I do not understand what ColumnSpace is doing in Maple, as I can't get it to match my hand solution and the book. It must be using different definition which is not the book standard.

In the textbook, it says to find column space of matrix A, is to find the Echelon form, then lookup the pivot columns. Then pick the corresponding columns from the original A. These are the column space.

I'll show the book example, and Maple code to try to get same answer.

Here is an example from the book

Notice the columns space are the first, second and fourth columns of A. Since these correspond to the pivot columns of the Echelon form.

In Maple, the Echelon is found using LinearAlgebra:-GaussianElimination(A,'method'='FractionFree'); which gives the same pivot columns as the book. So far so good (the numbers are not the same, but this is normal, as Echelon form is not unique. Only reduced Echelon form is unique), but as long as the pviot columns are correct, that is the important part. 

But when I do LinearAlgebra:-ColumnSpace(A); I get completely different result. The only thing I could see in help, is that it says The Vectors are returned in canonical form with leading entries 1.

I do not underand what that means or how to convert Maple answer to make it match the expeted result since the result is completely different and I do not see the mapping needed.

restart;
A:=Matrix([[1,2,1,3,2],[3,4,9,0,7],[2,3,5,1,8],[2,2,8,-3,5]]);
LinearAlgebra:-GaussianElimination(A,'method'='FractionFree');
LinearAlgebra:-ColumnSpace(A);

I also tried this on Wolfram Alpha, and got same answer as the book

Maple 2020.1

reference: Differential Equations and linear algebra, 4th ed., Edwards, Penney and Calvis. Pearson. 2017. page 247.

Hi,

I do not understand why this simple procedure evaluates so differently depending on the type of its second parameter?

A typo somewhere or a bug?


 

Download I_am_lost.mw

(2x+a^2-3a)=a(x-1) with all the real values of a

I have a system of six ODEs. i solve them numerically using dolve,numeric command. the problem is setting step size for example stepsize=1e-5 or minstep=500 lead to a result of order 1e-2; But without using this option, results are of order integer numbers. could any one help? tnx in advance

EDITED
 

Download ode_problem.mw

Hello,

I'm trying to fit experimental data to a model in which a variable is the result of an equation.

In a procedure I solved the equation and found lambda22 for each of the data lambda11 I have. It depends on a a parameter I need to identify to fit the data to the model.

But then when I use DataFit from DirectSearch package, I get this error :

Is it impossible to specify more than one independant variable ? Or is it a problem coming from the procedure ?

Thanks in advance,

Manon

data.xlsx

restart;

data:=ExcelTools:-Import("D:/data.xlsx") : # importer les donnees
with(LinearAlgebra): 
lambdaX:=Column(data,1): 
PK1X:=Column(data,2):


tensF:=Matrix([[lambda11,0,0],[0,lambda22,0],[0,0,lambda22]]):
tensFbar:=Matrix([[(lambda11/lambda22)^(2/3),0,0],[0,1/sqrt((lambda11/lambda22)^(2/3)),0],[0,0,1/sqrt((lambda11/lambda22)^(2/3))]]):
tensBbar:=Multiply(tensFbar,Transpose(tensFbar)):

kappa:=100:
tensPK:=Multiply(Transpose(MatrixInverse(tensF)),(2/Determinant(tensF))*a*(tensBbar-(1/3)*Trace(tensBbar)*Matrix(3,shape=identity))+kappa*(Determinant(tensF)-1)*Matrix(3,shape=identity)):
PK1:=tensPK[1,1]:
eq:=0=tensPK[2,2]:


lambda22sol:=proc(aValue) 
   global __old_a;
   local res, i, eq1, sol1, lambda22Est;
   if not [aValue]::list(numeric) then
      return 'procname'(args);
   end if;
   lambda22Est:=Vector(1..RowDimension(lambdaX)):
   eq1:=Array(1..RowDimension(lambdaX)):
   if __old_a<>aValue then
      __old_a:= aValue;
     for i from 1 to RowDimension(lambdaX) do
     	eq1[i]:=eval(eq,[a=aValue,lambda11=lambdaX[i]]):
		sol1:=solve(eq1[i], lambda22);
		lambda22Est[i]:=sol1[1]:
	end do:
	res:=lambda22Est;
   end if;
end proc:

matrix1:=<lambdaX|lambda22sol(0.9)|PK1X>;
DS1:=CodeTools:-Usage(
        DirectSearch:-DataFit(PK1,matrix1,[lambda11,lambda22]));
res2:=eval(PK1,DS1[2]):

Hi,

Using Maple 2020 I can't seem to get a trigonometric equation (fairly simple at that) to solve properly.

solve_err.mw

Here's a capture :

Expected answer is 1/(25*sqrt(3))

Adding +2*Pi on the RHS to match the sign doesn't work either.

Hi everybody, I have a set from fsolve command. As shown in below image

When I want to obtain only numbers, I write cozum[1] but maple gives me c-4=1.591497964.. As you understand I want only numbers. How can I do this? 

int(1/(x-1)+sum((k+1)*x^k,k=0..2018)/sum(x^k,k=0..2019),x);

Direct integration produces a large, messy statement, including numbers like 2017, 2016,...,1

Looking at the parts, the numbers 2020, 2019, etc. are in each.

(**) sum((k+1)*x^k,k=0..2018);
x^2019*(2019*x-2020)/(x-1)^2+1/(x-1)^2

(**) sum(x^k,k=0..2019);
x^2020/(x-1)-1/(x-1)

Looks like this should reduct to some simple expession????

I made a mistake in entering this expression:  2020 should have been 2018.

int(1/(x-1)+sum((k+1)*x^k,k=0..2020)/sum(x^k,k=0..2019),x);

realloc(): invalid old size

Process Maple aborted (core dumped)

Nothing I can do from the keyboard should cause a core dump.

Looking at #6 in http://www.mit.edu/~pax/pdf/qualifying_round_2020_test.pdf

How do I entter the continued fraction in Maple?  I am using the command line interface (actually Emacs).

Hi,

I want to solve a set of equation (four equations) to plot variation of u[i10],u[i20],phi[d0] versus delta[d] for delta[i]=0.5 and different values of alpha (for example, alpha=0, 0.1,0.2)

How can I do that with Maple?

(It should be noted that depending on the sign of phi[d0], one must use Eq. 2a or 2b. The other equations are Eqs. 1,3,4)
I expect to plot figures like the attached figure.

1.zip

I am looking to find a distribution function (both PDF and CDF) for the distance between two points on the unit square.

Each point will be uniformly distributed on the [0,1] interval for both the x and y axes.  The distance between these points (d_ij) will of course be:

d[i, j] := sqrt((x[i] - x[j])^2 + (y[i] - y[j])^2)

I think that using a convolution may be required, but this is over my head.  If anybody can show me (preferably via a Maple worksheet), I would be very appreciative.

Thank you.

Hello,

I have a question about fitting a function to experiemental data. The experimental data is Di = X(zi) +I*Y(zi) for a zi = [z1 z1+dz ...z1+dz*N], where zi are real values. I want to fit a complex function f(z,a,b,c) to the data zi and Di , where a,b,c are complex-valued parameters. Does someone know how to fit a complex function to a complex data? The Maple function that I found work only with real functions and data.

Hello,

I want to define an orthonormal tetrad basis of my choice in a spacetime having a metric given in some system of coordinates. My problem is that Maple automatically proposes an orthonormal metric but this is not the one that suits my requirements. So, I would like to specify the tetrad basis manually. As an example, I am trying to reproduce the calculations in sections 6 and 7 of the article https://arxiv.org/abs/gr-qc/0510083 . Here, the metric $g$ is given by the line element $ds^2 = - (c(t,r)^2 - v(t,r)^2) dt^2 + 2 v(t,r) dr dt + dr^2 + r^2 (d\theta^2 + sin(\theta)^2 d\phi^2)$ in $(t, r, \theta, \phi)$ coordinates. My chosen signature is (- + + +). Let, us adopt the convention used by Maple and denote spacetime indices by Greek alphabets and tetrad indices by lowercase Latin letters. Now, I would like to define a tetrad $e_a = (V, S, \Theta, \Phi)$ (as in section 7 of the article referred to above) where:

V^\mu = \frac{1}{c\sqrt{1-\beta(t,r)^2}}[1, - (v + c \beta), 0, 0] \\

S^\mu = \frac{1}{c\sqrt{1-\beta^2}}[-\beta, c + v \beta, 0, 0] \\

\Theta^\mu = [0,0,1,0]

\Phi^\mu = [0,0,0,1].

Here, $|\beta(t,r)| < 1$. I do not know how I may specify this in my worksheet. This may come of use somewhere later. Now, with this choice of the tetrad, we know that $g(e_a, e_b) = \eta_{ab}$ with $\eta$ being the Minkowski metric in spherical coordinates. After defining this tetrad basis, I finally want to calculate Einstein tensor, components of energy-momentum tensr etc. I have problem with constructing this orthonormal tetrad basis myself. It would be great if you could help me with this.

An additional curiosity: when we work with multiple tetrad bases, is it possible to denote the the tetrad indices by hatted tetrad labels themselves, as in $\eta_{\hat V, \hat \Theta}$?

Thank you.
 

Download dynBH.mw

#include<iostream>
#include<vector>
#include<cmath>
#define NODE 8

using namespace std;
int graph[NODE][NODE] = {
   {0,1,1,0,0,0,0,0},
   {1,0,1,1,1,0,0,0},
   {1,1,0,1,0,1,0,0},
   {0,1,1,0,0,0,0,0},
   {0,1,0,0,0,1,1,1},
   {0,0,1,0,1,0,1,1},
   {0,0,0,0,1,1,0,0},
   {0,0,0,0,1,1,0,0}
};
int tempGraph[NODE][NODE];
int findStartVert() {
   for(int i = 0; i<NODE; i++) {
      int deg = 0;
      for(int j = 0; j<NODE; j++) {
         if(tempGraph[i][j])
            deg++; //increase degree, when connected edge found
      }
      if(deg % 2 != 0) //when degree of vertices are odd
      return i; //i is node with odd degree
   }
   return 0; //when all vertices have even degree, start from 0
}
int dfs(int prev, int start, bool visited[]){
   int count = 1;
   visited[start] = true;
   for(int u = 0; u<NODE; u++){
      if(prev != u){
         if(!visited[u]){
            if(tempGraph[start][u]){
               count += dfs(start, u, visited);
            }
         }
      }
   }
   return count;
}
bool isBridge(int u, int v) {
   int deg = 0;
   for(int i = 0; i<NODE; i++)
      if(tempGraph[v][i])
   deg++;
   if(deg>1) {
      return false; //the edge is not forming bridge
   }
   return true; //edge forming a bridge
}
int edgeCount() {
   int count = 0;
   for(int i = 0; i<NODE; i++)
      for(int j = i; j<NODE; j++)
         if(tempGraph[i][j])
   count++;
   return count;
}
void fleuryAlgorithm(int start) {
   static int edge = edgeCount();
   static int v_count = NODE;
   for(int v = 0; v<NODE; v++) {
      if(tempGraph[start][v]) {
         bool visited[NODE] = {false};
         if(isBridge(start, v)){
            v_count--;
         }
         int cnt = dfs(start, v, visited);
         if(abs(v_count-cnt) <= 2){
            cout << start << "--" << v << " ";
            if(isBridge(v, start)){
               v_count--;
            }
            tempGraph[start][v] = tempGraph[v][start] = 0; //remove edge from graph
            edge--;
            fleuryAlgorithm(v);
         }
      }
   }
}
int main() {
   for(int i = 0; i<NODE; i++) //copy main graph to tempGraph
   for(int j = 0; j<NODE; j++)
      tempGraph[i][j] = graph[i][j];
   cout << "Euler Path Or Circuit: ";
   fleuryAlgorithm(findStartVert());
}

Kind help