Running MATSLISE at the command line

In some cases it may be interesting to run MATSLISE at the command line, not using the Graphical User Interface. This page discusses some of the functions and methods that can be called from the command line directly (or from a Matlab .m-file). Before the methods discussed below can be called, the MATSLISE directory and its subdirectories must be added to the search path. This can be done using the Set Path-dialog from the Matlab File-menu or using the Matlab command addpath.

A Schrödinger object is created by:

      sch = schrod(V,a,b,a0,b0,a1,b1)
      sch = schrod(V,a,b,a0,b0,a1,b1,var)

where V is a string representing the potential function. The double precision constants a,b,a0,b0,a1,b1 specify the integration interval and the boundary conditions. It is allowed to enter inf/-inf for the two input parameters a and b. Good truncated endpoints are then determined automatically for every eigenvalue. If there are less then eight input arguments, then the independent variable is supposed to be x, otherwise the independent variable is the character or string in var and V is then V(var). The method schrod(...) is the constructor of the class schrod.

In a very analogous manner a Sturm-Liouville equation is specified:

      sl = slp(p,q,w,a,b,a0,b0,a1,b1)
      sl = slp(p,q,w,a,b,a0,b0,a1,b1,var)

where p,q and w are strings representing the coefficient functions p(x),q(x) and w(x) (or p(var), q(var) and w(var)) and the double precision numbers a,b,a0,b0,a1,b1 are the endpoints of the integration interval and the coefficients of the boundary conditions. Again inf-values are allowed for a and b.

An object representing an equation with a distorted Coulomb potential is produced by

      d = distorted_coulomb(l,S,R,xmax)
      d = distorted_coulomb(l,S,R,xmax,var)

where S and R are two strings (representing S(x) and R(x)) and the orbital quantum number l is a double. xmax is the x-value where the integration interval is truncated. When xmax is equal to inf a good truncation point is automatically determined for every requested eigenvalue.

After the problem is specified, the user can call one of the classes which implement the actual Piecewise Perturbation algorithms included in MATSLISE (cpm12_10, cpm14_12, cpm16_14 or lpm10) e.g.:

      cp = cpm16_14(schrod,tol)
      cp = cpm16_14(slp,tol)
      cp = cpm16_14(distorted_coulomb,tol)
or
      lp = lpm10(schrod,tol)
      lp = lpm10(slp,tol)
      lp = lpm10(distorted_coulomb,tol)

with tol a positive constant representing the accuracy requested in the results. These function-calls start the construction of the partition. This partition and some related information is returned in the output argument (cp or lp).

In this stage the eigenvalues, in a range fixed by the user, are calculated. The user starts the calculation by calling a method from the ppm class:

E = get_eigenvalues(pp_child,pmin,pmax)
E = get_eigenvalues(pp_child,pmin,pmax,indices)
[E,pp_child] = get_eigenvalues(pp_child,pmin,pmax,indices) %(for infinite problems)

where pp_child is an instance of one of the child classes of the cpm class or lpm class(i.e. the output argument cp or lp from the previous section). If indices is true, the eigenvalues E_k (k=0,1,...) with indices k between pmin and pmax are calculated; if indices is false or omitted the eigenvalues in the range [pmin,pmax] are calculated. The method returns a structure E, in which all information related to the calculated eigenvalue(s) is stored. The fields E.eigenvalues, E.indices and E.errors are three vectors. E.eigenvalues contains the calculated eigenvalues in ascending order. The associated indices are collected in E.indices and E.errors holds an estimation of the error for each eigenvalue. The field E.success is false when the PP method was not able to obtain the data due to an error, e.g. when there is no eigenvalue in the interval [pmin,pmax]. In some cases a warning is generated in the method get_eigenvalues: e.g. when two eigenvalues are so close that double precision accuracy is not sufficient to differentiate between them adequately.

Another method in the ppm class which is visible to the user, is the method which allows the calculation of the eigenfunction associated with a certain eigenvalue e:

V = get_eigenfunction(pp_child,e)
V = get_eigenfunction(pp_child,e,n)
V = get_eigenfunction(pp_child,e,ap,bp,n)

where again pp_child is an instance of one of the child classes of ppm. To obtain a good approximation for the eigenfunction the eigenvalue e must be sufficiently accurate (e.g. by choosing a small value for tol). If n is omitted, then the eigenfunction is evaluated only in the meshpoints of the partition. If n > 0, then the interval [ap,bp] is taken and further partitioned in n intervals. On these intervals the additional potential-dependent expressions are calculated and the propagation matrix algorithm is applied to produce the eigenfunction. Only the part of the eigenfunction corresponding with the n+1 points in [ap,bp] is returned. When the input-arguments ap and bp are omitted, then the whole interval [a,b] is considered. The structure V has three fields: the three vectors V.x, V.y and V.yprime of which the meaning is clear.

The ppm class contains one more function accessible for users: the plot_partiton function. It can be called as follows:

plot_partition(pp_child)

where pp_child is again an instance of the cpm12_10, cpm14_12, cpm16_14 or lpm10 class.
The plot_partition function makes a plot of the partition: i.e. of the meshpoints and the reference potential.

Some properties of an object defined in MATSLISE can be accessed using the get-method of the form get(object_name,'property_name'). Some properties of a schrodinger problem which can be obtained in this way are e.g.: V (the potential function), min and max (the endpoints of the integration interval). But maybe the most interesting one is infostring which returns a string containing information on the classification of a problem. This classification information is obtained using the classification algorithm for Sturm-Liouville problems by C. Fulton, S. Pruess and Y. Xie.

sch = schrod('x^2',0,inf,1,0,1,0);
get(sch,'infostring')

The spectrum is simple, purely discrete and bounded below.
There is no continuous spectrum.

At endpoint A
   The problem is regular.
At endpoint B
   The problem is singular.
   It is nonoscillatory for all eigenvalues.
   It is limit point.
   The Friedrichs boundary conditions are 1y(b)+ 0y'(b)=0

This classification information can also be obtained for a distorted-Coulomb or a Sturm-Liouville object.

The MATSLISE examples-directory contains some example .m files. These examples illustrate the use of the functions discussed above to solve a Schrodinger or Sturm-Liouville problem, i.e. to calculate the eigenvalues and eigenfunctions.