May 2002 - PURRS-devel - BUGSENG Mailman

purrs/src alg_eq_solver.cc
by Tatiana Zolo 10 May '02

10 May '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zolo(a)cs.unipr.it 2002-05-10 10:03:41 Modified files: src : alg_eq_solver.cc Log message: Rewrote the function solve_equation_4, because the old one gave wrong answers (exchanged imaginary parts between couples of solutions in some, but not all, cases). Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/alg_eq_solver.cc.diff?c…

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function has
by Tatiana Zolo 09 May '02

09 May '02

Hi, the behavior of function 'has' in ginsh seems inexplicable: ginsh - GiNaC Interactive Shell (GiNaC V1.0.8) __, _______ Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, (__) * | Germany. This is free software with ABSOLUTELY NO WARRANTY. ._) i N a C | You are welcome to redistribute it under certain conditions. <-------------' For details type `warranty;'. Type ?? for a list of help topics. > has(b*x*x^a, $0*$1*$1^$2); 1 > has(c*x*x^a, $0*$1*$1^$2); 1 > has(d*x*x^a, $0*$1*$1^$2); 1 > has(e*x*x^a, $0*$1*$1^$2); 0 > has(f*x*x^a, $0*$1*$1^$2); 1 > has(g*x*x^a, $0*$1*$1^$2); 1 > has(h*x*x^a, $0*$1*$1^$2); 1 > has(i*x*x^a, $0*$1*$1^$2); 0 > has(l*x*x^a, $0*$1*$1^$2); 1 > exit; Then I cosed and opened again ginsh and ... Type ?? for a list of help topics. > has(e*x*x^a, $0*$1*$1^$2); 1 > has(f*x*x^a, $0*$1*$1^$2); 1 > has(g*x*x^a, $0*$1*$1^$2); 1 > has(h*x*x^a, $0*$1*$1^$2); 0 > has(i*x*x^a, $0*$1*$1^$2); 1 > has(l*x*x^a, $0*$1*$1^$2); 1 > has(m*x*x^a, $0*$1*$1^$2); 1 > has(n*x*x^a, $0*$1*$1^$2); 0 > has(o*x*x^a, $0*$1*$1^$2); 1 > exit; [zolo@sandbox zolo]$ ginsh ginsh - GiNaC Interactive Shell (GiNaC V1.0.8) __, _______ Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, (__) * | Germany. This is free software with ABSOLUTELY NO WARRANTY. ._) i N a C | You are welcome to redistribute it under certain conditions. <-------------' For details type `warranty;'. Type ?? for a list of help topics. > has(h*x*x^a, $0*$1*$1^$2); 1 > has(i*x*x^a, $0*$1*$1^$2); 1 > has(l*x*x^a, $0*$1*$1^$2); 1 > has(m*x*x^a, $0*$1*$1^$2); 0 > has(n*x*x^a, $0*$1*$1^$2); 1 > Furthermore, with the number it says always false, is it right? > has(2*x*x^a, $0*$1*$1^$2); 0 > has((-2)*x*x^a, $0*$1*$1^$2); 0 > has(sqrt(2)*x*x^a, $0*$1*$1^$2); 0 Thank you very much. Best regards Tatiana Zolo

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method 'subs()'
by Tatiana Zolo 08 May '02

08 May '02

Hi all, I don't understand the behavior of method 'subs()' in the following case: > subs(a^x*b^x, $1^$2*$3^$2==($1*$3)^$2); (a*b)^x > subs(a^5*b^5, $1^$2*$3^$2==($1*$3)^$2); b^5*a^5 I also check it with a little program, that I attach, and the output is Expression 1: b^x*a^x List of match: {$1==b,$2==x,$3==a} Expression 1 after subs: (b*a)^x Expression 2: b^5*a^5 List of match: {$1==b,$2==5,$3==a} Expression 2 after subs: b^5*a^5 Why is different the behavior of subs() when it is applied on b^x*a^x or on b^5*a^5? Best regards Tatiana Zolo #include <iostream> #include <ginac/ginac.h> using namespace std; using namespace GiNaC; typedef GiNaC::ex GExpr; typedef GiNaC::symbol GSymbol; typedef GiNaC::numeric GNumber; typedef GiNaC::lst GList; int main() { GSymbol x("x"); GSymbol a("a"); GSymbol b("b"); GList l_1, l_2; GExpr p = pow(a, x) * pow(b, x); cout << endl << "Expression 1: " << p << endl; match(p, pow(wild(1), wild(2)) * pow(wild(3), wild(2)), l_1); cout << "List of match: " << l_1 << endl; p = p.subs(pow(wild(1), wild(2)) * pow(wild(3), wild(2)) == pow(wild(1)*wild(3), wild(2))); cout << "Expression 1 after subs: " << p << endl << endl; GExpr q = pow(a, 5) * pow(b, 5); cout << "Expression 2: " << q << endl; match(q, pow(wild(1), wild(2)) * pow(wild(3), wild(2)), l_2); cout << "List of match: " << l_2 << endl; q = q.subs(pow(wild(1), wild(2)) * pow(wild(3), wild(2)) == pow(wild(1)*wild(3), wild(2))); cout << "Expression 2 after subs: " << q << endl << endl; return 0; }

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Extended arithmetic
by Tatiana Zolo 08 May '02

08 May '02

Dear Markus, thank you for your availability. We are studying recurrence relations and, consequently, the polynomial equations that derive from them. For instance: x_n = -3*x_{n-1} + x_{n-4} -> x^4 + 3*x^3 - 1 = 0 x_n = -3 * x_{n-10} -> x^10 + 3 = 0 For the polynomial equations of degree 2, 3 or 4 there is a formula (which may involve the computation of square roots, or root of higher index) but for the polynomial equations of degree 5 or more the general formula does not exist and then we only want an approximation of the roots. In conclusion, we always have to deal with square roots or roots with higher index, and the elementary functions sin, cos: we need to evaluate them at points of complex plane which may be everywhere, including the negative real axis. At the moment we want to find a (possibly complex) interval that contains the solution of any polynomial equation (to consider polynomial equation will be probably only a first step of our work...) ex: we consider again x^4 + 3*x^3 - 1 = 0. We find the solution x_1 = -3/4+1/4*sqrt(9+4*(-9/2+sqrt(2443/108))^(1/3)-4* *(9/2+sqrt(2443/108))^(1/3))+1/2*sqrt((3/2-1/2*sqrt(9+4*(-9/2+ +sqrt(2443/108))^(1/3)-4*(9/2+sqrt(2443/108))^(1/3)))^2-2*sqrt(4+ +((-9/2+sqrt(2443/108))^(1/3)-(9/2+sqrt(2443/108))^(1/3))^2)-2* *(-9/2+sqrt(2443/108))^(1/3)+2*(9/2+sqrt(2443/108))^(1/3)) If we estimate this solution we get x_1 = -0.3068848065239627712+0.64263790881274810124*I and, using your library, we can not find the interval because it throws an exception. Thank you. Best regards Tatiana Zolo

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purrs/doc purrs.tex
by Alessandro Zaccagnini 07 May '02

07 May '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zaccagnini(a)cs.unipr.it 2002-05-07 10:54:31 Modified files: doc : purrs.tex Log message: 1. Slighly improved use of LaTeX features; 2. Added preliminary section on containment set for complex intervals; 3. Checked spelling: corrected some misprints; 4. Added preliminary section on normal form for irrational numbers. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/doc/purrs.tex.diff?cvsroot=…

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purrs/src rr_solver.cc
by Tatiana Zolo 07 May '02

07 May '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zolo(a)cs.unipr.it 2002-05-07 10:28:29 Modified files: src : rr_solver.cc Log message: Changes in according to the bug fixed by GiNaC 1.0.6. about 'subs'. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/rr_solver.cc.diff?cvsro…

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filib-CoStLy
by Tatiana Zolo 02 May '02

02 May '02

Dear Sirs, we understand your problem to preserve the Brouwer Fixed Point Theorem but we want explain to you our requirements. Our aim is to use your libraries about real and complex intervals also for our needs. Our applications are based on idea of sound approximation. Suppose we have two partially known real values a and b and we want to approximate their ratio. If a is approximated by the intervals I(a) and b by the interval I(b), we want to approximate the ratio I(a)/I(b). The semantic of ratio's operation in which we are interested is: if b is not zero then a/b is approximated by I(a)/I(b) (we specify that into our applications the condition 'if b is not zero' it is always guaranteed by others tools). If we have understood correctly the ratio's operation among intervals defined in your libraries, it throws an exception when I(b) contains zero. Emphasizing the fact that we would not like to develop another library for the manipulation of real and complex intervals, we would like to know how we can obtain what we are interested in from the behavior given by filib and CoStLy. In the case of the ratio, that is one of the simpler cases, we could split the interval I(b) in two parts: [inf I(b), 0) and (0, sup I(b)], but this would be possible only if filib and CoStLy support both closed and open intervals: this seems not to be possible with your libraries. We would like your advice: how would you suggest to achieve our goal? Do you think that we could easily obtain what we need from filib and CoStLy, or do you suggest to use another library? We are novices to the field of intervals and we need your expert opinion. Thank you very much in advance, Roberto Bagnara Alessandro Zaccagnini Tatiana Zolo

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