# Worksheet to generate the random examples in Gao, Kaltofen, May, Yang,
# & Zhi:
# 'Approximate Factorization of Multivariate polynomials via
# differential Equations'
> 
# The routine used to generate random polynomials 
# This routine generates a polynomial with random dense factors of the
# given degrees with integer coefficents smaller than N.  Another random
# dense polynomial (noise) with the same degree as the product and
# coeffients smaller than 10^(-e) is generated then added to the
# product.
> randp:=proc(vars::list               # the variables 
>            ,list_deg_factor::list    # the list of the degree of the
> factors   
>            ,N::integer               # absolute maximum of
> coefficients of the factors
>            ,e::integer               # the perturbation size: 10^(-e)
>            ,dn)                      # Optional: number between 0 and
> 1 giving the how dense the perturbation should be compared to product
> default = 1.
>  
>        local i,f0,fac,pert_pol,res, deg_pert, dd;
> 
> if nops([args]) = 4 then dd:=1 else dd:=dn end if;
> 
>        deg_pert := convert(list_deg_factor, `+`);
>        
>        f0:=1;
>        for i from 1 to nops(list_deg_factor) do                       
>                                                 
> fac:=randpoly(vars,degree=list_deg_factor[i],coeffs=rand(-N..N),dense)
> ;
>           f0:=f0*fac;
>        end do;
> 
> dd:=round(nops(convert(expand(f0),list))*dd);
> 
> print('noise_terms' = dd);
>       
> pert_pol:=randpoly(vars,degree=deg_pert,coeffs=rand(-10^(e)..10^(e)),
> terms=dd); 
> 
>        return(f0,'`*`'(10^(-e)*norm(expand(f0),2)/norm(pert_pol,
> 2),pert_pol));
>        #return(res);
> 
> end proc:
# The Random Examples
# Set the path to the subdirectory where the example files will be
# stored
> path:="examples\/";

                         path := "examples/"

# Example 6: Three Factors (6, 6, 10), Moderate Noise -5
# This example is a polynomial with three factors with coefficients in
# [-5, 5], two factors with total degree 6 and one with total degree 10;
# the added noise is of order 10^(-5).
#  
> cleanF6, noiseF6:=randp([x,y],[6,6,10],5,5,.25):
> 
> save cleanF6, noiseF6, cat(path,"exF06");
> 

                           noise_terms = 69

# Example 7: Two Factors (9, 7),  Moderate Noise -4
# This example is a polynomial with two factors with coefficients in
# [-5, 5], one factor with total degree 9and one with total degree 7;
# the added noise is of order 10^(-4).
# 
> cleanF7, noiseF7:=randp([x,y],[9,7],5,4,.25):

                           noise_terms = 38

> save cleanF7, noiseF7, cat(path,"exF07");
> 
#  Example 8: Five Factors (4,4,4,4,4), Moderate Noise -5
# This example is a polynomial with five  factors with coefficients in
# [-5, 5] all of total degree 4;  the added noise is of order 10^(-5).
# 
> cleanF8, noiseF8:=randp([x,y],[4,4,4,4,4],5,5,.25):

                           noise_terms = 57

> save cleanF8, noiseF8, cat(path,"exF08");
> 
# Example 9: Three Factors (3,3,3), Large Noise -1
# This example is a polynomial with three  factors with coefficients in
# [-5, 5] all of total degree 3;  the added noise is of order 10^(-1).
# 
> cleanF9, noiseF9:=randp([x,y],[3,3,3],5,1,.25):

                           noise_terms = 14

> save cleanF9, noiseF9, cat(path,"exF09");
> 
#  Examples 10, 11 & 12: One polynomial (12, 7, 5) with perturbations
# added:
# This example we generate a polynomial with three factors, then add the
# same noise first with order -2 then with order -1 to generate examples
# 11 and 12 respectively.
# 
# Example 10: Three Factors (12, 7, 5), Large Noise -5
> cleanF10, noiseF10 := randp([x,y], [12,7,5], 5, 5, .05):

                           noise_terms = 16

> save cleanF10, noiseF10, cat(path,"exF10");
> 
# Example 11: Three Factors (12, 7, 5), Large Noise -5
> cleanF11, noiseF11 := randp([x,y], [12,7,5], 5, 5, .50):

                          noise_terms = 161

> cleanF11 := cleanF10:
> save cleanF11, noiseF11, cat(path,"exF11");
> 
# Example 12: Three Factors (12, 7, 5), Larger Noise -3
> cleanF12 := cleanF11:
> noiseF12 := '`*`'(10^2, noiseF11):
> 
> save cleanF12, noiseF12, cat(path,"exF12");
> 
# Example 13: Two Factors (5, 5^2), Moderate Noise -5
# This example is a polynomial with two  factors with coefficients in
# [-5, 5] both of total degree 8;  the added noise is of order 10^(-1).
# 
> cleanF13, noiseF13:=randp([x,y],[5,5,5],5,5,.25):

                           noise_terms = 34

> cleanF13:=op(1,cleanF13)*op(2,cleanF13)^2:
> save cleanF13, noiseF13, cat(path,"exF13");
> 
# Example 15: Two Factors (5, 5),  Moderate Noise -5, three vars
> cleanF15, noiseF15 := randp([x,y,z], [5,5], 5, 5, .25):

                           noise_terms = 70

> 
> save cleanF15, noiseF15, cat(path,"exF15");
> 
# Example 16: Two Factors (18, 18),  Moderate Noise -6
# This example is a polynomial with two  factors with coefficients in
# [-5, 5] both of total degree 18;  the added noise is of order 10^(-6).
# 
> cleanF16, noiseF16:=randp([x,y],[18,18],5,6,.25):

                          noise_terms = 175

> save cleanF16, noiseF16, cat(path,"exF16");
# Example 17: Two Factors (18, 18),  Moderate Noise -6 in 99% of the
# terms
# This example is a polynomial with two  factors with coefficients in
# [-5, 5] both of total degree 18;  the added noise is of order 10^(-6).
# 
> cleanF17, noiseF17:=randp([x,y],[18,18],5,6,.99):

                          noise_terms = 691

> save cleanF17, noiseF17, cat(path,"exF17");
> 
# Example 18: Complex example : Two Factors(6,6), Moderate Noise -6
> fa18:=randpoly([x,y],degree=6,coeffs=rand(-5..5),dense)
>       +I*randpoly([x,y],degree=6,coeffs=rand(-5..5),dense):
> ga18:=randpoly([x,y],degree=6,coeffs=rand(-5..5),dense)
>       +I*randpoly([x,y],degree=6,coeffs=rand(-5..5),dense):
> cleanF18:=fa18*ga18:
> noiseF18:='`*`'(10^(-6),randpoly([x,y],degree=12,coeffs=rand(-5..5),de
> nse)
>            +I*randpoly([x,y],degree=12,coeffs=rand(-5..5),dense)):
> save cleanF18, noiseF18, cat(path,"exF18");
> 
#