Maple Questions and Posts

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Please explain a method (including corresponding commands) to find first positive real root(s) of a large matrix determinant, briefly. Since each entry or element of the matrix involves with the large formula, probably Newton Iterative method is suitable. The main question is, how it is possible to define some matrices like the Hessian or gradient matrices, when calculating parametric solution of determinant is a very time consuming procedure.

Let assume that only one independent variable exists. After evaluating matrix value at a certain amount of independent variable, the determinant can be calculated swiftly. The problem is that the derivative of determinant in each step will be calculated slowly. How one can rectify this deficiency in Newton Iterative method? For example, is there another iterative method to find derivative of determinant at a certain amount of independent variable to use it in the Newton iterative method?

 

Hello dear users!

I have a problem, that I can not explain, and solve by myself. I have a function. It is an integral function, looks like this:

If I looking for a value such as LL(4) or LL(5) I got different value what i except from the plot of the function:

while, for example: evalf( LL(3))=1.6829

 

Can somebody explain to me, what is going on?

 

Also, if i want to solve the functiuon backwards with fsolve (so im looking for the phi values with the added LL(phi) value, sometimes I got results and sometimes not. (and not where the distruption is, because I know there the values go from -infinity to infinity). Can somebody help me in this too?

Thank you very much!

 

 

Hello

I have a question respective to the display of calculated numbers.

Is there a way to display the "correct" value of a calculated float or absurd fraction?

I tried it with evalf(), convert(rational) but without success.

 

To be concrete I calculate the Given Rotation matrices of

A := Matrix(3, 3, [[-3, 32/5, 4], [4, 24/5, 3], [5, 6*sqrt(2), 5*sqrt(2)]])

G := Matrix(3, 3, [[-3/5, 4/5, 0], [-4/5, -3/5, 0], [0, 0, 1]])

which results in the followin (correct) matrix

Matrix(3, 3, {(1, 1) = 5, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = -8, (2, 3) = -5, (3, 1) = 5, (3, 2) = 6*2^(1/2), (3, 3) = 5*2^(1/2)})

 

But using the method

Given1 := GivensRotationMatrix(A[1], 1, 2);

I get following weird approximated matrix

Matrix(3, 3, [[-(15/1249)*sqrt(1249), (32/1249)*sqrt(1249), 0], [-(32/1249)*sqrt(1249), -(15/1249)*sqrt(1249), 0], [0, 0, 1]])

Just look at the the cell a11 which should be -3/5 but with GivensRotationmatrix() I get this totaly

weird fraction -15*sqrt(1249)*(1/1249). I tried simpliefed, evalf, convert(ration) but never get the result of -3/5.

Maybe someone can help me? :)

 

EDIT

Writing maple code in this editor is a pain in the ass.

Isnt "Maple Math" wortking?!

I am trying to get Maple to understand, and evaluate the following integral analytically AND numerically.  
The original integrand to be evaluated is given by 

(w + m2)(1/4) (p2 + mu)(1/4) / ( (w2 +a2)2 (p2 +a2) (q2 + b2 2)2 ((p - q + w)2 + b12) )

which is to be integrated simultaneously with respect to dw dp dq (where w, p, q are 3-d vectors).  
Limits of integration are (in cartesian co-ordinates) from -Infinity to + Infinity

All variables contained in the integrand are Real and >0.
a, m, mu, b1, b2 are constants and >0.  
dp = d3p, dq = d3q, dw = d3w
p2=p2, w2=w2, q2=q2

It is of course possible to help Maple along by noting that 
(p - q + w)= p2 + q2 + w2 - (2 p q Cos(\theta_pq) ) + (2 p w Cos(\theta_pw) ) - (2 q w Cos(\theta_wq) ) + b12 
where \theta_pq is the angle between vectors p and q, theta_pw is the angle between vectors p and w, etc. 

However, using the expanded form of (p - q + w)2 would imply that the integral would now need to be written out in tems of 
dwradial, dqradial, dpradial, d\theta_w, d\theta_q, d\theta_p, and d\phi_w, d\phi_q, d\phi_p.
Now, the integral in spherical co-ordinates would be given by the pseudocode below (note that now there are no longer any vector symbols, and the limits of integration have changed)

int ( Sin(\theta_w) d\theta_w Limits {0, pi}) int ( w2 dwradial Limits:{0, Infinity} )
int ( Sin(\theta_p) d\theta_p Limits {0, pi}) int ( p2 dpradial Limits:{0, Infinity} )
int ( Sin(\theta_q) d\theta_q Limits {0, pi}) int ( q2 dqradial Limits:{0, Infinity} )

where we have omitted the integration over \phi_w, \phi_p, \phi_q, assuming that there is no dependence on the \phi angles, resulting in all the \phi angles being integrated over, with the result being a constant multiplied to the rest of the remaining integrals. No additional information on the vectors is available, e.g. the angle between p and q, p and w, q and w, are not specified, the magnitude of the vectors is not specified. 

Any ideas if Maple can understand and execute the integral with p, q, w are input as vectors in the integrand directly, as shown earlier?
If it's easier for Maple to evaluate the integral in spherical co-ordinates, any suggestions on setting up the input, i.e. set up that the variables and constants are real, setup the limits of the theta and radial components etc. would also be appreciated. 

Supposing I have a n x n matrix and I want to convert it from linear to subscript indices

I have this expression which I want to integrate numerically, but it is exponentially slow. I tried methods like _d01ajc but it does not help.

 

restart;

f2 := proc (m) options operator, arrow;

evalf(Int(exp(-(2*m-4)*exp(t)+t*(m+1))*(t-2*exp(t)), t = 0 .. infinity))

end proc;

 

plot([f2], 3 .. 10)

Good day all, I have a yet challanging integration problem. The problem is to find the numerical solution of the problem:
Eq1 := x(t) = x(b)+c*t+k*x(b)*t-k*(int((t-s)*x(s), s = 0 .. t))+2*(int(sqrt(t-s)*x(s)^3, s = 0 .. t))/sqrt(Pi)

I differentiated Eq1 wrt x as follows:Eq2 := diff(Eq1, t)

Lastly, on typing sol := dsolve({Eq2, x(0) = x(b)}, x(t), numeric)

I got an error messag. Please all the valuable suggestions given to me on a similar problem could not solve this one as this appears to be a little complex when compared to my previous question. Please I need your help. Thanks

 

 

Is it possible to increase the execution speed of the dsolve command? For example, is it possible to increase speed with the use of Multithreaded tools?

Thank you for your help

Dear All,

I am trying to calculate a product of a complex function and its complex conjugate, for instance, conjugate(exp(I*phi))*exp(I*phi), and integrate phi from 0 to 2*pi. The product is supposed to be 1 and the integral should be 2*pi. However, with the following code:

phi::real; simplify(conjugate(exp(I*phi))*exp(I*phi))

I obtain the result as shown in the attached figure. It still gives a bar over one phi and does not give 1. Could you please tell me how I can fix this problem? Any of your hellp is appreciated.

 

Best regards,

Toby

 

Hi

I write a code to read the matrix elements and return information about row and column number of some specific data. This code is lengthy.

Is there a command to give this information directly, without necessity of developing a code?

For instance, assume M=`<|>`(`<,>`(1, 3, 2), `<,>`(4, 1, 3))

I need a command to show that each element how many times is repeated and where is located. For example, the number 3  is repeated 2 times in locations [2,1] and [3,2].

Hi everyone

I want to define N martices in maple with for loop but I don`t how can I do that in maple. In Matlab one may use the cell concept simply. For example if one has to define 10 matrices then he/she can do as following (of course in matlab):

for k=1:10

A{k}=[k,k^2;2k,2k^2];   % matrix experssion;

end

where A{1} is A1 which is 2x2 matrix, A{2} is A2 which is 2x2 matrix and so on. Now I want do this in maple. What should I do?

After solve this problem I want to define a block matrix whom elelments are these N mtrices. I will be thankfull if one help me in this case.

I need to import a large collection of other alphabets into maple for a project for example, how would i import the below collection and have them available on the pallette?

 

Hi

I have a complex function (i.e. f) with real values for real domain (real amounts of independent variable omega). If the previous calculations are conducted correctly, the minimum positive real root of mentioned function must be equal to 15.4182/sqrt(12). The formula of f is inserted as a plain text.

Formula.txt

The plot of f is depicted as follows:

It seems that the root is between 4 and 5. The command fsolve does not work.

fsolve(simplify(f),omega=4..5)

Also, the Newton iterative method is useless since the imaginary part exists.

f:=simplify(f):

x[1]:=4:

for i while or(i=10, abs(eval(f,omega=x[i]))<1E-3) do

x[i+1]:=x[i]-eval(f,omega=x[i])/eval(diff(f,omega),omega=x[i])

end do:

x[i];

Please propose a way to find first positive real root of above function.

Moreover, I sometimes see discordant results when I use plot command to seek an approximate range for fsolve. I have another function (i.e. f(x)) that fsolve gives the root x=4, which is true but the plot command shows that the root is not occurred at x=4.

 

when using slideshow tool (F11), it just show the first page and arrows for going to other pages does not work. is it a bug?

Dear All. I am a beginner of using Maple. My calculation gives a complex expression as shonw in Line (23) in the attached figure. I used "simplify" to simplify it. However, the result is not the simplest form. The numerator can be divided by the denominator. Could you please tell me what command I can use to further simplify the expression? Any of your help is highly appreciated!

Best regards,

Toby

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