restart;
Digits:=30:

f:=proc(n)
	x[n]-y[n];
	
end proc:


e1:=y[n] = (15592/1575)*h*f(n+5)+(35618816/99225)*h*f(n+9/2)-(4391496/15925)*h*f(n+13/3)-(2035368/13475)*h*f(n+14/3)-(212552/121275)*h*f(n+1)+(10016/11025)*h*f(n+2)-(31672/4725)*h*f(n+3)+(19454/315)*h*f(n+4)-(351518/1289925)*h*f(n)+y[n+4]:
e2:=y[n+1] = -(34107/22400)*h*f(n+5)-(212224/3675)*h*f(n+9/2)+(92569149/2038400)*h*f(n+13/3)+(82333989/3449600)*h*f(n+14/3)-(568893/1724800)*h*f(n+1)-(459807/313600)*h*f(n+2)+(1189/22400)*h*f(n+3)-(50499/4480)*h*f(n+4)+(32951/6115200)*h*f(n)+y[n+4]:
e3:=y[n+2] = (69/175)*h*f(n+5)+(1466368/99225)*h*f(n+9/2)-(13851/1225)*h*f(n+13/3)-(60507/9800)*h*f(n+14/3)+(43/3675)*h*f(n+1)-(3509/9800)*h*f(n+2)-(6701/4725)*h*f(n+3)+(871/420)*h*f(n+4)-(247/396900)*h*f(n)+y[n+4]:
e4:=y[n+3] = -(31411/201600)*h*f(n+5)-(745216/99225)*h*f(n+9/2)+(13557213/2038400)*h*f(n+13/3)+(9737253/3449600)*h*f(n+14/3)-(20869/15523200)*h*f(n+1)+(36329/2822400)*h*f(n+2)-(202169/604800)*h*f(n+3)-(100187/40320)*h*f(n+4)+(14669/165110400)*h*f(n)+y[n+4]:
e5:=y[n+13/3] = -(3364243/1322697600)*h*f(n+5)-(134364928/651015225)*h*f(n+9/2)+(19955023/55036800)*h*f(n+13/3)+(5577703/93139200)*h*f(n+14/3)-(910757/101847715200)*h*f(n+1)+(1336457/18517766400)*h*f(n+2)-(2512217/3968092800)*h*f(n+3)+(31844549/264539520)*h*f(n+4)+(690797/1083289334400)*h*f(n)+y[n+4]:
e6:=y[n+14/3] = -(29107/10333575)*h*f(n+5)+(7757824/651015225)*h*f(n+9/2)+(180667/429975)*h*f(n+13/3)+(342733/2910600)*h*f(n+14/3)-(7253/795685275)*h*f(n+1)+(42467/578680200)*h*f(n+2)-(19853/31000725)*h*f(n+3)+(993749/8266860)*h*f(n+4)+(22037/33852791700)*h*f(n)+y[n+4]:
e7:=y[n+9/2] = -(115447/51609600)*h*f(n+5)-(21389/198450)*h*f(n+9/2)+(231041241/521830400)*h*f(n+13/3)+(43797591/883097600)*h*f(n+14/3)-(32833/3973939200)*h*f(n+1)+(48323/722534400)*h*f(n+2)-(91493/154828800)*h*f(n+3)+(1220071/10321920)*h*f(n+4)+(24863/42268262400)*h*f(n)+y[n+4]:
e8:=y[n+5] = (1989/22400)*h*f(n+5)-(61184/99225)*h*f(n+9/2)+(1496637/2038400)*h*f(n+13/3)+(2458917/3449600)*h*f(n+14/3)+(73/5174400)*h*f(n+1)-(31/313600)*h*f(n+2)+(359/604800)*h*f(n+3)+(1079/13440)*h*f(n+4)-(179/165110400)*h*f(n)+y[n+4]:



h:=0.01:
N:=solve(h*p = 8/8, p):
#N := 10:
#n:=0:
#exy:= [seq](eval(i+exp(-i)-1), i=h..N,h):
c:=1:
inx:=0:
iny:=0:

mx := proc(t,n):
   t + 0.01*n:
end proc:

exy := (x - 1.0 + exp(-x)):

vars := y[n+1],y[n+2],y[n+3],y[n+4],y[n+13/3],y[n+14/3],y[n+9/2],y[n+5]:

printf("%6s%20s%20s%20s\n", "h","numy1","Exact", "Error");
#for k from 1 to N/8 do
for c from 1 to N do

	par1:=x[n]=map(mx,(inx,0)),x[n+1]=map(mx,(inx,1)),
		x[n+2]=map(mx,(inx,2)),x[n+3]=map(mx,(inx,3)),
		x[n+4]=map(mx,(inx,4)),x[n+5]=map(mx,(inx,5)),
		x[n+13/3]=map(mx,(inx,13/3)),x[n+14/3]=map(mx,(inx,14/3)),
		x[n+9/2]=map(mx,(inx,9/2)):
	par2:=y[n]=iny:
	res:=eval(<vars>, fsolve(eval({e||(1..8)},[par1,par2]), {vars}));
	
	printf("%7.3f%22.10f%20.10f%17.3g\n", 
		h*c,res[8],(exy,[x=c*h]),abs(res[8]-eval(exy,[x=c*h]))):
		#c:=c+1:
	
	iny:=res[8]:
	inx:=map(mx,(inx,5)):
end do:

Dear all,

Please Kindly help to correct or modify the code above

Thank you and best regards
 

The rational expression at the beginning of the code is approximant. p1-p5 are conditions imposed on t, and are to be solved simultaneously to obtain a[0]-a[4] which are then substituted into t to obtain Cf. S1-S4 are to be obtained from Cf and its derivative.

However, I observed that Cf is not providing the desired results. What have I done wrong? Please Can someone be of help?

Thank you and kind regards

restart:
t:=sum(a[j]*x^j,j=0..2)/sum(a[j]*x^j,j=3..4):
F:=diff(t,x,x):
p1:=simplify(eval(t,x=q))=y[n]:
p2:=simplify(eval(t,x=q+h))=y[n+1]:
p3:=simplify(eval(F,x=q))=f[n]:
p4:=simplify(eval(F,x=q+h))=f[n+1]:
p5:=simplify(eval(F,x=q+2*h))=f[n+2]:
vars:= seq(a[i],i=0..4):
Cc:=eval(<vars>, solve({p||(1..5)}, {vars}));
for i from 1 to 5 do
	a[i-1]:=Cc[i]:
end do:
Cf:=t;
M:=diff(Cf,x):
s4:=y[n+2]=collect(simplify(eval(Cf,x=q+2*h)),[y[n],y[n+1],f[n],f[n+1],f[n+2]], recursive);
s3:=h*delta[n]=collect(h*simplify(eval(M,x=q)),{y[n],y[n+1],f[n],f[n+1],f[n+2]},factor);
s2:=h*delta[n+1]=collect(h*simplify(eval(M,x=q+h)),{y[n],y[n+1],f[n],f[n+1],f[n+2]},factor):
s1:=h*delta[n+1]=collect(h*simplify(eval(M,x=q+2*h)),{y[n],y[n+1],f[n],f[n+1],f[n+2]},factor):

Good day, all. I hope we are staying safe.

Please I need help with the regard to the following codes.

The major issue is with rho:=x[n]/h. The rho needs to change values as x[n] changes in the computation. This I think I have gotten wrong. Your professional modifications or/and corrections to the code would be appreciated.

Thank you and kind regards.

restart;
Digits:=20:
#assume(alpha>0,alpha < 1):
f:=proc(n)
	-y[n]-x[n]+(x[n])^2+(2*(x[n])^(2-alpha))/GAMMA(3-alpha)+((x[n])^(1-alpha))/GAMMA(2-alpha):
end proc:

e2:=y[n+2] = y[n]+(1/2)*((alpha^2-alpha)*rho^2+(2*alpha^2-2)*rho+(4/3)*alpha^2-(2/3)*alpha)*(rho*h)^alpha*GAMMA(2-alpha)*f(n)/rho-alpha*GAMMA(2-alpha)*((-1+alpha)*rho^2+(2*alpha-2)*rho+(4/3)*alpha-8/3)*(h*(rho+1))^alpha*f(n+1)/(rho+1)+(1/2)*((alpha^2-alpha)*rho^2+2*(-1+alpha)^2*rho+(4/3)*alpha^2-(14/3)*alpha+4)*(h*(rho+2))^alpha*f(n+2)*GAMMA(2-alpha)/(rho+2):
e1:=y[n+1] = y[n]+(1/4)*((alpha^2-alpha)*rho^2+(alpha^2+3*alpha-4)*rho+(1/3)*alpha^2+(4/3)*alpha)*GAMMA(2-alpha)*f(n)/((rho*h)^(-alpha)*rho)-(1/2)*GAMMA(2-alpha)*((alpha^2-alpha)*rho^2+(alpha^2+alpha-2)*rho+(1/3)*alpha^2+(1/3)*alpha-2)*f(n+1)/((h*(rho+1))^(-alpha)*(rho+1))+(1/4)*((-1+alpha)*rho^2+(-1+alpha)*rho+(1/3)*alpha-2/3)*alpha*GAMMA(2-alpha)*f(n+2)/((h*(rho+2))^(-alpha)*(rho+2)):


alpha:=0.25:
inx:=0:
iny:=0:
#x[0]:=0:
h:=1/20:
N:=solve(h*p = 1, p):
n:=0:
c:=1:
rho:=x[n]/h: 
err := Vector(round(N)):
exy_lst := Vector(round(N)):

for j from 0 to 2 do
	t[j]:=inx+j*h:
end do:
vars:=y[n+1],y[n+2]:

step := [seq](eval(x, x=c*h), c=1..N):

printf("%4s%15s%15s%16s\n", 
	"h","Num.y","Ex.y","Error y");
st := time():
for k from 1 to N/2 do

	par1:=x[n]=t[0],x[n+1]=t[1],x[n+2]=t[2]:
	par2:=y[n]=iny:
	res:=eval(<vars>, fsolve(eval({e||(1..2)},[par1,par2]), {vars}));

	for i from 1 to 2 do
		exy:=eval(-c*h+(c*h)^2):
		printf("%5.3f%17.9f%15.9f%15.5g\n", 
		h*c,res[i],exy,abs(res[i]-exy)):
		
		err[c] := abs(evalf(res[i]-exy)):
		exy_lst[c] := exy:
		numerical_y1[c] := res[i]:
		c:=c+1:
		rho:=rho+1:
	end do:
	iny:=res[2]:
	inx:=t[2]:
	for j from 0 to 2 do
		t[j]:=inx + j*h:
	end do:
end do:
v:=time() - st;
printf("Maximum error is %.13g\n", max(err));

numerical_array_y1 := [seq(numerical_y1[i], i = 1 .. N)]:

time_t := [seq](step[i], i = 1 .. N):

with(plots):
numerical_plot_y1 := plot(time_t, numerical_array_y1, style = [point], symbol = [asterisk],
				color = [blue,blue],symbolsize = 15, title="y numerical",legend = ["Numerical"]);
exact_plot_y1 := plot(time_t, exy_lst, style = [line], symbol = [box], 
				color = [red,red], symbolsize = 10, title="y exact",legend = ["Exact"]);

display({numerical_plot_y1, exact_plot_y1}, title = "Exact and Numerical Solution of Example 1 ");

I am trying to run this code but this error message is coming up "Error, (in fprintf) number expected for floating point format
". What have I done wrong and how do I correct it.

Thank you and best regards.

restart;
Digits:=30:

f:=proc(n)
	-((sin(x[n]))/(2-sin(x[n])))*(2+sin(x[n]-Pi)):
end proc:

e1:=y[n+1]=h*delta[n]-(1/2)*h^2*(-u^2*sin((1/2)*u)+2*cos((1/2)*u)*u-2*cos(u)*u-4*sin((1/2)*u)+2*sin(u))*f(n)/(u^2*(2*sin((1/2)*u)-sin(u)))+(1/2)*h^2*(u^2*sin((1/2)*u)+2*cos((1/2)*u)*u-4*sin((1/2)*u)+2*sin(u)-2*u)*f(n+1)/(u^2*(2*sin((1/2)*u)-sin(u)))-(1/2)*h^2*(u*sin(u)+2*cos(u)-2)*f(n+1/2)/(u*(2*sin((1/2)*u)-sin(u)))+y[n]:
e2:=h*delta[n+1]=h*delta[n]+h^2*(u*sin((1/2)*u)+cos(u)-1)*f(n)/(u*(2*sin((1/2)*u)-sin(u)))+h^2*(u*sin((1/2)*u)+cos(u)-1)*f(n+1)/(u*(2*sin((1/2)*u)-sin(u)))-h^2*(u*sin(u)+2*cos(u)-2)*f(n+1/2)/(u*(2*sin((1/2)*u)-sin(u))):
e3:=y[n+1/2]=(1/2)*h*delta[n]-(1/8)*h^2*(-u^2*sin((1/2)*u)+4*cos((1/2)*u)*u-4*cos(u)*u-16*sin((1/2)*u)+8*sin(u))*f(n)/(u^2*(2*sin((1/2)*u)-sin(u)))+(1/8)*h^2*(u*sin((1/2)*u)+4*cos((1/2)*u)-4)*f(n+1)/(u*(2*sin((1/2)*u)-sin(u)))-(1/8)*h^2*(u^2*sin(u)+4*cos(u)*u+16*sin((1/2)*u)-8*sin(u)-4*u)*f(n+1/2)/(u^2*(2*sin((1/2)*u)-sin(u)))+y[n]:
e4:=h*delta[n+1/2]=h*delta[n]-(1/2)*h^2*(-u*sin((1/2)*u)+4*cos((1/2)*u)-2*cos(u)-2)*f(n)/(u*(2*sin((1/2)*u)-sin(u)))+(1/2)*h^2*(u*sin((1/2)*u)+4*cos((1/2)*u)-4)*f(n+1)/(u*(2*sin((1/2)*u)-sin(u)))-(1/2)*h^2*(u*sin(u)+2*cos(u)-2)*f(n+1/2)/(u*(2*sin((1/2)*u)-sin(u))):


inx:=0:
ind:=1:
iny:=2:
h:=Pi/4:
n:=0:
omega:=1:
u:=omega*h:
N:=solve(h*p = 8*Pi, p):

c:=1:
for j from 0 to 1 by 1/2 do
	t[j]:=inx+j*h:
end do:

vars:=y[n+1/2],y[n+1],delta[n+1/2],delta[n+1]:

step := [seq](eval(x, x=c*h), c=1..N):
printf("%6s%15s%15s%16s%15s%15s%15s\n", 
	"h","Num.y","Num.z","Ex.y","Ex.z","Error y","Error z");
#eval(<vars>, solve({e||(1..4)},{vars}));


st := time():
for k from 1 to N do

	par1:=x[0]=t[0],x[1/2]=t[1/2],x[1]=t[1]:
	par2:=y[n]=iny,delta[n]=ind:
	res:=eval(<vars>, fsolve(eval({e||(1..4)},[par1,par2]), {vars}));

	for i from 1 to 2 do
		exy:=eval(2+sin(c*h/2)):
		exz:=eval(-sin(c*h/2)+10*epsilon*cos(10*c*h)):
		printf("%6.5f%17.9f%15.9f%15.9f%15.9f %13.5g%15.5g\n", 
		h*c,res[i],res[i+2],exy,exz,abs(res[i]-exy),
		abs(res[i+2]-exz)):
		
		c:=c+1:

	end do:
	iny:=res[2]:
	ind:=res[4]:
	inx:=t[1]:
	for j from 0 to 1 by 1/2 do
		t[j]:=inx + j*h:
	end do:
end do:

In this code, there are 8 vectors, each vector has 5 points except vectors 7 and 8 with 4 points. I made their  (7 and 8) respective fifth point a string. However, this error statement pops up "Error, invalid input: log[10] expects its 1st argument, x, to be of type algebraic, but received 2.51E-10".

Can someone help me with the correction?

Thank you all and kind regards

restart;

B:=<<601,1201,2401,4801,9601>|<2.87E-19,7.94E-21,7.94E-23,1.03E-24,5.91E-26>|
    <2001,4001,8001,16001,32001>|<3.30E-2,5.37E-4,8.47E-6,1.33E-7,2.08E-9>|
    <183,365,728,1476,2910>|<4.58E0,7.54E-8,2.20E-11,4.95E-14,9.15E-15>|
    <1621,3020,6166,12022,"2222">|<2.95E-3,8.51E-6,3.39E-8,2.51E-10,"2.51E-10">>:

for i from 1 to 5 do
   B[i, 2] := log[10](B[i, 2]):                      
   B[i, 4] := log[10](B[i, 4]):         
   B[i, 6] := log[10](B[i, 6]):         
   B[i, 8] := log[10](B[i, 8]):

   B[i, 1] := log[10](B[i, 1]):                      
   B[i, 3] := log[10](B[i, 3]):         
   B[i, 5] := log[10](B[i, 5]):         
   B[i, 7] := log[10](B[i, 7]):

end do:  # computing the log of the max-error
B: # This is the table of values we'll plot.

T:=plot([B[..,[1, 2]],B[1..1,[1, 2]], B[.., [3, 4]],B[1..1,[3, 4]], 
     B[..,[5, 6]],B[1..1,[5, 6]],B[.., [7, 8]],B[1..1,[7, 8]]], 
    legend = ["","BFFM","", "BNM","", "BHM","", "ARKN"],
    #title="Efficiency Curve for Example 5",
    style = ["pointline","point","pointline","point","pointline","point","pointline","point"], 
    symbolsize = 15,axes = framed, 
    symbol = [box,box, circle,circle,solidbox, solidbox,solidcircle, solidcircle],
    color=[ red, red, gold,gold, blue, blue,black, black], 
    axis = [gridlines = [colour = green, majorlines = 1,linestyle = dot]], 
    labels = [typeset(log__10(`NFE`)), typeset(log__10(`Max Err`))]);