# Worksheet to execute the examples in Gao, Kaltofen, May, Yang, & Zhi:
# 
# 'Approximate Factorization of Multivariate polynomials via
# differential Equations'
> # Factorization routines:
> read("multifac_1.2.mpl"):
> # rand_path:="debug_exs\/": # for debugging the worksheet
> rand_path:="examples\/": # examples in the paper
> infolevel[multifac]:=2;

                       infolevel[multifac] := 2

# A note about backward_error(F, G, [vars]):
# We compute c so that ||F-c*G||_2 is as small as possible.  The output
# is ||F - c*G||_2 / ||F||_2.
# 
# Example 16 has been moved to it's own worksheet.
# Examples 10, 11 & 12: One polynomial (12, 7, 5) with perturbations of
# different orders (-5 and -1) added:
# Example 10: Three Factors (12, 7, 5), Large Noise -5, very sparse
# noise: 5%
> read(cat(rand_path,"exF10")):
> F10 := expand(evalf(cleanF10 + noiseF10)):
> norm(evalf(noiseF10),2) / norm(F10,2);
> ceil(log10(%));

                                         -5
                          0.9999995252 10


                                  -5

> t10:=time():
> facF10:=appfac(F10,[x,y]):
> t10:=time()-t10;
` the biggest gap, the last r-th singular values and the number of
factors `, 658.3729016, .8558217297e-6, 3
` The time for computing the number of factors*****`, 57.864
`The time for the entire factorization******`, 1092.101

                           t10 := 1092.141

> nops(facF10);

                                  3

> berror10:=backward_error(F10, convert(facF10, `*`), [x,y]);
> ceil(log10(%));

                           c = -1064.869729


                     berror10 := 0.0001564296376


                                  -3

> save facF10, t10, berror10, cat(rand_path,"exF10-factors");
# Example 11: Three Factors (12, 7, 5), Large Noise -5
> read(cat(rand_path,"exF11")):
> F11 := expand(evalf(cleanF11 + noiseF11)):
> norm(evalf(noiseF11),2) / norm(F11,2);
> ceil(log10(%));

                           0.00001021859647


                                  -4

> t11:=time():
> facF11:=appfac(F11,[x,y]):
> t11:=time()-t11;
` the biggest gap, the last r-th singular values and the number of
factors `, 833.5367031, .6758979173e-6, 3
` The time for computing the number of factors*****`, 50.202
`The time for the entire factorization******`, 1659.746

                           t11 := 1659.786

> nops(facF11);

                                  3

> berror11:=backward_error(F11, convert(facF11, `*`), [x,y]);
> ceil(log10(%));

                           c = 3400.278174


                     berror11 := 0.0002870445337


                                  -3

> save facF11, t11, berror11, cat(rand_path,"exF11-factors");
# Example 12: Three Factors (12, 7, 5), Larger Noise -3
> #read("exF12"):
> F12 := expand(evalf(cleanF11 + 10.^2*noiseF11)):
> norm(evalf(10.^2.*noiseF11),2) / norm(F12,2);
> ceil(log10(%));

                            0.001021825208


                                  -2

> t12:=time():
> facF12:=appfac(F12,[x,y]):
> t12:=time()-t12;
` the biggest gap, the last r-th singular values and the number of
factors `, 8.340144257, .6756157841e-4, 3
` The time for computing the number of factors*****`, 51.323
`The time for the entire factorization******`, 3163.960

                           t12 := 3164.010

> nops(facF12);

                                  3

> berror12:=backward_error(F12, convert(facF12, `*`), [x,y]);
> ceil(log10(%));

                           c = 838.1890412


                      berror12 := 0.05478797560


                                  -1

> save facF12, t12, berror12, cat(rand_path,"exF12-factors");
> 
# 
#