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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 77 "Worksheet to generate the \+
random examples in Gao, Kaltofen, May, Yang, & Zhi:" }}{PARA 3 "" 0 "
" {TEXT -1 82 "'Approximate Factorization of Multivariate polynomials \+
via differential Equations'" }{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 48 "The routine used to generate random polynomials " }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 276 "This routine generates a polynomi
al with random dense factors of the given degrees with integer coeffic
ents smaller than N.  Another random dense polynomial (noise) with the
 same degree as the product and coeffients smaller than 10^(-e) is gen
erated then added to the product." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1126 "randp:=proc(vars::list               # the variable
s \n           ,list_deg_factor::list    # the list of the degree of t
he factors   \n           ,N::integer               # absolute maximum
 of coefficients of the factors\n           ,e::integer               \+
# the perturbation size: 10^(-e)\n           ,dn)                     \+
 # Optional: number between 0 and 1 giving the how dense the perturbat
ion should be compared to product default = 1.\n \n       local i,f0,f
ac,pert_pol,res, deg_pert, dd;\n\nif nops([args]) = 4 then dd:=1 else \+
dd:=dn end if;\n\n       deg_pert := convert(list_deg_factor, `+`);\n \+
      \n       f0:=1;\n       for i from 1 to nops(list_deg_factor) do
                                                                      \+
   fac:=randpoly(vars,degree=list_deg_factor[i],coeffs=rand(-N..N),den
se);\n          f0:=f0*fac;\n       end do;\n\ndd:=round(nops(convert(
expand(f0),list))*dd);\n\nprint('noise_terms' = dd);\n       pert_pol:
=randpoly(vars,degree=deg_pert,coeffs=rand(-10^(e)..10^(e)), terms=dd)
; \n\n       return(f0,'`*`'(10^(-e)*norm(expand(f0),2)/norm(pert_pol,
 2),pert_pol));\n       #return(res);\n\nend proc:" }}}}{SECT 0 {PARA 
4 "" 0 "" {TEXT 264 10 "The Random" }{TEXT -1 1 " " }{TEXT 256 8 "Exam
ples" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 71 "Set the path to the subdi
rectory where the example files will be stored" }}{EXCHG {PARA 260 "> \+
" 0 "" {MPLTEXT 1 0 19 "path:=\"examples\\/\";" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%pathGQ*examples/6\"" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT 257 54 "Example 6: Three Factors (6, 6, 10), Moderate Noise -5" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "This example is a polynomial wi
th 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).\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF6, noiseF6:=randp
([x,y],[6,6,10],5,5,.25):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "save
 cleanF6, noiseF6, cat(path,\"exF06\");" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#p" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT 258 9 "Example 7" }{TEXT -1 40 ": Two Fa
ctors (9, 7),  Moderate Noise -4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
172 "This example is a polynomial with two factors with coefficients i
n [-5, 5], one factor with total degree 9and one with total degree 7; \+
the added noise is of order 10^(-4).\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "cleanF7, noiseF7:=randp([x,y],[9,7],5,4,.25):" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#Q" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 41 "save cleanF7, noiseF7, cat(path,\"exF07\"
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 
"" 0 "" {TEXT -1 1 " " }{TEXT 259 54 "Example 8: Five Factors (4,4,4,4
,4), Moderate Noise -5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "This ex
ample is a polynomial with five  factors with coefficients in [-5, 5] \+
all of total degree 4;  the added noise is of order 10^(-5).\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cleanF8, noiseF8:=randp([x,y
],[4,4,4,4,4],5,5,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_te
rmsG\"#d" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "save cleanF8, n
oiseF8, cat(path,\"exF08\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 260 9 "Example 9" }{TEXT -1 
39 ": Three Factors (3,3,3), Large Noise -1" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 139 "This example is a polynomial with three  factors with co
efficients in [-5, 5] all of total degree 3;  the added noise is of or
der 10^(-1).\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cleanF9, \+
noiseF9:=randp([x,y],[3,3,3],5,1,.25):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%,noise_termsG\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "save cleanF9, noiseF9, cat(path,\"exF09\");" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 74 " Exa
mples 10, 11 & 12: One polynomial (12, 7, 5) with perturbations added:
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "This example we generate a po
lynomial with three factors, then add the same noise first with order \+
-2 then with order -1 to generate examples 11 and 12 respectively.\n" 
}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 52 "Example 10: Three Factors (12,
 7, 5), Large Noise -5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "cl
eanF10, noiseF10 := randp([x,y], [12,7,5], 5, 5, .05):" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%,noise_termsG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "save cleanF10, noiseF10, cat(path,\"exF10\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "
" {TEXT 261 52 "Example 11: Three Factors (12, 7, 5), Large Noise -5" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "cleanF11, noiseF11 := rand
p([x,y], [12,7,5], 5, 5, .50):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,n
oise_termsG\"$h\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cleanF
11 := cleanF10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cle
anF11, noiseF11, cat(path,\"exF11\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 262 53 "Example 12
: Three Factors (12, 7, 5), Larger Noise -3" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 56 "cleanF12 := cleanF11:\nnoiseF12 := '`*`'(10^2, nois
eF11):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "save cleanF12, noiseF12, cat(path,\"exF12\");" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT 263 10 "Example 13" }{TEXT -1 41 ": Two Factors (5, 5^2), M
oderate Noise -5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "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).\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF13, noiseF13:=randp([x,y],[5,
5,5],5,5,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#M
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "cleanF13:=op(1,cleanF13
)*op(2,cleanF13)^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save
 cleanF13, noiseF13, cat(path,\"exF13\");" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 62 "Example 1
5: Two Factors (5, 5),  Moderate Noise -5, three vars" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 55 "cleanF15, noiseF15 := randp([x,y,z], [5,5
], 5, 5, .25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#q
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "save cleanF15, noiseF15, cat(path,\"exF15\");
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "
" 0 "" {TEXT 265 10 "Example 16" }{TEXT -1 42 ": Two Factors (18, 18),
  Moderate Noise -6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "This examp
le is a polynomial with two  factors with coefficients in [-5, 5] both
 of total degree 18;  the added noise is of order 10^(-6).\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF16, noiseF16:=randp([x
,y],[18,18],5,6,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_term
sG\"$v\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF16, \+
noiseF16, cat(path,\"exF16\");" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 266 
10 "Example 17" }{TEXT -1 62 ": Two Factors (18, 18),  Moderate Noise \+
-6 in 99% of the terms" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "This ex
ample is a polynomial with two  factors with coefficients in [-5, 5] b
oth of total degree 18;  the added noise is of order 10^(-6).\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF17, noiseF17:=randp([x
,y],[18,18],5,6,.99):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_term
sG\"$\"p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF17, \+
noiseF17, cat(path,\"exF17\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 65 "Example 18: Complex \+
example : Two Factors(6,6), Moderate Noise -6" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 115 "fa18:=randpoly([x,y],degree=6,coeffs=rand(-5..5
),dense)\n      +I*randpoly([x,y],degree=6,coeffs=rand(-5..5),dense):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "ga18:=randpoly([x,y],d
egree=6,coeffs=rand(-5..5),dense)\n      +I*randpoly([x,y],degree=6,co
effs=rand(-5..5),dense):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"cleanF18:=fa18*ga18:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "n
oiseF18:='`*`'(10^(-6),randpoly([x,y],degree=12,coeffs=rand(-5..5),den
se)\n           +I*randpoly([x,y],degree=12,coeffs=rand(-5..5),dense))
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF18, noiseF1
8, cat(path,\"exF18\");" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "2 12 0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }