Lecture 5

Question 1

Relative error

Answer 1

er = |x-^x|/|x|

Independent of magnitude (≠ to absolute errors)

Question 2

Significant digits/figures

Answer 2

digits carrying meaningful information

0. 00350 = 3 digits
1. 14159 = 6 digits
2. 00035 = 2 digits

Question 3

Relation relative error / accurate digits

Answer 3

rel_er <= 10^{-n+1}, n is the number of accurate significant digits

Question 4

You know a tree is 170 ft tall, but your tool has a rel_err=10%. What’s the max measurement to expect?

Answer 4

187 ft

^x = x(1 ± err)

Question 5

You measure 170 ft tall with a measurement tool rel_err=10%, what’s the actual min height?

Answer 5

155 ft

x = ^x / (1 ± err)

Question 6

After rounding, a number has 5 digits, what’s an estimate of the upper bound of the relative error?

Answer 6

10^-4 (n=5, rel_err <= 10^{-n+1})

Question 7

Sources of error

Answer 7

• rounding (1/3 = 0.333…)

- truncation (approx. math exp cos/sin…)

Question 8

f(x) = 2x^2 + 27x + 1000

Big O with x→0, x→∞

Answer 8

1. |f(x)| <= M*1, f(x) = O(1)

2. |f(x)| <= Mx^2, f(x)=O(x^2)

Question 9

Taylor approximation x_0

Answer 9

f(x) = ∑i=0→∞ f^(i)(x_0)(x-x_0)^i/i!

Question 10

Taylor approximation error

Answer 10

h=x-x_0
error = |f(x) - ∑(0→n) f^(i)(x0)h^i/i!| = |∑(n+1→∞) f^(i)(x0)h^i/i!|
Order n → error O(h^{n+1}) – exponent of the dominant term of the remainder (which is the error)
If ξ \in (x0,x), then error <= f^(n+1)(ξ)/(n+1)! (h)^{n+1} = M(h)^{n+1}

Question 11

Degree of Taylor app?

Answer 11

Terms from 0→n (first n+1) gives approximation of degree n (in range(n+1)…) that is s.t. x^n is in the expression

Question 12

Find error / order on a graph f(x) = error (loglog)

Answer 12

e = ax^n
log(e) = log(a)+n.log(x)
n=(y2-y1)/(x2-x1) is the slope → error O(x^4)
Approximation order n-1!

Question 13

Factorial python

Answer 13

from math import factorial

Question 14

Calculate python (sin(√x))^(2)(0)

Answer 14

import sympy as sp
sp.init_printing()
sp.var("x")
expr = sp.sin(sp.sqrt(x))
expr.diff(x,2).subs(x,0).evalf()

Question 15

Generate 1000 points between -1 and 1

Answer 15

np.linspace(-1, 1, 1000)

Question 16

The measured distance between Champaign and Chicago is 277.5 miles with a relative error of 2%, and the measured distance between Chicago and Milwaukee is 225.1 miles with a relative error of 6%

The distance between Champaign to Milwaukee can be given by the sumation of both distances. In the worst case, what is the relative error (in percent) of the distance between Champaign to Milwaukee?

Answer 16

err_{X+Y} = \frac{|\hat{x}/(1-err_X) + \hat{y}/(1-err_Y) - (\hat{x}+\hat{y})|}{|\hat{x}/(1-err_X) + \hat{y}/(1-err_Y)|} = 3.83%

Question 17

If the error in a computation behaves as O(h^6) depending on a mesh size parameter h, and if the error for h=0.5 is 0.01, what will the error be for h=0.7?

Answer 17

e2 = e1.(h2/h1)^6 = 0.0752

Question 18

If the error for h=0.01 is 0.08 and the error for h=0.005 is 0.01, what power of h would you expect for this calculation?

Answer 18

n = log(e2/e1)/log(h2/h1)

Question 19

If the algorithm takes 8 seconds with n=1200 and it takes 2 seconds with n=600, what is the order of the time cost?

Answer 19

8 = c.1200^a
2 = c.600^a
a = log(8/2)/log(1200/600)

Question 20

def taylor_exp(n, x0)

Answer 20

import sympy as sp
from math import factorial
sp.init_printing()
sp.var(“x”)

def taylor_exp(n, x0):
    taylor = 0
    for i in range(n+1):
        taylor += f.diff(x, i).subs(x, x0)/factorial(i) * (x-x0)**i
    return taylor