Lab 7: False Position Lab · Numerical Methods

Each step draws the chord between the current endpoints. Watch which endpoint moves.

Choose bisection or false position, type your equation and interval, and build the table step by step.

False position

Try this

Run it. Compare rows 1–3 with the Problem 1 table.
Change f to e^x − 3x on [0, 1] (Python: import math).
Add the Illinois fix: when the same endpoint is kept twice in a row, halve its stored f value.

Bisection vs false position

Try this

Run it. Which method wins on each function?
Add cos(x) − x on [0, 1]. Prediction first, then run.

Q1 Derive the false position formula.

The chord through

(a, f(a))

and

(b, f(b))

is

y - f(b) = \frac{f(b)-f(a)}{b-a}(x - b)

. Setting

y = 0

:

c = b - f(b)\frac{b-a}{f(b)-f(a)} = \frac{af(b) - bf(a)}{f(b) - f(a)}

.

Q2 Find the root of

x^3 - 5x + 1 = 0

in

[0, 1]

by false position, 3 iterations.

f(0) = 1

,

f(1) = -3

.

x_1 = 0.25

,

f = -0.234375

;

x_2 = 0.202532

,

f = -0.004351

;

x_3 = 0.201654

,

f = -0.000072

. Root ≈ 0.20164 (true 0.2016397).

Q3 Find a root of

x e^{x} = 1

between 0 and 1 by Regula Falsi, 3 iterations.

f(x) = xe^x - 1

:

f(0) = -1

,

f(1) = 1.718282

.

x_1 = 0.367879

(

f = -0.468536

),

x_2 = 0.503314

(

f = -0.167420

),

x_3 = 0.547412

(

f = -0.053649

). Root 0.5671433.

Q4 Why does the bracket in false position usually not shrink to zero?

If

f

is convex (or concave) on the bracket, every chord lies on the same side of the curve, so every new point falls on the same side of the root. One endpoint is replaced every time; the other never moves.

Q5 Compare false position and bisection (three points).

Both bracket the root and always converge. Bisection uses only signs, has the guaranteed bound

(b-a)/2^n

, and is slow but steady. False position uses function values, is usually faster, but has no simple bound and can stall when one endpoint gets stuck.