Carl Love

Carl Love

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9 years, 353 days
Natick, Massachusetts, United States
My name was formerly Carl Devore.

MaplePrimes Activity


These are answers submitted by Carl Love

I can't help you with the palettes, but if you're willing to type your commands then do

Z:= Statistics:-Distribution(Normal(0,1));
Z:-CDF(.87) - Z:-CDF(0);
evalf(%);

The following answers apply to the transcendental RootOf. The situation for the polynomial RootOf is slightly different.

Q1)  How can I evaluate the RootOf?

You can substitute values for the variables and apply commands to it just as you can with any other expression. Just don't substitute for the "with respect to" variable (the _Z). That wouldn't make sense, just like substituting a nonatomic expression for x in int(f(x), x) or solve(f(x), x) wouldn't make sense.

Q2) In other words, What is it saying?

A ?RootOf is Maple's way of expressing an inverse function (usually with multiple branches) for which it otherwise has no expression. Another way of thinking about it is that it's Maple's way of expressing an implicitly defined function (usually one with multiple branches).

Q3) Is there any way that I can get an expression without RootOf for the variable that I am solving for?

For an exact symbolic solution, generally, no, not unless you are aware of some solving technique which Maple is not using. A floating-point solution is usually possible if you supply numeric values to the parameters. In either case, you can try to apply the command ?allvalues. That is the command specifically designed for expressions containing RootOf. Especially, try to reduce the RootOf expression with allvalues if you have assigned numeric values to some of the parameters. If you've applied assumptions to some of the parameters, then try simplify.

Q4) Is there any way that I can get an analytical expression for the variable that I am solving for?

I am guessing that what you really meant by this question is what I answered in Q3 above. But I separated out this question so that I coud point out that an expression with RootOf is an analytic expression. You can generally apply to it all the Maple commands that you would apply to any other analytic expression, including diff, series, simplify.

Q5)  Or from here we can only get numerical means to solve for the variable. If so, how to proceed?

Use ?eval to supply numeric values to the parameters, then apply allvalues. At this point, you may need to select a branch. Then apply evalf. There are cases where not all branches will satisfy the original equation, so plug back into the original to check.

First, you have a small error. It has no effect for this current question, but you should fix it. The command restart must be in lowercase letters.

You could apply a DataTable to this situation, as in your previous question. But I'll show you another way. Define q[1] with a variable as one of the numeric components, and still include the units. For example, I replaced the numerator 14.6 with a variable Luftmængde. Then execute the rest of the commands. Some of them will have symbolic answers expressed in terms of Luftmængde, but the units will carry through. After the end, you can use the command

eval(P[dyn1], Luftmængde= 14.6);

Of course, you'll be able to use any number that you want.

There are many other ways to do what you want, and if the above is not satisfactory, then I can show you another.

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);

`Maple 16.02, X86 64 WINDOWS, Nov 18 2012, Build ID 788210`

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

surd(2*x-x^(3/2)-1, 3)+x^(1/2)+surd(1-2*x, 3)

solve(expr = 0, x);

0, 1, 1/2, (1/4)*(5^(1/2)+1)^2


Download surd_solve.mw

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.

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!

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.

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)));

exp(-r)*r*_C2+(2*exp(-r)*Ei(1, -2*r)*r+exp(r))*_C1-(8/9)*((1/4)*(Int((2*Ei(1, -2*r)*r+exp(2*r))*(4*r*exp(r)-9)*exp(-3*r), r))*r*exp(3*r)+((r+1/2+r^2)*exp(r)-1/2-(3/2)*r)*r*Ei(1, -2*r)+((1/2)*r^2+(1/2)*r+1/4)*exp(3*r)-(3/4)*exp(2*r)*(r+1/3))*exp(-4*r)

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

exp(-r)*r*_C2+(-2*exp(-r)*Ei(2*r)*r+exp(r))*_C1-(8/9)*((1/4)*(Int((-2*Ei(2*r)*r+exp(2*r))*(4*r*exp(r)-9)*exp(-3*r), r))*r*exp(3*r)-((r+1/2+r^2)*exp(r)-1/2-(3/2)*r)*r*Ei(2*r)+((1/2)*r^2+(1/2)*r+1/4)*exp(3*r)-(3/4)*exp(2*r)*(r+1/3))*exp(-4*r)

value(%);

exp(-r)*r*_C2+(-2*exp(-r)*Ei(2*r)*r+exp(r))*_C1-(8/9)*((1/4)*(3*exp(-r)+(1/2)*Ei(2*r)*(8*r^2*exp(-2*r)+8*r*exp(-2*r)+4*exp(-2*r)-12*r*exp(-3*r)-4*exp(-3*r))-4*r-2*ln(2*r)-2*Ei(1, r))*r*exp(3*r)-((r+1/2+r^2)*exp(r)-1/2-(3/2)*r)*r*Ei(2*r)+((1/2)*r^2+(1/2)*r+1/4)*exp(3*r)-(3/4)*exp(2*r)*(r+1/3))*exp(-4*r)

simplify(%);

exp(-r)*r*_C2-2*exp(-r)*Ei(2*r)*_C1*r+exp(r)*_C1+(4/9)*exp(-r)*ln(r)*r+(4/9)*exp(-r)*ln(2)*r+(4/9)*exp(-r)*Ei(1, r)*r+(4/9)*r^2*exp(-r)-(4/9)*exp(-r)*r-(2/9)*exp(-r)+(2/9)*exp(-2*r)

YY:= %:

Verify that it satifies the original equation.

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

2*(4/9-exp(-r)/r)*r*exp(-r)+(4/9)*exp(-r)/r+(8/9)*exp(-r)-(4/9)*(exp(-r))^2/r+2*(exp(-r)*r*_C2-2*exp(-r)*Ei(2*r)*_C1*r+exp(r)*_C1+(4/9)*exp(-r)*ln(r)*r+(4/9)*exp(-r)*ln(2)*r+(4/9)*exp(-r)*Ei(1, r)*r+(4/9)*r^2*exp(-r)-(4/9)*exp(-r)*r-(2/9)*exp(-r)+(2/9)*exp(-2*r))/r-2*exp(-r)*_C2-(8/9)*exp(-r)*ln(r)-(8/9)*exp(-r)*ln(2)-(8/9)*exp(-r)*Ei(1, r)+4*exp(-r)*Ei(2*r)*_C1-2*exp(-r)*exp(2*r)*_C1/r+(2/3)*exp(-2*r)+(4/3)*(exp(-r))^2-(16/9)*exp(-r)*r

simplify(%);

0

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))[];

Int((2*Ei(1, -2*r)*r+exp(2*r))*(4*r*exp(r)-9)*exp(-3*r), r)

with(IntegrationTools):

simplify(Expand(expand(J)));

8*(Int(exp(-2*r)*Ei(1, -2*r)*r^2, r))-18*(Int(exp(-3*r)*Ei(1, -2*r)*r, r))+4*(Int(r, r))-9*(Int(exp(-r), r))

JJ:= value(%);

8*(int(exp(-2*r)*Ei(1, -2*r)*r^2, r))-18*(int(exp(-3*r)*Ei(1, -2*r)*r, r))+2*r^2+9*exp(-r)

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

-2*(2*r^2+2*r+1)*exp(-2*r)*Ei(1, -2*r)-4*r-2*ln(r)+2*(3*r+1)*exp(-3*r)*Ei(1, -2*r)-6/exp(r)-2*Ei(1, r)+9*exp(-r)

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

exp(-r)*r*_C2+2*exp(-r)*Ei(1, -2*r)*_C1*r+exp(r)*_C1+(4/9)*exp(-r)*ln(r)*r+(4/9)*exp(-r)*Ei(1, r)*r+(4/9)*r^2*exp(-r)-(4/9)*exp(-r)*r-(2/9)*exp(-r)+(2/9)*exp(-2*r)

Verify that it satifies the original equation.

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

2*(4/9-exp(-r)/r)*r*exp(-r)+(4/9)*exp(-r)/r+(8/9)*exp(-r)-(4/9)*(exp(-r))^2/r-2*exp(-r)*_C2-(8/9)*exp(-r)*ln(r)-(8/9)*exp(-r)*Ei(1, r)+2*(exp(-r)*r*_C2+2*exp(-r)*Ei(1, -2*r)*_C1*r+exp(r)*_C1+(4/9)*exp(-r)*ln(r)*r+(4/9)*exp(-r)*Ei(1, r)*r+(4/9)*r^2*exp(-r)-(4/9)*exp(-r)*r-(2/9)*exp(-r)+(2/9)*exp(-2*r))/r-2*exp(-r)*exp(2*r)*_C1/r-4*exp(-r)*Ei(1, -2*r)*_C1+(2/3)*exp(-2*r)+(4/3)*(exp(-r))^2-(16/9)*exp(-r)*r

simplify(%);

0

 


Download Ei_Integration.mw

Check out command ?RootFinding,Analytic.

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) > >;

M := Matrix(2, 2, {(1, 1) = cos(theta), (1, 2) = -sin(theta), (2, 1) = sin(theta), (2, 2) = 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);

[0, (1/4)*Pi, (1/2)*Pi, (3/4)*Pi, Pi, (5/4)*Pi, (3/2)*Pi, (7/4)*Pi]

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

Mlist:= Constructor ~ (Angles);

Mlist := [Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = (1/2)*sqrt(2), (1, 2) = -(1/2)*sqrt(2), (2, 1) = (1/2)*sqrt(2), (2, 2) = (1/2)*sqrt(2)}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = -1, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = -(1/2)*sqrt(2), (1, 2) = -(1/2)*sqrt(2), (2, 1) = (1/2)*sqrt(2), (2, 2) = -(1/2)*sqrt(2)}), Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), Matrix(2, 2, {(1, 1) = -(1/2)*sqrt(2), (1, 2) = (1/2)*sqrt(2), (2, 1) = -(1/2)*sqrt(2), (2, 2) = -(1/2)*sqrt(2)}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = -1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = (1/2)*sqrt(2), (1, 2) = (1/2)*sqrt(2), (2, 1) = -(1/2)*sqrt(2), (2, 2) = (1/2)*sqrt(2)})]

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

Mlist:= evalf[4](Mlist);

Mlist := [Matrix(2, 2, {(1, 1) = 1., (1, 2) = 0., (2, 1) = 0., (2, 2) = 1.}), Matrix(2, 2, {(1, 1) = .7070, (1, 2) = -.7070, (2, 1) = .7070, (2, 2) = .7070}), Matrix(2, 2, {(1, 1) = 0., (1, 2) = -1., (2, 1) = 1., (2, 2) = 0.}), Matrix(2, 2, {(1, 1) = -.7070, (1, 2) = -.7070, (2, 1) = .7070, (2, 2) = -.7070}), Matrix(2, 2, {(1, 1) = -1., (1, 2) = 0., (2, 1) = 0., (2, 2) = -1.}), Matrix(2, 2, {(1, 1) = -.7070, (1, 2) = .7070, (2, 1) = -.7070, (2, 2) = -.7070}), Matrix(2, 2, {(1, 1) = 0., (1, 2) = 1., (2, 1) = -1., (2, 2) = 0.}), Matrix(2, 2, {(1, 1) = .7070, (1, 2) = .7070, (2, 1) = -.7070, (2, 2) = .7070})]

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

Mlist[4];

Matrix(2, 2, {(1, 1) = -.7070, (1, 2) = -.7070, (2, 1) = .7070, (2, 2) = -.7070})

 

Download Matrix_function.mw

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.

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

Grade:= proc(mark::And(nonnegative,numeric))

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.

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.

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));

 

Download poly.mw

 

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