MATSLISE: The directory "predefined_problems" |
The directory "predefined_problems"
The MATSLISE package includes a directory "predefined_problems".
This directory contains several examples (saved as .mat
-files.) The problems included in this directory
can be listed by typing showPredefinedProblems
at the command line:
>> showPredefinedProblems
airy.mat (/Schrodinger/infinite_interval)
anharm_oscillator.mat (/Schrodinger/infinite_interval)
bender_orszag.mat (/Schrodinger/infinite_interval)
bessel.mat (/Sturm-Liouville/regular)
bessel_normalform.mat (/distorted_Coulomb/finite_interval)
bessel_order0.mat (/distorted_Coulomb/finite_interval)
biswas.mat (/Schrodinger/infinite_interval)
close_eigenvalues.mat (/Schrodinger/regular/close_eigenvalues)
close_eigenvalues.mat (/Schrodinger/infinite_interval/close_eigenvalues)
coffey_evans.mat (/Schrodinger/regular/close_eigenvalues)
collatz.mat (/Sturm-Liouville/regular)
coulomb.mat (/distorted_Coulomb/infinite_interval)
expon_cosine_part_screening.mat (/distorted_Coulomb/infinite_interval)
expon_cosine_screening.mat (/distorted_Coulomb/infinite_interval)
gelfand_levitan_truncated.mat (/Schrodinger/regular)
harmonic_oscillator.mat (/Schrodinger/infinite_interval)
hr_anharm_oscillator.mat (/Schrodinger/infinite_interval)
hulthen_part_screening.mat (/distorted_Coulomb/infinite_interval)
hulthen_screening.mat (/distorted_Coulomb/infinite_interval)
hydrogen.mat (/distorted_Coulomb/infinite_interval)
hydrogen_truncated.mat (/distorted_Coulomb/finite_interval)
klotter.mat (/Sturm-Liouville/regular)
laguerre.mat (/distorted_Coulomb/infinite_interval)
mathieu.mat (/Schrodinger/regular)
mathieu_version.mat (/Schrodinger/regular/close_eigenvalues)
morse1.mat (/Schrodinger/infinite_interval)
morse2.mat (/Schrodinger/infinite_interval)
paine1.mat (/Schrodinger/regular)
paine2.mat (/Schrodinger/regular)
paine_slp.mat (/Sturm-Liouville/regular)
parameter_example1.mat (/Schrodinger/parameter_problems)
parameter_example2.mat (/Schrodinger/parameter_problems)
parameter_example3.mat (/Schrodinger/parameter_problems)
parameter_example_slp.mat (/Sturm-Liouville/parameter_problems)
parameter_problem_dc.mat (/distorted_Coulomb/parameter_problems)
pruess_fulton133.mat (/Schrodinger/regular)
pruess_fulton19.mat (/Sturm-Liouville/regular)
pryce33.mat (/Sturm-Liouville/infinite_interval)
pryce36.mat (/distorted_Coulomb/infinite_interval)
pryce42.mat (/distorted_Coulomb/infinite_interval)
pryce43.mat (/distorted_Coulomb/finite_interval)
pryce60.mat (/Schrodinger/infinite_interval)
pure_coulomb.mat (/distorted_Coulomb/infinite_interval)
quartic_anharm_oscillator.mat (/Schrodinger/infinite_interval)
razavy.mat (/Schrodinger/infinite_interval)
simple_slp1.mat (/Sturm-Liouville/regular)
simple_slp2.mat (/Sturm-Liouville/regular)
symm_double_well.mat (/Schrodinger/infinite_interval)
wicke_harris.mat (/Schrodinger/infinite_interval)
woods-saxon.mat (/distorted_Coulomb/infinite_interval)
Examples of Schrödinger problems
- Regular Schrödinger problems: The partitioning process has to be performed only once at the very
beginning of the run. Several batches of eigenvalues (even at high energies) can be calculated one after the other,
without recalculating the partition. This makes the CP-methods (and MATSLISE) very fast and efficient for
this kind of problems. More concrete: the calculation of the eigenvalue with index one thousand, takes approximately
the same amount of time as the first eigenvalue.
- Gelfand-Levitan truncated (Pryce #6)
- Mathieu (Pryce #2)
- Paine 1
- Paine 2 (Pryce #1)
- Pruess-Fulton 133 (Pryce #11)
- Close eigenvalues problems: For certain parameter-values clustering of the lower eigenvalues
appears. Numerically, clustering causes the eigenfunctions to be very ill-conditioned. This ill-conditioning
causes some difficulties in the calculation of the eigenvalues and eigenfunctions. For symmetric double
well problems (e.g. Coffey-Evans), the halfrange reduction option may make the problem more tractable.
- Coffey-Evans (Pryce #7)
- Mathieu version (Pryce #5)
- Schrödinger problems with an infinite integration interval:
These problems are solved by regularization. This means
that the problem is approximated by a regular problem on a truncated interval. MATSLISE uses some kind of
two-pass process. The reason for this is that there is no way to tell where to truncate without having some
approximation of the eigenvalue. The first pass is done on a very coarse mesh; on this coarse mesh a first
approximation of the eigenvalue is obtained. This approximation is then used to determine good truncation points.
A more accurate eigenvalue-approximation is then obtained on the truncated integration interval (second pass).
It is clear that the solution of these problems with infinite integration interval needs some additional computation time
for the determination of appropriate truncation points (first pass). Moreover the eigenvalue-independence of the partition is
lost, since the truncated endpoints are not the same for every eigenvalue, i.e. a higher eigenvalue needs a larger truncated
integration interval than a lower eigenvalue. This explains why the calculation of a higher eigenvalue requires now more time
than the calculation of a lower eigenvalue (in contradiction to the regular problems).
- Airy (Pryce #27)
- Anharmonic Oscillator
- Bender_Orszag (Pryce #14)
- Biswas
- Close-Eigenvalues (Pryce #38)
- Harmonic Oscillator (Pryce #28)
- Half-Range Anharmonic Oscillator (Pryce #17)
- Morse1 (Pryce #35)
- Morse2 (Pryce #39)
- (Pryce #60)
- Quartic Anharmonic Oscillator (Pryce #37)
- Razavy
- Symm. double-well
- Wicke-Harris (Pryce #40)
- The use of parameters in Schrödinger problems: These examples show how a parameter can be used
to study the behaviour of the solution when there are some changes in the potential function or boundary conditions.
- parameter_example1
- parameter_example2
- parameter_example3
Examples of Sturm-Liouville problems
- Regular Sturm-Liouville problems: As for regular Schrödinger problems, MATSLISE performs well on
these regular problems. Again the partition has to be calculated only once and is never modified further on, but now
the Liouville-transformation requires some small amount of additional time, both in the calculation of the partition and in
the eigenfunction-computations.
- Bessel (Pryce #19)
- Collatz
- Klotter (Pryce #3))
- Paine_slp
- Pruess-Fulton 19 (Pryce #25)
- Simple_slp1
- Simple_slp2
- Sturm-Liouville problems with an infinite integration interval: Same remarks as for the Schrödinger problems with
an infinite integration interval.
- The use of parameters in Sturm-Liouville problems:
Examples of Distorted Coulomb problems
- Distorted Coulomb problems with a finite integration interval: Around the origin a numerical method
is used which is consistent with the singular nature of the problem. Again the same partition is used
for all eigenvalues.
- Bessel_normalform (Pryce #13)
- Bessel_order0 (Pryce #18)
- Hydrogen truncated (Pryce #4)
- (Pryce #43)
- Distorted Coulomb problems with an infinite integration interval: The same procedure is used for the truncation of the
infinite endpoints as for the Schrödinger and Sturm-Liouville problems.
- Coulomb (Pryce #30)
- Exponential Cosine potential
- Hulthén potential
- Hydrogen (Pryce #29)
- Laguerre (Pryce #32)
- (Pryce #36) (Morse)
- (Pryce #42)
- Pure Coulomb
- Woods-Saxon
- The use of parameters in distorted Coulomb problems:
Most of these problems are defined as a testproblem by Pryce in the Appendix B of [1].
[1] J. D. Pryce, Numerical Solution of Sturm-Liouville Problems (Oxford Univ. Press, Oxford, 1993).