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### Why does maple compute $\prod_{k=3}^{\infty} \cos ... February 25 2014 1 11 I have computed the infinite multiplication using Maple,$\Pi_{k=3}^{\infty} ( \cos (\frac{\pi}{k} ))$, as follows, but it resulted in 0! I wonder why this happened, although maple was using exact arithmetic. P := Product(cos(Pi/k), k = 3 .. infinity) value(P) Note that if I use floating-point instead, it gives me the right answer, 0.1149420449. evalf(P) By the way, I did not expect such a situation! Exact arithmetic should be exact! I am multiplying non-zero numbers to each other, starting from$1/2$to$1$as$n \rightarrow \infty$. So it should not be zero! Why such a thing happened? ### Taylor Expansion using Maple... February 25 2014 0 7 Dear all; Special thanks for all the member who help me in Maple. My last question is: Write a maple procedure that solves for y(1) in the initial value problem y'(x)=f(y), y(0)=1 using a Numerical stencil based on the n^{th] order taylor series expansion of y. The procedure arguments include an arbitrary function f, an integrer n, representing the accuracy of the taylor series expansion, and N representing the number of steps between x=0 and x=1. ### Function that diagonalises a 2x2 complex matrix... February 25 2014 0 1 Given a 2x2 matrix I am struggling to write a function that would return a list (a,b, a1, a2) of 2 complex numbers followed by 2 vectors such that the set of the 2 vectors is a basis for CxC and also Ab1=ab1, Ab2=Bb2 if these exist Any ideas would be greatly appreciated ### how to convert decimal to fraction without simplif... February 25 2014 1 13 how to convert decimal to fraction without simplify for example convert(0.25, fraction) expect 25/100, but not 1/4 ### How do I solve a differential equation system in M... February 25 2014 1 3 Hi, Please help me in solving system consist the three differential equations with three unknowns. I did already a few attempts, but I can not finish. Once in the final result was got RootOf and do not know what to do. I tried also numerically. I very very ask for some suggestions: ( with the boundary condition parameters A,B,C are constans 1) qa1 := A1*(diff(Tg(x), x, x))+A2*(diff(Tg(x), x))+(A3+A4)*Tg+A3*Tz+A4*Tw = 0; eqa2 := B1*(diff(Tw(x), x, x))+B2*(diff(Tw(x), x))+(B3+B4)*Tw+B3*Tz+B4*Tg = 0; eqa3 := C1*(diff(Tz(x), x, x))+(C3+C4)*Tz+C3*Tg+C4*Tw = 0 2) On paper, the system of three equations with three unknowns I changed to system of two equations with two unknowns but still nothing. A[1] := 2, eqa1 := A[1]*C[1]*(diff(z(x), x, x, x, x))/C[3]+A[2]*C[1]*(diff(z(x), x, x, x))/C[3]+(A[1]*C[3]+A[1]*C[4]+A[3]*C[1]+A[4]*C[1])*(diff(z(x), x, x))/C[3]+(A[1]*C[3]+A[1]*C[4])*(diff(z(x), x))/C[3]+(A[3]*C[3]+A[3]*C[4]+A[4]*C[3]+A[4]*C[4]+A[3]*C[3])*z(x)/C[3]+A[1]*C[4]*(diff(y(x), x, x))/C[3]+A[2]*C[4]*(diff(y(x), x))/C[3]+(A[3]*C[4]+A[4]*C[4]+A[4]*C[3])*y(x)/C[3] = 0; eqa2 := B[4]*C[1]*(diff(z(x), x, x))/C[3]+(B[4]*C[3]+B[4]*C[4]+B[3]*C[3])*z(x)/C[3]+B[1]*(diff(y(x), x, x))+B[2]*(diff(y(x), x))+(B[4]*C[3]+B[4]*C[4]+B[3]*C[3])*y(x)/C[3] = 0; row := eqa1, eqa2; sol := dsolve({row}, {y(x), z(x)}); Thank you very much for your help. Ewa. ### error between exact and approximation... February 24 2014 0 6 Dear all, I need to compute the error, How to define the error between the exact and approximation. d --- y(x) = -y(x) dx y(x) = exp(-x) I have a problem in this code, my goal is to compute the error between the approximate solution obtained by RK3 and Exact and E ( approximation by RK3). How to definie the error and prouve that the error is O(h^4) ( with one step) and the global error is O(h^3). Thank you for helping me. ### A problem with dsolve... February 24 2014 0 4 Hi, I have a problem with dsolve in the following code restart; > n:=20; m:=1; cc:=-200; zzeta:=0.1; sefr1:=0.3; sefr:=0.2; MM:=0; lambda:=0.1; Br:=1; nn:=3; > > #u(tau):=tau; u(tau):=421.7129935*tau-2217.587728*tau^2+8897.376593*tau^3-27612.59182*tau^4+64248.00336*tau^5-1.083977605*10^5*tau^6-10.57029600-1.080951714*10^6*tau^13+7.999517316*10^5*tau^14-4.788741005*10^5*tau^15+2.309563748*10^5*tau^16+26511.11102*tau^18-5959.001794*tau^19+1.148523882*10^5*tau^7-95.23809524*tau^21+4.545454545*tau^22-9435.563781*tau^8-2.587683745*10^5*tau^9+6.473880128*10^5*tau^10+948.0272727*tau^20-88660.41892*tau^17-1.008692404*10^6*tau^11+1.175504242*10^6*tau^12; > > B := 1+(2*(1-zzeta))*Br*(int(tau*(diff(u(tau), tau))^2, tau = (1/2)*zzeta*(1-zzeta) .. 1/2*(1-zzeta))); eq4 := 4*B*u(tau)-(1+zzeta)*(diff(tau*(diff(theta(tau), tau)), tau))/tau-(1+zzeta)*Br*(diff(u(tau), tau))^2; theta(tau):=sum(p^ii*theta[ii](tau),ii=0..nn); HH:= p*((4*(1+(2*(1-zzeta))*Br*(int(tau*(diff(u(tau), tau))^2, tau = (1/2)*zzeta*(1-zzeta) .. 1/2-(1/2)*zzeta))))*u(tau)-(1+zzeta)*(diff(theta(tau), tau)+tau*(diff(theta(tau), tau, tau)))/tau-(1+zzeta)*Br*(diff(u(tau), tau))^2)+(1-p)*(diff(theta(tau),tau$2)):
eq5:=simplify(HH):
eq6:=collect(expand(eq5),p);

eq7:=
convert(series(collect(expand(eq5), p), p, nn+1), 'polynom');

for ii to nn do
ss[ii] := (coeff(eq7, p^ii)) ;
print (ii);
end do;

ss[0]:=diff(theta[0](tau), tau, tau);

icss[0]:=theta[0](zzeta/(2*(1-zzeta)))=0, D(theta[0])(1/(2*(1-zzeta)))=1;

dsolve({ss[0], icss[0]});
theta[0](tau):= rhs(%);

for ii to nn do
ss[ii]:=evalf[5](ss[ii]);
icss[ii]:=theta[ii](zzeta/(2*(1-zzeta)))=0, D(theta[ii])(1/(2*(1-zzeta)))=0;
dsolve({ss[ii], icss[ii]});
theta[ii](tau):=rhs(%);
end do;

I would be most grateful if you help me to find this problem.

### Tangent line calculus ...

February 24 2014
0 2

How to find the equation of the tangent line to f(x)=6x/squareroot(x^2+12) at the point (2,3) ?

### Error, invalid arrow procedure...

February 24 2014
1 3

Dear all,

I write in maple this line but  I get an error,

> Butcher, map(x->if (x=0) then  else x end if,Butcher);
Error, invalid arrow procedure

### Billiards with Maple...

February 24 2014
0 15

Is it possible to describe billiards in the unit square (see http://en.wikipedia.org/wiki/Dynamical_billiards for info) in terms of events and if? For example, let the dynamical system
{x''(t)=0,y''(t)=0, x(0)=0.5, x'(0)=0.2, y(0)=0.5, y'(0)= sqrt(3)/5}
be given, where the trajectory [x(t), y(t)] satisfies the usual laws of reflection http://en.wikipedia.org/wiki/Reflection_%28physics%29 when reaching the boundary {x=0}, {x=1}, {y=0}, {y=1}. The same question about billiards in the unit disk.

### Solve by Newton Raphson Method...

February 24 2014
0 4

hi everybody

I want to solve this system of equations

t*((p-.764*z-2.194768)^2-1.170308549+.529948*(z-.382)^2)-(1-t)*p
t*((p+.382*z+.661624*y-1.907568)^2-1.018097144+.529984*(-.866*y-.5*z-.382)^2)+(1-t)*y
t*((p+.382*z-.661624*y-2.470348)^2-1.31636154+.529984*(.866*y-.5*z-.382)^2)+(1-t)*z

while t varies from 0 to 1 by 0.0001 interval. Using newton raphson method, the inital value for each step is the result of the previous step for y,z,p. the very initial values are y=1,z=1,p=1

### Memory Leak...

February 24 2014
1 3

I'm having a weird memory leak problem when I use Compiler, and numerical NAG routines for integration - at least I think it's that combination that's causing it. To see it run this:

First define the functions to use:

F_:=Compiler:-Compile(codegen[prep2trans](codegen[optimize](codegen[makeproc](convert(sinh(x)*sin(x),float),[x]))),optimize=true);

F:=proc(k)
if not (k::numeric) then return 'procname'(args) fi;
F_(k);
end proc;

INT:=proc(X) if not X::numeric then return 'procname'(x) fi; int(F(x),x=0..X,numeric,method=_d01ajc); end proc;

Then evaluate lots of times:

for i to 10^6 do a:=rand()/1e12; INT(a);  od:

And watch your computer memory go wild. I'm using Maple 17 on a new mac with 8Gig RAM, using Mavericks. It doesn't take more than a few minutes to grind the system to a halt. A restart command does not release the memory in the mserver process. [I've noticed similar problems in the past, so it's not locallised to this OS I think.]

On a related note, there's a problem with nested NAG integrals - in the do loop try this:

int('INT'(XX),XX=0..a,numeric,method=_d01akc)-int('INT'(XX),XX=0..a,numeric,method=_d01ajc);

it never gives zero! The problem lies in using the same NAG routine twice [here _d01ajc] for a nested integral. I've never found a way around this bug!

Any help would be much appreciated!

### How to perform dynamic geometry program in Maple 1...

February 24 2014
0 0

Good afternoon sir.

I am working on problems related to functions which require dynamic geometry program or

the Geometers sketch pad. I request to you to kindly suggest me with regard to the above cited query.

With thanks & Regards

M.Anand

Assistant Professor in Mathematics

SR International Institute of Technology,

### Graphing the cube root of x...

February 23 2014
1 2

I am trying to graph the cube root of x, but when I do as

plot(x^1/3,thickness=5)

I get a graph with domain of non negative reals.  I need the entire graph, across all reals and I'm not sure how to handle it.  Thanks

### getting parametric equations...

February 23 2014
0 4

Hello Maple users friends,

I have two lines in the space (x,y,z) described by the equations in L1 and L2:

L1:= {4*x + 3*y + z = 0, x + y - z - 15 = 0}:

L2:={12*x + 5*y + 7*z -13 = 0, 9*x + y -3*z - 5 = 0}:

I would like the get the parametric (with z=t) equations P1 and P2 of the two lines..

I see the "form" of such parametric equations P1 and P2 using "solve"

solve(L1, {x, y}); solve(L2, {x,y});

but I do not know how to use those values to get my parametric equations P1 and P2 to continue with additional computation (area, volume etc).

Thanks for your attention and help.

JJ

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