Polynomy a racionalni funkce
Polynomy a racionalni funkce
> restart;
> p1:=-3*x+7*x^2-3*x^3+7*x^4;#col
lected form
:=p1 - + - +3 x 7 x2
3 x3
7 x4
> type(p1, 'polynom');
true
> whattype(p1);
+
> ved:=lcoeff(p1);
:=ved 7
> stupen:=degree(p1);
:=stupen 4
Scitani a nasobeni polynomu
> p2:=5*x^5+3*x^3+x^2-2*x+1;#expa
nded canonical form
:=p2 + + - +5 x5
3 x3
x2
2 x 1
> 2*p1-3*p2+3;
- + -11 x2
15 x3
14 x4
15 x5
> p1*p2;
( )- + - +3 x 7 x2
3 x3
7 x4
( )+ + - +5 x5
3 x3
x2
2 x 1
> v1:=expand(%);
v1 17 x6
11 x4
20 x3
13 x2
3 x- + - + -:=
56 x7
4 x5
15 x8
35 x9
+ + - +
Maple neprovadi automaticky roznasobeni, coz
vypada jako nevyhoda, ale ve skutecnosti neni.
> (3*x+5)^10;
( )+3 x 5 10
> expand((3*x+5)^10);
59049 x10
984150 x9
7381125 x8
+ +
32805000 x7
95681250 x6
+ +
191362500 x5
265781250 x4
+ +
253125000 x3
158203125 x2
+ +
58593750 x 9765625+ +
Cleny polynomu nejsou automaticky usporadany
(z pametovych duvodu). Usporadani provedeme
pomoci procedury sort (sestupne vzhledem ke
stupni polynomu).
> sort(v1);
35 x9
15 x8
56 x7
17 x6
4 x5
11 x4
- + - + +
20 x3
13 x2
3 x- + Sort
meni interni datovou strukturu.
> restart;
> p:=1+x+x^3+x^2;
:=p + + +1 x x3
x2
> x^3+x^2+x+1;
+ + +1 x x3
x2
> q:=(x-1)*(x^3+x^2+x+1);
:=q ( )-x 1 ( )+ + +1 x x3
x2
> sort(p);
+ + +x3
x2
x 1
> q;
( )-x 1 ( )+ + +x3
x2
x 1
> ?sort
Urcovani koeficientu.
> p1:=-3*x+7*x^2-3*x^3+7*x^4;p2:=
5*x^5+3*x^3+x^2-2*x+1;
:=p1 - + - +3 x 7 x2
3 x3
7 x4
:=p2 + + - +5 x5
3 x3
x2
2 x 1
> coeff(p2,x^3);
3
> coeffs(p2, x, 'pow');pow;
, , , ,1 5 3 1 -2
, , , ,1 x5
x3
x2
x
> coeff(x^2-x*(x-1), x);
1
Prikaz coeffs pozaduje polynom v roznasobenem
tvaru (collected form).
> ?coeffs
> p:=x^3-(x-3)*(x^2+x)+1;
:=p - +x3
( )-x 3 ( )+x2
x 1
> coeffs(p);
Error, invalid arguments to coeffs
> coeffs(expand(p));
, ,1 2 3
Jednou ze zakladnich operaci pro polynomy je
deleni se zbytkem. Maple ma k tomuto ucelu dve
procedury: quo a rem.
> q:=quo(p2,p1,x, 'r');
:=q +
5 x
7
15
49
> r;
- + -1
53
49
x3
x2
53
49
x
> testeq(p2=(q*p1+r));
true
> rem(p2,p1,x,'q');
- + -1
53
49
x3
x2
53
49
x
> q;
+
5 x
7
15
49
> gcd(p1,p2); #nejvetsi spolecny
delitel polynomu p1 a p2
+x2
1
> pol:=expand(p1*p2);
pol 17 x6
11 x4
20 x3
13 x2
3 x- + - + -:=
56 x7
4 x5
15 x8
35 x9
+ + - +
> expand(sqrt(2+x)*sqrt(3+x));
+2 x +3 x
> expand(combine(sqrt(2+x)*sqrt(3
+x), symbolic));
+ +6 x2
5 x
Komplikovanejsi jsou algoritmy pro rozklad
polynomu na soucin. Procedura factor zapisuje
polynom s racionalnimi koeficienty ve tvaru
soucinu ireducibilnich polynomu nad Q.
> factor(pol);
x ( )-7 x 3 ( )- +5 x3
2 x 1 ( )+x2
1
2
Zapis factor(polynom, pole) provadi rozklad nad
algebraickym polem.
> factor(pol, I);
x ( )- +5 x3
2 x 1 ( )-7 x 3 ( )-x I 2
( )+x I 2
> p:=x^2+1;
:=p +x2
1
> factor(p);
+x2
1
> irreduc(p);
true
> factor(p,I);
( )-x I ( )+x I
> irreduc(p,I);
false
Silnejsim nastrojem pro rozklady je procedura
split z knihovny polytools.
> pol:=8*x^3-12*x;
:=pol -8 x3
12 x
> factor(pol);
4 x ( )-2 x2
3
> polytools[split](pol,x);
8





+x
1
2
( )RootOf -_Z2
6 x





-x
1
2
( )RootOf -_Z2
6
> convert(%,'radical');
8





+x
6
2
x





-x
6
2
> polytools[split](x^2+1,x);
( )-x ( )RootOf +_Z2
1
( )+x ( )RootOf +_Z2
1
> convert(%,'radical');
( )-x I ( )+x I
Polynomy vice promennych
> pol:=6*x*y^5+12*y^4+14*y^3*x^3
-15*x^2*y^3 + 9*x^3*y^2 -
30*x*y^2 - 35*x^4*y + 18*y*x^2
+21*x^5;
pol 6 x y5
12 y4
14 y3
x3
15 x2
y3
+ + -:=
9 x3
y2
30 x y2
35 x4
y 18 y x2
21 x5
+ - - + +
> sort(pol, [x,y], 'plex'); #Pure
LEXicographic ordering
21 x5
35 x4
y 14 x3
y3
9 x3
y2
15 x2
y3
- + + -
18 x2
y 6 x y5
30 x y2
12 y4
+ + - +
> sort(pol, [y,x], 'plex'); #Pure
LEXicographic ordering
6 y5
x 12 y4
14 y3
x3
15 y3
x2
9 y2
x3
+ + - +
30 y2
x 35 y x4
18 y x2
21 x5
- - + +
> sort(pol, [x,y]); #total degree
term ordering
14 x3
y3
6 x y5
21 x5
35 x4
y 9 x3
y2
+ + - +
15 x2
y3
12 y4
18 x2
y 30 x y2
- + + Nebo
se muzeme na predchazejici polynom divat
jako na polynom v promenne x, polynomy v y
jsou pak koeficienty.
> collect(pol, x);
21 x5
35 x4
y ( )+14 y3
9 y2
x3
- +
( )-18 y 15 y3
x2
( )- +30 y2
6 y5
x 12 y4
+ + +
A obracene:
> collect(pol, y);
6 x y5
12 y4
( )- +15 x2
14 x3
y3
+ +
( )-9 x3
30 x y2
( )- +35 x4
18 x2
y 21 x5
+ + +
Priklady na praci s polynomy vice promennych.
> coeff(pol, x^3);
+14 y3
9 y2
> coeffs(pol, x, 'powers');
powers;
12 y4
-35 y 21 +14 y3
9 y2
-18 y 15 y3
, , , , ,
- +30 y2
6 y5
, , , , ,1 x4
x5
x3
x2
x
> settime:=time():
> factor(pol);
( )- +3 x2
5 x y 2 y3
( )+ +7 x3
6 y 3 x y2
> cpu_time:=time()-settime;
:=cpu_time 0.016
Racionalni funkce
> r:=(x^2+3*x+2)/ (x^2+5*x+6);
:=r
+ +x2
3 x 2
+ +6 x2
5 x
> type(r, 'ratpoly');
true
> whattype(r);
*
> numer(r), denom(r); #citatel a
jmenovatel
,+ +x2
3 x 2 + +6 x2
5 x
Narozdil od racionalnich cisel Maple neprovadi
automaticke zjednoduseni.
Zjednoduseni provedeme prikazem normal (tak,
ze gcd(citatel, jmenovatel)=1).
> r;
+ +x2
3 x 2
+ +6 x2
5 x
> normal(r);
+x 1
+3 x
Zjednoduseni se provede automaticky pouze v
pripade, ze Maple okamzite pozna spolecne
cleny.
> ff:=(x-1)*numer(r);
:=ff ( )-x 1 ( )+ +x2
3 x 2
> gg:=(x-1)*denom(r);
:=gg ( )-x 1 ( )+ +6 x2
5 x
> ff/gg;
+ +x2
3 x 2
+ +6 x2
5 x
> expand(ff)/gg;
+ - -x3
2 x2
x 2
( )-x 1 ( )+ +6 x2
5 x
> (x^(100)-1)/(x-1);
-x100
1
-x 1
> normal(%);
1 x4
x5
x3
x2
x x6
x7
x8
x9
+ + + + + + + + +
x87
x86
x85
x90
x89
x88
x96
x95
+ + + + + + + +
x93
x84
x83
x82
x92
x91
x94
x99
+ + + + + + + +
x97
x98
x81
x79
x78
x77
x80
x76
+ + + + + + + +
x74
x73
x72
x71
x70
x75
x69
x67
+ + + + + + + +
x66
x65
x64
x63
x62
x61
x60
x68
+ + + + + + + +
x59
x57
x56
x55
x54
x53
x52
x51
+ + + + + + + +
x50
x49
x48
x47
x46
x58
x45
x43
+ + + + + + + +
x42
x41
x40
x39
x38
x37
x36
x35
+ + + + + + + +
x34
x33
x32
x31
x30
x29
x28
x27
+ + + + + + + +
x26
x44
x25
x23
x22
x21
x20
x19
+ + + + + + + +
x18
x17
x16
x15
x14
x13
x12
x11
+ + + + + + + +
x10
x24
+ +
> f:=161*y^3+333*x*y^2+184*y^2+16
2*x^2*y+144*x*y+77*y+99*x+88:
> g:=49*y^2+28*x^2*y+63*x*y+147*y
+36*x^3+32*x^2+117*x+104:
> racfce:=f/g;
racfce 161 y3
333 x y2
184 y2
+ +(:=
162 x2
y 144 x y 77 y 99 x 88+ + + + + ) (
49 y2
28 x2
y 63 x y 147 y 36 x3
32 x2
+ + + + +
117 x 104+ + )
> normal(racfce);
+ +18 x y 23 y2
11
+ +4 x2
7 y 13
Rozklad na parcialni zlomky
> q:=(x^3+x^2-x+1)/p1;
:=q
+ - +x3
x2
x 1
- + - +3 x 7 x2
3 x3
7 x4
> convert(q, 'parfrac', x);
- + +
1
3 x
143
87 ( )-7 x 3
+7 x 3
29 ( )+x2
1
> convert(q,'parfrac',x,real);
0.2348111658
-x 0.4285714286
+0.1034482759 0.2413793105 x
+x2
1.
+
0.3333333334
x
>
convert(%, rational);
+ -
143
609





-x
3
7
+
7 x
29
3
29
+x2
1
1
3 x
> convert(q,'parfrac',x,I);
+ - +
+
7
58
3
58
I
+x I
-
7
58
3
58
I
-x I
1
3 x
143
87 ( )-7 x 3
> convert(q, 'fullparfrac', x);
1
3 x
143
609





-x
3
7
- +















=_ ( )RootOf +_Z
2
1
- +
3 _
58
7
58
-x _
+
> convert(%, radical);
- + + +
1
3 x
143
609





-x
3
7
-
7
58
3
58
I
-x I
+
7
58
3
58
I
+x I
> 1/(x^4-5*x^2+6);
1
- +x4
5 x2
6
> convert(%,parfrac,x);
-
1
-x2
3
1
-x2
2
> convert(%,parfrac,x,sqrt(2));
+ -
2
4 ( )+x 2
1
-x2
3
2
4 ( )-x 2
> convert(%,parfrac,x,{sqrt(2),sq
rt(3)});
2
4 ( )+x 2
3
6 ( )+x 3
2
4 ( )-x 2
- -
3
6 ( )-x 3
+
> ratfun:=(x-a)/(x^5+b*x^4-c*x^2-
b*c*x);
:=ratfun
-x a
+ - -x5
b x4
c x2
b c x
> convert(ratfun, 'parfrac', x);
- - - + +c x2
x2
b2
a b c x x c a b2
c a b c
( )-x3
c ( )+b3
c c
- -b a
( )+x b b ( )+b3
c
a
x b c
+ +
Usmerneni:
> 2/(2-sqrt(2));
2
-2 2
> rationalize(%);
+2 2
> z/(1+sqrt(x));
z
+1 x
> rationalize(%);
z ( )- +1 x
-x 1
Poznamky k manipulaci s polynomy a
racionalnimi funkcemi
> souc:=(x^2-x)*(x^2+2*x+1);
:=souc ( )-x2
x ( )+ +x2
2 x 1
> expform:=expand(souc);
:=expform + - -x4
x3
x2
x
> soucin:=(a+b)*(c+d);
:=soucin ( )+b a ( )+c d
> expand(soucin);
+ + +b c b d c a a d
Pokud nechceme roznasobovat (c+d), musime to
Maplu sdelit uvedenim parametru v procedure
expand.
> expform:=expand(soucin, c+d);
:=expform +( )+c d b ( )+c d a
> (x+1)^3;
( )+x 1 3
> expand(%);
+ + +x3
3 x2
3 x 1
> power:=(x+1)^(-2);
:=power
1
( )+x 1 2
> expand(power);
1
( )+x 1 2
Zaporne mocniny Maple neexpanduje. Musime
provest umocneni jmenovatele zvlast.
> numer(power)/expand(denom(power
));
1
+ +x2
2 x 1
> (x+1)^2/((x^2+x)*x);
( )+x 1 2
( )+x2
x x
Vsimneme si efektu pouziti expand na racionalni
lomenou funkci.
> expand(%);
+ +
x
+x2
x
2
+x2
x
1
( )+x2
x x
> expand(numer(%%))/expand(denom(
%%));
+ +x2
2 x 1
+x3
x2
FACTOR
Factor provadi rozklad polynomu na soucin
korenovych cinitelu nad racionalnimi cisly.
Zapis factor(polynom, pole) provadi rozklad nad
algebraickym polem.
> q:=x^2+9/25;
:=q +x2
9
25
> factor(q,I);
( )-5 x 3 I ( )+5 x 3 I
25
> pol:=8*x^3-12*x;
:=pol -8 x3
12 x
Silnejsim nastrojem je procedura split (z
knihovny polytools):
> polytools[split](pol,x);
8





+x
1
2
( )RootOf -_Z2
6 x





-x
1
2
( )RootOf -_Z2
6
> convert(%,'radical');
8





+x
6
2
x





-x
6
2
> factor(pol,sqrt(6));
2 x ( )+2 x 6 ( )-2 x 6
> polytools[split](x^2+1,x);
( )-x ( )RootOf +_Z2
1
( )+x ( )RootOf +_Z2
1
> convert(%,'radical');
( )-x I ( )+x I
NORMAL
> (x-1)*(x+2)/((x+1)*x)+(x-1)/(1+
x)^2;
+
( )-x 1 ( )+2 x
x ( )+x 1
-x 1
( )+x 1 2
Vykraceni spolecnych clenu z citatele a
jmenovatele, prevod na spolecneho jmenovatele:
> normal(%);
( )-x 1 ( )+ +2 4 x x2
x ( )+x 1 2
> normal(%, expanded);
- + + -2 x 3 x2
x3
2
+ +x3
2 x2
x
> racfce:=(x^4+x^3-4*x^2-4*x)/(x^
3+x^2-x-1);
:=racfce
+ - -x4
x3
4 x2
4 x
+ - -x3
x2
x 1
Vsimnete si efektu pouziti prikazu expand,
normal, factor v nasledujici posloupnosti
prikazu:
> factor(racfce);
x ( )-x 2 ( )+2 x
( )-x 1 ( )+x 1
> factor(numer(racfce))/sort(expa
nd(denom(racfce)));
x ( )-x 2 ( )+2 x ( )+x 1
+ - -x3
x2
x 1
> sort(expand(numer(racfce)))/fac
tor(denom(racfce));
+ - -x4
x3
4 x2
4 x
( )-x 1 ( )+x 1 2
> sort(normal(racfce,
'expanded'));
-x3
4 x
-x2
1
Nekolik poznamek k praci se systemem
> restart;
Vice informaci o tom, jak system pracuje,
dosahneme nastavenim promenne printlevel.
Default nastaveni je 1. Zaporna hodnota
znamena bez doplnujicich komentaru.
> integrate(1/(sin(x)^2+1),
x=0..Pi);
 2
2
> printlevel;
1
> printlevel:=100;
:=printlevel 100
> integrate(1/(sin(x)^2+1),
x=0..Pi);
value remembered (at top level): sin(x) -> sin(x)
{--> enter int, args = 1/(sin(x)^2+1), x = 0 .. Pi
value remembered (in int): int/int([1/(sin(x)^2+1), x = 0 .. Pi], 10,
_EnvCauchyPrincipalValue, _EnvAllSolutions, _EnvContinuous) -> 1/2*Pi*2^(1/2)
:=answer
 2
2
 2
2
<-- exit int (now at top level) = 1/2*Pi*2^(1/2)}
 2
2
> printlevel:=1;
:=printlevel 1
> interface(prettyprint=false):
> solve(a*x^2+b*x+c, x);
1/2/a*(-b+(b^2-4*a*c)^(1/2)), -1/2*(b+(b^2-4*a*c)^(1/2))/a
> interface(prettyprint=true):#im
plicitni nastaveni
> interface(prettyprint=1):
> solve(a*x^2+b*x+c, x);
2 1/2 2 1/2
-b + (b - 4 a c) b + (b - 4 a c)
--------------------, - -------------------
2 a 2 a
> interface(prettyprint=2):
> solve(a*x^2+b*x+c, x);
,
- +b -b2
4 a c
2 a
+b
-b2
4 a c
2 a
> sol:=solve(a*x^2+b*x+c, x):
> print(sol);
,
- +b -b2
4 a c
2 a
+b
-b2
4 a c
2 a
> lprint(sol);
1/2/a*(-b+(b^2-4*a*c)^(1/2)), -1/2*(b+(b^2-4*a*c)^(1/2))/a
> interface(verboseproc=2):
> print(unassign);
nomproc( )
"remove an assignment from a\description
n assigned expression"
...
end proc
Nacteni knihovny
> with(student);
D Diff Doubleint Int Limit Lineint, , , , , ,[
Product Sum Tripleint changevar, , , ,
completesquare distance equate integrand, , , ,
intercept intparts leftbox leftsum makeproc, , , , ,
middlebox middlesum midpoint powsubs, , , ,
rightbox rightsum showtangent simpson, , , ,
slope summand trapezoid, , ]
Pokud chceme pouzit pouze jednu konkretni
funkci z dane knihovny, muzeme provest jeji
volani takto:
> student[distance]([1,1],[3,4]);
13
Seznam knihoven ziskame prikazem
> ?index[packages];
Napovedu ke konkretni knihovne prikazem
> ?student;
Definice synonym (pouzivame, kdyz se chceme
vyhnout dlouhym jmenum):
> restart;
> alias(D = student[distance]);
D
> D([1,1],[3,4]);
13
> alias(D=D); #odstrani alias
Definice zkratek:
> macro(D = student[distance]);
D
> D([1,1],[3,4]);
13
> macro(D=D);
Alias ovlivnuje vstup i vystup, zatimco makro
pouze vstup.
> alias(c=a^2+b^2);
c
> c;
c
> 1/(a^2+b^2);c;
1
c
c
> a^2+b^2;
c
> alias(c=c);
> macro(c=a^2+b^2);
c
> 1/(a^2+b^2);
1
+a2
b2
> a^2+b^2;
+a2
b2
> c;
+a2
b2
> macro(c=c);
> restart;
Ukladani a nacitani dat
Save - uklada ve formatu, ktery se da pozdeji
opetne nacist do mapleovskeho zapisniku.
Muzeme ukladat v internim formatu Maplu (.m)
nebo v textovem formatu:
> pol:=x^2+2*x+1; `cislo ctyri`
:=4;
:=pol + +x2
2 x 1
:=cislo ctyri 4
> save pol, `cislo ctyri`,
`datafile.m`; #nutne uzavreni
do levych apostrofu!
> save pol, `cislo ctyri`,
datafile;
> restart;
> pol;
pol
> read datafile;
:=pol + +x2
2 x 1
:=cislo ctyri 4
> pol;
+ +x2
2 x 1
> restart;
> read `datafile.m`;
> pol;
+ +x2
2 x 1
V pripade nacteni souboru datafile jsou instrukce
opet zobrazeny, v pripade nacteni souboru
datafile.m tomu tak neni (nacitani souboru
datafile.m je efektivnejsi).
Maple po startu hleda soubor .mapleinit ve
Vasem domovskem adresari - zde muzeme zadat
prikazy, ktere chceme provadet pri kazdem startu
Maplu- napriklad nacteni casto pouzivanych
knihoven.
Adresare, ve kterych Maple hleda nacitane
knihovny, jsou urcovany promennou libname.
> libname;
"/usr/local/maple95/lib"
> libname:=`/home_zam/plch/maple`
,libname;
libname "/home_zam/plch/maple",:=
"/usr/local/maple95/lib"
> libname;
"/home_zam/plch/maple",
"/usr/local/maple95/lib"
Procedura latex generuje zdrojovy kod LaTeXu
pro zadany vzorec (vzorce):
> latex((x^2+y^2)/(x^2-y^2));
{\frac {{x}^{2}+{y}^{2}}{{x}^{2}-{y}^{2}}}
> (x^2+y^2)/(x^2-y^2);
+x2
y2
-x2
y2
> latex(%, `vzorec1.tex`);
> polyeq:=x^3-a*x=1;
:=polyeq =-x3
a x 1
> sols:=solve(polyeq,x);
sols
( )+108 12 - +12 a3
81
( )/1 3
6
:=
2 a
( )+108 12 - +12 a3
81
( )/1 3
+ ,
( )+108 12 - +12 a3
81
( )/1 3
12
-
a
( )+108 12 - +12 a3
81
( )/1 3
1
2
I 3





- +
( )+108 12 - +12 a3
81
( )/1 3
6
2 a
( )+108 12 - +12 a3
81
( )/1 3
-





,
( )+108 12 - +12 a3
81
( )/1 3
12
-
a
( )+108 12 - +12 a3
81
( )/1 3
1
2
I 3





- (
)+108 12 - +12 a3
81
( )/1 3
6
2 a
( )+108 12 - +12 a3
81
( )/1 3
-





> sol1:=sols[1];
sol1
( )+108 12 - +12 a3
81
( )/1 3
6
:=
2 a
( )+108 12 - +12 a3
81
( )/1 3
+
> CodeGeneration[C](sol1);
cg0 = pow(0.108e3 + 0.12e2 * sqrt(-0.12e2 * pow(a, 0.3e1) + 0.81e2), 0.1e1 / 0.
3e1) / 0.6e1 + 0.2e1 * a * pow(0.108e3 + 0.12e2 * sqrt(-0.12e2 * pow(a, 0.3e1)
+ 0.81e2), -0.1e1 / 0.3e1);
> codegen[C](sol1,
filename=`vystup.c`);
> with(CodeGeneration);
C Fortran IntermediateCode Java, , , ,[
LanguageDefinition Matlab Names Save, , , ,
Translate VisualBasic, ]
> Java(sol1);
cg2 = Math.pow(0.108e3 + 0.12e2 * Math.sqrt(-0.12e2 * Math.pow(a, 0.3e1) + 0.81
e2), 0.1e1 / 0.3e1) / 0.6e1 + 0.2e1 * a * Math.pow(0.108e3 + 0.12e2 * Math.sqrt
(-0.12e2 * Math.pow(a, 0.3e1) + 0.81e2), -0.1e1 / 0.3e1);
> Matlab(sol1);
cg3 = (0.108e3 + 0.12e2 * sqrt((-12 * a ^ 3 + 81))) ^ (0.1e1 / 0.3e1) / 0.6e1 +
0.2e1 * a * (0.108e3 + 0.12e2 * sqrt((-12 * a ^ 3 + 81))) ^ (-0.1e1 / 0.3e1);
>