suppose i have an ODE dx/dt=x/t how do i draw isoclines? thanks

I have a Maple code which I wrote on a Windows machine. Now I need to run it on a Linux machine. I set the directory by the command

currentdir("C:/example"):

How should I change this if I want it to be /home/user/example ?

and I use

fopen("C:/mydir/examp/example.dat", WRITE):

How should I change this line in Linux if I want it to be in /usr/local/mydir/examp/example.dat?

In a comment to a Mapleprimes thread, Jacques mentioned an old suggestion of Kahan's that numerical computations should return an estimate of conditioning alongside a result.

I mentioned in this comment an approach for numerical estimation of (all) roots of a univariate polynomial with real or complex numeric coeffficients that is based upon computing eigenvalues of a companion matrix. Here below is some rough code to inplement that idea, but which also returns condition number estimates associated with the eigenvalues.

I include an example of the badly conditioned Wilkinson's polynomial. It is possible that better results could be obtained by using a Lagrange basis representation of that polynomial, but I didn't try to figure out how that would work in an analogous way. The standard Maple utility, fsolve, has no problem with this example.

This forum question led to a discussion of a bitwise magazine review that compared Mathematica 5.2 and Maple 10. In that review the author struggled to get the following numeric integral to compute accurately and quickly in Maple.

evalf(Int(BesselJ(0, 50001*x)*x*exp(I*(355*x^2*1/2)), x = .35 .. 1));

Below, I reproduce an attempt at computing an accurate result quickly in Maple. I'm copying it here because that thread got quite long and messy.

hello i was asked to draw direction field and some isoclines for some ODEs Now i have some problems: the equation is t*diff(x(t),t)-x(t)=0 and i use contourplot(x(t)/t,t=-10..10,x=-10..10); to draw the isoclines... however, the output is not quite as i expected, it looks like some curve because if i draw it by hand, i put dx/dt=x/t and set x/t=c ( c is some constants) then the graph should be some straight line what am i doing wrong??

The problem I'm having is that I have a large system of linear equations, the solution to which I know must be integers between 1 and 9. I have tried applying bounds etc., but the problem is that when I use solve, it said 'evaluating' for around half an hour, and still came out with nothing. Does anyone have any ideas as to how to make this faster, because I suspect that it's trying to find all solutions, and then eliminating the ones of the wrong form, rather than only looking for solutions where each variable is an integer between 1 and 9.

Hello Maple experts,

I'm currently trying to get more familiar with semidefinite programs (SDP), i.e. I want to minimize/maximize a linear objective matrix function over linear equality/inequality constraints with a positive semidefinite matrix condition.

It seems that there are a number of Matlab packages available but so far I couldn't really see how I could do this with Maple (which I would prefer to Matlab if there is a reasonable solution...)

I am in a mode with a red arrow on the left.

How do I define f to be an algebraic function of x over C, the complex numbers, such as

f(x)=x+7+I

I want to be able to plug in complex numbers for x and have a complex number

returned, say

f(9-5*I)=16-4*I

In the blog  Plots of twisted ribbons, the author gave an interesting description of plotting twisted ribbons. In this blog , we give a similar description of twisted ribbons and give the geometrical interpretations of this definition.

 

Let r(phi)=[a*cos(phi), a*sin(phi), 0] (phi=0..2*Pi) be a circle in the xy-plane, and P be a point on the circle. Let QR be a line segment (with length of 2) passing through the point P and let P be the middle point of QR. Also, QR is coplanar with the z-axis.

Now let P rotate about the z-axis at the angular velocity of phi, where phi is the angle between OP and the x-axis. At the same time, the line segment QR is rotating about its middle point, P, at the angular velocity of theta (where theta is the angle between PQ and the z-axis and theta is dependent on phi, eg, theta=k*phi). In the whole process, QR will remain coplanar with the z-axis.

 

Apparently, the locus of the line segment QR is a twisted ribbon. When theta=phi/2, we have the Mobius strip.

2. The equation of the twisted ribbon

 

Now we try to find the equation of the surface. Clearly,

vector(OP)= [a*cos(phi), a*sin(phi), 0].

And with some geometrical manipulations, we have

vector(PQ)=[sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)].

So the vector equation of the twisted ribbon is

V(phi, t)=vector(OP)+t*vector(PQ)

And the parametric equation is
x=a*cos(phi)+t*sin(theta)*cos(phi),
y=a*sin(phi)+t*sin(theta)*sin(phi)

z=t*cos(theta)

(where theta=k*phi (k a constant), phi=0..2*Pi and t=-b..b (b determines the width of the ribbon.))

Or
x=(a+t*sin(k*phi))*cos(phi),
y=(a+t*sin(k*phi))*sin(phi)

z=t*cos(k*phi)

(where k a constant, phi=0..2*Pi and t= -b..b)

When we take k=1/2, 1, 3/2, 2,…, we have different twisted ribbons.

When k=1/2, we have the equation of the Mobius strip:
x=(a+t*sin(phi/2))*cos(phi),
y=(a+t*sin(phi/2))*sin(phi)

z=t*cos(ph)/2

(phi=0..2*Pi and t= -b..b) .

 

k=1

 

k=2

Is there a way to change the background color of a plot viewing area different from the white?

 

 

Thanks!

John Fredsted suggested using the following procedure (slightly modified) to determine whether an expression was deeply algebraic.

isDeeplyAlgebraic := proc(x)
	if not type(x,'algebraic') then false
	elif type(x,'atomic') then true
	else andmap(procname,x)
	end if
end proc:

TypeTools:-AddType('DeeplyAlgebraic1'
                   , proc(x)
                         if not x :: 'algebraic' then false;
                         elif x :: 'atomic' then true;
                         else andmap(type,x,'DeeplyAlgebraic1');
                         end if;
                     end proc);

The first is a completely flat expression, the second is deeply nested.  The following graphs plot the time required to determine whether each expression is "deeply algebraic" as n increases, with each approach. The graph on the left is the time required to evaluate expr1, the graph on right is for expr2.  The red plot corresponds to the procedure, the green plot corresponds to the type. As you can see, for both flat and nested expressions, the procedure is significantly faster than the type.

 

I then did some testing to see whether the type matching could be improved.  A more efficient use of the type mechanism is to use a structured type rather than a procedure.  Alas, I don't believe that it is possible to create a purely structured type, with no use of 'satisfies', that is equivalent to what we want.  Here is the best I could come up

TypeTools:-AddType('DeeplyAlgebraic3'
                   , 'And'('algebraic'
                           , 'Or'('atomic'
                                  , 'satisfies'(x->andmap(type,x,'DeeplyAlgebraic3'))
                                 )));

Adding that to each graph gives the following two plots

 

The yellow line (p3) corresponds to this new type.  For expr1, the flat expression, it is significantly faster than the previous type, and approaches the speed of the standalone procedure.  Alas, for expr2, the nested expression, it is even slower than the previous type.  However, the reason it is slower is that with a nested expression the 'satisfies' part of the type has to be evaluated, which generates a call to a user-level procedure.

This observation suggests that if the type could be expressed as a structured type, with no use of 'satisfies', it might be significantly more efficient.  While I can see no way to do that with the desired predicate, it is possible to construct a type specific to these two examples:

TypeTools:-AddType('nestedF', 'And(algebraic,Or(atomic,function(Or(atomic,nestedF))))');

This isn't equivalent to the original predicate because it only works with functions, not operators (+, *, etc).

Here are the results with that type

Now we are making progress.  The blue plot (p4) is the time for this restricted type.  It is significantly faster than even the standalone procedure.

Note that if there were a structured type, say allops, that returned true if all of the operands of an expression match the given type, then we would be able to construct a purely structured type that matches our original predicate, that is,

TypeTools:-AddType('DeeplyAlgebraic4'
                   , 'And(algebraic
                          , Or(atomic
                               , allops(DeeplyAlgebraic4)))');

Hey,

I am trying to animate this code, I have a loop that makes a matrix of random numbers
 

with(RandomTools):for i from 1 to 20 do
>   for j from 1 to 2 do
>   p[i, j] := i+Generate(float, 0..1);
>   od;
> od;   
 

and if I try to plot the following this works such as

points:= {seq([p[T,1], p[T, 2], T],T=1..20)};
pointplot3d(points, symbol=box);

This gives me all 20 points on the plot

Hello,

I am trying to do some basic plots with maple, but when I plot them they look strange and do not look like they should, what am I doing wrong?

The plot I tried to do is:

plot(cos(20*Pi*t), t = -5 .. 5)

I am having this problem with most of the plots I try to do, they do not come out symmetrical and have strange anomalies in them.

 

Thanks,

 

Hi all,

I have the following function:

phiS := (C,q) -> w*(int(a*f*g(x), x=0..a*f) + int(x*g(x), x=a*f..((w+pi2)*q-pi2*C)/w) + int(q*g(x), x=((w+pi2)*q-pi2*C)/w..infinity)) - s*q - pi1*(f-C) - pi2*int((C-q)*g(x), x=((w+pi2)*q-pi2*C)/w..infinity);

when I type the command:

simplify(diff(phiS(C,q),C));

I get the correct output (I've differentiated phiS on paper): pi1- pi2*int(g(x), x=((w+pi2)*q-pi2*C)/w..infinity);

***************

eqn1:= 1.x+1.y+1.z=0

eqn2:= a.x + b.y + c.z=0

eqn3:= b.c.x+c.a.y+a.b.z=1

When I try to solve this system of equations as:

solve( {eq1,eqn2,eqn3},{x,y,z} )

I get the message "Warning, solutions may have been lost"

Can you please help me to solve these equations and express my solutions in "factored form" ?