radionica_polinomi
automatic_names(True)
R.<x>=QQ[]
R1=PolynomialRing(QQ,x)
R,R1
f=R.lagrange_polynomial([(1,3),(2,16),(-1,1),(3,45)])
f
plot(f,-2,3.5)+point(((1,3),(2,16),(-1,1),(3,45)),rgbcolor=(1,0,0),size=50,faceted=True)
point2d.options
import pylab import numpy
x1=numpy.arange(-2.0,3.5,0.01) y1=map(f,x1)
tocke_x = [1,2,-1,3] tocke_y = [3,16,1,45]
#pylab.clf() pylab.axis([-2,4, -10, 70]) pylab.grid(True) pylab.plot(x1,y1,'b-',tocke_x,tocke_y,'ro',markersize=8,linewidth=2) pylab.savefig('lagrange.png')
f=x^5-3*x^3-5*x g=x^2-x+1
f//g
f%g
f=x^77+x^55+x^33+x^11+1 g=x^2-1
f%g
time f//g
A=x^4+6*x^3+17*x^2+24*x+12 B=x^3-2*x^2-13*x-10
A.gcd(B)
gcd(A,B)
A.xgcd(B)
xgcd(A,B)
f=5*x^3-9*x^2-x-2
f.real_root_intervals()
plot(f,-1,3,figsize=[5,4])
f.roots()
f.roots(RR)
f.roots(CC)
f.roots(ComplexField(200))
solve(5*u^3-9*u^2-u-2,u)
P=5*u^3-9*u^2-u-2
P.series(u==-2,4)
g=x^2-2
g.roots()
g.roots(QQ)
g.roots(RR)
g.roots(RealField(300))
f=2*x^4+13*x^3+25*x^2+15*x+9
f.real_root_intervals()
f.real_root_intervals()[0][0]
[n(i,digits=50) for i in f.real_root_intervals()[0][0]]
plot(f,-5,1,figsize=[5,3])
f.roots(RR)
f.roots(CC)
solve(2*u^4+13*u^3+25*u^2+15*u+9,u)
f.factor()
f=x^3+3*x^2-9*x-20
f.real_root_intervals()
plot(f,-5,4,figsize=[5,4])
f.roots(RR)
solve(u^3+3*u^2-9*u-20,u)
vađenje kompleksnog korijena
from sage.rings.polynomial.complex_roots import *
a=CC(-I)
a.nth_root(3,all=True)
a2=ComplexField(200)(-I)
a2.nth_root(3,all=True)
a3=ComplexField(200)(2-3*I)
a3.nth_root(3,all=True)
a4=CC(2,-3)
a4.nth_root(5,all=True)
polinomi nad proizvoljnim poljem ili prstenom
R.<x>=PolynomialRing(QQ)
f=x^3+1 g=x^2+x+1
f//g
f%g
R.<x>=PolynomialRing(GF(2))
f=x^3+1 g=x^2+x+1
f//g
f%g
R.<x>=PolynomialRing(GF(5))
f=x^3+1 g=x^2+x+1
f//g
f%g
R.<x>=PolynomialRing(ZZ)
f=x^3+1 g=5*x^2+x-2
f//g
f%g
f=x^3+1 g=x^2+x-2
f//g
f%g
() [] {}