Maple 12 Questions and Posts

These are Posts and Questions associated with the product, Maple 12

Even and odd complement each other

how to find other sequence which complement each other?

such as 3 sequences divided integers or 5 sequences divided whole integers

is there monomials creation method such that solve result about coefficient and power are integers when right side columns are sequences?

 

i find even multiplication numbers are always solved into integers coefficient and power.

is there any more other sequences?

if I choose six multiplication table sequence, 

what is this complement of six multiplication table sequence?

How to solve system of convolution equations in maple?

When x axis range is large, the graph display still small , how to expand the graph display in large size?

Excel tool import can not be used in cmaple.exe

how to massive install 100 maple into amazon to do distributed computing or cluster computing?

and how to write this distributed function to run into 100 maple instance?

assume there is a function with two parameter which is the range of array

and the total number of times of calculations is from 1 to 9^(9^2) and divided this range into many range by  each range length 100,000

how can it write the output into a list of length 9^(9^2) ?

will there limit in length of list or result?

and 100 instance of virtual machine may not enough to calculate, how to allocate the next 100,000 calculations to idle instance which finished previous 100,000 calculations?

 

is there example data that can verify maxwell equations?

How to calculate determinant of a cube matrix ?

is there function to calculate 3x 3x 3 cube determinant?

How to proof classical and quantum both p summation equal to one in maple?

aab := int(exp(-a*x)*cos(b*x), x=0..infinity);
simplify(eval(aab));
 

why above equation is not equal to a/(a^2+b^2) ?

 

and how to solve below equation

z assuming complex;
solve(z^n - z = 0, z);

it only return 1 or 0, but not complex equations in terms of n

 

If decimal place is 1 such as 123.1 as input matrix and output matrix is eigenvector

is there a easy way to see changes of eigenvector in graph ?

because 3 x 3 matrix has 9 elements 

and eigenvector each vector may not fix in place,

how do mathematician observe the perturbation in this case?

would like to find which changes can cause different in eigenvectors

 

I can calculate real number eigenvector with QR algorithm from rotation matrix

but this real number is just magnitude of complex number

what is the rotation matrix for calculating complex number?

what algorithm do maple use to calculate fraction eigenvector?

are complex decimal and complex fraction eigenvectors are two different algorithm?

with(LinearAlgebra):
A := Matrix([[1,2,3],[4,5,6],[7,8,9]]);
v, e := Eigenvectors(A):
evalf(e);
A := Matrix([[1,2,3+0*0.1],[4,5,6+0*0.1],[7,8,9+0*0.1]]);
v, e := Eigenvectors(A):
evalf(e);

Rayleigh's identity is listed below:


              infinity                                           
               -----                                             
                \                                                
                 )             2      /      2        1      1  \
                /        |f(k)|  = int||f(t)| , t = - - T .. - T|
               -----                  \               2      2  /
            k = -infinity                                        

 

sum(abs(f(k))^2, k = -infinity .. infinity) = int(abs(f(t))^2, t = -(1/2)*T .. (1/2)*T);

This identity is an extension from Parseval's theorem for the case where the function of interest is periodic.  The link below provides a worksheet that confirms for a finite series that Rayleigh's identity is valid to within so many significant figures as the frequency parameter, k, increases for CASE 1.  However, for CASE 2 concurrence between the integral and the finite series is not that great.  I suspect I have an error somewhere that is causing the discrepancy.  I thought it might be useful if I get other sets of eyes on this to help isolate the discrepancy.  How I came up with Ck for CASE 2 I can create another worksheet with that derivation if requested.

Rayleighs_identity.mw

Appreciate any useful feedback

 

in maple 12,

i follow https://en.wikipedia.org/wiki/Eigenvalue_algorithm

it said QR return eigenvector but after compare with eigenvector function, they are not the result from eigenvector function

with(LinearAlgebra):
M := Matrix([[1,2,3],[4,5,6],[7,8,9]]);
A := HessenbergForm(M);
Q1, R1 := QRDecomposition(A);

Q1, R1 := QRDecomposition(M);

v, e := Eigenvectors(M);
v, e := evalf(Eigenvectors(M));
 

in maple 2015

if start from GivensRotationMatrix , how many times of 

GivensRotationMatrix in order to calculate the result of QR decomposition?

can this QR decomposition calculate the result of eigenvector?

MAPLE will evaluate the 1st integral below, but not the 2nd.  Is it possible to get MAPLE to do the evaluation?  I know it can be done numerically, but what about analytically?

int(sin(Pi*x*n/T), x);

 

int(sin(Pi*x*n/T), x)

int(sin(Pi*x*n/T)*cos(Pi*x*n/T)/(sin(Pi*x/T)*cos(Pi*x/T)), x);

int(sin(Pi*x*n/T)*cos(Pi*x*n/T)/(sin(Pi*x/T)*cos(Pi*x/T)), x)

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