The wonderful bouncing balls are from
http://javascript.internet.com.
Understanding that there are many areas of overlap, the
commented index is loosely grouped into the following categories:
Date: Mon, 24 Jan 2000 15:54:56 -0500
From: The MathWorks
The new MATLAB Student Version, available directly from The
MathWorks was announced in the fall. MATLAB Student Version
includes: unlimited matrix size, Simulink, and the Symbolic Math
Toolbox at a price of $99. Students can also purchase additional
toolboxes directly from our Web store.
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| 1. Calculus
1.a. Numerical integration
Compare left, right, trapezoid, midpoint, Simpson
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| directory listing |
| egg.m
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Numerical estimation of moment of inertia of an egg
(enter data from slice, numerical integration).
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leftrite.m
| script to generate lots of data to compare
left/rite/mid/trap sums for int(exp(-x^2),x=0..1) with n
from n=1 to n=1,048,576.
(Use this to generate data -- but don't worry about the details.
The code effectively reuses function values that already have
been computed -- but at the price of making the code less transparent.)
(See this old
class-material for more explanations, data tables generated, and images.)
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| 2. Ordinary Differential Equations
2.a. Graphical focus
Slopefields versus vector fields,
overlay solution curves
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| directory listing |
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slopefld.m
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Modification of the MATLAB file quiver.m that
draws slopefield for a differential equation with line segments of
normalized lengths. (A file from the 1995 DE class.)
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| RC.m |
sample function file for use w/ ode23: first order diff eqn of RC circuit.
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|
intrDE1.m |
Demonstration how to set the path, how to get help, how to use ode23.
Includes plotting the soln and overlaying many solns for simultaneously
solved initial value problems.
Uses the file
RC.m which should be saved first in suitablle directory.
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|
intrDE3.m |
Demonstration how to overlay the direction field (quiver.m) and
solution curves of a first order DE. Alternative: Use the homemade
slopefld.m instead of the built-in quiver.m.
Uses the file
RC.m.
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|
intrDE4.m |
Demonstration how to overlay the vector field (phase portrait) (using
quiver.m) and solution curves of a second order autonomous DE.
Uses the file
damped.m.
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|
intrDE.m |
Concatenation of the
intrDE1.m
intrDE2.m
intrDE3.m
intrDE4.m
The script files intrDEn.m call the examples RC.m, forced.m,
and damped.m -- download these first and save them in a subdirectory
in the search path....
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2.b. Numerical integration
Euler's method, Runge-Kutta schemes
Numerical stability
transition to chaos
| directory listing |
| euler.m |
Commented simple implementation of Euler's method
for the unstable DE y'=t*sin(y).
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| unstable.m |
sample function file for use w/ ode23: good example to demonstrate
unstable DE y'=t*sin(y) -- which solver lasts longer?
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| euler1.m |
Minimalist implementation of Euler's method for the unstable DE y'=t*sin(y).
damped2.m
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peneuler.m |
Simple pendulum using Euler's method.
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penHeun.m |
Simple pendulum using Heun's method.
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penRK4.m |
Simple pendulum using a 4th order Runge Kutta method.
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2.c. Second order systems
Nonlinear versus linearized pendulum
Several useful templates in the format expected by ode23
| directory listing |
|
cascade.m |
sample function file for use w/ ode23: 2d-cascade.
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forced.m |
sample function file for use w/ ode23: Forced, damped, harmonic oscillator.
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intrDE2.m |
Demonstration how to use ode23 to solve second order DE. Uses the example
forced.m
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oscill.m |
same as forced.m.
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damped.m |
similar to forced.m, but nonlinear oscillator and without forcing.
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damped2.m |
sample m-file (for use with ode23 etc.) to simultaneously integrate
a nonlinear oscillator and its linearization.
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compare.m |
sample script for simultaneous integration of a nonlinear oscillator
and its linearization.
Extensive comments for MATLAB-beginners (from setting the path,
opening m-files for editing, calling ode23, and overlaying the plots).
Good place to copy from, and
edit commands for further explorations. Uses the file
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pencompare.m |
Comparison of the simple pendulum and the linearized pendulum.
For the latter both exact solutions and numerical RK4 approximation
are supplied.
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dpend.m
July 2005 |
Animation of double pendulum, and graphs of solution.
sample image.
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| 3. Linear Algebra
3.a. Gauss Jordan elimination
Systems of linear equations,
reduced row echelon form,
matrix algebra
LU-factorization
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| directory listing |
| rnd.m |
Random matrices with integer entries in [0,N] or [-N,N]. (MK July 2008).
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| fitpoly.m |
Polynomial interpolation (set up as system of linear
equations, solve, plot).
See hello.m
and hellow.m
for similar scripts with many more comments. |
| hello.m |
Very similar to fitpoly.m (this
from summer 1999), but with more comments for MATLAB beginners.
|
| hellow.m |
Analogous to hello.m,
but with extensive comments for first time users. |
strangIIIe_2_3_17.m
2007 |
Start w/ quadratic fit of 3 points, extend to n-points,
and generic basis functions, incl. plots.
Emphasis: Column picture, functions as columns.
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strangIIIe_2_4_26.m
2007 |
Matrix multiplication as sum of outer products.
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strangIIIe_2_4_27.m
2007 |
Product of two upper triangular matrices as sum of outer products.
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| exa1.m |
A semi-automatic example for Gauss-Jordan elimination.
Uses r1.m, r2.m, and
r3.m. |
| r1.m,
r2.m,
r3.m
|
Perform elementary row operations (mulitply by scalar,
interchange two rows, add multiple of one row to another) |
| rrefmovy.m
|
Like MATLAB-built-in rrefmovie, but does not clear
screen at each step. Allows for scrolling back.
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| hw1.m |
Sample solution related to rref. |
| hw11.m |
Sample solution related to rref. |
| hw12.m |
Sample solution related to rref. |
| mastquiz.m |
rrefmovy applied to example of mastery quiz |
| matlabintro.html
|
Beginner's tasks related to matrix multiplication and using MATLAB
|
| EM1.m,
EM2.m,
EM3.m
|
Creates elementary matrices corresponding to elmentary row-operations.
Typical use: construct LU-factorization via product of EM's.
|
| trigrank.m |
Exploration of the linear dependence (numerical rank) of the set
of functions {1,cos(x),cos(2x),cos^2(x)}. |
Several old MS-WORD files using MATLAB plug-in (from
a 1996 linear algebra class).
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3.b. Linear Transformations
(linear combinations,
polygons)
| directory listing |
| lincomb.m |
Express any vector as a linear combination
of two basis vectors. Draws grid corresponding to
basis and overlays parallelogram. |
| drawfig3.m |
Draw a polygon creating an 3 x n matrix with ones in third row.
Used as basic figure to explore affine trafo's. Reinforces concept
of block-muliplication (each column is one vertex).
Use with ffill.m
and pplot.m.
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| drawfig.m |
Draw a polygon creating an 2 x n matrix. Used as
basic figure to explore linear trafo's. Reinforces concept
of block-muliplication (each column is one vertex).
Use with ffill.m
and pplot.m.
|
| ffill.m |
Similar to fill.m, but using different data format:
Vertices' coord's are columns of 2xn matrix as
generated by drawfig.m |
| pplot.m |
Similar to plot.m, but using different data format:
Vertices' coord's are columns of 2xn matrix as
generated by drawfig.m |
| lt2d.m |
Overlays a polygon p (2 x n matrix) with its image A*p.
Requires: ffill.m and
pplot.m | , typically also uses
drawfig.m to create polygon.
(July 2000).
| twoDexa.m |
Defines a list of some 30 basic matrices to practice
linking the numbers to the graphical roles. Typically
used as initialization before using
lt2d.m together with
drawfig.m.
(July 2000) |
| rot.m |
Generates 2x2 rotation matrix for given angle t.
Depending on whether |t| is <5 or > 5, the angle is
interpreted in radians or degrees.i
This intelligent version of the file
is apparently misplaced
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| lintrafo.m |
Simple script that demonstrates (products of) matrices
and the corresponding (compositions of) linear transformations
of the plane. Uses simple polygons, then calls
catt.m. |
| catt.m |
Using a cat's face, demonstrates linear transfo's of the
plane that corresponds to given matrix (or to randomly
generated matrix if called w/o parameters).
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3.c. Polygons (graphical tools, fun)
"Morphing" one polygon into another, e.g. a walking dinosaur
Polygons as ample regions for lin trafos,
and polygons as vector in 2n-dimensions!
| directory listing |
| morph.m |
Creates an animation that "morphs" one polygon
into another (make sure numbers of vertices agree).
Used as fun/relaxation when introducing more abstract
vectors (each polygon is a point/vector in R^(2n)). |
| dino.mat |
.mat file from walking dinosaur. |
| dino2.mat |
|
| Xmas.m |
Defines data for a polygon used as an exam polygon.
Includes several comments. |
| Xmassoln.m |
Commented sample solution -- find the affine transformation
that transforms one given X-mas tree into another one. |
| tree.txt |
E-mail correspondence w/ journal article that started the tree fractals... |
| origtree.m |
Old: fractals and polygon's that look like real trees. |
| origtri.m |
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| basetree.m |
Old: fractals and polygon's that look like real trees. |
| tree.m |
Old: fractals and polygon's that look like real trees. |
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3.d. Inner products:
Gram-Schmidt process,
Least squares approximations for "measured data points",
See section linalg/fourier for Fourier analysis of real data.
See images/compression for image compression via
svd (singular value decomposition).
| directory listing |
| gs3.m |
Graphical illustration of Gram-Schmidt process: Starting
with basis for R^3, show step by step each projection
and each change of basis. (Updated Nov. 2003: fatter lines)
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mygs1.m
mygs2.m
mygs3.m
mygs4.m
mygs5.m
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A sequence of increasingly more sophisticated implementations of the
Gram-Schmidt process or QR-factorization. First script is simple transcript
of almost manual computaion w/ 3 vectors in 3-space. Subsequent versions
add for-loops, matrix products for projections, go from script to function
file, and finally also return the R-matrix that records the successive
column operations. Note, in general no error checking (e.g. full col rank)
is performed.
Main use: In a class, as first programming experience -- reference after the fact!
Compare
in-class exercise from
MAT 342, fall 2003.
with basis for R^3, show step by step each projection
and each change of basis. (Updated Nov. 2003: fatter lines)
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| leastsq.m |
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| uspop.m |
Sample data: US-population each decade from 1790 to 1990.
May be used for least squares fits. Note, quite surprisingly,
while (dP/dt)/P is far from nearly constant,
(dP/dt)/T is almost linear. |
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| directory listing |
| projectfourier.m |
First example of projection, going from 3D to first Fourier example
by identifying functions with long column vectors (size: 21 x 3).
July 2008.
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| fake.m |
A quick demo of how the fft command works on a random 5th order trig polynomial.
Illustrates the time and frequency (spectrum) domains graphically.
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|
heartbeat.html |
General description of naive Fourier approximation of electro-cardiogram.
Compare parallel
EXCEL worksheet more suitable for first explorations by students.
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| ecg.tif |
Sample image (rather poor quality) of 1 period of
an electro-cardiogram. |
| ecgbw.tif |
Sample grey-tone image (rather poor quality) of 1 period of
an electro-cardiogram.
Use with ecgdigi.m to create
data set for function approximation. |
| ecgdigi.m |
Graphical tool to digitize the (almost) hardcopy
ecgbw.tif of an electro cardiogram
as used by ecgfft.m. |
| ecgdata.m |
Sample digitization of electro-cardiogram picture, used
for function approximation with
ecgfft.m. |
| ecgfft.m |
Reasonably fast (FFT-based ideas!) Fourier approximation of
electro cardiogram. Uses ecgdata.m,
and emphasizes graphical images of successive approximations.
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|
eric.m |
Data for a real sound signal. Displays the sound graphically w/ scrolling
window and exploratory FFT analysis.
Nontrivial, real-world problem of shifting base frequencidea of windowingy!
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|
win.m |
A leftover (?) looking at different windows of the sound data???
eric.m |
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ericfft.m |
FFT exploration of real sound data provided by EricBrewe.
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fourierpix.m
| Naive implementation of Fourier approximation of a grey-scale
image. Uses default images
einstein.gif
i(or in color: me.tif), and
displays side-by-side the grey-scale image and the associated
graph of a function of 2 variables.
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3.f. Diagonalization
Eigenvectors and eigenvalues.
See linalg/lintrafo for more eigenvalues in 2D-plane.
See images/compression for svd used for image compression.
| directory listing |
eigplot.m
(1995) 2004 |
Visualizing eigen vectors of 2 by 2 matrix: Appends respective images
to the tip of large number of unit vectors. |
eignmovie.m
(1995) 2004 |
For 2 by 2 matrix animates a unit vector tracing out a circle, while
its image traces out an ellipse. Vectors meet at eigen-directions. |
| nodet.m |
Demonstration how to obtain the char poly w/o determinants, but rather
from extracting dependency relations between iterates of random vector. |
| eig3plot.m |
An unfinished attempt to create an three dimensional version of
eigplot.m using buckyball-vertices
for equally spaced unit vectors on the sphere. |
| hw61.m |
Sample solution related to diagonalization. |
| hw61a.m |
Sample solution related to diagonalization. |
| hw62.m |
Sample solution related to diagonalization. |
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| 4. Working with images
4.a. Image compression
SVD compression.
See linalg/fourier section for Fourier approximation of image.
See pde/heateq section for blurring/sharpening images.
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|
Naive implementations -- emphasis on demonstrating
that matrices / linear algebra lurk everywhere. Images
are those vectors in R^(m x n) whose entries lie in [0,1].....
| directory listing |
| rose.tif |
Sample image: A red rose as 64x56x3 .tif |
einstein.gif
einstein.tif
| Sample image: Einstein's portrait as an 80 x 60 graysclae image.
Used in several of the following examples as default image. |
| me.tif |
Sample image: My portrait as 84x60x3 .tif.
Used in several of the following as default image.
Problem: The image is still too large --> hang-ups.... |
fasthaar2.m
invfasthaar2.m
Aug 2008
| "fast" Haar wavelet transform in 2D -- unfinished, 1st experiments --
overwritiung the original matrix with its transform. Inverse is almost
the same code (one transpose, and reversed loop) -- as should be expected!
For illustration use a 32 x 32 central section of
einstein.gif, and
play with the files (unfinished -- need to add proper scaling):
demohaar2.m
hwv.m
allhwvf.m
allhwv.m
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|
svdpix.m
| Image compression using singular decomposition.
Uses default image einstein.gif, and
displays side-by-side the original gray-scale image and a sequence of
increasingly better approaximations using naive SVD approximations.
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|
rasterpic.m
| Image compression using rastering. Very crude implementation,
having lots of trouble with line breaks in file ????
Use to initiate discussion of "linear trafo", and matrix representations.
Uses default image einstein.gif, and
displays side-by-side the gray-scale original and rastered images.
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directory
ipmovy.m
ip.m
basismovie.m
basismovie.m
maxcon.m
vblock.m
hblock.m
dblock.m
|
The first incomplete trial to create and visually display a set of matrices
that are piecewise constant" (with very few pieces), and that form
an intuitive orthonormal basis -- so far the creation of the matrices and
display work, but they are not ONB... need to do alittle math first; still
the display in
basismovy.m
might be inspiring. This file uses
hblock.m
and
vblock.m
to create the matrices, and
maxcon.m
to rescale and shift them back to the usual colormap range and display them.
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| 5. Partial differential equations
5.a. Numerical soln of Laplace and heat equation
Discretization schemes,
blurring/sharpening images
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|
| directory listing |
| blur.m |
A very naive implementation of a discretization of the heat or
diffusion equation. Used to "blur" an image by smearing out the
grey values. Uses me.tif as the default
sample image. Use together with the MS-WORD template
blurme.doc to get a feeling for
the structure of the very sparse matrix. |
| randblur.m |
Creates a random grey-scale image and "blurs" and "unblurs"
the image via naive matrix multiplication (by inverse).
Side-by-side images.
Requires: blur.m (July 2000). |
| blurpix.m
| Illustration of (deterministically) blurring
an image. Very naive implementation that does not use the
sparse matrix structure, but rather tries to emphasize the
character of this effect as a linear transformation.
Requires: blur.m and
default image einstein.gif.
Main problem: Need to clip small region like blurme(30,20)
as most computers hang-up with full image that is too large.
(July 2000, August 2004) |
sharpenme.m
sharpenpix.m |
Under reconstruction.
Complements blurme.m, exploring
the inverse of the blurring matrix ("solving the heat eqn
backwards"). Clips grey values at black/white but usually
still crashes quite quickly.
Requires: blur.m and
default image me.tif.
(July 2000) |
| atcm98.gif |
The official logo for the
AsianTechConference 1998 in Japan:
Starting from "atcm 1998" in any ordinary font, saved as a tiff image,
I used MATLAB to solve the diffusion equation iteratively on the
"complement" of the letters. The boundary data were u=0 at the edges of
a rectangular box and u=1 on the letters -- some magic coloring and
overlay of a customized contourplot made a dramatic image which
was widely seen on banners and is shown on the front-cover of the
proceedings in full color. When I find a free minute I will provide
documentation how to make the image.... |
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5.b. PDEs using separation of variables and Fourier expansions
Fourier expansions
See also Fourier analysis in Linear Algebra section
| directory listing |
current.m
matlab1.html
sep7.m
sep7a.m |
Real current data from industrial laser, see
also the background
EXCEL-spreadsheet.
Summary of first day w/ MATLAB in class.
Sample by-hand numerical integration of Fourier coefficients.
(Sept.2000), with
sep7.m very simple code, and
sep7a.m using a little slicker use of matrix-products.
|
step.m,
saw.m,
hat.m,
myfun.m
|
Sample function files whose names can be passed to the procedures below. |
simpson.m,
fourcoeffs.m,
fintegrand.m,
fourpol.m,
fapp.m,
fsamples.m
|
Several procedures to calculate Fourier coefficients and Fourier
approximations for functions that are given by formulas (as opposed to
functions given as data vectors in which case fft, ifft, etc. do the
corresponding job much more efficiently). The routines are not optimized,
instead meant to be reasonably transparent.
fourcoeffs.m calculates the Fourier coefficients
using Simpson's rule which is coded in simpson.m
(This circumvents the warnings in quad8() when that hits discontinuities).
It requires the file fintegrand.m to
create certain products...
fourpol.m generates the corresponding Fourier
approximation.
fapp.m, creates a Fourier approximation
(combining calculation of Fourier coefficients and constructing
the associted trig polynomial).
Sample uses, incl. plots of spectra and calling sequences
are provided in fsamples.m.
All of these are very new,
and NOT YET thoroughly tested.
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|
heat1.m
|
Solution of heat equation via separtion of variables,
animation of the Fourier approximations.
(still unfinished) |
simpson.m,
|
See also Fourier analysis in Linear Algebra section
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5.c. Wave and Laplace equation
d'Alembert's traveling wave soln of 1dim wave eqn
| directory listing |
| wave1.m
| Animations (slices of 3d graph u(x,t) and simultaneous animation
of (f(x-ct), f(x+ct), u(x,t) as functions of x paramterized by t.
m-file takes as input any u(x,0)=f(x) data vector.
|
| wave2.m
| Side-by-side animations of eigenmodes and combinations of linear
combinations of a vibrating membrane (2d-wave equation on rectangle).
Input matrix of Fourier coefficients b(i,j) and up to two special
modes that are diaplayed in separate windows. Main animation plays
soln sum_(i,j) b(i,j)sin(i*x)sin(j*y)cos(sqrt(i^2+j^2)t).
Special attraction:
Overlay two simple modes and get NONPERIODIV vibration.
|
| wave2disk.m
| Very similar to wave2.m, but animating
vibrations of a circular (disk) membrane.
Experimental -- have not double-checked that all frequencies are
indeed correct. Uses table of first 10 zeros of BesselJ(0, . )
through BesselJ(10, . ).
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| 6. Visualization in advanced math
6.a. Complex analysis
Color coding,
Convergence of Laurent series
Essential singularities
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|
| directory listing |
ccf.m
dcf.m
cdemo.m
|
Some small updates April 2006 to ensure animation show in fixed window.
Addition of a driver to run usual examples.
Updated (2002, September) m-file to plot color-maps for functions of a complex variable;
input either as formula (caution w/ array-operations), or as 1 x n or 2 x n numeric
array for polynomial (taylor!) or rational (Laurent) functions.
Many samples (incl. animations of convergence, and root-locus plots) on help-page.
See also Publications #39 for discussion in
recent conference article.
The new file dcf.m complements the color-plots of
functions of a complex variable by visualizing their derivatives as fields of
linear maps that act by "ampli-twists" (c.f. Needham) on small squares.
Facilities to zoom, and color coding to detect/reject analyticity (conformality).
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|
|
icat.m
ccat.m
cfill.m
cplot.m
|
Updated (2002, September) adaptations of similar files for visualization
of linear transformations from real plane (linear algebra, s.a.) to the
complex plane, w/ some new interactivity -- e.g. select complex "multiplier"
by a mouse-click (in real case a 2x2 matrix need be entered via keyboard).
Still missing, an analogous adaptation of the file drawfig.m to generate
(complex) polygons via mouse-clicks....
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|
laurentdemo.html
|
Sample images computed in MATLAB showing the convergence of a
Laurent expansion of a rational function. The code might sometime
be cleaned up ... Send e-mail
if you are interested in the implementation in MATLAB.
|
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6.x A special treat from
Tony Roberts
from the University of Southern Queensland.
Return to the top
6.b. Differential Geometry
DiffGeom Spring 2000
AdvMathTech Spring 1998
|
Some classes (that utilized largely MAPLE) required every once in a while
much faster numerics and faster graphics -- and MATLAB is the best choice here.
However, at this time I do not have any organized repository of these files
available for public use... The main MATLAB use was for generating imagery
similar to the
laurent-demo, but related to geodesic spheres etc. My goal is to
sometime code most of this in JAVA so that anyone can run it on the WWW,
w/o having to understand MATLAB code that was written for a one-time-use.
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6.c. control
Nonholonomic mechanical systems
attitude control of satellite,
planar robot ("falling cat")
Currently the focus is to develop similar tools in JAVA accessible to
everyone, hoping that this will invite more students to participate in
the advanced studies where MATLAB becomes the tool of choice.
|
Click here or on image
for larger view of sample screen.
|
|
ftp-directory
|
Several files provide tools for
a very nice graphical interface allowing to enter
the control by drawing a curve, they display the lifted curves (holonomy)
and the physical motion of the robot.
|
pp.m
pp1.m
pp2.m
pp3.m
|
Dynmaic output feedback stabilization of Lotka Volterra predator prey system.
Very naive implementation -- main purpose is illustration of the concepts,
and serve as first utility to allow playing w/ paramaters.
pp1.m linear feedback (incl. "restocking"),
pp3.m piecewise linear feedback (u=0 or u=-kz),
pp2.m bangbang feedback (u=0 or u=-k).
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|
curv.m
|
(Not yet public). Color-plot curvature for optimal control and overlay
animation of geodesic spheres (initialized by mouse-click).
Compare the sample images shown shown here.
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