Re: [fricas-devel] real? sqrt (-sqrt 2)
On Wed, Nov 22, 2023 at 08:29:11AM +0800, Qian Yun wrote:
> Now I understand most of your points.
>
> So what are the possible improvements regarding algebraic number
> in the future?
I do not think about deep changes to AlgebraicNumber or Expression,
at least in reasonably near future. Rather, I think that extra
functionality should go into separate packages.
> For example, in a "locally breaking symmetry" use case, a "real?"
> function to determine if an algebraic number is real, is that useful?
> (e.g. (sqrt(3+4*sqrt(-1))+sqrt(3-4*sqrt(-1
Recognizing that the above is real would be useful and does not
look very hard: it is symmetric in 'sqrt(-1)', so invariant
under complex conjugations (that is assuming that branches are
choosen in symmetric way). Assuming principal branch one
could even deduce that it is positive (real part of principal
branch square root is nonnegative).
Certainly I would like to have stronger methods of determining sign.
CAD may help, but is quite heavy. Some folks suggested splitting
(writing) expressions as sums of squares. In 2008 Strzeboński
proposed a simple method for functions build from 'x' and 'exp(x)'
(it generalizes Sturm sequences).
> Also, there's no function to transform a radical ('nthRoot) to
> 'rootOf currently, right?
No specific function, but 'subst' can tranform any expression
given mapping on kernels. And for kernels transformation is
essentially trivial.
--
Waldek Hebisch
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Re: [fricas-devel] real? sqrt (-sqrt 2)
Now I understand most of your points.
So what are the possible improvements regarding algebraic number
in the future?
For example, in a "locally breaking symmetry" use case, a "real?"
function to determine if an algebraic number is real, is that useful?
(e.g. (sqrt(3+4*sqrt(-1))+sqrt(3-4*sqrt(-1
Also, there's no function to transform a radical ('nthRoot) to
'rootOf currently, right?
- Qian
On 11/21/23 23:39, Waldek Hebisch wrote:
On Tue, Nov 21, 2023 at 08:38:49PM +0800, Qian Yun wrote:
On 11/21/23 20:26, Waldek Hebisch wrote:
On Tue, Nov 21, 2023 at 06:35:57PM +0800, Qian Yun wrote:
If so, I want to confirm that is 'nthRoot already a kind of
algebraic number?
I am not sure what you want to say here: 'nthRoot is in
AlgebraicNumber which under right conditions models
algebraic numbers and in Expression, where we have
things more general than numbers.
OK, to give a specific example, what's the relationship
between sqrt(2) (a 'nthRoot) and %x0 (a 'rootOf):
(1) -> sqrt(2)
+-+
(1) \|2
(2) -> rootsOf(x^2-2)
(2) [%x0, - %x0]
Using them together is user error. Argument to rootOf is supposed
to be irreducible polynomial. Once you add 'sqrt(2)' polynomial
'x^2 - 2' factors, so considering both togehter is user error.
Both could be used to represent "the same" element of AlgebraicNumber.
To put it differently, you should be able to replace one by the
other in all places and get correct result. Due to the way
in which AlgebraicNumber works, if you get one as input you
should return the same in the output. OTOH, internally one
can be replaced by the other if computation is more convenient
with different representation. Of course, after computation
is done we need to do backsubstitution, so that kernels
are preserved in the output.
Ralf's code allows you to have both in single computation
and either you have sqrt(2) = rootsOf(x^2-2) or
sqrt(2) = -rootsOf(x^2-2). Which of the two equalites holds
is depends on implementation in a way which for practical
purposes should be considered random.
Note that '>' (order) is not defined for AlgebraicNumber,
so it makes no sense to ask if "sqrt(2) > 0".
To put it differently, Galois theory says that at algebraic
level role of 'sqrt(2)' and '-sqrt(2)' is entirely symmetric.
You (and lot ot other folks) want to break this symmetry.
IMO breaking symmetry for AlgebraicNumber and for Expression
as domains is undesirable. Simply, it requires a lot of effort
and breaking symmetry is dangerous as it may break algorithms
that depend on symmetry. OTOH, locally we may do some things
that break symmetry, but we need to be careful since rest of
algbera will _not_ cooperate and each place when we break
symmetry need some justitication that it will not introduce
contradictions.
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Re: [fricas-devel] real? sqrt (-sqrt 2)
On Tue, Nov 21, 2023 at 08:38:49PM +0800, Qian Yun wrote: > > > On 11/21/23 20:26, Waldek Hebisch wrote: > > On Tue, Nov 21, 2023 at 06:35:57PM +0800, Qian Yun wrote: > > > > > > If so, I want to confirm that is 'nthRoot already a kind of > > > algebraic number? > > > > I am not sure what you want to say here: 'nthRoot is in > > AlgebraicNumber which under right conditions models > > algebraic numbers and in Expression, where we have > > things more general than numbers. > > > > OK, to give a specific example, what's the relationship > between sqrt(2) (a 'nthRoot) and %x0 (a 'rootOf): > > (1) -> sqrt(2) > > +-+ >(1) \|2 > > (2) -> rootsOf(x^2-2) > >(2) [%x0, - %x0] Using them together is user error. Argument to rootOf is supposed to be irreducible polynomial. Once you add 'sqrt(2)' polynomial 'x^2 - 2' factors, so considering both togehter is user error. Both could be used to represent "the same" element of AlgebraicNumber. To put it differently, you should be able to replace one by the other in all places and get correct result. Due to the way in which AlgebraicNumber works, if you get one as input you should return the same in the output. OTOH, internally one can be replaced by the other if computation is more convenient with different representation. Of course, after computation is done we need to do backsubstitution, so that kernels are preserved in the output. Ralf's code allows you to have both in single computation and either you have sqrt(2) = rootsOf(x^2-2) or sqrt(2) = -rootsOf(x^2-2). Which of the two equalites holds is depends on implementation in a way which for practical purposes should be considered random. Note that '>' (order) is not defined for AlgebraicNumber, so it makes no sense to ask if "sqrt(2) > 0". To put it differently, Galois theory says that at algebraic level role of 'sqrt(2)' and '-sqrt(2)' is entirely symmetric. You (and lot ot other folks) want to break this symmetry. IMO breaking symmetry for AlgebraicNumber and for Expression as domains is undesirable. Simply, it requires a lot of effort and breaking symmetry is dangerous as it may break algorithms that depend on symmetry. OTOH, locally we may do some things that break symmetry, but we need to be careful since rest of algbera will _not_ cooperate and each place when we break symmetry need some justitication that it will not introduce contradictions. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVzPHgJaD7axtsuL%40fricas.org.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On 11/21/23 20:26, Waldek Hebisch wrote: On Tue, Nov 21, 2023 at 06:35:57PM +0800, Qian Yun wrote: If so, I want to confirm that is 'nthRoot already a kind of algebraic number? I am not sure what you want to say here: 'nthRoot is in AlgebraicNumber which under right conditions models algebraic numbers and in Expression, where we have things more general than numbers. OK, to give a specific example, what's the relationship between sqrt(2) (a 'nthRoot) and %x0 (a 'rootOf): (1) -> sqrt(2) +-+ (1) \|2 (2) -> rootsOf(x^2-2) (2) [%x0, - %x0] - Qian -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/d76a52b6-3fc9-4e2c-807b-8905386d9e2e%40gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Tue, Nov 21, 2023 at 06:35:57PM +0800, Qian Yun wrote: > > If so, I want to confirm that is 'nthRoot already a kind of > algebraic number? I am not sure what you want to say here: 'nthRoot is in AlgebraicNumber which under right conditions models algebraic numbers and in Expression, where we have things more general than numbers. > 'nthRoot constructs radicals, a subset of algebraic numbers. > And clearly code in different places treats 'nthRoot by > using its real valued branch, different than 'rootOf. Well, part of original motivation for domains and packages was that each domain and package can have its own assumptions. For example one place may assume that square root has nonnegative argument and nonnegative value and different place can make different assumbption. There was hope that modularity (that is domains and packages) will prevent contradictions. This does not work as well as originally hoped for. In integrator we deal with rather general roots and we should not depend on "reality" of expressions for correctness (this is the reason for removing calls to 'real'). We can use "reality" arguments in advisory way, so that result is correct independent of "reality", but if determination of "reality" is correct then we get nicer looking result. AFAICS this applies to basic Rioboo algorithm: result is always a valid antiderivative, but it removes poles and imaginaries only when determination of real parts was correct. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVyiB4uxINmWM7Dl%40fricas.org.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On 11/21/23 10:02, Waldek Hebisch wrote: On Tue, Nov 21, 2023 at 08:24:39AM +0800, Qian Yun wrote: On 11/20/23 20:48, Waldek Hebisch wrote: Extra remark: it should be possible to get much of effect of RealClosure by considering pairs of AlgebraicNumber and floating point approximation. Namely, we can compute minimal polynomial of algebraic number and moderately accurate floating point approximation uniquely determines corresponding exact root. In case when we get reducible polynomials we could use floating point approximations to find right factor. Does this mean the right thing to do is to have two kinds of algebraic numbers? For first kind, its position is not fixed (like current status of 'rootOf), for the other kind, the position is fixed (you described above). Because some computation doesn't care which root is which (for example sum of all roots or a polynomial), while other computation cares. We may have more kinds. Current AlgebraicNumber may run into trouble if you create depenednt roots. Ralf posted code where roots have consistent but unkown "order". That is it avoid troubles due to dependent roots. There is correspondence with roots in a finite field, and relation of roots in finite field and roots in complex numbers is hard to compute. Roots in finite field have rather simple relation to roots in completition of p-adic numbers, to we can get several competing variants of algebraic numbers. If so, I want to confirm that is 'nthRoot already a kind of algebraic number? 'nthRoot constructs radicals, a subset of algebraic numbers. And clearly code in different places treats 'nthRoot by using its real valued branch, different than 'rootOf. - Qian -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/2a4e7b83-0f87-4dbe-a353-90293137f224%40gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Tue, Nov 21, 2023 at 08:24:39AM +0800, Qian Yun wrote: > > > On 11/20/23 20:48, Waldek Hebisch wrote: > > > > Extra remark: it should be possible to get much of effect > > of RealClosure by considering pairs of AlgebraicNumber > > and floating point approximation. Namely, we can compute > > minimal polynomial of algebraic number and moderately > > accurate floating point approximation uniquely determines > > corresponding exact root. In case when we get reducible > > polynomials we could use floating point approximations to > > find right factor. > > > > Does this mean the right thing to do is to have two kinds > of algebraic numbers? For first kind, its position is not > fixed (like current status of 'rootOf), for the other kind, > the position is fixed (you described above). > > Because some computation doesn't care which root is which > (for example sum of all roots or a polynomial), while other > computation cares. We may have more kinds. Current AlgebraicNumber may run into trouble if you create depenednt roots. Ralf posted code where roots have consistent but unkown "order". That is it avoid troubles due to dependent roots. There is correspondence with roots in a finite field, and relation of roots in finite field and roots in complex numbers is hard to compute. Roots in finite field have rather simple relation to roots in completition of p-adic numbers, to we can get several competing variants of algebraic numbers. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVwPqD-OKVnsKEM-%40fricas.org.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On 11/20/23 20:48, Waldek Hebisch wrote: Extra remark: it should be possible to get much of effect of RealClosure by considering pairs of AlgebraicNumber and floating point approximation. Namely, we can compute minimal polynomial of algebraic number and moderately accurate floating point approximation uniquely determines corresponding exact root. In case when we get reducible polynomials we could use floating point approximations to find right factor. Does this mean the right thing to do is to have two kinds of algebraic numbers? For first kind, its position is not fixed (like current status of 'rootOf), for the other kind, the position is fixed (you described above). Because some computation doesn't care which root is which (for example sum of all roots or a polynomial), while other computation cares. - Qian -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/906ea4a1-4f1a-4137-bef4-c6c8993b6de4%40gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Mon, Nov 20, 2023 at 12:48 PM Waldek Hebisch wrote: > > On Mon, Nov 20, 2023 at 06:20:45PM +0800, Qian Yun wrote: > > Answer to myself: some real algebraic number need sqrt(-1) to > > represent them (e.g. some root of degree 3 polynomial). > > So RealClosure can't handle them I guess. > > No, this is not a problem. RealClosure can handle real roots of > polynomials without going into complex domain. sqrt(-1) appears > when one wants expression in radicals, but RealClosure represent > roots in more general way. > > > - Qian > > > > On 11/20/23 07:45, Qian Yun wrote: > > > Does RealClosure work with Expression? (e.g doing integration) > > > Currently we can't do so in FriCAS. Is that possible in theory? > > Currently you can have Expression(RealClosure(Fraction(Integer))): > > (8) -> sqrt(2)$eR > > +-+ >(8) \|2 > Type: Expression(RealClosure(Fraction(Integer))) > (9) -> kernels(sqrt(2)$eR) > >(9) [] >Type: List(Kernel(Expression(RealClosure(Fraction(Integer) > (10) -> sqrt(-1)$eR > > +---+ >(10) \|- 1 > Type: Expression(RealClosure(Fraction(Integer))) > (11) -> kernels(sqrt(-1)$eR) > >+---+ >(11) [\|- 1 ] >Type: List(Kernel(Expression(RealClosure(Fraction(Integer) > (12) -> kernels(sqrt(2 + sqrt(-1)$eR)) > >+--+ >| +---+ >(12) [\|\|- 1 + 2 ] >Type: List(Kernel(Expression(RealClosure(Fraction(Integer) > > (12) indicate possible confusion, mathematically the same thing may be > expressed in different ways leading to troubles with unrecoginized > zeros. > > For integration we need PolynomialFactorizationExplicit and that > is absent for RealClosure. To say the truth, it is not entirely > clear what PolynomialFactorizationExplicit should do for > RealClosure. Interpreting it literally would mean splitting > univariate polynomials into quadratic and linear factors, which > is likely to be quite expensive. After that we would have to > implement multivariate factorization on top of this. I guess > that for multivariate factorization we probably should do > absolute factorization first and then only one variable > splitting. > > Note that methods used by RealClosure in principle could > be generalized to elementary constants. But they will > fail in presence of parameters, that is for general > expressions. > > Extra remark: it should be possible to get much of effect > of RealClosure by considering pairs of AlgebraicNumber > and floating point approximation. Namely, we can compute > minimal polynomial of algebraic number and moderately > accurate floating point approximation uniquely determines > corresponding exact root. In case when we get reducible > polynomials we could use floating point approximations to > find right factor. the corresponding exact root will also be uniquely determined by the signs of k-th order derivatives, for all k, of the minimal polynomial at the root. This is called Thom encoding (cf e.g. https://arxiv.org/pdf/1609.02879.pdf). The advantage is that it still works in a parametrised setting, and that it does not need any approximations. Dima > > -- > Waldek Hebisch > > -- > You received this message because you are subscribed to the Google Groups > "FriCAS - computer algebra system" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to fricas-devel+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/fricas-devel/ZVtVlGr6rtdJqSSX%40fricas.org. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/CAAWYfq0TTzKBrmHFDgWxefOKeg0Prcn1unD-hfwji7_kvAYHZg%40mail.gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Mon, Nov 20, 2023 at 06:20:45PM +0800, Qian Yun wrote: > Answer to myself: some real algebraic number need sqrt(-1) to > represent them (e.g. some root of degree 3 polynomial). > So RealClosure can't handle them I guess. No, this is not a problem. RealClosure can handle real roots of polynomials without going into complex domain. sqrt(-1) appears when one wants expression in radicals, but RealClosure represent roots in more general way. > - Qian > > On 11/20/23 07:45, Qian Yun wrote: > > Does RealClosure work with Expression? (e.g doing integration) > > Currently we can't do so in FriCAS. Is that possible in theory? Currently you can have Expression(RealClosure(Fraction(Integer))): (8) -> sqrt(2)$eR +-+ (8) \|2 Type: Expression(RealClosure(Fraction(Integer))) (9) -> kernels(sqrt(2)$eR) (9) [] Type: List(Kernel(Expression(RealClosure(Fraction(Integer) (10) -> sqrt(-1)$eR +---+ (10) \|- 1 Type: Expression(RealClosure(Fraction(Integer))) (11) -> kernels(sqrt(-1)$eR) +---+ (11) [\|- 1 ] Type: List(Kernel(Expression(RealClosure(Fraction(Integer) (12) -> kernels(sqrt(2 + sqrt(-1)$eR)) +--+ | +---+ (12) [\|\|- 1 + 2 ] Type: List(Kernel(Expression(RealClosure(Fraction(Integer) (12) indicate possible confusion, mathematically the same thing may be expressed in different ways leading to troubles with unrecoginized zeros. For integration we need PolynomialFactorizationExplicit and that is absent for RealClosure. To say the truth, it is not entirely clear what PolynomialFactorizationExplicit should do for RealClosure. Interpreting it literally would mean splitting univariate polynomials into quadratic and linear factors, which is likely to be quite expensive. After that we would have to implement multivariate factorization on top of this. I guess that for multivariate factorization we probably should do absolute factorization first and then only one variable splitting. Note that methods used by RealClosure in principle could be generalized to elementary constants. But they will fail in presence of parameters, that is for general expressions. Extra remark: it should be possible to get much of effect of RealClosure by considering pairs of AlgebraicNumber and floating point approximation. Namely, we can compute minimal polynomial of algebraic number and moderately accurate floating point approximation uniquely determines corresponding exact root. In case when we get reducible polynomials we could use floating point approximations to find right factor. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVtVlGr6rtdJqSSX%40fricas.org.
Re: [fricas-devel] real? sqrt (-sqrt 2)
Answer to myself: some real algebraic number need sqrt(-1) to represent them (e.g. some root of degree 3 polynomial). So RealClosure can't handle them I guess. - Qian On 11/20/23 07:45, Qian Yun wrote: Does RealClosure work with Expression? (e.g doing integration) Currently we can't do so in FriCAS. Is that possible in theory? - Qian On 11/19/23 23:12, Waldek Hebisch wrote: On Sun, Nov 19, 2023 at 10:50:17PM +0800, oldk1331 wrote: Then is there a way to specify the algebraic number which square is 2 and is near +1.41421? (1) -> rC := RealClosure(Fraction(Integer)) (1) RealClosure(Fraction(Integer)) Type: Type (2) -> s2 := sqrt(2)$rC +-+ (2) \|2 Type: RealClosure(Fraction(Integer)) (3) -> approximate(s2, 1/1) 46341 (3) - 32768 Type: Fraction(Integer) (4) -> 2 - (%)^2 4633 (4) - -- 1073741824 Type: Fraction(Integer) (5) -> sqrt(-s2)$rC >> Error detected within library code: no roots ATM roots in Complex(rC) remain unimplemented. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/8daa2af7-4251-495e-bc96-aece11925256%40gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
Does RealClosure work with Expression? (e.g doing integration) Currently we can't do so in FriCAS. Is that possible in theory? - Qian On 11/19/23 23:12, Waldek Hebisch wrote: On Sun, Nov 19, 2023 at 10:50:17PM +0800, oldk1331 wrote: Then is there a way to specify the algebraic number which square is 2 and is near +1.41421? (1) -> rC := RealClosure(Fraction(Integer)) (1) RealClosure(Fraction(Integer)) Type: Type (2) -> s2 := sqrt(2)$rC +-+ (2) \|2 Type: RealClosure(Fraction(Integer)) (3) -> approximate(s2, 1/1) 46341 (3) - 32768 Type: Fraction(Integer) (4) -> 2 - (%)^2 4633 (4) - -- 1073741824 Type: Fraction(Integer) (5) -> sqrt(-s2)$rC >> Error detected within library code: no roots ATM roots in Complex(rC) remain unimplemented. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/4d2e3436-9f51-4f4c-86b2-1eef1f42112f%40gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Sun, Nov 19, 2023 at 10:50:17PM +0800, oldk1331 wrote: > Then is there a way to specify the algebraic number > which square is 2 and is near +1.41421? (1) -> rC := RealClosure(Fraction(Integer)) (1) RealClosure(Fraction(Integer)) Type: Type (2) -> s2 := sqrt(2)$rC +-+ (2) \|2 Type: RealClosure(Fraction(Integer)) (3) -> approximate(s2, 1/1) 46341 (3) - 32768 Type: Fraction(Integer) (4) -> 2 - (%)^2 4633 (4) - -- 1073741824 Type: Fraction(Integer) (5) -> sqrt(-s2)$rC >> Error detected within library code: no roots ATM roots in Complex(rC) remain unimplemented. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVol91cfpkR1hpNu%40fricas.org.
Re: [fricas-devel] real? sqrt (-sqrt 2)
Then is there a way to specify the algebraic number which square is 2 and is near +1.41421? - Qian -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/CAGBJN90C4wJ5mT3ZkBii7viHCQuXKw6ChnhfaWp4%2BrETBFi7nQ%40mail.gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Sun, Nov 19, 2023 at 09:49:10PM +0800, Qian Yun wrote: > > > On 11/19/23 21:36, Waldek Hebisch wrote: > > On Sun, Nov 19, 2023 at 07:58:50PM +0800, Qian Yun wrote: > > > "real? sqrt (-sqrt 2)" returns "true" by default. > > > > You mean should return false? We get this when working > > with expressions: > > > > (2) -> real? sqrt (-sqrt 2::EXPR(INT)) > > > > (2) false > > > > > Because interpreter prefers CTRIGMNP over TRIGMNIP > > > as shown by ")set mes bot on": > > > > > > Function Selection for real? > > >Arguments: AN > > > > > > [1] signature: EXPR(COMPLEX(INT)) -> BOOLEAN > > >implemented: slot (Boolean)(Expression (Complex (Integer))) from > > > CTRIGMNP(INT,EXPR(COMPLEX(INT))) > > > [2] signature: EXPR(INT) -> BOOLEAN > > >implemented: slot (Boolean)(Expression (Integer)) from > > > TRIGMNIP(INT,EXPR(INT)) > > > > > > > > > Shall we consider this a bug? > > > > I do not think so. IIUC 'real?' in CTRIGMNP is quite different > > function than 'real?' in TRIGMNIP. Namely, CTRIGMNP deals > > with COMPLEX(EXPR(INT)), which is formal linear combination > > 'a + %i*b' where 'a' and 'b' are expressions and 'real?' > > here tests if 'b' equals 0, which AFAICS is true. > > I don't think 'real?' in CTRIGMNP is wrong. > > I think the interpreter choose EXPR(COMPLEX(INT)) over EXPR(INT) > for AN is wrong. Isn't AN the same as EXPR(INT) in Rep? > So I think the interpreter auto coercion should prefer EXPR(INT). Interpreter has no idea what is Rep of AN. One can argue that COMPLEX(INT) is more complex than INT, so should be preferable. OTOH letter C is before letter I and letter T. Consequently, interpreter sorting puts complex variant first in search list. >From somewhat different point of view, one can ask if the two functions should share the name. Or if CTRIGMNP and TRIGMNIP should be exposed. Or maybe more things should be hidden in ITRIGMNP. > > 'real?' in TRIGMNIP is making reasonable effort to find out if > > "real" transformations of functions will give correct results. > > Your expression is affected by trouble, what is meant by > > 'sqrt'? '-sqrt(2)' have two possible values, positive and negative. > > So numerical value depends on choice of branches. > > I think for sqrt which is 'nthRoot, should be interpreted as the > positive branch. > > While the algebraic number generated by "rootsOf(x^2-2)", which > is 'rootOf, have two possible values. Well, even for AlgebraicNumber you almost immediately get complex numbers. And no, there is no deep difference between 'nthRoot and 'rootOf. > > One possible > > meaning of 'real?' could be "argument is real for all possible choices > > of branches". Another could be "argument is real for some choice > > of branches". Yet another (and it is more or less what we use) > > is that argument is "formally real", that is consists of > > applications of "real" functions to "formally real" argument > > with exception made for roots: even root of negative constant > > is considered not-real. Each meaning have some sense. The first > > two are problemantic, because in general branch questions are > > undecidable. The third one is decidable and useful in some > > contexts. But may be confusing for some users. And your case > > is borderline for this definition: if sqrt(2) is considered as function > > applied to a constant, than correct answer is "true". If > > sqrt(2) is considered to be a positive constant, then > > answer is "false". > > I think we should have "sign", "real?" well defined for AlgebraicNumber. Well, if you want then well defined use RealClosure(Fraction(Integer)) for real things and Complex(RealClosure(Fraction(Integer))) in general case. But this really do not generalize to expressions, so you are basically throwing away current expression-based AlgebraicNumber. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVoeaTvVdIN2wJIO%40fricas.org.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On 11/19/23 21:36, Waldek Hebisch wrote: On Sun, Nov 19, 2023 at 07:58:50PM +0800, Qian Yun wrote: "real? sqrt (-sqrt 2)" returns "true" by default. You mean should return false? We get this when working with expressions: (2) -> real? sqrt (-sqrt 2::EXPR(INT)) (2) false Because interpreter prefers CTRIGMNP over TRIGMNIP as shown by ")set mes bot on": Function Selection for real? Arguments: AN [1] signature: EXPR(COMPLEX(INT)) -> BOOLEAN implemented: slot (Boolean)(Expression (Complex (Integer))) from CTRIGMNP(INT,EXPR(COMPLEX(INT))) [2] signature: EXPR(INT) -> BOOLEAN implemented: slot (Boolean)(Expression (Integer)) from TRIGMNIP(INT,EXPR(INT)) Shall we consider this a bug? I do not think so. IIUC 'real?' in CTRIGMNP is quite different function than 'real?' in TRIGMNIP. Namely, CTRIGMNP deals with COMPLEX(EXPR(INT)), which is formal linear combination 'a + %i*b' where 'a' and 'b' are expressions and 'real?' here tests if 'b' equals 0, which AFAICS is true. I don't think 'real?' in CTRIGMNP is wrong. I think the interpreter choose EXPR(COMPLEX(INT)) over EXPR(INT) for AN is wrong. Isn't AN the same as EXPR(INT) in Rep? So I think the interpreter auto coercion should prefer EXPR(INT). 'real?' in TRIGMNIP is making reasonable effort to find out if "real" transformations of functions will give correct results. Your expression is affected by trouble, what is meant by 'sqrt'? '-sqrt(2)' have two possible values, positive and negative. So numerical value depends on choice of branches. I think for sqrt which is 'nthRoot, should be interpreted as the positive branch. While the algebraic number generated by "rootsOf(x^2-2)", which is 'rootOf, have two possible values. One possible meaning of 'real?' could be "argument is real for all possible choices of branches". Another could be "argument is real for some choice of branches". Yet another (and it is more or less what we use) is that argument is "formally real", that is consists of applications of "real" functions to "formally real" argument with exception made for roots: even root of negative constant is considered not-real. Each meaning have some sense. The first two are problemantic, because in general branch questions are undecidable. The third one is decidable and useful in some contexts. But may be confusing for some users. And your case is borderline for this definition: if sqrt(2) is considered as function applied to a constant, than correct answer is "true". If sqrt(2) is considered to be a positive constant, then answer is "false". I think we should have "sign", "real?" well defined for AlgebraicNumber. - Qian For our purposes it does not matter much. We just need to be consistent: tests used in various places should be "uniformly dumb", otherwise there is danger that something which we consider real will be transformed into something visibly complex by "too smart" transformation, leading to possible bugs. And ideally we should not depend on 'real?' at all: when we really care about expressions having real values we should use stronger (or at least more reliable) tests. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/6bddcc85-f58c-4c44-bbc5-c3354ab6608a%40gmail.com.
Re: [fricas-devel] real? sqrt (-sqrt 2)
On Sun, Nov 19, 2023 at 07:58:50PM +0800, Qian Yun wrote: > "real? sqrt (-sqrt 2)" returns "true" by default. You mean should return false? We get this when working with expressions: (2) -> real? sqrt (-sqrt 2::EXPR(INT)) (2) false > Because interpreter prefers CTRIGMNP over TRIGMNIP > as shown by ")set mes bot on": > > Function Selection for real? > Arguments: AN > > [1] signature: EXPR(COMPLEX(INT)) -> BOOLEAN > implemented: slot (Boolean)(Expression (Complex (Integer))) from > CTRIGMNP(INT,EXPR(COMPLEX(INT))) > [2] signature: EXPR(INT) -> BOOLEAN > implemented: slot (Boolean)(Expression (Integer)) from > TRIGMNIP(INT,EXPR(INT)) > > > Shall we consider this a bug? I do not think so. IIUC 'real?' in CTRIGMNP is quite different function than 'real?' in TRIGMNIP. Namely, CTRIGMNP deals with COMPLEX(EXPR(INT)), which is formal linear combination 'a + %i*b' where 'a' and 'b' are expressions and 'real?' here tests if 'b' equals 0, which AFAICS is true. 'real?' in TRIGMNIP is making reasonable effort to find out if "real" transformations of functions will give correct results. Your expression is affected by trouble, what is meant by 'sqrt'? '-sqrt(2)' have two possible values, positive and negative. So numerical value depends on choice of branches. One possible meaning of 'real?' could be "argument is real for all possible choices of branches". Another could be "argument is real for some choice of branches". Yet another (and it is more or less what we use) is that argument is "formally real", that is consists of applications of "real" functions to "formally real" argument with exception made for roots: even root of negative constant is considered not-real. Each meaning have some sense. The first two are problemantic, because in general branch questions are undecidable. The third one is decidable and useful in some contexts. But may be confusing for some users. And your case is borderline for this definition: if sqrt(2) is considered as function applied to a constant, than correct answer is "true". If sqrt(2) is considered to be a positive constant, then answer is "false". For our purposes it does not matter much. We just need to be consistent: tests used in various places should be "uniformly dumb", otherwise there is danger that something which we consider real will be transformed into something visibly complex by "too smart" transformation, leading to possible bugs. And ideally we should not depend on 'real?' at all: when we really care about expressions having real values we should use stronger (or at least more reliable) tests. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/ZVoPc5_hFCIV8a1F%40fricas.org.
[fricas-devel] real? sqrt (-sqrt 2)
"real? sqrt (-sqrt 2)" returns "true" by default. Because interpreter prefers CTRIGMNP over TRIGMNIP as shown by ")set mes bot on": Function Selection for real? Arguments: AN [1] signature: EXPR(COMPLEX(INT)) -> BOOLEAN implemented: slot (Boolean)(Expression (Complex (Integer))) from CTRIGMNP(INT,EXPR(COMPLEX(INT))) [2] signature: EXPR(INT) -> BOOLEAN implemented: slot (Boolean)(Expression (Integer)) from TRIGMNIP(INT,EXPR(INT)) Shall we consider this a bug? - Qian -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/fricas-devel/3221f108-fb2c-4057-b9c2-08fc3e2a620b%40gmail.com.
