dualaxisplot(listplot(sols[2], color = red), listplot(sols[1], color = blue), style = line, gridlines = false);

How can I define general matrices in Maple and do symbolic manipulation - for example specifying a matirx M to be of dimension n x m where n and m are integers ?

The permanent of a square matrix is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns. However, where the determinant weights each of these products with a ±1 sign based on the parity of the set, the permanent weights them all with a + 1 sign.While the determinant can be computed in polynomial time by Gaussian elimination, the permanent cannot. Valiant has showed that computing permanents is #P-hard, and even #P-complete for matrices in which all entries are 0 or 1 [L.G. Valiant, The Complexity of Computing the Permanent, Theoretical Computer Science 8 (1979) 189–201]. The development of both exact and approximate algorithms for computing the permanent of a matrix is an active area of research. (see Wikipedia: computing the permanent, more details on permanent)

The best known general exact algorithm is due to H. J. Ryser (1963) [Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus mathematical monographs, The Mathematical Association of America. Ryser formula ]. Ryser's formula can be evaluated using O(2^{n}*n^2) arithmetic operations, whereas O(n!*n) arithmetic operations if we compute the permanent using the definition.

There have been codes of Ryser's algorithm for computing the permanent written in C language and Matlab. I cannot find a Ryser's procedure in Maple. So I try to write a Maple procedure to compute the permanent using Ryser's algorithm. The code is as follows:

permanentRyser := proc (M::Matrix) 
      local S, T, B, m, n, s, i, j, rowsum, sum, prod, perm;
      m, n := op(1, M);
      if m <> n then
               error "expecting a square Matrix, got dimensions %1, %2", m, n
      end if;
      rowsum := 0;
      sum := 0;
      prod := 1; 
      S := combinat:-subsets([seq(1 .. m)]);
      S_nextvalue();
      while not S_finished do
             T := S_nextvalue();
             s := numelems(T);
             B := LinearAlgebra:-SubMatrix(M, [1 .. m], T);
             for i to m do
                  for j to s do
                          rowsum := rowsum+B[i][j];
                 end do;
                 prod := prod*rowsum; 
                 rowsum := 0;
             end do; 
            prod := (-1)^s*prod;
            sum := sum+prod;
            prod := 1 ;
      end do;
      perm := expand((-1)^m*sum) ;
end proc:
The last second statement "perm := expand((-1)^m*sum) ;" I mean to compute the permanent of  a matrix containing a variable, e.g. the characteristic matrix. If the matrix is numeric, then "expand" should be deleted.

Now I have two questions:

1. Suppose that A is random matrix of order 20, the time(permanentRyser(A)) is about 716 seconds and time(LinearAlgebra:-Permanent(A)) is about 66 seconds. We can see that LinearAlgebra:-Permanent(A) is much faster than permanentRyser(A).  I don't know whether the code I written is accurate and efficient. Thanks to anyone who gives a more efficient procedure of computing the permanent using Ryser's formular.

2. The source code of the function permanent is as follows.

proc (A::(`~Matrix`(square)), ` $`)
     option `Copyright (c) 1999 Waterloo Maple Inc. All rights reserved.`;
     local m, n, t1;
     m, n := op(1, A);
     if m <> n then
              error "expecting a square Matrix, got dimensions %1, %2", m, n
     end if;
     if n = 1 then t1 := A[1, 1] ;
     else
            tools/ClearRememberTable(Permanent/pminor`);
            t1 := [`$`(1 .. n)];
            t1 := Permanent/pminor(A, t1, t1, n)
      end if
end proc

I cannot understand the source code. Does the source code compute the permanent by Laplacian expansion? If the LinearAlgebra:-Permanent() compute the permanent by Laplacian expansion, then it should take more time than by using Ryser's algorithm.

Thanks all.

1-Suppose that after applying the command "factor(f)" the "f "takes the form:

f=(BesselJ(0,r))*(A very lengthy term)

Is there a way to force maple to show f like below:

f=(A very lengthy term)*(BesselJ(0,r))


2-Suppose f is written as follows:

f=a*b*x+a*b*y

Can you suggest a way (without using "op" command) to write f as:

f=a*(b*x+b*y)

Hi:

i will solve the three equations below with numerical method,how?

eq1 := -2.517407096*10^12*q[1](t)^2-5.292771429*10^12*q[1](t)-1.888055322*10^12*q[2](t) = 0
eq2 := 2.246321962*10^12*q[1](t)^2+1.684741471*10^12*q[2](t)+8.110113889*10^12*q[1](t)-7.480938859*10^10*q[3](t) = 0
eq3 := int((-3.826000000*10^11*q[2](t)*cos(Pi*x)*Pi^2-3.826000000*10^11*q[1](t)^2*cos(Pi*x)*Pi^3*sin(Pi*x)+3.414000000*10^11*q[1](t)^2*sin(Pi*x)^2*Pi^4-3.414000000*10^11*q[1](t)^2*cos(Pi*x)^2*Pi^4+7*(int(exp(10*tau), tau = -infinity .. t))+q(x, t))*sin(Pi*x), x = 0 .. 1) = 0

Hello everybody,

I'm trying to solve for a challenging problem : a moving inclined plane with a block

I want to solve for the acceleration components for the block and the plane and the normal force acting on the block.

Let O=(0,0) be an external origin.

Let h be the upper left height of the inclined plane.

Let x₁ be the x-position of the center of gravity of the inclined plane.

Let x₂ be the x-postion of the center of gravity of the block.

Let y be the y-position of the center of gravity of the block.

Let m₁ be the mass of the plane. Let m₂ be the mass of the block.

Let   be the coeffiction of kinetic friction between the bottom of the inclined plane and the level surface.

Let  be the coeffiction of kinetic friction between the block and the upper surface of the inclined plane.

Let  be the angle of the plane with the horizontal.

Let F_p a force applied to the inclined plane.

With those defined variables, I make two separable free body diagrams for the block and for the inclined plane, indicating all of the external forces acting on each. It then comes those two vectorial equations :

Block : m₂a₂=W_{weight of block}+F_{plan acting on block}+F_{friction from plan to block}+N_{normal from plan to block}

Plane : m₁a₁=W_{weight of plane}+F_{pushing force}+F_{block acting on plane}+F_{friction from level to plan}+N_{normal from level to plane}+F_{friction from block to plane}+N_{normal from block to plane}

I am quite not sure whether I should include the F_{friction from block to plane} and the N_{normal from block to plane} into the plane's acceleration calculation. Am I right ?

I notice that from the geometry of the figure, I can write down the relation : tan()=(h-y)/(x₂-x₁)

This implies the relation : -a_2y=tan()(a_2x-a_1x) (equation 1)

Writing down the equations for the x- and y- components of the accelerations of the block and of the plane , this yields :

( equation 2) : m₂a_2x=m₁ sqrt(a_1x²+a_1y²) cos() -     N₁ sin() +N₁ cos() 

(equation 3) : m₂a_2y= m₂g+m₁ sqrt(a_1x²+a_1y²) sin() +   N₁ cos() +N₁ sin() 

(equation 4) : m₁a_1x=Fp - m₂ sqrt(a_2x²+a_2y²)  sin() +   N₁ cos() - N₁ cos() 

(equation 5) : m₁a_1y=-m₁g  - m₂ sqrt(a_2x²+a_2y²)  cos() -  N₁ + N₁ -   N₁ sin() -   N₁ cos()

Since N₁=m₁g,  equation 5 becomes : m₁a_1y= - m₂ sqrt(a_2x²+a_2y²)  cos()   -   N₁ sin() -   N₁ cos()

I am confused at this stage because a_1y=0, that is to say, the plane remains at the ground level surface.

Where am I wrong ? Does this comes from my previous question ?

I want to solve this problem with Maple and plot the solutions. Thank you for any answer !

Hello,

I could obtain the simulation of my multibody with kinematic closed chain (CKC).

However, it seems that from a specific time (around 12s) in my model I believe that I have some numerical instabilities. Indeed, I could compare my simulation results with another mulbody software. I obtain the same simulation until 12s and after in MapleSim, it appears many perturbations as you can see on the figures belows.

So, I think that I tune the numerical solver. This numerical solver must solve DAEs equations since my model contains 4 kinematic closed loops.

If i read correctly the help menu, there are the following methods to solve the DAEs :

- use specific DAE numerical solver (3 differents solvers are used : ck45 method, RKF45 method and Rosenbrock method

- use reformulation equations techniques (Baumgarte, Projection) which can be associated (I believe) with a classic solver like (RK4).

For the moment, I have obtained my results with the rosenbrock solver with error absolute : 1.0*10^(-4) and eror relative :error absolute : 1.0*10^(-4) 

Do you have some ideas or advices so as to find a better method to solve my multibody systems with kinematic closed loops ? This method should  prevent the creation of numerical instabilities.

Thanks a lot for your help

Hi Everyone

Is there a built-in way to fit a regression using generalized least squares in Maple ?

Hello,

I would like to apply a torque directly on a revolute joint. I use a torque modulated by a signal source which is modified chirp signal.

I have done like in the figure below :

After having added this torque, my model no longer runs. The error message is only "simulation interrupted".

Do I correctly plug the torque ? Should I plug the torque element more in the flange a or b of the considered revolute joint ?

Do I tick "use support flange" ? Anyway, with or without, my model crashes.

Thanks a lot for your help.

Hi,

I am using densityplot function to produce different plots of a function. The function depends on two variables q and yi .

densityplot(f, q = 0 .. 1, yi = 0 .. 1, grid = [100, 100], style = patchnogrid, color = "Red", axes = boxed, axesfont = [Times, Roman, 15], labelfont = [Times, Roman, 15])

I pass the command style = patchnogrid, but still I get these white lines which I believe are grid lines. 
In the example provided in maple tutorial, when they used this command, all gridlines were removed. Why are they not removed in my plots?

if the plot is not visible, click the link

https://www.dropbox.com/s/7xxfefv8n1arr4q/Example.jpg?dl=0

I have used the command "op" in a code that I have written with Maple 17. When I restart the maple server and run the whole code again, the result of the "op" command changes! why is this happening?

Hi all

How can I produce following matrix by maple?

the order of matrix should be {N*(m+1)}*{N*(m+1)} where N is known constant

thanks for any guide

Mahmood   Dadkhah

Ph.D Candidate

Applied Mathematics Department

How do you use algcurves and Puiseux series in Maple?

Dear all,

I've a very complicated matrix, and I'm unable to read the result by maple. It should be a 6x6 matrix.

What mean the following expression ?

Thank you for your help.

Hi,

I trying to simulate a force sensor on robot arm, but every time I try something, I get nothing from my sensor, can you help me?

Here it's my "design":

Also, if I add a rigid body I get this error:

Thanks!