Hi all:
Is there a Sage command similar to Maple 11's 'simplify/siderels'
which simplifies an expression with respect to given relations? I
couldn't find mention of such a command in the Sage documentation.
For more details, here's the Maple 11 help documentation.
Alex
==================================================
simplify/siderels - simplify with respect to side relations
Calling Sequence
simplify(expr, eqns)
simplify(expr, eqns, vars)
simplify(expr, eqns, tord)
simplify(expr, eqns, 'mindeg')
Parameters
expr - expression
eqns - set or list of equations (an expression e is understood as
the equation e=0)
vars - (optional) set or list of variables appearing in eqns
tord - (optional) a monomial order in the variables appearing in
eqns
Description
Simplification of expr with respect to the side relations eqns is
performed. The result is an expression which is mathematically
equivalent to expr but which is in normal form with respect to the
specified side relations.
In all cases simplify computes a Groebner basis for the set of side
relations. This basis contains all the algebraic consequences of eqns,
so that if a expr can be rewritten in terms of the relations then
simplify will reduce it to zero. Otherwise, simplify returns a
remainder in which none of the terms are reducible.
To ensure a finite division process simplify must impose a total order
on terms which can appear. This order, called a monomial order, is
used for both the Groebner basis computation and the division. The
monomial order is specified by an optional third argument, and
different orders can drastically affect a non-zero result:
If vars is a list then pure lexicographic (plex) order is used and the
leftmost variables are eliminated first.
If vars is a set then a total degree (tdeg) order is used and terms
with the highest degree in all the variables are eliminated first.
Terms with the same total degree are ordered heuristically.
If no third argument was specified then lexicographic order is used
and variables appearing on the left hand side of an equation (in eqns)
are eliminated first.
If a monomial order tord is specified, then that order is used for all
computations. See Groebner[MonomialOrders] for a list of the available
monomial orders.
If expr is a rational expression then one of three algorithms can be
used. If the optional third argument vars is specified (or omitted)
then simplification is applied to the numerator and denominator
separately. This method is very fast, however no attempt is made to
simplify the ratio of the numerator and denominator so simpler
representations of the fraction may be missed.
If a monomial order tord is specified then the largest monomial
appearing in the numerator and denominator is minimized with respect
to tord. The result is a simplified canonical form with respect to the
monomial order. Note that in some cases a common factor may be
introduced to satisfy this requirement, as in example 2 below. If the
denominator is invertible or if it divides the numerator modulo the
side relations then a quotient is computed and returned instead.
The optional argument 'mindeg' minimizes the sum of the degrees of the
resulting fraction. It uses a global search to find an equivalent
fraction a/b with deg(a)+deg(b) minimal. The result is not canonical,
however in some cases the output may be preferable to a canonical form
(see example 3). In general the search is exponential time in deg(a)
+deg(b).
If the expression is neither a polynomial nor a rational function in
the variables, simplify will recurse on the input expr and apply the
normalization to any subexpressions which are rational functions in
the variables.
Examples
Example 1: trigonometric polynomials
eqns := {sin(x)^2 + cos(x)^2 = 1}:
e := sin(x)^3-11*sin(x)^2*cos(x)+3*cos(x)^3-sin(x)*cos(x)+2;
3 2 3
e := sin(x) - 11 sin(x) cos(x) + 3 cos(x) - sin(x) cos(x) + 2
simplify(e, eqns);
2 3
2 - sin(x) cos(x) - 14 sin(x) cos(x) + sin(x) + 3 cos(x)
simplify(e, eqns, [sin(x),cos(x)]); # eliminate sin(x) first
3
2
2 - sin(x) cos(x) + 14 cos(x) - 11 cos(x) + sin(x) - sin(x)
cos(x)
simplify(e, eqns, [cos(x),sin(x)]); # eliminate cos(x) first
2 3
2 - sin(x) cos(x) - 14 sin(x) cos(x) + sin(x) + 3 cos(x)
f := 8*sin(x)^4*cos(x)+15*sin(x)^2*cos(x)^3-15*sin(x)^2*cos(x)
+7*cos(x)^5-14*cos(x)^3+7*cos(x);
4 2 3
2 5
f := 8 sin(x) cos(x) + 15 sin(x) cos(x) - 15 sin(x) cos(x) + 7
cos(x)
3
- 14 cos(x) + 7 cos(x)
simplify(f, eqns);
0
r := (1-cos(x)^2+sin(x)*cos(x))/(sin(x)*cos(x)+cos(x)^2);
2
1 - cos(x) + sin(x) cos(x)
r := ---------------------------
2
sin(x) cos(x) + cos(x)
simplify(r, eqns); # numerator and denominator reduced seperately
2
sin(x) cos(x) + sin(x)
---------------------------
2
sin(x) cos(x) + 1 - sin(x)
simplify(r, eqns, 'mindeg'); # their ratio is simplified
sin(x)
------
cos(x)
Example 2: a non trivial example
eqns2 := {x*y^5-x-y};
/ 5 \
eqns2 := { x y - x - y }
\ /
r := (y^5+x+y)/(x-y);
5
y + x + y
r := ----------
x - y
simplify(r, eqns2, tdeg(x,y)); # a common factor of x is introduced
2
x + x y + x + y
----------------
2
x - x y
simplify(r, eqns2, plex(x,y)); # highest degree in x is minimized
5 9 4
y + y - y
------------
5
-y + 2
simplify(r, eqns2, 'mindeg'); # sum of degrees is minimized
2
x + x y + x + y
----------------
2
x - x y
r2 := (-x^3-y*x^2-x^4-y*x^3+2*x*y^4+2*y^4+y^5+x+y)/
(y*x^3-x^4+2*x*y^4-4+x-y);
3 2 4 3 4 4 5
-x - y x - x - y x + 2 x y + 2 y + y + x + y
r2 := ---------------------------------------------------
3 4 4
y x - x + 2 x y - 4 + x - y
simplify(r2,eqns2); # numerator and denominator are reduced
seperately
3 2 4 3 4 4 5
x + y x + x + y x - 2 x y - 2 y - y - x - y
--------------------------------------------------
3 4 4
-y x + x - 2 x y + 4 - x + y
simplify(r2,eqns2,tdeg(x,y)); # fraction is simplified to a canonical
form
2
x + x y + x + y
----------------
2
x - x y
Example 3: the canonical form algorithm may increase total degree
eqns3 := {x^5+x*y-1};
/ 5 \
eqns3 := { x + x y - 1 }
\ /
r := (x^3*y^3-x^4+x-1)/(x^2-y^2+1);
3 3 4
y x - x + x - 1
r := ------------------
2 2
x - y + 1
simplify(r, eqns3, tdeg(x,y)); # smaller monomials are obtained at the
cost of degree
4 3 2 3 2
x y - x - y x - y + x + x
------------------------------
4 2 2 2
-x + x y - x
simplify(r, eqns3, 'mindeg'); # the input had minimal degree
3 3 4
y x - x + x - 1
------------------
2 2
x - y + 1
Example 4: the side relations define a field
eqns4 := {z^3-z^2-z*y+2*y^2=1, z^3+y^2=1, z^2+z*y-y^2=0, x+y=z};
/ 3 2 2 3 2 2
2
eqns4 := { z - z - z y + 2 y = 1, z + y = 1, z + z y - y = 0, x
+ y = z
\
\
}
/
h1 := 36*z^4*y^2+36*z*y^4-36*z*y^2-1/2*z^2+z*y-1/2*y^2+1/2*x*z-1/2*x*y
+2/3*z^4+4/3*z^3*y-2/3*z^2*y^2-4/3*z*y^3+2/3*y^4;
4 2 4 2 1 2 1 2 1
1 2 4
h1 := 36 z y + 36 z y - 36 z y - - z + z y - - y + - x z - - x y
+ - z
2 2 2
2 3
4 3 2 2 2 4 3 2 4
+ - z y - - z y - - z y + - y
3 3 3 3
simplify(h1, eqns4);
0
h2 := z*y^2+z^3*y^3-3*z^2*y
+23/3+5/6*z^2+7/3*z*y-11/6*y^2+1/2*x*z-1/2*x*y
+2/3*z^4+4/3*z^3*y-2/3*z^2*y^2-4/3*z*y^3+2/3*y^4;
2 3 3 2 23 5 2 7 11 2 1
1 2 4
h2 := z y + z y - 3 z y + -- + - z + - z y - -- y + - x z - - x
y + - z
3 6 3 6 2
2 3
4 3 2 2 2 4 3 2 4
+ - z y - - z y - - z y + - y
3 3 3 3
simplify(h2, eqns4, {x,y,z});
34 2 2
- -- + 8 z + 13 y + 16 z - 34 z y + 14 z y
3
simplify(h2, eqns4, lexdeg([x,z],[y]));
29 5 2
-- - y - 2 y
3
h2inv := simplify(1/h2, eqns4, tdeg(x,z,y)); # inverse is computed
4473752523 236183994 321405507
36881532
h2inv := ----------- + ----------- z + ----------- y + -----------
z y
42772565971 42772565971 42772565971
42772565971
815384160 2 252548433 3
+ ----------- y + ----------- y
42772565971 42772565971
simplify(h2*h2inv, eqns4);
1
Example 5: a difficult example that can not be handled by other
methods
eqns5 := {y^2-x^3+x*y-1};
/ 2 3 \
eqns5 := { y - x + x y - 1 }
\ /
algcurves[parametrization](op(eqns5), x, y, t); # can not map into
Q(t)
Error, (in algcurves/ratpar) genus is not zero
r := (-y^3+1-2*y*x^2-x^2*y^5-2*x*y^2-y^6*x-y^7+y^6)/(y^3-2*y^2+1);
3 2 2 5 2 6 7 6
-y + 1 - 2 y x - x y - 2 x y - x y - y + y
r := --------------------------------------------------
3 2
y - 2 y + 1
simplify(r,eqns5,tdeg(x,y));
5
y + x + y
----------
x - y
See Also
algcurves[parametrization], Groebner[Basis], Groebner[NormalForm],
Groebner[MonomialOrders]
References
D. Cox, J. Little, D. O'Shea. Ideals, Varieties, and Algorithms.
Second Edition. Springer-Verlag, 1997
M. Monagan, R. Pearce. Rational Simplification Modulo a Polynomial
Ideal. ISSAC 2006 proceedings.
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