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I have asked this question over here but I got no answer.

I am trying to do some iterative calculations where each time SAGE constructs a fraction and lists the coefficients in the partial fraction decomposition of that fraction. I realized that doing everything symbolically, SAGE wants to keep everything as integer as possible. So what it does is to simplify fractions like 1/(x-3/2) as 2/(2x-3) and then when I ask for

f.coefficient(x-3/2,-1)

it returns 0, while I expect it to return 1.

I have tried to solve things numerically, but there are two problems:

1. The errors get really big after each iteration
2. It takes much much longer to calculate it

Any suggestions to get SAGE to solve this is greatly appreciated.

I included the complete code at the end of the post.

In a for loop I generate a list of functions, then I take their derivatives with respect to different variables. For example, in the follwoing, Y[0,0] is a 'MPolynomial_polydict', and is equal to

x_0_1*x_0_1 + x_0_0*x_1_1 + x_0_2*x_0_2 + x_1_2*x_1_2 + x_0_0*x_2_2 + x_1_1*x_2_2 - 1.00000000000000

and list_of_variables[0] = x_0_1. When I take the derivative of the mentioned polynomial with respect to the mentioned variable I should get 2*x_0_1, but instead I get:

sage: derivative(Y[0,0],list_of_variables[0])
x_0_1

When I try this manually, it gives me the correct answer though:

sage: derivative(x_0_1*x_0_1 + x_0_0*x_1_1 + x_0_2*x_0_2 + x_1_2*x_1_2 + x_0_0*x_2_2 + x_1_1*x_2_2 - 1.00000000000000,x_0_1)
2*x_0_1

So, for some reason it considers the two x_0_1's in the first term of my polynomial as independent variables when I construct them automatically. Any ideas on why this problem arises, and how to fix this?

Edit:

This is how I build my vaiables:

names = [ [[] for i in range(n)] for j in range(n) ]

for i in range(n):
    for j in range(i+1):
        names[i][j] = (SR('x_' + str(j) + '_' + str(i)))
        names[j][i] = (SR('x_' + str(j) + '_' + str(i)))

R = PolynomialRing(CC,names[i][j] for i in range(n) for j in range(n)])
R.gens()
R.inject_variables()

More edit: Here is a the complete code:

#########################################################################
# Build variables, and the matrix corresponding to it
#########################################################################

def build_variables(n):
    names = [ [[] for i in range(n)] for j in range(n) ]

    for i in range(n):
        for j in range(i+1):
            names[i][j] = (SR('x_' + str(j) + '_' + str(i)))
            names[j][i] = (SR('x_' + str(j) + '_' + str(i)))

    return(names)


#########################################################################
# Define the function f that maps a matrix to the coefficients of its characteristic polynomial
#########################################################################
def CharPoly(Mat):
    X = matrix(Mat)
    n = X.ncols()
    C_X = X.characteristic_polynomial()
    Y = []
    for i in range(n):
        Y.append(C_X[i])
    return(Y)


############################################################################
# This solves that lambda problem
############################################################################
def lambda_siep(G,L,iter=100,epsilon = .1):
                        # G is any graph on n vertices
                        # L is the list of n desired distinct eigenvalues
                        # m is the number of itterations of the Newton's method
                        # epsilon: the off-diagonal entries will be equal to epsilon

    n = G.order()
    my_variables = build_variables(n)

    R = PolynomialRing(CC,[my_variables[i][j] for i in range(n) for j in range(n)])
    R.gens()
    R.inject_variables()


    X = [ [ R.gens()[n*i+j] for i in range(n) ] for j in range(n) ]

    Y = matrix(CharPoly(X)) - matrix(CharPoly(diagonal_matrix(L)))

    J = matrix(R,n)
    for i in range(n):
        for j in range(n):
            J[i,j] = derivative(Y[0][i],my_variables[j][j])

    B = diagonal_matrix(L) + epsilon * G.adjacency_matrix()

    count = 0
    while count < iter:

        T = [ B[i,j] for i in range(n) for j in range(n)]

        C = (J(T)).solve_right(Y(T).transpose())
        LC = list(C)

        B = B - diagonal_matrix([LC[i][0] for i in range(n)])

        count = count + 1

    return(B)

I worked out a Newton's method base technique and got some solutions that works for me, but the code is really ugly, so I am waiting to see if there are any better solutions out here.

A Gorebner basis is not going to work for my problem, since my actual problem is much bigger, and even on 10 equations and 10 variables the Groebner basis gets really ugly. Thanks for the suggestions though.

This is what I did:

sage -n '~/Dropbox/Sage'

It creates a folder called Sage.sagenb and it works like your local sagenb folder in Sage installation folder. Then it asks for admin password if it's the first time you're running it in that folder.

You can do something like this

sage: f = 1/(x-3/2)
sage: g = f.denominator()
sage: f.coeff(f.denominator(), -1) / g.leading_coeff(x)
1

I have a set of polynomial equations and I want to find one of its real solutions close to some point, and I need only one solution. Here is an example:

This is the list of equations and variables:

Equations = [x_0*x_1*x_2*x_3 - x_0*x_1 - x_0*x_2 - x_0*x_3 - x_1*x_2 - x_1*x_3 + 2*x_0 + 2*x_1 - 448, -x_0*x_1*x_2 - x_0*x_1*x_3 - x_0*x_2*x_3 - x_1*x_2*x_3 + 3*x_0 +
3*x_1 + 2*x_2 + 2*x_3 + 452, x_0*x_1 + x_0*x_2 + x_0*x_3 + x_1*x_2 + x_1*x_3 + x_2*x_3 - 159, -x_0 - x_1 - x_2 - x_3 + 21]

Variables = [x_0, x_1, x_2, x_3]

If I ask Sage to solve this

S = solve(Equations,Variables)

it returns a bunch of solutions. But in some cases it doesn't give me any real solutions. I can prove that the above set of equations has a real solution close to [2,4,7,8]. Is there any way that I can perform an algorithm like the Newton's method with the start point [2,4,7,8], and find that real solution?

Hi,

As you can see with the output of ls, some of the directory within .sage are owned by root, namely ipython-0.12, sage_notebook.sagenb and tmp. I don't know how such a thing could happen ! You can change the permissions with the command

sudo chown k1:k1 -R .sage

The command chown is for modifying users, k1:k1 means that you change the user and the group to k1 and -R is for recursive.

After that, it may work.

Vincent

How do one go to close a thread?

Just wrote

sage -n 'directory'

and it worked.

easy enough. Thanks for the help. I just used `sage -n '/home/k1/Dropbox/Math/Scripts/Sage'` and it worked.

I want to change the default directory of notebook. I've learned that I can run

notebook("directory")

to have sage save and open worksheets from that directory, but I generally run

sage -notebook

from terminal. Is there a way to change the default directory for sage? or run it in a directory directly from terminal?

PS: My main reason for doing this is to connect the sage to my Dropbox in order to have a backup of all my worksheets. I've recently lost a whole bunch of them due some problems.

Thank you very much. It worked great.

Hi,

I would prefer to have a method .real() for a complex matrix as there exists one for complex number::

sage: CC(1,0.5).real()
1.00000000000000

But it does not exist yet! Moreover, using .real() ignore all imaginary part. Here is a function that makes the job:

trunc_small_imag = lambda x: x.real() if abs(x.imag()) < .000001 else x
mat_trunc_small_imag = lambda m: m.parent()(map(trunc_small_imag, m.list()))

And here is an example:

sage: m = matrix([[CC(1,.0000001), CC(2,-0.5)],[CC(2,0.4),CC(1.1,-.000000004)]])
sage: mat_trunc_small_imag(m)
[                      1.00000000000000 2.00000000000000 - 0.500000000000000*I]
[2.00000000000000 + 0.400000000000000*I                       1.10000000000000]

I am generating a matrix that its entries come from roots of a polynomial which are all real. Due to calculation errors sage returns entries such as: -2.8 + 2.2e-16*I

How can I ask sage to ignore the small imaginary part and return -2.8?

BTW, is there an easy to understand instruction on how to apply patches? I found [this](http://www.sagemath.org/doc/developer/patching_spkgs.html) but it doesn't seem like what I need.

Here is the output: "bash: /home/k1/sage5.8/spkg/bin:/home/k1/sage5.8/local/bin:/usr/lib/lightdm/lightdm:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/home/k1/.rvm/bin: No such file or directory" But it doesn't mean much to me!