OpenXM/Risa/Asir-Contrib

OpenXM/Risa/Asir-Contrib User's Manual (English Edition)

Edition 1.3.1 for OpenXM/Asir2000

February 2010

by OpenXM Developing Team

Introduction

The computer algebra system asir can use servers, which support the OpenXM protocols (Open message eXchange for Mathematics, http://www.openxm.org), as components. The interface functions to call these servers are loaded by loading the file `OpenXM/rc/asirrc'. This file is automatically loaded in "Risa/Asir(OpenXM distribution)", which we call OpenXM/Risa/Asir in this document. This document explains these interface functions for asir and several mathematical and utility functions written in the user languages of Risa/Asir. These mathematical and utilitiy functions are outcome of the Asir-contrib project.

As to technical details on the OpenXM protocols, see `openxm-en.tex' at `$(OpenXM_HOME)/doc/OpenXM-specs'.

Enjoy mathematics on your computer.

List of contributors:

See OpenXM/Copyright for the copyright notice.

How to load Asir/Contrib

With loading `OpenXM/rc/asirrc', we can use most functions in Asir/Contrib. The OpenXM/Risa/Asir reads this file, which is specified by the ASIR_CONFIG environmental variable, when it starts. The file `names.rr' is the top level file of the Asir/Contrib. Most other files are loaded from `names.rr'. Some packages are not loaded from `names.rr' and they must be loaded individually.

A sample of `asirrc' to use Asir/Contrib. 

load("gr")$
load("primdec")$
load("katsura")$
load("bfct")$
load("names.rr")$
load("oxrfc103.rr")$
User_asirrc=which(getenv("HOME")+"/.asirrc")$
if (type(User_asirrc)!=0) {
   if (!ctrl("quiet_mode")) print("Loading ~/.asirrc")$
   load(User_asirrc)$
}else{ }$
end$

Function Names in Asir Contrib

Not yet written.

Not yet written.

Asir-contrib for Windows

A part of Asir-contrib works on Windows. The following functions and components work on windows; the outer component sm1 and functions in asir-contrib which do not call outer components. In the cygwin environement, the outer components sm1, phc work. The other outer components do not work.

The following functions do not work on Windows. Some of them work in the cygwin environment of Windows.

Basic (Standard Functions)

base_cancel

base_cancel(S)
: It simplifies S by canceling the common factors of denominators and numerators.

Example: 

 base_cancel([(x-1)/(x^2-1), (x-1)/(x^3-1)]);

base_choose

base_choose(L,M)
: It returns the list of the order M subsets of L.

Example: 

 base_choose([1,2,3],2);

base_flatten

base_flatten(S)
: It flattens a nested list S.

Example: 

 base_flatten([[1,2,3],4]);

base_intersection

base_intersection(A,B)
: It returns the intersection of A and B as a set.

Example: 

 base_intersection([1,2,3],[2,3,5,[6,5]]);

base_memberq

base_memberq(A,S)
: It returns 1 if A is a member of the set S else returns 0.

Example: 

 base_memberq(2,[1,2,3]);

base_permutation

base_permutation(L)
: It outputs all permutations of L. BUG; it uses a slow algorithm.

Example: 

 base_permutation([1,2,3,4]);

base_position

base_position(A,S)
: It returns the position of A in S.

Example: 

 base_position("cat",["dog","cat","monkey"]);

base_prune

base_prune(A,S)
: It returns a list in which A is removed from S.

Example: 

 base_prune("cat",["dog","cat","monkey"]);

base_replace

base_replace(S,Rule)
: It rewrites S by using the rule Rule

Example: 

 base_replace(x^2+y^2,[[x,a+1],[y,b]]);

x is replaced by a+1 and y is replaced by b in x^2+y^2.

base_replace_n

base_replace_n(S,Rule)
: It rewrites S by using the rule Rule. It is used only for specializing variables to numbers and faster than base_replace.

Example: 

 base_replace_n(x^2+y^2,[[x,1/2],[y,2.0+3*@i]]);

x is replaced by 1/2 and y is replaced by 2.0+3*@i in x^2+y^2.

base_set_minus

base_set_minus(A,B)
:

Example: 

 base_set_minus([1,2,3],[3,4,5]);

base_set_union

base_set_union(A,B)
:

Example: 

 base_set_union([1,2,3],[3,4,5]);

base_subsetq

base_subsetq(A,B)
:

Example: 

 base_subsetq([1,2],[1,2,3,4,5]);

base_subsets_of_size

base_subsets_of_size(K,S)
: It outputs all subsets of S of the size K. BUG; it uses a slow algorithm. Do not input a large S.

Example: 

 base_subsets_of_size(2,[3,5,3,2]);

Numbers (Standard Mathematical Functions)

number_abs

number_abs(X)
:

Example: 

 number_abs(-3);@end example




number_ceiling







number_ceiling(X)


:  




Example:


 number_abs(1.5);@end example




number_factor







number_factor(X)


:   It factors the given integer X. 




Example:


 number_factor(20);@end example




number_floor







number_floor(X)


:  




Example:


 number_floor(1.5);@end example




number_imaginary_part







number_imaginary_part(X)


:  




Example:


 number_imaginary_part(1+2*@i);@end example




number_is_integer







number_is_integer(X)


:  




Example:


 number_is_integer(2/3);@end example




number_real_part







number_real_part(X)


:  




Example:


 number_real_part(1+2*@i);@end example




Calculus (Standard Mathematical Functions)





Series (Standard Mathematical Functions)





Special Functions (Standard Mathematical Functions)




Not yet written





Matrix (Standard Mathematical Functions)







matrix_clone







matrix_clone(M)


: 
 It generates the clone of the matrix M. 




Example:


 matrix_clone(matrix_list_to_matrix([[1,1],[0,1]]));






matrix_det








matrix_det(M)


: 
 It returns the determinant of the matrix M. 




Example:


 poly_factor(matrix_det([[1,x,x^2],[1,y,y^2],[1,z,z^2]]));






matrix_diagonal_matrix








matrix_diagonal_matrix(L)


: 
 It returns the diagonal matrix with diagonal entries L. 




Example:


 matrix_diagonal_matrix([1,2,3]);





References:





 matrix_list_to_matrix 






matrix_eigenavalues








matrix_eigenavalues(M)


: 
 It returns the eigenvalues of the matrix M. 




Example:


 matrix_eigenvalues([[x,1],[0,y]]);






matrix_identity_matrix








matrix_identity_matrix(N)


: 
 It returns the identity matrix of the size N. 




Example:


 matrix_identity_matrix(5);





References:





 matrix_diagonal_matrix 






matrix_image








matrix_image(M)


: 
 It computes the image of M.
 Redundant vectors are removed. 




Example:


 matrix_image([[1,2,3],[2,4,6],[1,0,0]]);





References:





 matrix_kernel 






matrix_inner_product








matrix_inner_product(A,B)


: 
 It returns the inner product of two vectors A and B. 




Example:


 matrix_inner_product([1,2],[x,y]);






matrix_inverse








matrix_inverse(M)


: 
 It returns the inverse of the matrix M. 




Example:


 matrix_inverse([[1,2],[0,1]]);






matrix_kernel








matrix_kernel(M)


: 
 It returns the basis of the kernel of the matrix M. 




Example:


 matrix_kernel([[1,1,1,1],[0,1,3,4]]);






matrix_list_to_matrix








matrix_list_to_matrix(M)


: 
 It translates the list M to a matrix. 




Example:


 print_xdvi_form(matrix_list_to_matrix([[1,1],[0,2]]));





References:





 matrix_matrix_to_list 






matrix_matrix_to_list








matrix_matrix_to_list(M)


: 
 It translates the matrix M to a list. 





References:





 matrix_list_to_matrix 






matrix_rank








matrix_rank(M)


: 
 It returns the rank of the matrix M. 




Example:


 matrix_rank([[1,1,1,1],[0,1,3,4]]);






matrix_solve_linear








matrix_solve_linear(M,X,B)


: 
 It solves the system of linear equations M X = B 




Example:


 matrix_solve_linear([[1,2],[0,1]],[x,y],[1,2]);






matrix_submatrix








matrix_submatrix(M,Ind)


: 
 It returns the submatrix of M defined by the index set Ind. 




Example:


 matrix_submatrix([[0,1],[2,3],[4,5]],[1,2]);






matrix_transpose








matrix_transpose(M)


: 
 It returns the transpose of the matrix M. 





References:





 matrix_list_to_matrix 






Graphic (Standard Mathematical Functions)




Not yet written.





Print (Standard Mathematical Functions)







print_dvi_form








print_dvi_form(S)


: 
 It outputs S to a dvi file. 




Example:


 print_dvi_form(x^2-1);





References:





 print_xdvi_form , 
 print_tex_form 






print_em








print_em(S)


: 
 It outputs S by a font to emphasize it. 




Example:


 print_em(x^2-1);






print_gif_form








print_gif_form(S)


: 
 It outputs S to a file of the gif format. 

print_gif_form(S | table=key0)


: This function allows optional variables 
  table




Example:


 print_gif_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));





References:





 print_tex_form 






print_input_form








print_input_form(S)


: 
 It transforms S to a string which can be parsed by asir. 




Example:


 print_input_form(quote(x^3-1));






print_open_math_tfb_form








print_open_math_tfb_form(S)


: 
 It transforms S to a tfb format of OpenMath XML. 





Description:





 It is experimental. You need to load taka_print_tfb.rr 
 to call it.
 





Example:


 print_open_math_tfb_form(quote(f(x,1/(y+1))+2));






print_open_math_xml_form








print_open_math_xml_form(S)


: 
 It transforms S to a string which is compliant to OpenMath(1999). 




Example:


 print_open_math_xml_form(x^3-1);





References:





 www.openmath.org 






print_output








print_output(Obj)


: 
 It outputs the object Obj to a file. 
 If the optional variable file is set, then it outputs the Obj to the
 specified file, else it outputs it to "asir_output_tmp.txt".
 If the optional variable mode is set to "w", 
 then the file is newly created.
 If the optional variable is not set, the Obj is appended to the file. 

print_output(Obj | file=key0,mode=key1)


: This function allows optional variables 
  file, mode




Example:


 print_output("Hello"|file="test.txt");





References:





 glib_tops , 
 ( , 
 ) 






print_ox_rfc100_xml_form








print_ox_rfc100_xml_form(S)


: 
 It transforms S to a string which is compliant to OpenXM RFC 100. 




Example:


 print_ox_rfc100_xml_form(x^3-1);





References:





 www.openxm.org 






print_png_form








print_png_form(S)


: 
 It transforms S to a file of the format png.
 dvipng should be installed. 




Example:


 print_png_form(x^3-1);





References:





 print_tex_form 






print_terminal_form








print_terminal_form(S)


: 
 It transforms S to the terminal form??? 






print_tex_form








print_tex_form(S)


: 
 It transforms S to a string of the LaTeX format. 

print_tex_form(S | table=key0,raw=key1)


: This function allows optional variables 
  table, raw





Description:






 The global variable Print_tex_form_fraction_format
 takes the values "auto", "frac",  or "/".
 The global variable Print_tex_form_no_automatic_subscript
 takes the values 0 or 1.
 BUG; A large input S cannot be translated.
 





Example:


 print_tex_form(x*dx+1 | table=[["dx","\\partial_x"]]);





 The optional variable table is used to give a translation
 table of asir symbols and tex symbols.
 




References:





 print_xdvi_form 






print_tfb_form








print_tfb_form(S)


: 
 It transforms S to the tfb format. 




Example:


 print_tfb_form(x+1);






print_xdvi_form








print_xdvi_form(S)


: 
 It transforms S to a xdvi file and previews the file by xdvi. 




Example 0:


 print_xdvi_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));





Example 1:


 print_xdvi_form(print_tex_form(1/2));





References:





 print_tex_form , 
 print_dvi_form 






print_xv_form








print_xv_form(S)


: 
 It transforms S to a gif file and previews the file by xv. 

print_xv_form(S | input=key0,format=key1)


: This function allows optional variables 
  input, format




Example 0:


 print_xv_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));





Example 1:


 print_xv_form(x+y | format="png");





 If the optional variable format="png" is set,
  png format will be used to generate an input for xv.
 




References:





 print_tex_form , 
 print_gif_form 






Polynomials (Standard Mathematical Functions)







poly_degree








poly_degree(F)


: 
 It returns the degree of F with respect to the given weight vector. 

poly_degree(F | weight=key0,v=key1)


: This function allows optional variables 
  weight, v





Description:






 The weight is given by the optional variable weight w.
 It returns
 





Example:


 poly_degree(x^2+y^2-4 |weight=[100,1],v=[x,y]);






poly_elimination_ideal








poly_elimination_ideal(I,VV)


: 
 It computes the ideal intersection of I and the monomial ideal
 generated by VV. 

poly_elimination_ideal(I,VV | grobner_basis=key0,v=key1)


: This function allows optional variables 
  grobner_basis, v





Description:






 If grobner_basis is "yes", I is assumed to be a Grobner basis.
 The optional variable v is a list of variables which defines the ring 
 of polynomials.
 





Example 0:


 poly_elimination_ideal([x^2+y^2-4,x*y-1],[x]);





Example 1:


 A = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=2,v=[y,x]);
          poly_elimination_ideal(A,[x]|grobner_basis="yes");





References:





 gr , 
 hgr , 
 gr_mod , 
 dp_* 






poly_expand








poly_expand(F)


: 
  This is an alias of poly_sort. 





References:





 poly_sort 






poly_factor








poly_factor(F)


: 
 It factorizes the polynomial F. 




Example:


 poly_factor(x^10-y^10);






poly_gcd








poly_gcd(F,G)


: 
 It computes the polynomial GCD of F and G. 




Example:


 poly_gcd(x^10-y^10,x^25-y^25);






poly_grobner_basis








poly_grobner_basis(I)


: 
 It returns the Grobner basis of I. 

poly_grobner_basis(I | order=key0,v=key1)


: This function allows optional variables 
  order, v





Description:






 The optional variable v is a list of variables which defines the ring 
 of polynomials.
 





Example:


 A = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=2,v=[y,x]);






poly_hilbert_polynomial








poly_hilbert_polynomial(I)


: 
 It returns the Hilbert polynomial of the ideal I. 

poly_hilbert_polynomial(I | s=key0,v=key1)


: This function allows optional variables 
  s, v





Description:






 The optional variable v is a list of variables.
 





Example:


 poly_hilbert_polynomial([x1*y1,x1*y2,x2*y1,x2*y2]|s=k,v=[x1,x2,y1,y2]);






poly_initial








poly_initial(I)


: 
 It returns the initial ideal of I with respect to the given order. 

poly_initial(I | order=key0,v=key1)


: This function allows optional variables 
  order, v





Description:






 The optional variable v is a list of variables. This function computes 
 





Example:


 poly_initial([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);






poly_initial_coefficients








poly_initial_coefficients(I)


: 
 It computes the coefficients of the initial ideal of I with respect to the given order. 

poly_initial_coefficients(I | order=key0,v=key1)


: This function allows optional variables 
  order, v





Description:






 The optional variable v is a list of variables. The order is specified by the
 optional variable order
 





Example:


 poly_initial_coefficients([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);






poly_initial_term








poly_initial_term(F)


: 
 It returns the initial term of a polynomial F 
 with respect to the given weight vector. 

poly_initial_term(F | weight=key0,order=key1,v=key2)


: This function allows optional variables 
  weight, order, v





Description:






 The weight is given by the optional variable weight w.
 It returns
 





Example:


 poly_initial_term( x^2+y^2-4 |weight=[100,1],v=[x,y]);






poly_solve_linear








poly_solve_linear(Eqs,V)


: 
 It solves the system of linear equations Eqs with respect to the set of
 variables V. 




Example:


 poly_solve_linear([2*x+3*y-z-2, x+y+z-1], [x,y,z]);






poly_sort








poly_sort(F)


: 
 It expands F with a given variables v=V and a given weight w=W.
 It returns a quote object.
 If trucate option is set, the expansion is truncated at the given degree. 

poly_sort(F | v=key0,w=key1,truncate=key2)


: This function allows optional variables 
  v, w, truncate




Example:


 poly_sort((x-y-a)^3 | v=[x,y], w=[-1,-1])  
    returns a series expansion in terms of x and y.






Complex (Standard Mathematical Functions)





Graphic Library (2 dimensional)




The library glib  provides a simple interface like old BASIC
to the graphic primitive (draw_obj) of Risa/Asir.








glib_clear








glib_clear()


: 
  Clear the screen. 






glib_flush








glib_flush()


: ;
  Flush the output. 
 (Cfep only. It also set initGL to 1.). 






glib_line








glib_line(X0,Y0,X1,Y1)


: 
  It draws the line [X0,Y0]-- [X1,Y1] with color and shape 

glib_line(X0,Y0,X1,Y1 | color=key0,shape=key1)


: This function allows optional variables 
  color, shape




Example:


 glib_line(0,0,5,3/2 | color=0xff00ff);
           glib_line(0,0,10,0 | shape=arrow);






glib_open








glib_open()


: 
  It starts the ox_plot server and opens a canvas.
  The canvas size is set to Glib_canvas_x X Glib_canvas_y
  (the default value is 400). 
  This function is automatically called when the user calls glib
  functions. 






glib_plot








glib_plot(F)


: 
  It plots an object F on the glib canvas. 




Example 0:


 glib_plot([[0,1],[0.1,0.9],[0.2,0.7],[0.3,0.5],[0.4,0.8]]);





Example 1:


 glib_plot(tan(x));






glib_print








glib_print(X,Y,Text)


: 
  It put a string Text at [X,Y] on the glib canvas. 

glib_print(X,Y,Text | color=key0)


: This function allows optional variables 
  color




Example:


 glib_print(100,100,"Hello Worlds" | color=0xff0000);






glib_ps_form








glib_ps_form(S)


: 
  It returns the PS code generated by executing S
  (experimental). 




Example 0:


 glib_ps_form(quote( glib_line(0,0,100,100) ));





Example 1:


 glib_ps_form(quote([glib_line(0,0,100,100),glib_line(100,0,0,100)]));





References:





 glib_tops 






glib_putpixel








glib_putpixel(X,Y)


: 
  It puts a pixel at [X,Y] with color 

glib_putpixel(X,Y | color=key0)


: This function allows optional variables 
  color




Example:


 glib_putpixel(1,2 | color=0xffff00);






glib_remove_last








glib_remove_last()


: 
  Remove the last object.  glib_flush() should also be called to remove
  the last object.
 (cfep only). 






glib_set_pixel_size








glib_set_pixel_size(P)


: 
  Set the size of putpixel to P. 1.0 is the default.
 (cfep only). 






glib_tops








glib_tops()


: 
  If Glib_ps is set to 1, 
  it returns a postscript program to draw the picture on the canvas. 





References:





 print_output 






glib_window








glib_window(Xmin,Ymin,Xmax,Ymax)


: 
  It generates a window with the left top corner [Xmin,Ymin] and
  the right bottom corner [Xmax,Ymax].
  If the global variable Glib_math_coordinate is set to 1, mathematical
  coordinate system will be employed, i.e., the left top
  corner will have the coordinate [Xmin,Ymax]. 




Example:


 glib_window(-1,-1,10,10);






OpenXM-Contrib General Functions




Functions






ox_check_errors2








ox_check_errors2(p)


::  get a list of error objects on the statck of the server p.







return


List

p


Number





    

  • It gets a list of error objects on the server stack.


  • It does not pop the error objects.






[219] P=sm1.start();
0
[220] sm1.sm1(P," 0 get ");
0
[221] ox_check_errors2(P);
[error([7,4294967295,executeString: Usage:get])]
Error on the server of the process number = 1
To clean the stack of the ox server,
type in ox_pops(P,N) (P: process number, N: the number of data you need to pop)
out of the debug mode.
If you like to automatically clean data on the server stack,
set XM_debug=0;







OXshell Functions




OXshell is a system to execute system commands from ox servers.
As to details, see the files
OpenXM/src/kan96xx/Doc/oxshell.oxw and
OpenXM/doc/Papers/rims-2003-12-16-ja.tex.






oxshell.get_value








oxshell.get_value(NAME,V)


: 
  It get the value of the variable NAME on the server ox_shell. 




Example:


 oxshell.set_value("abc","Hello world!");
           oxshell.oxshell(["cp", "stringIn://abc", "stringOut://result"]);
           oxshell.get_value("result");





References:





 oxshell.oxshell , 
 oxshell.set_value 






oxshell.oxshell








oxshell.oxshell(L)


: 
  It executes command L on a ox_shell server.
  L must be an array.
  The result is the outputs to stdout and stderr. 




Example:


 oxshell.oxshell(["ls"]);





References:





 ox_shell , 
 oxshell.set_value , 
 oxshell.get_value 






oxshell.set_value








oxshell.set_value(NAME,V)


: 
  It set the value V to the variable Name on the server ox_shell. 




Example:


 oxshell.set_value("abc","Hello world!");
           oxshell.oxshell(["cat", "stringIn://abc"]);





References:





 oxshell.oxshell , 
 oxshell.get_value 






Utility Functions




Utility functions provide some usuful functions to access to
the system and to process strings.






util_filter








util_filter(Command,Input)


: 
 It executes the filter program Command with the Input
 and returns the output of the filter as a string. 

util_filter(Command,Input | env=key0)


: This function allows optional variables 
  env




Example:


 util_filter("sort","cat\ndog\ncentipede\n");






util_find_and_replace








util_find_and_replace(W,S,Wnew)


: 
 It replaces W in S by Wnew.
 Arguments must be a list of ascii codes. 






util_find_substr








util_find_substr(W,S)


: 
 It returns the position of W in S.
 If W cannot be found, it returns -1.
 Arguments must be a list of ascii codes. 






util_index








util_index(V)


: 
  It returns the name part and the index part of V. 




Example:


 util_index(x_2_3)





References:





 util_v 






util_load_file_as_a_string








util_load_file_as_a_string(F)


: 
 It reads a file F as a string. 






util_part








util_part(S,P,Q)


: 
 It returns from Pth element to Qth element of S. 






util_read_file_as_a_string








util_read_file_as_a_string(F)


: 
 It reads a file F as a string. 






util_remove_cr








util_remove_cr(S)


: 
 It removes cr/lf/tabs from S.
 Arguments must be a list of ascii codes. 






util_timing








util_timing(Q)


: 
 Show the timing data to execute Q. 




Example:


 util_timing( quote( fctr(x^50-y^50) ));






util_v








util_v(V,L)


: 
  It returns a variable indexed by L. 




Example:


 util_v("x",[1,3]);





References:





 util_index 






util_write_string_to_a_file








util_write_string_to_a_file(Fname,S)


: 
 It writes a string S to a file Fname. 






Other Manuals



This section introduces other manuals in the asir-contrib project.




This section also describes functions that have not yet been classifed.
These will be moved to independent sections in a future.





dsolv (Solving the initial ideal for holonomic systems)



  ../dsolv-html/dsolv-en.html



ok_diff (Okutani's library for differential operators)



  ../ok_diff-html/ok_diff-en.html



ok_dmodule (Okutani's library for D-modules)



  ../ok_dmodule-html/ok_dmodule-en.html



(Plucker relations)



  ../plucker-html/plucker-en.html



pfpcoh (Ohara's library for homology/cohomology groups for p F q )



  ../pfpcoh-html/pfpcoh-en.html



(gnuplot ox server for graphics)



  ../gnuplot-html/gnuplot-en.html



mathematica (Mathematica (TM) ox server)



  ../mathematica-html/mathematica-en.html



om (om (java) ox server for translating CMO and OpenMath)



  ../om-html/om-en.html



phc (PHC ox server for solving systems of algebraic equations by the homotopy method)



  ../phc-html/phc-en.html



sm1 (Kan/sm1 ox server for the ring of differential operators)



  ../sm1-html/sm1-en.html



tigers (tigers ox server for toric universal Grobner bases)



  ../tigers-html/tigers-en.html



f_res (Comuting resultant)



  ../f_res-html/f_res-en.html



mt_graph (3D grapher)



  ../mk_graph-html/mk_graph-en.html



noro_mwl (Mordel Weil Lattice)



  ../noro_mwl-html/noro_mwl-en.html



nn_ndbf (local b-function)



  ../nn_ndbf-html/nn_ndbf-en.html





[[todo_parametrize]]



   todo_parametrize/todo_parametrize_toc




With loading the file todo_parametrize/todo_parametrize.rr
the function paramerize is installed.
The function finds a parametric expression of a given rational curve.
As to details, see
See section `' in A package for algebraic curves (in Japanese).




[1205] load("todo_parametrize/todo_parametrize.rr");
1
[1425] parametrize(y^2-x^3);
[155*t^2+20*t+1,720*t^4+1044*t^3+580*t^2,155*t^4+20*t^3+t^2,(-x)/(y)]
[1426] parametrize(y^2+x^3);
[-t,1,t^3,(-x)/(y)]






taji_alc




With loading the file taji_alc.rr
functions for algebraic local cohomology groups in one variable are imported.




import("taji_alc.rr");
taji_alc.laurent_expansion(x,(x-1)^3);






Manual and papers which are not written in texinfo.




Links to manuals and papers related to files and commands in asir-contrib are
at 
OpenXM documents.





Index



Jump to:
b
-
g
-
m
-
n
-
o
-
p
-
u



b



  • base_cancel

  • base_choose

  • base_flatten

  • base_intersection

  • base_memberq

  • base_permutation

  • base_position

  • base_prune

  • base_replace

  • base_replace_n

  • base_set_minus

  • base_set_union

  • base_subsetq

  • base_subsets_of_size

    • 

    g
    


  • glib_clear

  • glib_flush

  • glib_line

  • glib_open

  • glib_plot

  • glib_print

  • glib_ps_form

  • glib_putpixel

  • glib_remove_last

  • glib_set_pixel_size

  • glib_tops

  • glib_window

    • 

    m
    


  • matrix_clone

  • matrix_det

  • matrix_diagonal_matrix

  • matrix_eigenavalues

  • matrix_identity_matrix

  • matrix_image

  • matrix_inner_product

  • matrix_inverse

  • matrix_kernel

  • matrix_list_to_matrix

  • matrix_matrix_to_list

  • matrix_rank

  • matrix_solve_linear

  • matrix_submatrix

  • matrix_transpose

    • 

    n
    


  • number_abs

  • number_ceiling

  • number_factor

  • number_floor

  • number_imaginary_part

  • number_is_integer

  • number_real_part

    • 

    o
    


  • ox_check_errors2

  • oxshell.get_value

  • oxshell.oxshell

  • oxshell.set_value

    • 

    p
    


  • poly_degree

  • poly_elimination_ideal

  • poly_expand

  • poly_factor

  • poly_gcd

  • poly_grobner_basis

  • poly_hilbert_polynomial

  • poly_initial

  • poly_initial_coefficients

  • poly_initial_term

  • poly_solve_linear

  • poly_sort

  • print_dvi_form

  • print_em

  • print_gif_form

  • print_input_form

  • print_open_math_tfb_form

  • print_open_math_xml_form

  • print_output

  • print_ox_rfc100_xml_form

  • print_png_form

  • print_terminal_form

  • print_tex_form

  • print_tfb_form

  • print_xdvi_form

  • print_xv_form

    • 

    u
    


  • util_filter

  • util_find_and_replace

  • util_find_substr

  • util_index

  • util_load_file_as_a_string

  • util_part

  • util_read_file_as_a_string

  • util_remove_cr

  • util_timing

  • util_v

  • util_write_string_to_a_file

    • 
Jump to:

    
@vfill @eject

    
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