[i-1]+y[i+1]-2*y[i]-h^2*
(3*y[i]^2/2-x[i]*(y[i+1]-y[i-1])/(2*h)),i=1..N-1)];" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "for i from 0 to N do y[i]:=2.+i*h; od:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Digits:=4;print(y=[seq(evalf(y[i])
,i=0..N)]);Digits:=10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "kl:=0:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "apu:=[seq(1,i=1..N-1)]:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "while norm(apu)>0.00001 do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "     kl:=kl+1;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "     apu:=tridiag(l,d,u,-b,N-1);" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 43 "     for i to N-1 do y[i]:=y[i]+apu[i]; od;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "     Digits:=4:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "     print('kl'=kl,y=[seq(evalf(y[i]),i=0..N)]);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "     Digits:=10:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 
"dN:=pointplot([seq([a+i*h,y[i]],i=0..N)],symbol=cross,color=red):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots):display(\{dN,dj\});" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 27 }
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