How to use the MATSLISE GUI

How to use the MATSLISE GUI
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How to use the MATSLISE
GUI
MATSLISE can be used to solve differential equation eigenvalue problems that can be put in the
general form of the classical Sturm-Liouville problem
on a finite or infinite interval with boundary conditions at the ends as appropriate. A value of the
eigenparameter E for which there is a nontrivial solution - that is, not identically zero- subject to
the boundary conditions, is called an eigenvalue, and the solution is the corresponding
eigenfunction. We will illustrate how the MATSLISE GUI works by using it to compute the
eigenvalues and eigenfunctions of the following problem:
which has known analytic forms for its eigenvalues:
This problem is included as an example problem which can be loaded into the MATSLISE GUI.
Use the option Open from the File menu to open Klotter.mat from the MATSLISE examples
directory.
The GUI is simply opened by typing the command matslise at the Matlab command line:
>> matslise
Problem Specification
file://C:\helpMatslise\gui.html
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The figure shows the problem-specification fields for our example problem:
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p(x): a stricly positive continuous function.
q(x): a continuous function.
w(x): a stricly positive continuous function.
xmin,xmax: the integration interval (real values)
The boundary conditions
Parameters
In all fields Matlab notation needs to be used. Note that the user is allowed to enter -inf or inf in
the integration-interval-fields (xmin and xmax) in order to specify an infinite interval.
The Plot-button can be used to generate a plot of the coefficient functions. The clear-button
clears all the problem specification fields.
It is possible to use a parameter in the input fields for the coefficient functions, the integration
interval or the boundary conditions. When any of these input fields contains a parameter the user
must enter the name of the parameter and the value in the Parameters input field.
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The parameter-fields can also be used to replace rather lengthy subexpressions in one of the
coefficient functions by a parameter
Multiple parameters are separated by a comma:
Eigenvalue Computations
The Eigenvalues-part of the MATSLISE GUI allows the user to calculate the eigenvalues of the
problem he/she specified. There are two options: (i) one can calculate a set of eigenvalues with
indices between a user-specified lower-and upper limit (ii) one can use MATSLISE to calculate
eigenvalues between a lower and upper limit. Pressing the Compute-button starts the
computations. When the computations have finished the results appear in the results listbox. The
computed indices and eigenvalues are shown. Also for each eigenvalue approximation an
(absolute) error estimation is given, which is the difference between this eigenvalue
approximation and a reference approximation computed by a reference method (usually a higher
order method). A positive double precision constant representing the accuracy requested by the
user can be entered in the Accuracy input field.
Indices between...and...
The user enters a lower and upper index. The eigenvalues with index between this lower and
upper index will be calculated. One single eigenvalue is calculated when the upper index is equal
to the lower index. The eigenvalues are indexed from 0, not 1.
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Eigenvalues between...and...
The user specifies the lower and upper limits of the interval to be scanned for eigenvalues, e.g.
when the user wants to calculate the eigenvalues larger than 20 and smaller than 50, he enters 20
and 50 in the two inputfields.
Plotting the results
The Plot button creates a graphical representation of the computed eigenvalues. A MATLAB
figure window is opened with a plot of the eigenvalues with respect to their indices. The figure
can be saved and edited like any MATLAB figure.
If the problem is in Liouville normal form (i.e. p(x)=w(x)=1) a second figure window is opened
showing the eigenvalues together with the potential function q(x).
Save results into MATLAB workspace
The Workspace button allows the user to make the results accessible from the MATLAB
workspace. In this way the eigenvalue approximations can be easily used in further computations.
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The user enters variable names for the vectors of indices, eigenvalues and their estimated errors.
Eigenfunction Computations
After the computation of the eigenvalue approximations, one can use MATSLISE to compute the
associated (normalized) eigenfunctions. Of course reliable evaluations of the eigenfunctions can
only be obtained if the approximated eigenvalues are sufficiently accurate, i.e. the input accuracy
tolerance should be sufficiently low and the eigenvalues are well-separated.
There are two popupmenus. The user can select an eigenfunction from the first popupmenu: only
eigenfunctions for which an eigenvalue was computed can be selected. In the second popupmenu,
the user has to specify if the eigenfunction needs to be evaluated only in the mesh points or in a
vector of user-specified x-values.
The Compute-button starts the eigenfunction computations and shows the obtained results in the
listbox. Also a regular MATLAB plot of the eigenfunction y and its (pseudo)derivative py' is
generated. The red dots correspond to the eigenfunction results which are listed in the listbox.
Also a more smooth graphical representation of the eigenfunction is displayed in blue, which is
the eigenfunction computed over a larger number of evaluation points spread over the problem
domain. The figure can be edited like any MATLAB figure e.g. by using the figure toolbar.
There are many possible uses for eigenfunction data. In order to allow the user to use the
MATSLISE eigenfunction results in other applications, the eigenfunction evaluations computed in
the MATSLISE GUI can be saved into workspace variables (vectors) using the Workspace-button
Evaluation of an eigenfunction in the mesh points
In this case, a click on the Compute-button leads to the computation and the display of the
eigenfunction y(x) and the derivative p(x)y'(x) in the mesh points as red dots in the figure
windows. These mesh points are automatically generated by MATSLISE during the eigenvalue
computations.
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Evaluation of an eigenfunction in user-specified points
The number of mesh points needed by the CP algorithm is usually quite small. Evaluations of the
eigenfunction in only these mesh points may be unsufficient to have a clear idea about the shape
of the eigenfunction. Moreover there are many possible uses for eigenfunctions. In many cases, it
may then be convenient for the user to be able to provide an array of x points and having the
corresponding eigenfunction values returned. The array of x points is a vector variable in the
MATLAB workspace, the MATSLISE user is asked for the name of this variable.
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Saving and loading problems in the MATSLISE GUI
Using the Open and Save As options in the File menu, the user can save a problem specification
into a MAT-file and reopen it later.
The MATSLISE package contains a directory examples with predefined problems which can be
loaded into the MATSLISE GUI. This MATSLISE test set collects problems which are commonly
used to test Sturm-Liouville software, some examples from physical applications and some
classical mathematical problems. Many examples have been taken from earlier test sets:
• the list of test problems provided by John D. Pryce in the book Numerical Solution of SturmLiouville Problems, Oxford University Press, 1993 and the SLTSTPAK package: A test
package for Sturm-Liouville solvers, ACM Trans. Math. Software 25 (1999).
• the catalogue of Sturm-Liouville problems composed by W.N. Everitt: A Catalogue of SturmLiouville differential equations, in: Sturm-Liouville Theory. Past and Present, W.O. Amrein,
A.M. Hinz, D.B. Pearson (eds), Birkhauser-Verlag 2005.
What is MATSLISE?
file://C:\helpMatslise\gui.html
Using MATSLISE via the command line?
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