In [1]:
# Lineárna, kvadratická a polynomiálna interpolácia
# Programové kódy: SageMath v.9.x(Python 3.0)
# Prerekvizity: Polynómy, Python, LaTex
#
# Použitá literatúra:
# [1] R. E. Mezei, An introduction to SAGE programming, John Wiley,and Sons,2016
# ISBN: 978-1-119-12280-7.
#
# [2] Gregory V.Bard. Sage for Undergraduates:Second Edition,Compatible with Python3.
# American Mathematical Society,2022.ISBN978-1-4704-6155-3.
# URL https://bookstore.ams.org/mbk-143/ - Sage pre pokročilých
#
# [3] Paul, Zimmermann and Alexandre, Casamayou and Nathann, Cohen and Guillaume Con-
# nan and Thierry Dumont and Laurent Fousse and François Maltey and Matthias Meulien
# and Marc Mezzarobba and Clément Pernet and Nicolas M. Thiéry and Erik Bray and
# John Cremona and Marcelo Forets and Alexandru Ghitza and Hugh Thomas:
# Computational Mathematics with SageMath, 2018, ISBN978-1-611975-45-1.
# https://my.siam.org/Store/Product/viewproduct/?ProductId=30174951,
# https://www.sagemath.org/sagebook/english.html
#
# E-kurz: P. Szabó: Numerická matematika, 2022
# Spúšťanie programov:
# 1) Otvoriť lokalitu https://sagecell.sagemath.org/
# 2) Kopírovať obsah príkladu In[n] do SageMath boxu
# 3) Kliknutie na "Evaluate"
# Príloha publikácie:
# [4] Peter Szabó a Eva Baranová: Integrálny počet a numerická matematika, 2022, TUKE
# ISBN:978-80-553-4185-9
# Modifikované: 10.12.2022
In [2]:
# Lineárna interpolácia
# Vstupy: interaktívna množina bodov "points" [xi,yi]
# Výstup: p(x) - Lineárna lomená čiara, p(xi)=yi
def Linear_Spline(points):
var("x")
#first we plot the given points
p = list_plot( points, color = "red", size = 50)
#compute the number of subintervals
n = len(points)-1
#then compute and plot each branch of the linear spline
for k in [1,..,n]:
#we find the equation y=mx+b for the kth subinterval
m = (points[k][1]-points[k-1][1])/ \
(points[k][0]-points[k-1][0])
b = points[k][1]-m*points[k][0]
#then we plot that line on the interval [xk-1, xk]
p += plot(m*x+b, x,points[k-1][0],points[k][0],
color = "green", title = "linear spline interpolation")
#display the linear spline and the given points
p.show()
@interact
def LinearSplineInteract(
points = input_box(default = [(-3,-10),(-1,-5),(1,1),(2,10),
(4,15)]) ):
Linear_Spline(points)
In [3]:
# Lineárna interpolácia
# Vstupy: množina bodov "points" [xi,yi], i=0,1,..,n
# Výstup: p(x) - Lineárna lomená čiara, p(xi)=yi, i=0,1,..,n
def Linear_Spline(points):
var("x")
#first we plot the given points
p = list_plot( points, color = "red", size = 50)
#compute the number of subintervals
n = len(points)-1
#then compute and plot each branch of the linear spline
for k in [1,..,n]:
#we find the equation y=mx+b for the kth subinterval
m = (points[k][1]-points[k-1][1])/ \
(points[k][0]-points[k-1][0])
b = points[k][1]-m*points[k][0]
#then we plot that line on the interval [xk-1, xk]
p += plot(m*x+b, x,points[k-1][0],points[k][0],
color = "green", title = "linear spline interpolation")
#display the linear spline and the given points
p.show()
points = [(-3,-10),(-1,-5),(1,1),(2,10),(4,15)]
Linear_Spline(points)
In [4]:
# Lagrangeova (polynomiálna) interpolácia
# Vstupy: interaktívna množina bodov "points" [xi,yi], i=0,1,..,n
#
# Výstup: q(x) polynóm n-tého stupňa, q(xi)=yi, i=0,1,..,n
# Lagrangeove bázické polynómy
def Lagrange_Basis(i, points):
var("x")
n = len(points)-1
Li=1
for j in [0,1,..,n] :
if(i!=j):
Li *= (x-points[j][0])/(points[i][0]-points[j][0])
return Li
def Lagrange_Polynomial(points):
var("x")
n = len(points)-1
p=0*x
for i in [0,1,..,n]:
p += points[i][1]*Lagrange_Basis(i,points)
return p.full_simplify()
@interact
def LagrangeInterpolationInteract2(
points = input_box(default = [(-3,-15),(-1,-5),(0,1),(2,10),
(3,15)]) ):
n = len(points)-1
a = min([points[i][0] for i in [0,1,..,n]])
b = max([points[i][0] for i in [0,1,..,n]])
p = plot(Lagrange_Polynomial(points) , x, a-1, b + 1)
p = p + list_plot( points, color = "red", size = 70)
q(x)=Lagrange_Polynomial(points)
show (q)
In [5]:
# Lagrangeova (polynomiálna) interpolácia
# Vstupy: množina bodov "points" [xi,yi], i=0,1,..,n
#
# Výstup: q(x) polynóm n-tého stupňa, q(xi)=yi, i=0,1,..,n
# Lagrangeove bázické polynómy
def Lagrange_Basis(i, points):
var("x")
n = len(points)-1
Li=1
for j in [0,1,..,n] :
if(i!=j):
Li *= (x-points[j][0])/(points[i][0]-points[j][0])
return Li
def Lagrange_Polynomial(points):
var("x")
n = len(points)-1
p=0*x
for i in [0,1,..,n]:
p += points[i][1]*Lagrange_Basis(i,points)
return p.full_simplify()
points = [(-3,-15),(-1,-5),(0,1),(2,10),(3,15)]
n = len(points)-1
a = min([points[i][0] for i in [0,1,..,n]])
b = max([points[i][0] for i in [0,1,..,n]])
p = plot(Lagrange_Polynomial(points) , x, a-1, b + 1)
p = p + list_plot( points, color = "red", size = 70)
q(x)=Lagrange_Polynomial(points)
show(p)
show (q)
In [6]:
# Lineárna, kvadratická a Lagrangeova (polynomiálna) interpolácia
# Vstupy: interaktívna množina bodov "points" [xi,yi], i=0,1,..,n
# Výber typu interpolácie: linear, cubic a polynomial
# Výstupy: p(x) polynóm n-tého stupňa,
# lineárna lomená čiara l(x),
# Polynóm druhého stupňa q(x),
# p(xi)=l(xi)=q(xi)=yi, i=0,1,..,n.
# Poznámka: použitie knižnice "scipy"
import scipy
from scipy.interpolate import interp1d
def Lagrange_Basis(i, points):
var("x")
n = len(points)-1
Li=1
for j in [0,1,..,n] :
if(i!=j):
Li *= (x-points[j][0])/(points[i][0]-points[j][0])
return Li
def Lagrange_Polynomial(points):
var("x")
n = len(points)-1
p=0*x
for i in [0,1,..,n]:
p += points[i][1]*Lagrange_Basis(i,points)
return p.full_simplify()
def DrawInterpolations(points, plotPoly, plotLinear, plotCubic):
#needed if you want to use x in the input
# like [(x, sin(x)) for x in [0,1,..,4]]
var("x")
#compute the size of the given array
n = len(points)
#get the x values of points
xvals = [points[i][0] for i in [0,1,..,n-1]]
#get the y values of points
yvals = [points[i][1] for i in [0,1,..,n-1]]
#plot the points
p = list_plot( points, color = "black", size = 50)
#find the interval [a,b]
a = min([points[i][0] for i in [0,1,..,n-1]])
b = max([points[i][0] for i in [0,1,..,n-1]])
if(plotPoly == True):
fp=Lagrange_Polynomial(points)
p += plot(fp, x, a, b, color = "red",
legend_label = "polynomial interpolation")
if(plotLinear == True):
f1=interp1d(xvals, yvals, kind = "linear")
p += plot(f1, x, a, b, color = "green",
legend_label = "linear spline interpolation")
if (plotCubic == True):
f3=interp1d(xvals, yvals, kind = "cubic")
p += plot(f3, x, a, b, color = "blue",
legend_label = "cubic spline interpolation")
#return the plot object
return p
@interact
def InterpolationsInteract(
points = input_box(default = [(-3,-10),(-1,-5),(1,1),(2,10),
(4,15)]) ,
plot_poly = ("Polynomial Interpolation", true),
plot_linear = ("Linear Spline", true),
plot_cubic = ("Cubic Spline", true) ):
DrawInterpolations(points, plot_poly, plot_linear, plot_cubic).show()
In [7]:
# Lineárna, kvadratická a Lagrangeova (polynomiálna) interpolácia
# Vstupy: množina bodov "points" [xi,yi], i=0,1,..,n
# Výstupy: p(x) polynóm n-tého stupňa,
# lineárna lomená čiara l(x),
# Polynóm druhého stupňa q(x),
# p(xi)=l(xi)=q(xi)=yi, i=0,1,..,n.
# Poznámka: použitie knižnice "scipy"
import scipy
from scipy.interpolate import interp1d
def Lagrange_Basis(i, points):
var("x")
n = len(points)-1
Li=1
for j in [0,1,..,n] :
if(i!=j):
Li *= (x-points[j][0])/(points[i][0]-points[j][0])
return Li
def Lagrange_Polynomial(points):
var("x")
n = len(points)-1
p=0*x
for i in [0,1,..,n]:
p += points[i][1]*Lagrange_Basis(i,points)
return p.full_simplify()
def DrawInterpolations(points, plotPoly, plotLinear, plotCubic):
#needed if you want to use x in the input
# like [(x, sin(x)) for x in [0,1,..,4]]
var("x")
#compute the size of the given array
n = len(points)
#get the x values of points
xvals = [points[i][0] for i in [0,1,..,n-1]]
#get the y values of points
yvals = [points[i][1] for i in [0,1,..,n-1]]
#plot the points
p = list_plot( points, color = "black", size = 50)
#find the interval [a,b]
a = min([points[i][0] for i in [0,1,..,n-1]])
b = max([points[i][0] for i in [0,1,..,n-1]])
if(plotPoly == True):
fp=Lagrange_Polynomial(points)
p += plot(fp, x, a, b, color = "red",
legend_label = "polynomial interpolation")
if(plotLinear == True):
f1=interp1d(xvals, yvals, kind = "linear")
p += plot(f1, x, a, b, color = "green",
legend_label = "linear spline interpolation")
if (plotCubic == True):
f3=interp1d(xvals, yvals, kind = "cubic")
p += plot(f3, x, a, b, color = "blue",
legend_label = "cubic spline interpolation")
#return the plot object
return p
points = [(-3,-10),(-1,-5),(1,1),(2,10),(4,15)]
plot_poly = true
plot_linear = true
plot_cubic = true
DrawInterpolations(points, plot_poly, plot_linear, plot_cubic).show()
