(maxima.info)Functions and Variables for Affine

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24.2 Functions and Variables for Affine
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 -- Function: fast_linsolve ([<expr_1>, ..., <expr_m>], [<x_1>, ...,
          <x_n>])
     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.

 -- Function: grobner_basis ([<expr_1>, ..., <expr_m>])
     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.

          grobner_basis ([3*x^2+1, y*x])$

          polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2

     '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.

 -- Function: set_up_dot_simplifications
          set_up_dot_simplifications (<eqns>, <check_through_degree>)
          set_up_dot_simplifications (<eqns>)

     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.

 -- Function: declare_weights (<x_1>, <w_1>, ..., <x_n>, <w_n>)
     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.

 -- Function: nc_degree (<p>)
     Returns the degree of a noncommutative polynomial <p>.  See
     'declare_weights'.

     'load("affine")' loads this function.

 -- Function: dotsimp (<f>)
     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.

 -- Function: fast_central_elements ([<x_1>, ..., <x_n>], <n>)
     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.

 -- Function: check_overlaps (<n>, <add_to_simps>)
     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.

 -- Function: mono ([<x_1>, ..., <x_n>], <n>)
     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.

 -- Function: monomial_dimensions (<n>)
     Compute the Hilbert series through degree <n> for the current
     algebra.

     'load("affine")' loads this function.

 -- Function: extract_linear_equations ([<p_1>, ..., <p_n>], [<m_1>,
          ..., <m_n>])

     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.

 -- Function: list_nc_monomials
          list_nc_monomials ([<p_1>, ..., <p_n>])
          list_nc_monomials (<p>)

     Returns a list of the non commutative monomials occurring in a
     polynomial <p> or a list of polynomials <p_1>, ..., <p_n>.

     'load("affine")' loads this function.

 -- Option variable: all_dotsimp_denoms
     Default value: 'false'

     When 'all_dotsimp_denoms' is a list, the denominators encountered
     by 'dotsimp' are appended to the list.  'all_dotsimp_denoms' may be
     initialized to an empty list '[]' before calling 'dotsimp'.

     By default, denominators are not collected by 'dotsimp'.