Next: Project Oscillations
Up: Matrix Algebra
Previous: Sparse Matrices and the
Problems
- Poisson's equation
is usually differenced in 2 dimensions as
where
and
.
The difference equation can be written as a matrix equation
where
and
.
Write down the matrix
for
. Assume the boundaries are at
zero potential, i.e.
iff
or
.
- The equation in question 1 can be solved
by the Gauss-Seidel method. If
and
otherwise,
find
and
if
for
all
and
, where
is the value of
after
iterations.
- Work out the factors, L and U, in the LU decomposition
of the matrix,
Hence,
- Solve the simultaneous equations,
for a variety of right hand sides,
.
- Evaluate
- Find
- Show that the Jacobi method (3.14) is
stable as long as,
, the eigenvalue of largest modulus of
is less than unity.
- Find both eigenvalues and the corresponding
left and right handed eigenvectors of the matrix
- The vibrational modes of a certain molecule are
described by an equation of motion in the form
where
and
are the mass and the displacement respectively
of the
th atom and the real symmetric matrix
describes the
interactions between the atoms.
Show that this problem can be represented in the form of a generalised
eigenvalue problem:
in which the
matrix
is positive definite.
By considering the transformation
, show how to
transform this into a simple eigenvalue problem in which the matrix is
real symmetric.
- Write down the types of matrices which occur in the
following problems:
- A simple discretisation (as in question1) of Poisson's
equation in 1 dimension.
- The same but in more than 1D.
- A simple discretisation (as above) of Schrödinger's equation
in 3 dimensions.
- Schrödinger's equation for a molecule written in terms of atomic basis
functions.
- Schrödinger's equation for a crystal at a general
point in the Brillouin zone.
Next: Project Oscillations
Up: Matrix Algebra
Previous: Sparse Matrices and the