[sage-support] Re: groebner bases of polynomials with complex coefficients
Thank you for the reply. I am now trying the computation with a somewhat more involved example and I am getting the error. The example I am trying to solve is below: R = QQ[sqrt(-1)] RI = R.gens()[0] S.x1,x2,x3,x4,x5,x6,x7,x8 = PolynomialRing(R,order='lex') SI = S.ideal(4*RI*x4+2*x1-2.52*x3-8*RI,3*x4+2*x1-3.5*x2-6,-4*RI*x8+2*x5-2.52*x7+8*RI,3*x8+2*x5-3.5*x6-6,x1*x5-1,x2*x6-1,x3*x7-1,x4*x8-1) The last line ends up with an error: TypeError: unsupported operand parent(s) for '*': 'Real Field with 53 bits of precision' and 'Multivariate Polynomial Ring in x1, x2, x3, x4, x5, x6, x7, x8 over Number Field in I with defining polynomial x^2 + 1' I tried the other two methods discussed in this thread and I am obtaining similar type errors. I am not sure why this example is causing this error. Is there any solution to this error? Any help is well appreciated. Best Regards, On Monday, February 17, 2014 3:47:24 PM UTC-5, john_perry_usm wrote: The actual computation I had in mind requires a somewhat more convoluted: sage: R = QQ[sqrt(-1)] sage: RI = R.gens()[0] # necessary, since Sage's I is symbolic, and causes issues sage: S.x,y = PolynomialRing(R,order='lex') sage: SI = S.ideal((1+RI)*x+y,x+(1-RI)*y-(1-RI)) sage: SI.groebner_basis() [x + (-I + 1), y - 2] Sorry for my earlier imprecision: when I did it the first time, I didn't realize that I was defined as something other than sqrt(-1) in my current session. is the variable x in the first line a dummy one, i.e. has nothing to do with the x in the second line? I wouldn't say it's a dummy; it's used to define the number field. But, it is unrelated to the second x. I don't use CC much myself, but it certainly has its uses (e.g., approximation). john perry On Monday, February 17, 2014 12:39:46 PM UTC-6, sahi...@gmail.com wrote: Thank you, I get the solution by using N.i = NumberField(x^2+1) S.x,y = PolynomialRing(QQ,order='lex') is the variable x in the first line a dummy one, i.e. has nothing to do with the x in the second line? Sorry, I am new to Sage and sometimes I get confused. If CC is not appropriate for this kind of problems we are discussing, for what computational reason can CC be used in sage or any other computer algebra system? Best Regards, On Monday, February 17, 2014 1:08:30 PM UTC-5, luisfe wrote: On Monday, February 17, 2014 6:39:38 PM UTC+1, sahi...@gmail.com wrote: OK, I tried the following: S.i,x,y = PolynomialRing(QQ,order='lex') I = ideal(i^2+1,(1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() G would give me [i - x - 1, x^2 + 2*x + 2, y - 2] which are the results. But I am confused; why I can't get the result when I try to get a polynomial ring in the field of complex numbers implemented by sage? Also, does adding i**2+1=0 really extend the rational numbers to complex number field? The problem with CC is that it is an *inexact field. *If you do computations with coefficients in CC, you will end up with roundup errors. For instance, buchberger algorithm to compute Grobner basis would yield the ideal (1) with high probability. In the case, you are not computing on the complex numbers, only on the gaussian rationals. Essentially, you are working on QQ[i] without naming it. In this case your solutions live on QQ[i] so it is not a problem. Consider the following example: system x^2+i+y^3, y^4-x sage: S.i,x,y=PolynomialRing(QQ,order='lex') sage: I=Ideal(x^2+i+y^3, y^4-x) sage: I.groebner_basis() [i + y^8 + y^3, x - y^4] Then, y is any of the 8 roots of the polynomial *'i + y^8 + y^3*', and for each one of these roots, *x=y^4*. So you get 8 pairs (x,y) of solutions. By the way, the suggestion given by John Perry is to do: sage: N.i = NumberField(x^2+1) sage: S.x,y=PolynomialRing(QQ,order='lex') if you do this, then sage: i^2 -1 you are really working on QQ[i] -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-support] Re: groebner bases of polynomials with complex coefficients
On 18/02/14 14:31, sahin...@gmail.com wrote: TypeError: unsupported operand parent(s) for '*': 'Real Field with 53 bits of precision' and 'Multivariate Polynomial Ring in x1, x2, x3, x4, x5, x6, x7, x8 over Number Field in I with defining polynomial x^2 + 1' This means that you are mixing elements from * the 'Real Field with 53 bits of precision' and * 'Multivariate Polynomial Ring in x1, x2, x3, x4, x5, x6, x7, x8 over Number Field in I with defining polynomial x^2 + 1' which you shouldn't do. In particular: 3.5*x2 is your problem. Replace 3.5 by 7/2 and you are good to go. Cheers, Martin PS: Those error messages are meant to be helpful regardless of how scary they might look :) You can also try to reduce your input in order to figure out which part exactly triggers the error. signature.asc Description: OpenPGP digital signature
Re: [sage-support] Re: groebner bases of polynomials with complex coefficients
I changed the decimals into fractions and I confirm that I am now getting results. Thank you so much for your help. You saved me out of stress and depression. Best Regards, On Tuesday, February 18, 2014 11:23:37 AM UTC-5, Martin R. Albrecht wrote: On 18/02/14 14:31, sahi...@gmail.com javascript: wrote: TypeError: unsupported operand parent(s) for '*': 'Real Field with 53 bits of precision' and 'Multivariate Polynomial Ring in x1, x2, x3, x4, x5, x6, x7, x8 over Number Field in I with defining polynomial x^2 + 1' This means that you are mixing elements from * the 'Real Field with 53 bits of precision' and * 'Multivariate Polynomial Ring in x1, x2, x3, x4, x5, x6, x7, x8 over Number Field in I with defining polynomial x^2 + 1' which you shouldn't do. In particular: 3.5*x2 is your problem. Replace 3.5 by 7/2 and you are good to go. Cheers, Martin PS: Those error messages are meant to be helpful regardless of how scary they might look :) You can also try to reduce your input in order to figure out which part exactly triggers the error. -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: groebner bases of polynomials with complex coefficients
Instead of CC, try using QQ[i]. That works for me, giving the basis [x + 4/25, y - 24/25] john perry On Monday, February 17, 2014 10:37:30 AM UTC-6, sahi...@gmail.com wrote: Hi: I am trying to obtain solution of a system of polynomial equations with complex coefficients without success. For example, when I try S.x,y = PolynomialRing(CC,order='lex') I = ideal((1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() I see this error: AttributeError: 'Ideal_generic' object has no attribute 'groebner_basis' I tried another ideal and I am getting the same error. What am I doing wrong? Any help would be well appreciated. Best Regards, -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: groebner bases of polynomials with complex coefficients
ACK! Make sure I=sqrt(-1) first! john perry On Monday, February 17, 2014 10:37:30 AM UTC-6, sahi...@gmail.com wrote: Hi: I am trying to obtain solution of a system of polynomial equations with complex coefficients without success. For example, when I try S.x,y = PolynomialRing(CC,order='lex') I = ideal((1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() I see this error: AttributeError: 'Ideal_generic' object has no attribute 'groebner_basis' I tried another ideal and I am getting the same error. What am I doing wrong? Any help would be well appreciated. Best Regards, -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: groebner bases of polynomials with complex coefficients
OK, I tried the following: S.i,x,y = PolynomialRing(QQ,order='lex') I = ideal(i^2+1,(1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() G would give me [i - x - 1, x^2 + 2*x + 2, y - 2] which are the results. But I am confused; why I can't get the result when I try to get a polynomial ring in the field of complex numbers implemented by sage? Also, does adding i**2+1=0 really extend the rational numbers to complex number field? Best Regards, On Monday, February 17, 2014 12:14:25 PM UTC-5, john_perry_usm wrote: ACK! Make sure I=sqrt(-1) first! john perry On Monday, February 17, 2014 10:37:30 AM UTC-6, sahi...@gmail.com wrote: Hi: I am trying to obtain solution of a system of polynomial equations with complex coefficients without success. For example, when I try S.x,y = PolynomialRing(CC,order='lex') I = ideal((1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() I see this error: AttributeError: 'Ideal_generic' object has no attribute 'groebner_basis' I tried another ideal and I am getting the same error. What am I doing wrong? Any help would be well appreciated. Best Regards, -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: groebner bases of polynomials with complex coefficients
On Monday, February 17, 2014 6:39:38 PM UTC+1, sahi...@gmail.com wrote: OK, I tried the following: S.i,x,y = PolynomialRing(QQ,order='lex') I = ideal(i^2+1,(1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() G would give me [i - x - 1, x^2 + 2*x + 2, y - 2] which are the results. But I am confused; why I can't get the result when I try to get a polynomial ring in the field of complex numbers implemented by sage? Also, does adding i**2+1=0 really extend the rational numbers to complex number field? The problem with CC is that it is an *inexact field. *If you do computations with coefficients in CC, you will end up with roundup errors. For instance, buchberger algorithm to compute Grobner basis would yield the ideal (1) with high probability. In the case, you are not computing on the complex numbers, only on the gaussian rationals. Essentially, you are working on QQ[i] without naming it. In this case your solutions live on QQ[i] so it is not a problem. Consider the following example: system x^2+i+y^3, y^4-x sage: S.i,x,y=PolynomialRing(QQ,order='lex') sage: I=Ideal(x^2+i+y^3, y^4-x) sage: I.groebner_basis() [i + y^8 + y^3, x - y^4] Then, y is any of the 8 roots of the polynomial *'i + y^8 + y^3*', and for each one of these roots, *x=y^4*. So you get 8 pairs (x,y) of solutions. By the way, the suggestion given by John Perry is to do: sage: N.i = NumberField(x^2+1) sage: S.x,y=PolynomialRing(QQ,order='lex') if you do this, then sage: i^2 -1 you are really working on QQ[i] -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: groebner bases of polynomials with complex coefficients
Thank you, I get the solution by using N.i = NumberField(x^2+1) S.x,y = PolynomialRing(QQ,order='lex') is the variable x in the first line a dummy one, i.e. has nothing to do with the x in the second line? Sorry, I am new to Sage and sometimes I get confused. If CC is not appropriate for this kind of problems we are discussing, for what computational reason can CC be used in sage or any other computer algebra system? Best Regards, On Monday, February 17, 2014 1:08:30 PM UTC-5, luisfe wrote: On Monday, February 17, 2014 6:39:38 PM UTC+1, sahi...@gmail.com wrote: OK, I tried the following: S.i,x,y = PolynomialRing(QQ,order='lex') I = ideal(i^2+1,(1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() G would give me [i - x - 1, x^2 + 2*x + 2, y - 2] which are the results. But I am confused; why I can't get the result when I try to get a polynomial ring in the field of complex numbers implemented by sage? Also, does adding i**2+1=0 really extend the rational numbers to complex number field? The problem with CC is that it is an *inexact field. *If you do computations with coefficients in CC, you will end up with roundup errors. For instance, buchberger algorithm to compute Grobner basis would yield the ideal (1) with high probability. In the case, you are not computing on the complex numbers, only on the gaussian rationals. Essentially, you are working on QQ[i] without naming it. In this case your solutions live on QQ[i] so it is not a problem. Consider the following example: system x^2+i+y^3, y^4-x sage: S.i,x,y=PolynomialRing(QQ,order='lex') sage: I=Ideal(x^2+i+y^3, y^4-x) sage: I.groebner_basis() [i + y^8 + y^3, x - y^4] Then, y is any of the 8 roots of the polynomial *'i + y^8 + y^3*', and for each one of these roots, *x=y^4*. So you get 8 pairs (x,y) of solutions. By the way, the suggestion given by John Perry is to do: sage: N.i = NumberField(x^2+1) sage: S.x,y=PolynomialRing(QQ,order='lex') if you do this, then sage: i^2 -1 you are really working on QQ[i] -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: groebner bases of polynomials with complex coefficients
The actual computation I had in mind requires a somewhat more convoluted: sage: R = QQ[sqrt(-1)] sage: RI = R.gens()[0] # necessary, since Sage's I is symbolic, and causes issues sage: S.x,y = PolynomialRing(R,order='lex') sage: SI = S.ideal((1+RI)*x+y,x+(1-RI)*y-(1-RI)) sage: SI.groebner_basis() [x + (-I + 1), y - 2] Sorry for my earlier imprecision: when I did it the first time, I didn't realize that I was defined as something other than sqrt(-1) in my current session. is the variable x in the first line a dummy one, i.e. has nothing to do with the x in the second line? I wouldn't say it's a dummy; it's used to define the number field. But, it is unrelated to the second x. I don't use CC much myself, but it certainly has its uses (e.g., approximation). john perry On Monday, February 17, 2014 12:39:46 PM UTC-6, sahi...@gmail.com wrote: Thank you, I get the solution by using N.i = NumberField(x^2+1) S.x,y = PolynomialRing(QQ,order='lex') is the variable x in the first line a dummy one, i.e. has nothing to do with the x in the second line? Sorry, I am new to Sage and sometimes I get confused. If CC is not appropriate for this kind of problems we are discussing, for what computational reason can CC be used in sage or any other computer algebra system? Best Regards, On Monday, February 17, 2014 1:08:30 PM UTC-5, luisfe wrote: On Monday, February 17, 2014 6:39:38 PM UTC+1, sahi...@gmail.com wrote: OK, I tried the following: S.i,x,y = PolynomialRing(QQ,order='lex') I = ideal(i^2+1,(1+i)*x+y,x+(1-i)*y-(1-i)) G = I.groebner_basis() G would give me [i - x - 1, x^2 + 2*x + 2, y - 2] which are the results. But I am confused; why I can't get the result when I try to get a polynomial ring in the field of complex numbers implemented by sage? Also, does adding i**2+1=0 really extend the rational numbers to complex number field? The problem with CC is that it is an *inexact field. *If you do computations with coefficients in CC, you will end up with roundup errors. For instance, buchberger algorithm to compute Grobner basis would yield the ideal (1) with high probability. In the case, you are not computing on the complex numbers, only on the gaussian rationals. Essentially, you are working on QQ[i] without naming it. In this case your solutions live on QQ[i] so it is not a problem. Consider the following example: system x^2+i+y^3, y^4-x sage: S.i,x,y=PolynomialRing(QQ,order='lex') sage: I=Ideal(x^2+i+y^3, y^4-x) sage: I.groebner_basis() [i + y^8 + y^3, x - y^4] Then, y is any of the 8 roots of the polynomial *'i + y^8 + y^3*', and for each one of these roots, *x=y^4*. So you get 8 pairs (x,y) of solutions. By the way, the suggestion given by John Perry is to do: sage: N.i = NumberField(x^2+1) sage: S.x,y=PolynomialRing(QQ,order='lex') if you do this, then sage: i^2 -1 you are really working on QQ[i] -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
