Carl Love

## 27224 Reputation

11 years, 345 days
Himself
Wayland, Massachusetts, United States
My name was formerly Carl Devore.

## convert to surd form...

You may have an older version of Maple. There have been some recent improvements in RealDomain. You may be able to solve your equation without using RealDomain if you first convert  it to surd form.

 > restart;
 > kernelopts(version);

 > expr:= convert((2*x-x*sqrt(x)-1)^(1/3)+sqrt(x)+(1-2*x)^(1/3), surd);

 > solve(expr = 0, x);

## Recall "how many routes in a 4x4 square?...

Go back to your question from one month ago, "how many routes in a 6x6 square?" Careully study the lengthy answers to that. I covered there many of the questions that are posed here, including

1. how to represent the vertices as pairs of integers
2. how to label the vertices with convenient two-character letter-number combinations while maintaining the underlying integer-pair representation
3. how to create the graph in GraphTheory using the simple criterion that vertices u and v are adjacent iff LinearAlgebra:-Norm( < u - v > ) = 1
4. how to use GraphTheory:-Departures to construct paths (words) in the graph
5. how to plot the board and the paths (words).

It would be much quicker to start with the above base, which is already coded in the post, than to work from the Mathematica. The project is already half done. It would be nice to know that my prior work on it was not for naught.

What needs doing?

1. Setting the distribution of letters. Counting the letters in a real Boggle game would help for this.
2. Assigning the letters to the vertices.
3. Dictionary lookups.
4. Scoring. Having the actual game would help with this.
5. Adding the letters to the plot.

Place your cursor in your worksheet in the spot that you want the table. Then from the Insert menu, select Spreadsheet. When you want to update the spreadsheet, right click in it and select Evaluate All. For programmatic access to spreadsheets, see ?Spread.

## radians and zip...

Before answering your main question, I want to point out that angles need to be in radians in Maple. You can enter them in degrees and then convert to radians, like this

Angles:= [30, 60, 90];
Angles:= Angles *~ Pi/180:

or

Angles:= map(convert, Angles *~ degrees, radians):

Second, in my previous post, I said that you'd probably want to use more than 4 digits in actual practice, but you kept it at 4. You should change all occurences of evalf[4] to evalf[15] or simply evalf (for reasons that I won't go into here, 15 digits usually gives the optimal balance of speed and accuracy).

Third, you should probably use different names for the two Constructor procedures: the one that constructs the Ts and the one that constructs the Ths.

Now for your main questions. You have two lists of eight matrices each, Ts and Ths, and a single matrix Q, and you want to compute a list of matrices Qb such that Qb[k] = Ts[k]^(-1).Q.Ths[k] for k from 1 to 8. Here are two ways do that:

Qb:= [seq](Ts[k]^(-1).Q.Ths[k], k= 1..8):

or

Qb:= zip((X,Y)-> X^(-1).Q.Y, Ts, Ths):

Finally, you want to construct a 6x6 matrix from 3x3 matrices A, B, and D, with pattern

A B
B D.

The command for this is

M:= < < A | B > , < B | D > >:

Let me know if that is all clear to you!

## The last errors in Comp_Simp...

Sorry that I didn't see this earlier. This system is very bad at notifying authors when there are responses to their answers.

So, I promised to help you again if you found two of the five errors. You found four, and correctly fixed three. The one that you incorrectly fixed is the definition of h. It should be

h:= (b-a)/2/n;   (or, equivalently, h:= (b-a)/(2*n);.)

The last error is that subtle one. It's another fencepost error. What is the sequence of coefficients of the evaluation points? If n = 2, then the coefficients are

1, 4, 2, 4, 1

and if n = 3, then the coefficients are

1, 4, 2, 4, 2, 4, 1.

Do you agree? So, there is always one more 4 than there are 2s. But your loop does the 4s and 2s in pairs so there are always the same number of them. You are missing the last 4. It occurs at x = b - h. So, your initialization of aa should be

aa:= f(a)+4*f(b-h)+f(b);

You should first test your procedures against Student:-Calculus1:-ApproximateInt (like I described in the previous post) before checking for convergence.

## Two ways to evaluate integrals of Ei...

I came up with two ways to evaluate the the integral of Ei in this problem: an Ei identity and integration by parts. The identity is Ei(1,x) = -Ei(-x).

 > restart;
 > DQ:= -((D@@2)(y))(r)-2*y(r)/r+y(r) = ((4/9-exp(-r)/r)*2)*r*exp(-r):
 > Y:= rhs(dsolve(DQ, y(r)));

 > subs(Ei(1,-2*r)= -Ei(2*r), %);

 > value(%);

 > simplify(%);

 > YY:= %:

Verify that it satifies the original equation.

 > eval((rhs-lhs)(DQ), [(D@@2)(y)(r)= diff(YY,r\$2), y(r)= YY]);

 > simplify(%);

The identity is not valid everywhere. We can do it without the identity by using integration by parts.

Extract the unevaluated integral:

 > J:= indets(Y, Int(algebraic\$2))[];

 > with(IntegrationTools):
 > simplify(Expand(expand(J)));

 > JJ:= value(%);

 > jj:= Parts(JJ, Ei(1,-2*r), applytoall);

 > YY:= simplify(eval(Y, J= jj));

Verify that it satifies the original equation.

 > eval((rhs-lhs)(DQ), [(D@@2)(y)(r)= diff(YY, r\$2), y(r)= YY]);

 > simplify(%);

 >

## RootFinding:-Analytic...

Check out command ?RootFinding,Analytic.

## unapply...

Here's an example. Here I use 2x2 matrices to save space. I'm sure that you'll be capable to generalize this. The essence is the use of unapply to turn one matrix with a parameter into a constructor for other matrices. (The ?unapply command is much more general than just being used for matrices.)

 > restart;

Make a Matrix that depends on a unspecified angle:

 > M:= < < cos(theta),sin(theta) > | < -sin(theta),cos(theta) > >;

Make that Matrix into a procedure that constructs more Matrices:

 > Constructor:= unapply(M, theta):

Make a list (not a vector) of 8 angles:

 > Angles:= [seq](Pi*k/4, k= 0..7);

Apply the Conctructor to the list of angles to create a list of Matrices:

 > Mlist:= Constructor ~ (Angles);

(Optional) Evaluate to floating point (in actual usage, you'd probably want more than 4 digits).

 > Mlist:= evalf[4](Mlist);

Individual Matrices in the list are specified by index, 1 to 8.

 > Mlist[4];

## Shift-enter or Code Edit Region...

Yes, the "problem" is caused by the lack of end proc. You can avoid the problem two ways:

1. When you want a blank line, use Shift-Enter (or Shift-Return) instead of Enter (or Return). Don't hit Enter until you're ready for the procedure to be processed ("parsed" is the lingo). If you do hit Enter prematurely, just ignore any error message and put the cursor back where you want it. This takes a bit of getting used to, which brings us to my preferred method...
2. From the Insert Menu, create a Code Edit Region. This gives you a box, a subwindow, that you can use like Microsoft Notepad or other primitive text editor. Here, Enter just gives you a new line. When you want the code to be parsed (and Executed, if it's more than just a procedure definition), right-click in the box and select Execute Code. You can resize the box by left-clicking in it and then dragging one of the blue square dots on the sides or corners. But the box doesn't need to be big enough to hold your code; it will scroll if needed.

Put a semicolon after the end proc.

## No semicolon for proc line...

The proc line should not end with a semicolon.

There is also a mistake in your type declaration, but the mistake will unfortunately not generate an error message. If you want a parameter to be restricted to having both type A and type B, then the type is And(A,B). So with that the proc line becomes

You should look up a user named Rupunzel on this forum. She got help about this very same homework problem in two separate threads a few days ago. There's a Users tab at the top of the page.

## A chart of character codes...

For me, in worksheet text mode, the alt codes generate different characters than the characters accessible programmatically by Maple. (I haven't explored the possibility of getting more characters with MathML or XML.) But, I see that you got a thorn for character 254, which is the same as Maple's standard font's 254. So, I can just plot the standard font for you:

plots:-display(
plots:-pointplot([[0,0],[16,16]], color= white),
plots:-textplot(
[seq([.5+irem(k,16), .5+iquo(k,16), convert([k],bytes)
,color= black, font= [times, roman, 24]
]
,k= 0..255
)
,seq([irem(k,16), .75+iquo(k,16), sprintf("%d",k)
,color= blue, font= [times, roman, 9], align= {above, right}
], k= 0..255
)
]
),
labels= ["low nybble", "high nybble"],
labeldirections= [horizontal,vertical],
axes= boxed,
axis= [gridlines= [\$(1..15)]],
tickmarks= [(1/2 +~ [seq](k, k= 0..15) =~ [\$(0..9),A,B,C,D,E,F]) \$ 2],
axesfont= [times, italic, 16],
labelfont= [HELVETICA, bolditalic, 16],
title= "Character codes 0-255 in the regular font",
titlefont= [HELVETICA, BOLDOBLIQUE, 24]
);

When you plot this in Maple, you'll need to enlarge the plot with your mouse so that the characters fit neatly in their boxes.

## Plot3d over multiple parametrized region...

As posted, you have overlapping ranges for x, so I think that you meant for the upper bound of x in the first line to start -2*Pi instead of 2*Pi, and my code below reflects that change. Note that the constant Pi is capitalized in Maple. That being said, what I get from your ranges is a nonagon, not a hexagon.

Plotting a surface over a complicated region in the plane involves the same steps as setting up a double integral over that region. You've already done the hardest part---defining the ranges. I chose an f, and here's the rest of the steps.

 > rt3:= sqrt(3):  C:= Pi/3/rt3:
 > R:= [[x= -4*C..-2*C, y= -rt3*(x-4*C)..rt3*(x-4*C)]     ,[x= -2*C..2*C, y= -2*Pi/3..2*Pi/3]     ,[x= 2*C..4*C, y= -rt3*(4*C-x)..rt3*(4*C-x)]     ]:
 > f:= (x,y)-> x^2+y^2:
 > plots:-display((r-> plot3d(f(x,y), r[])) ~ (R));

## copy(g) ?...

After executing the statement

gn:= g;

gn is identical to g, so any changes made to gn will automatically be made to g also. To avoid this, i.e. to create a separate independent matrix, use copy:

gn:= copy(g);

But I can't tell for sure if this is what you need because you never use gn after constructing it. So the only way that the code as posted makes sense is if the changes are made to g. Perhaps you didn't upload the whole code.

Also, the double if statement at the end can be replaced by

if not (j in s[i] xor n in s[i]) then g[j,n]:= 0 end if;

## ::`+`, nops, op, indets, and degree...

To avoid unintended conversion of monomials, you can check if the expression is of type `+`.

Example:

 > p:= randpoly([x,y,z]);

 > evalb(p::`+`);

 > `if`(p::`+`, nops, 1)(p);

 > L:= `if`(p::`+`, [op], `[]`)(p);

 > tL:= indets ~ (L);

 > nL:= nops ~ (tL);

 > [seq]({seq}(v=degree(L[k],v), v= tL[k]), k= 1..nops(L));