October 2002 - PURRS-devel - BUGSENG Mailman

purrs/src Expr.defs.hh Expr.inlines.hh
by Tatiana Zolo 11 Oct '02

11 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zolo(a)cs.unipr.it 2002-10-11 12:50:35 Modified files: src : Expr.defs.hh Expr.inlines.hh Log message: Added function `Expr factorial(const Expr& x)'. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/Expr.defs.hh.diff?cvsro… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/Expr.inlines.hh.diff?cv…

1 0

purrs/src alg_eq_solver.cc
by Tatiana Zolo 11 Oct '02

11 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zolo(a)cs.unipr.it 2002-10-11 12:00:54 Modified files: src : alg_eq_solver.cc Log message: Changed `int' to `unsigned' because `degree()' returns an unsigned. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/alg_eq_solver.cc.diff?c…

1 0

purrs/src Expr.defs.hh Expr.inlines.hh
by Tatiana Zolo 11 Oct '02

11 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zolo(a)cs.unipr.it 2002-10-11 11:58:55 Modified files: src : Expr.defs.hh Expr.inlines.hh Log message: Added constructors for integer expressions. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/Expr.defs.hh.diff?cvsro… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/Expr.inlines.hh.diff?cv…

1 0

[Fwd: Re: Extended arithmetic]
by Tatiana Zolo 11 Oct '02

11 Oct '02

-------- Original Message -------- Subject: Re: Extended arithmetic Date: Fri, 11 Oct 2002 09:48:56 +0200 From: Markus Neher <markus.neher(a)math.uni-karlsruhe.de> To: Tatiana Zolo <zolo(a)cs.unipr.it> References: <3CD8FA8A.97151456(a)cs.unipr.it> Dear Tatiana, I just ran across an old e-mail of yours, dating back to May 8. If you're still interested in an answer (I think I have not sent an answer so far), here are a few thoughts: > 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. I see your point, but I'm afraid there is a basic problem for which I have no good answer. There are three possible solutions to your problem, but each gets you in trouble. 1. As we did in CoStLy, you can restrict function values to principal values. Then log, sqrt and other functions are not defined everywhere in the complex plane. E.g., log(Z)is not defined for Z = [-2,-1] + i[-1,1]. This is not what you want to have. 2. Evaluating a standard function, you can choose some arbitrary branch. If you do this for log on the negative real axis, you can evaluate log(Z) for the above example, but log(Z) is then a union of two intervals. Thus, you have to write your function library in such a way, that you can enter a list of argument interval and get another list of intervals of function values, like (W_1, W_2, W_3, W_4, W_5) = log( Z_1, Z_2, Z_3). If you evaluate a composite function like log(Z) - sqrt(Z+1/Z), you have to add lists of intervals, which finally may produce a large list of intervals that contains few "true" solutions and many "ghost" solutions. 3. You can compute sets of all solutions instead of principal function values. Obviously, this won't work for log(log(Z)), as I pointed out in a former e-mail. If you try it for sqrt, it may work, but there are even more ghost solutions than with item 2. Here, you do not only have multiple solutions on the negative real axis, but everywhere in the complex plane. Consider your example: > 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 you compute two roots for each "sqrt", then the total evaluation of your expression yields dozens of solutions. I don't see a simple method to identify the 4 "true" roots of your 4th order polynomial among the ghost solutions. If you have a solution to this problem, I'll be interested to hear about it. Best regards Markus

1 0

Re: Now seriously: what is a polynomial?
by Roberto Bagnara 10 Oct '02

10 Oct '02

Christian Bauer wrote: >>The point here is to decide _exactly_ for which class of expressions >>are the functions degree(), ldegree(), coeff(), lcoeff(), tcoeff(), >>expand(), collect(), quo(), rem(), prem(), gcd(), lcm(), sqrfree() >>and so forth guaranteed to work. > > > An exact specification of the maximal classes of expressions that these > functions can operate on and the exact results they produce would require > more work than I'm willing to invest (that's partly because GiNaC was > designed for a bunch of physicists to "get some work done", not as a > collection of rigorously defined functions). > > I'd be more comfortable if you could propose a set of specifications for > these functions (for the cases you need), where I can say "Yes, this will > work in all future versions of GiNaC", even if these are not the 'maximal' > specifications. Here is a formal definition of what is, for our application, a polynomial in `x'. The first (auxiliary) function defines scalars for `x', the other function defines the real thing. Notice that this is written referring to the layer of software we have put on top of GiNaC, but the concepts should be easily recognizable anyway. //! \brief //! Returns <CODE>true</CODE> if \p e is a scalar rapresentation for \p x; //! returns <CODE>false</CODE> otherwise. /*! This function realizes the definition of scalar representation for \f$ x \f$, where \f$ x \f$ is any symbol. This is more briefly written scalar and defined inductively as follows: - every number is a scalar; - every symbolic constant is a scalar; - every parameter different from \f$ x \f$ is a scalar; - if \f$ f \f$ is any unary function and \f$ x \f$ is a scalar representation, then \f$ f(x) \f$ is a scalar; - if \f$ a \f$ and \f$ b \f$ are scalars then \f$ a+b \f$, \f$ a*b \f$, and \f$ a^b \f$ are scalars. */ bool is_scalar_representation(const Expr& e, const Symbol& x) { if (e.is_a_number()) return true; else if (e.is_a_constant()) return true; else if (e.is_a_symbol() && !e.is_equal(x)) return true; else if (e.is_a_power()) return is_scalar_representation(e.op(0), x) && is_scalar_representation(e.op(1), x); else if (e.is_a_function()) return is_scalar_representation(e.op(0), x); else if (e.is_a_add() || e.is_a_mul()) { for (unsigned i = e.nops(); i-- > 0; ) if (!is_scalar_representation(e.op(i), x)) return false; return true; } return false; } //! \brief //! Returns <CODE>true</CODE> if \p e is a polynomial in \p x; //! returns <CODE>false</CODE> otherwise. /*! This function realizes the definition of polynomial in \f$ x \f$, where \f$ x \f$ is any symbol. This is more briefly written polynomial and defined inductively as follows: - every scalar representation for \f$ x \f$ is a polynomial; - \f$ x \f$ is a polynomial; - if \f$ a \f$ is a polynomial in \f$ x \f$ and \f$ b \f$ is a positive integer, then \f$ a^b \f$ is a polynomial; - if \f$ a \f$ and \f$ b \f$ are polynomials then \f$ a + b \f$ and \f$ a * b \f$ are polynomials. */ bool is_polynomial(const Expr& e, const Symbol& x) { if (is_scalar_representation(e, x)) return true; else if (e.is_equal(x)) return true; else if (e.is_a_power()) { if (is_polynomial(e.op(0), x)) if (e.op(1).is_a_number()) { Number exponent = e.op(1).ex_to_number(); if (exponent.is_positive_integer()) return true; } } else if (e.is_a_add() || e.is_a_mul()) { for (unsigned i = e.nops(); i-- > 0; ) if (!is_polynomial(e.op(i), x)) return false; return true; } return false; } -- Prof. Roberto Bagnara Computer Science Group Department of Mathematics, University of Parma, Italy http://www.cs.unipr.it/~bagnara/ mailto:bagnara@cs.unipr.it

1 0

purrs/tests rrs_driver.cc
by Roberto Bagnara 10 Oct '02

10 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: roberto(a)cs.unipr.it 2002-10-10 19:50:20 Modified files: tests : rrs_driver.cc Log message: Accepts options so as to serve both as a debugging tool and as a test program for `make check'. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/tests/rrs_driver.cc.diff?cv…

1 0

purrs/tests Makefile.am ehandlers.hh ehandlers.cc
by Roberto Bagnara 10 Oct '02

10 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: roberto(a)cs.unipr.it 2002-10-10 19:49:16 Modified files: tests : Makefile.am Added files: tests : ehandlers.hh ehandlers.cc Log message: New installer of exception handlers useful for debugging purposes. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/tests/ehandlers.hh?cvsroot=… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/tests/ehandlers.cc?cvsroot=… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/tests/Makefile.am.diff?cvsr…

1 0

purrs/src gosper.cc sum_poly.cc util.cc
by Tatiana Zolo 10 Oct '02

10 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: zolo(a)cs.unipr.it 2002-10-10 19:16:41 Modified files: src : gosper.cc sum_poly.cc util.cc Log message: Restored types `unsigned' (and not `int') because now degree() and ldegree() return an `unsigned'. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/gosper.cc.diff?cvsroot=… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/sum_poly.cc.diff?cvsroo… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/util.cc.diff?cvsroot=pu…

1 0

purrs/tests Makefile.am
by Roberto Bagnara 10 Oct '02

10 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: roberto(a)cs.unipr.it 2002-10-10 19:08:21 Modified files: tests : Makefile.am Log message: Always compile the drivers, not only for `make check'. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/tests/Makefile.am.diff?cvsr…

1 0

purrs/src Expr.defs.hh Expr.inlines.hh
by Roberto Bagnara 10 Oct '02

10 Oct '02

CVSROOT: /cvs/purrs Module name: purrs Changes by: roberto(a)cs.unipr.it 2002-10-10 19:03:23 Modified files: src : Expr.defs.hh Expr.inlines.hh Log message: Now Expr::degree() and Expr::ldegree() return an unsigned. Patches: http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/Expr.defs.hh.diff?cvsro… http://www.cs.unipr.it/cgi-bin/cvsweb.cgi/purrs/src/Expr.inlines.hh.diff?cv…

1 0