Introduction to interval methods
Here are the demo programs for the
introductory
minicourse on interval methods.
Download imni.zip.
The programs are written in
Lua
and run under the
LÖVE framework,
which you need to download and install.
You also need to learn how to run a LÖVE program in your machine.
See
this tutorial.
Demo programs
The demo programs are interactive programs that illustrate interval solutions to some basic numerical problems.
The programs typically solve a problem at different resolutions.
Most of the code in these programs is for graphics,
which you can skip.
Read the code to understand and play with the numerical part, which is quite small.
The numerical core is provided by the libraries.
Programs for the same problem differ only in the numerical part.
Compare them with a visual diff tool.
All programs react to the keys below.
Some programs react to additional keys.
See love.keypressed in the code
and the brief description of individual programs below.
- q or esc
- quit the program
- s
- save a screenshot to your LÖVE directory.
On Mac OS X,this is ~/Library/Application Support/LOVE/imni/.
- ]
- increase the resolution
- [
- decrease the resolution
To run a program, copy or link it to main.lua.
On Mac OS X, just do make main P=foo.lua to select a program to run.
This will create a symbolic link main.lua to foo.lua.
Doing plain make always runs the program in main.lua.
That's how LÖVE works.
Lecture 1
- pi.lua
- Archimedes and the approximation of π
toggle drawing outer polygons with
o
- sqrt2.lua
- bisection method for computing sqrt(2)
- graph.lua
- graph plotting and sampling
toggle drawing colors, graph, points, uniform sampling, random sampling
c
g
p
u
r
- graph-ia.lua
- graph plotting with interval arithmetic
– uniform division
toggle drawing colors with
c
- graph-ia-adapt.lua
- graph plotting with interval arithmetic
– adaptive division
finds ranges within tolerance
- graph-ia-ad.lua
- graph plotting with interval arithmetic and automatic differentiation
– uniform division
finds monotone pieces
- graph-ia-ad-adapt.lua
- graph plotting with interval arithmetic and automatic differentiation
– adaptive division
finds monotone pieces
Lecture 2
- zero-ia.lua
- solving equations with interval arithmetic
– adaptive division
- zero-ad.lua
- solving equations with interval arithmetic and automatic differentiation
– adaptive division
finds monotone pieces and critical points
- zero-ia-ad.lua
- solving equations with interval arithmetic and automatic differentiation
– adaptive division stops at monotone pieces
- ineq-ia.lua
- solving inequalities with interval arithmetic
– adaptive division
- go-ia.lua
- global optimization with interval arithmetic
– uniform division
- go-ia-bb.lua
- global optimization with interval arithmetic
– adaptive division
with branch and bound
- go-ia-bb-2d.lua
- global optimization with interval arithmetic
– adaptive division
with branch and bound 2d
Lecture 3
- fp.lua
- distribution of floating-point numbers
- implicit-fp.lua
- implicit curve plotting via uniform sampling
toggle drawing boxes, using corners with
b
c
- implicit-ia.lua
- implicit curve plotting with interval arithmetic
– adaptive division
toggle drawing boxes, pruning tree with
b
p
Lecture 4
- implicit-aa.lua
- implicit curve plotting with affine arithmetic
– adaptive division
toggle drawing boxes, pruning tree with
b
p
- zero-aa.lua
- solving equations with affine arithmetic
- ineq-aa.lua
- solving inequalities with affine arithmetic
- go-aa.lua
- global optimization with affine arithmetic
– uniform division
- zonotope-1d.lua
- zonotope approximaton with affine arithmetic
toggle drawing boxes, graph, zonotopes with
b
g
z
move left and right with
◀
▶
- graph-aa.lua
- graph plotting with affine arithmetic
– uniform division
shows zonotopes
toggle drawing boxes, colors, zonotopes with
b
c
z
- curve-ia.lua
- curve plotting with interval arithmetic
– uniform division
toggle drawing boxes, curve, points with
b
c
p
- curve-aa.lua
- curve plotting with affine arithmetic
– uniform division
toggle drawing boxes, curve, points, zonotopes with
b
c
p
z
Libraries
The demo programs use these libraries.
The libraries for interval arithmetic and affine arithmetic export the same interface so that you can transparetly use affine arithmetic as a replacement for interval arithmetic.
The libraries are very simple and are for demo only.
They are not meant to be used in production.
- ia.lua
- interval arithmetic
- ad.lua
- automatic differentiation, one independent variable
- aa1.lua
- affine arithmetic, one independent variable
- aa2.lua
- affine arithmetic, two independent variables
Feedback
All feedback is welcome.
Please send questions, comments, suggestions, requests, bug reports, and new demos to me by email.