MA 531 HW 3 Sample Code.sagews

#3.4.13. Calculations check

#makes Sage simplify polynomials more easily sometimes
R = PolynomialRing(QQ, 'x')

#multiplying out the polynomials

p=7*1/168*(x-2)*(x-3)*(x-4)*(x-7)+10*-1/20*x*(x-3)*(x-4)*(x-7)+20*1/12*x*(x-2)*(x-4)*(x-7)+32*(-1/24)*x*(x-2)*(x-3)*(x-7)+1*(1/420)*x*(x-2)*(x-3)*(x-4)
p

#part of 3.5.26

#defines 31 equally spaced test points
#used m so it didn't conflict with n=10 at all
n=10
f(x)=1/(1+25*x^2)
points=[(-1+i*(2/n),f(-1+i*(2/n))) for i in [0,1,..,n]]


m=31

#defines function
f(x)=1/(1+25*(x^2))

#calculates interpolation polynomial p(x)
p(x)=Newton_Interpolation(x)
xvalues=[-1+i*(2/m) for i in [0,1,..,m]]

#evaluates f and p for 31 points
#notice that xvalues[i] selects the ith entry in the xvalues array (which is just like a row matrix with m columns)

ftruevalues=[f(xvalues[i]) for i in [0,1,..,m]]
pestvalues=[p(xvalues[i]) for i in [0,1,..,m]]

#calculates error between f and p, outputs maximum error of the 31 values (using 11 evenly spaced nodes to build p).
error=[abs(ftruevalues[i]-pestvalues[i]).n(20) for i in [0,1,..,m]]
error
max_error=max(error[i] for i in [0,1,..,m])
max_error