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I am trying to compute a double integral with respect to variables x and y. I first compute the integral with respect to the variable x:

first_int = integral(h, x, 1, 5)

which gives me as an output :

integrate(1/(log(abs(x^4 + 13*x^3*y - 30*x^2*y^2 + 2*x*y^3 - 3*x^2*y + x*y^2 + x^2 + x*y + y^2 + 2*x + 13))^log(abs(x^4 + 13*x^3*y - 30*x^2*y^2 + 2*x*y^3 - 3*x^2*y + x*y^2 + x^2 + x*y + y^2 + 2*x + 13))), x, 1, 5)

instead of an expression now only depending on the second variables y.

Then, when I try to integrate with respect to the second variable y by writing

second_int = first_int.integrate(y, 1, 5)

I get the following output:

integrate(integrate(log(abs(x^4 + 13*x^3*y - 30*x^2*y^2 + 2*x*y^3 - 3*x^2*y + x*y^2 + x^2 + x*y + y^2 + 2*x + 13))^(-log(abs(x^4 + 13*x^3*y - 30*x^2*y^2 + 2*x*y^3 - 3*x^2*y + x*y^2 + x^2 + x*y + y^2 + 2*x + 13))), x, 1, 5), y, 1, 5)

finally if I try to write (to get the numerical value)

second_int = n(first_int.integrate(y, 1, 5))

I get the error message ValueError: The function to be integrated depends on 2 variables (x, y), and so cannot be integrated in one dimension. Please fix additional variables with the 'params' argument

I have tried integrating the same type of function when there is only one variable and the integration works fine, so I am thinking something goes wrong in the double integration but I can't figure out what!

Thanks.

How can I generate a random polynomial with two variables x, y of fix degree which is not of type : <type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular' ?

I would like to compute the absolute value of this polynomial, but when I generate it using : R = ZZ['x','y']; F = R.random_element(4, 1000)

for example, I get : TypeError: bad operand type for abs(): 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'

What would be a good alternative to get |F(x,y)| ? Thanks

Thanks again for your answer. I have added an absolute value to my polynomial, so now I am integrating log(abs(F)). I thought this would solve the issue of having the log of a negative value, but it doesn't seem to change..... I need to fix the f =0 case.

This is an example for which the double integration of log(F) never ends: F = -2x^4 - x^2y^2 - 2xy^3 + x^3 + 5x^2y + 5xy^2 + 5y^3 + 2y^2 - 3*x + y - 21

Thank you. I tried copying your example and I run into the error :

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation may help (example of legal syntax is 'assume(y^2-2y+2>0)', see assume? for more details) Is y^2-2y+2 positive or negative?

I am running into various odd problems while trying to compute a double integral. My code is as follows:

x, y = var('x y')
##### Two different ways of generating random polynomials F(x,y) to test double-integration ########
#F = ZZ['x','y'].random_element(4, 1000)
F = randint(-1000, 1000)*x**4*y**4 + randint(-1000, 1000)*x**3*y + randint(-1000, 1000)*x**2*y**3 +randint(-1000, 1000)*x  + randint(-1000, 1000)*y**2
G = randint(0, 1000)*x**4*y + randint(0, 1000)*x**3 + randint(0, 1000)*x**2*y**2 +randint(0, 1000)*x  + randint(0, 1000)*y**4
###### Defining the function we want to integrate over dx dy ##########
f =  log(F)
####### Double integration  #####
g= integrate(f, x, 1, 2)
print 'after integration dx:', g
assume(y>0)
print n(g.integral(y, 1, 2))

Problems: 1) If I use the random polynomial F generated by .random_element, the integration generates the following error : TypeError: integral() takes exactly one argument (3 given). However, if I use the other one I manually generate, then it integrates okay. If I copy from the terminal the .random_element() polynomial and explicitly write it in my code, then it will also work. Why not directly from the .random_element() function then ?

2) If I set f = F or even f = FG, then the double integral works fine. However, as soon as I introduce the *log function, the calculation never terminates (if I work with an example in the terminal with a log function, then it does integrate properly).

Any idea what might cause these weird bugs ? Thanks!

I am running into various odd problems while trying to compute a double integral. My code is as follows:

x, y = var('x y')
##### Two different ways of generating random polynomials F(x,y) to test double-integration ########
#F = ZZ['x','y'].random_element(4, 1000)
F = randint(-1000, 1000)*x**4*y**4 + randint(-1000, 1000)*x**3*y + randint(-1000, 1000)*x**2*y**3 +randint(-1000, 1000)*x  + randint(-1000, 1000)*y**2
G = randint(0, 1000)*x**4*y + randint(0, 1000)*x**3 + randint(0, 1000)*x**2*y**2 +randint(0, 1000)*x  + randint(0, 1000)*y**4
###### Defining the function we want to integrate over dx dy ##########
f =  log(F)
####### Double integration  #####
g= integrate(f, x, 1, 2)
print 'after integration dx:', g
assume(y>0)
print n(g.integral(y, 1, 2))

Problems: 1) If I use the random polynomial F generated by .random_element, the integration generates the following error : TypeError: integral() takes exactly one argument (3 given). However, if I use the other one I manually generate, then it integrates okay. If I copy from the terminal the .random_element() polynomial and explicitly write it in my code, then it will also work. Why not directly from the .random_element() function then ?

2) If I set f = F or even f = FG, then the double integral works fine. However, as soon as I introduce the *log function, the calculation never terminates (if I work with an example in the terminal with a log function, then it does integrate properly).

Any idea what might cause these weird bugs ? Thanks!

My matrix is slightly different and has many zeros. Maybe this causes the ring R to be non-commutative. I will have a look. Thanks for your answer!

I consider a matrix $M$ which I transform into a system of equations similar as this (but with a different $M$)

sage: M=matrix(3,3,[1,2,3,4,5,6,7,8,9])

sage: P=PolynomialRing(GF(p),M.nrows(),names="x")

sage: (vector(P.gen(i) for i in range(3))*M).list()

[x0 + 4x1 + 7x2, 2x0 + 5x1 + 8x2, 3x0 + 6x1 + 9x2]

(Example taken from another equation where $p$ is a prime).

When I try to create the ideal generated by this system, I get the following error : TypeError: R must be a commutative ring. Any idea how I can fix this?

I have a list of integers $(a, b, c, d)$ and a system of equations, where each equation has two unknowns. My system looks like $(x0+x4, x1+x4, x2+x4, x3+x4).$ I am trying to evaluate the system in $(a,b,c,d)$. If I write $system(x0=1, x1=2, x3=1, x4=1)$, the system correctly outputs the result of each equation. Now, I would like to do something like $system(x0=a, x1=b, x3=c, x4=d)$ but without having to manually put the variables in. I have tried writing $system(list)$ but the number of variables and unknown doesn't match. Is there any way I can achieve this ?

I have an element $v$ of type 'sage.modules.free_module_element.FreeModuleElement_generic_dense', concretely I have $(x0, x1, 1)$, which I would like to multiply by a matrix $M$, i.e., have $M*v.$ \

I am trying to convert this element into a vector, but the naive attempt to do $vector(v)$ fails to give me a valid multiplication (TypeError: unsupported operand parent(s) for '*': 'Full MatrixSpace of 2 by 2 dense matrices over Integer Ring' and 'Ambient free module of rank 3 over the integral domain Multivariate Polynomial Ring in x0, x1 over Rational Field'). Is there any way I can do this multiplication ? thanks!

thank you ! I have two additional questions. It seems that your solution gives the column coefficient to a polynomial (and not the rows). Would it be possible to have the row coefficients? One way would be to simple transpose the matrix beforehand. I have tried using M.ncolumns() but it seems to generate an error. Also, is there anyway the last coefficient could be a constant variable (1) (in order to have $ax+by+c$ for example) ?

I have a $n \times n$ matrix that I would like to transform into a system of $n$ equations (for $n$ disctinct variables). I am not sure how to proceed (the end goal being to solve the system eventually). The number of variables is not fixed. Should I consider every row of the lattice and map each coefficient $c_{ij}$ to a string that I write as $"c_{11}x_1 + .....+ c_{1n-1}x_n + c_{1n}$. I am hoping there exists a more direct solution. Thanks!

I am trying to solve a linear system of equation modulo some prime $p$. I have a matrix which gives me the coefficients of the polynomials (i.e., first row would be ($a, c, b, d$) for $ax+by+cz+d =0$). I would usually use solve_right(), writing my system as $Ax=b$ mod $p$, but I am intrigued by the use of Groebner basis. I have read about their implementation but I am confused at the notion of ideal to define before solving the system. I would like to solve the system in the ring $Z/pZ$. Also, can I feed a matrix to the function or should I first convert the matrix in a bunch of equations ? Any clarification would be welcome !

great, thanks. I'll try the last option.

I am trying to solve a system of equations mod $p$, for $p$ prime. I have build matrices $A$ and $b$ and I am using the function solve_right() to solve $Ax = b$. But I need to solve $Ax = b$ mod $p$. Is there any functions that could do that ? thanks !