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MANE 3351

Lecture 9

Classroom Management

Agenda

  • Part 2: Bisection Search Error Analysis
  • False Position Method
  • Test 1
  • Laboratory Session 9 covers Lab 4

Resources

Handouts

Assignments

Schedule

Lecture/Lab Date Topic
10 9/27 Bisection Method Error Analysis, False Position (not on Test 1), lab session
11 10/2 Test 1 (lectures 1-7); no lab

Bisection Method: Error Analysis

Bisection Method Theorem

Cheney and Kincaid (2004)[^1] provide a definition of the bisection method

If the bisection algorithm is applied to a continuous function on an interval \([a,b]\), where \(f(a)f(b)<0\), then, after \(n\) steps, an approximate root will have been computed with error at most \((b-a)/2^{n+1}\).

This definition provides the following useful results:

  • \(|e_n|\leq \frac{1}{2^{n+1}}(b-a)\)
  • \(n> \frac{\log\left(b-a\right) -\log 2\varepsilon}{\log 2}\)

Percent Relative Error

Chapra and Canale (2015) [^2] provide a formula for the approximate percent relative error

  • \(\varepsilon_a=\left|\frac{x_r^{new}-x_r^{old}}{x_r^{new}}\right|\times 100\%\) where \(x_r^{new}\) is the root for the present iteration and \(x_r^{old}\) is the root from the previous iteration

False Position

  • Attempts to converge faster than bisection method by using functional information
  • Referred to as the bracketed secant method in textbook
  • The new root is found by
\[ x_r=x_u-\frac{f(x_u)(x_l-x_u)}{f(x_l)-f(x_u)} \]
  • Figure 5.12 (Chapra and Canale(2015)[^2] illustrates the false position method False Position Method

False Position Code

The pseudocode, shown below was taken from [^3]

False Position Pseudocode

False Position Error Analysis

  • Much harder to analyze than bisection method
  • Convergence to root depends on shape of \(f(x)\)
  • At times, false position may converge slowly!, Example from textbook[^4], shown below

False Position troublesome function

Loops in Python

  • Python only supports two types of loops: for and while
  • Pseudocode used do/while loop

Python Code for False Position

# False Position
from scipy import stats
import math
def f(x):
    return stats.norm.cdf(x)-.25
# input a & b
while True:
    a=float(input("enter the lower bound: "))
    b=float(input("enter the upper bound: "))
    if f(a)*f(b)<0.0:
        break
    print("incorrect bounds, please re-enter")
print("the lower bound is {}".format(a))
print("the upper bound is {}".format(b))
counter=0
# process loop
while True:
    m= a - ((a-b)*f(a))/(f(a)-f(b))
    if f(a)*f(b)<0.0:
        b=m
    else:
        a=m
    counter=counter+1
    if math.fabs(f(m)) <0.0005:
        break
print("the root is {}".format(m))   
print("the value at the root is {}".format(f(m)))
print("the number of steps were {}".format(counter))

Test One

  • You are allowed one 4 inch by 6 inch notecard containing handwritten notes
  • Calculator is needed; no programmable calculators that can calculate derivatives
  • Covers material from Lectures 1 - 7
  • Review Homework 1 and 2 and solutions
  • Review Test 1
  • Resistor information will be provided

References

[1]: Cheney, W. and Kincaid, D. (2004), Numerical Mathematics and Computing, 5th edition

[2]: Chapra, S. and Canale, R. (2015), Numerical Methods for Engineering, 7th edition

[3]: Source: https://www.codesansar.com/numerical-methods/regula-falsi-or-false-position-method-pseudocode.htm

[4]: Brin, L., (2020), Tea Time Numerical Analysis: Experiences in Mathematics, 3rd edition