Vorige: Introduction to Affine, Nach oben: affine [Inhalt][Index]
Solves the simultaneous linear equations expr_1, …, expr_m
for the variables x_1, …, x_n. Each expr_i may be an
equation or a general expression; if given as a general expression, it is
treated as an equation of the form expr_i = 0
.
The return value is a list of equations of the form [x_1 =
a_1, ..., x_n = a_n]
where a_1, …, a_n are
all free of x_1, …, x_n.
fast_linsolve
is faster than linsolve
for system of equations
which are sparse.
load("affine")
loads this function.
Returns a Groebner basis for the equations expr_1, …, expr_m.
The function polysimp
can then be used to simplify other functions
relative to the equations.
polysimp(f)
yields 0 if and only if f is in the ideal generated by
expr_1, …, expr_m, that is, if and only if f is a
polynomial combination of the elements of expr_1, …, expr_m.
load("affine")
loads this function.
Beispiel:
(%i1) load("affine")$ (%i2) grobner_basis ([3*x^2+1, y*x]); eliminated one . 0 . 0 2 (%o2)/R/ [- y, - 3 x - 1] (%i3) polysimp(y^2*x+x^3*9+2); (%o3)/R/ - 3 x + 2
The eqns are polynomial equations in non commutative variables. The
value of current_variables
is the list of variables used for computing
degrees. The equations must be homogeneous, in order for the procedure to
terminate.
If you have checked overlapping simplifications in dot_simplifications
above the degree of f, then the following is true:
dotsimp(f)
yields 0 if and only if f is in the ideal
generated by the equations, i.e., if and only if f is a polynomial
combination of the elements of the equations.
The degree is that returned by nc_degree
. This in turn is influenced
by the weights of individual variables.
load("affine")
loads this function.
Assigns weights w_1, …, w_n to x_1, …, x_n,
respectively. These are the weights used in computing nc_degree
.
load("affine")
loads this function.
Returns the degree of a noncommutative polynomial p.
See declare_weights
.
load("affine")
loads this function.
Returns 0 if and only if f is in the ideal generated by the equations, i.e., if and only if f is a polynomial combination of the elements of the equations.
load("affine")
loads this function.
If set_up_dot_simplifications
has been previously done, finds the
central polynomials in the variables x_1, …, x_n in the given
degree, n.
For example:
set_up_dot_simplifications ([y.x + x.y], 3); fast_central_elements ([x, y], 2); [y.y, x.x];
load("affine")
loads this function.
Checks the overlaps thru degree n, making sure that you have sufficient
simplification rules in each degree, for dotsimp
to work correctly.
This process can be speeded up if you know before hand what the dimension of
the space of monomials is. If it is of finite global dimension, then
hilbert
should be used. If you don’t know the monomial dimensions, do
not specify a rank_function
. An optional third argument reset
,
false
says don’t bother to query about resetting things.
load("affine")
loads this function.
Returns the list of independent monomials relative to the current dot simplifications of degree n in the variables x_1, …, x_n.
load("affine")
loads this function.
Compute the Hilbert series through degree n for the current algebra.
load("affine")
loads this function.
Makes a list of the coefficients of the noncommutative polynomials p_1,
…, p_n of the noncommutative monomials m_1, …,
m_n. The coefficients should be scalars. Use list_nc_monomials
to build the list of monomials.
load("affine")
loads this function.
Vorige: Introduction to Affine, Nach oben: affine [Inhalt][Index]