17-09-2012, 05:27 PM
A MOTIVATIONAL EXAMPLE FOR THE NUMERICAL SOLUTION OF THE ALGEBRAIC EIGENVALUE PROBLEM*
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Abstract.
This paper presents an example of an algebraic eigenvalue problem which can be used
to motivate the study of numerical techniques for solving such problems. The problem consists of
finding the axis and angle of rotation from a 3 x 3 rotation matrix and is referred to as the axis-angle
problem. The problem is used to demonstrate the inverse power method for finding eigenvectors.
The axis-angle problem is developed and numerical results are given.
Introduction.
In a previous note [I], an example was given to motivate the
study of the numerical solution of two-point boundary-value problems. This note
describes another example which can be used to motiviate the numerical solution
of the algebraic eigenvalue problem and, in particular, is a good application of the
inverse power method, also called inverse iteration or Wielandt iteration [9].
The example.
Consider the following situation which arises in computer
graphics. Usually objects in a scene are constructed in some reference coordinate
system, the local coordinates, and then rotated and translated into world coordinates,
the coordinates of the scene. This allows a particular object which may occur several
times in a scene to be created once in the local coordinates and then copied to the
various locations in the world coordinates of the scene. For example, if the scene is
a classroom there may be 30 of the same chairs in the room. The chair is modeled
once in local coordinates and then 30 occurrences are rotated and translated into
their positions in the room. Sometimes several rotations are performed to achieve the
correct orientation of the object. These rotations can be combined into one rotation
about some arbitrary axis of rotation by some angle of rotation. The rotation matrix
is determined by just multiplying the individual rotations together. However, it is
sometimes necessary to determine the axis and angle of rotation of the final rotation.
Let the axis of rotation be denoted by the unit vector a and the angle of rotation be
8.