Change the line

from i = 1 to N do

to

for i from 1 to N do

A dsolve(..., numeric) solution can be obtained very quickly. Just put the option numeric at the end of your dsolve command. Remove the evalf. To plot the solutions, do

plots:-odeplot(coz1, [x,X11(x)]);

and likewise for X22 and X33.

Let's define the error, e, as the absolute value of the difference between the true value and the computed approximation. Then for an iterative method we have a sequence of errors e[1], e[2], .... We say that the order of convergence is q if limit(e[n+1]/e[n]^q, n= infinity) approaches a positive constant u. (See the Wikipedia article "Rate of convergence".) It follows that for sufficiently large n, ln(e[n+1]) ~ ln(u) + q*ln(e[n]), where ~ denotes approximate equality. This is a linear equation in q, so we can use the e as the data for linear regression to approximate the "slope" q.

The warning is due to several of iterates having order of convergence exactly equal to 3 (as far as floating-point computation is concerned). The warning can be ignored.

Download Halley.mw

Welcome. I moved your Question from the Posts category to the Questions category.

Try typing this. Type it explicitly; do not use palettes or menus.

int(exp(2*x), x);

Using the hint, the second initial value is obviously 21. Indeed, I don't see that the syntax allows you to call Roots(..., method= secant) or Secant without specifying a second initial value. So, I don't know what "secant method" command you are currently using. You should also lower your tolerance. Here's my worksheet:

restart:
Digits:= 18: #Close to double precision.
f:= mul(x-k, k= 1..20) - 1e-8*x^19:
r1:= Student:-NumericalAnalysis:-Roots(
     f, x= [20,21],
     method= secant,
     maxiterations= 2^10, tolerance= 10^(2-Digits)
);


#Check residual
eval(f, x=r1);
      17.9

#Compare with fsolve results.
R:= [fsolve(f, complex, fulldigits)]:
#Find fsolve root closest to r1:
Min:= infinity:
for r in R do
     m:= abs(r-r1);
     if m < Min then  r2:= r; Min:= m  end if
end do:
 
r2;
      20.2402751265279542

eval(f, x= r2);

#So, the secant method beat fsolve for accuracy!

abs(r1-r2);

Download Roots.mw

I thought that this was already answered in the past week. Your function has no discontinuities, which is obvious if you plot it. If you don't understand this, then you need to study continuity more. I'd be happy to explain it. (There is a command discont for finding discontinuities, but it says that it cannot work on this function. I think that it is confounded by the max.) Your function does however have points of nondifferentiability. Both D and diff will find these without needing further prompting. These are very powerful symbolic commands.

f:= max(x^2, sqrt(abs(x))):
diff(f,x);

We'll need to take this step by step. The first thing is to correct the obvious syntax errors. You have

legend= sprintf("Total dose...

but you did not close the quotation marks.

People here would be more likely to debug your code if you used 1D input. Most of us OGs find working with 2D input very tiresome. 

Generally, the plots, if simple, are printed in the reverse order that they are listed. So, try

plots:-display([pts2, pts1, seq(p1[i], i= 3..1, -1)]);

However, this is confounded when the plots contain different types of material. The types are surfaces, curves, points, text, and axes. These types are printed in that order, and then the printing order within each type is the reverse of the order that they appear in the display. These rules apply to inline plots. Directly exported plots may be different; I'll need to research that.

It can be done using just columnwise indexing, like this:

(x,y):= 'lista[..,k]' $ k= 1..2;

or

(x,y):= seq(lista[..,k], k= 1..2);

No, your approach is not correct. Your limits of integration are wrong. [1/2 <= x*y <= 2, 1 <= x <= 3] means that the limits of integration should be [y= 1/(2*x)..2/x, x= 1..3]. The integral should be entered as

int(exp(1/(x*y))/(y^2*(x+1)^2), [y= 1/(2*x)..2/x, x= 1..3]);

Note that you should enter this using a single int to avoid an irrelevant warning message.

Note that 1/x*y is not the same as 1/(x*y). Indeed, the former is the same as y/x and the latter is the same as 1/x/y or 1/y/x. So I made it exp(1/(x*y)) because the other forms are not elementary integrals. This parentheses placement is just a guess on my part; you'll need to check it.

Maple's differential operator, D, can find the derivative of quite complicated procedurally defined functions.

f:=x->max(x^2, sqrt(abs(x))):
D(f);

eqn:= diff(y(x),x$4)-20*diff(y(x),x$3)+158*diff(y(x),x$2)-580*diff(y(x),x)+841*y(x);
eval(eqn, y(x)= x*exp(5*x)*cos(2*x));

The above returns 0, which verifies the solution.

The implicitdiff command that you gave works for me (Maple 16.02), so I wonder what version of Maple you are using. Try changing y(x) to y; that also works for me.

g:= (x,y)-> x^3+y^3=1:
implicitdiff(g(x,y), y, x);

Get second derivative:

implicitdiff(%, y, x);

string1:= cat(seq(sprintf("(%a,%a,)==", pt3[k,1], pt3[k,2]), k= 1..op([1,1], pt3)))[1..-3];

A few characters shorter and a bit more cryptic (but requiring no local variable) is

string1:= sprintf(cat("(%a,%a,)==" $ op([1,1], pt3)), convert(pt3^%T, list)[])[1..-3];

This is not available in Maple. I think that implementing it would require fundamental changes to the GUI.