Lesson 7.1 · 22 of 25 lessons

Conditioning & sensitivity

See how conditioning magnifies input error and how repeated computation converges toward one direction.

Open the interactive animation

01 · Story

What to verify in this scene

Follow the explanations and operations, then identify the mathematical evidence that remains in the final scene.

01
Two clearly separated lines have an intersection stable that stays stable under small changes
02
These two conditions are nearly parallel, so their intersection is sensitive to input error
03
Changing the constant from 201 to 202 moves the solution from (1,1) to (2,0)
The condition number measures this sensitivity

02 · Notebook

The runnable notebook

This source creates the mathematical objects and controls the order of operations and animation. Read it here, then play the same scene yourself.

2d
zoom 1/2
coordinates on
x + y = 2  #R1
x - y = 0  #R2
solution(R1, R2)  #stable@
// Two clearly separated lines have an intersection {{stable}} that stays stable under small changes
focus stable
checkpoint Conditioning & sensitivity · 1

focus -
clear
x + y = 2  #S1
101x + 100y = 201  #S2
solution(S1, S2)  #sensitive@
// These two conditions are nearly parallel, so their intersection is sensitive to input error
checkpoint Conditioning & sensitivity · 2

clear
S1 +
101x + 100y = 202  #T2
solution(S1, T2)  #moved@
// Changing the constant from 201 to 202 moves the solution from `(1,1)` to `(2,0)`\nThe **condition number** measures this sensitivity
focus moved

Open the interactive animation

03 · Unit 7

Lessons in this unit

Matrix computation & stability — See how conditioning magnifies input error and how repeated computation converges toward one direction.