HP 48g Graphing Calculator - Manual (Page 180)

14
Ill-Conditioned and Singular Matrices
A singular matrix is a square matrix that doesn’t have an inverse.
You normally get an error if you use [llx) to find the inverse of a
singular matrix—or use © to solve a system of linear equations having
a singular coefficient matrix.
The most common cause of singular matrices are equations within a
s}fstem of linear equations that are
linear combinations of one another.
That is, the coefficients of one equation can be computed exactly
from the coefficients of the other. Two equations thus related are
linearly dependent and the set of equations as a whole referred to as
dependent.
If a set of equations is independent, but small changes in their
coefficients would make them dependent, then the set of equations
(and their corresponding matrix A) are said to be ill-conditioned.
To determine if a matrix is ill-conditioned;
1. Enter the matrix onto the stack.
2. Compute its condition number: Press fMTH | MflTR N C i R M
C u N D . If it is large, then it is ill-conditioned. If the condition
number is on the order of 10^^, the HP 48 may not be able to
distinguish it from a singular matrix.
To use ill-conditioned matrices in solving systems of linear equa­
tions:
1. Set f l a g - 2 2 : Press MODES) 22 fV-) F'LflG :3F . This is
the Infinite Result Exception flag that will keep you from getting
an error using a singular matrix.
2. Solve the system of linear equations. The HP 48 perturbs the
singular matrix by an amount that’s usually small compared to the
rounding error. The calculated result corresponds to that for a
matrix close to the original, ill-conditioned matrix.
3. Determine the accuracy of the computed solution using the
condition number as you would for any ill-conditioned matrix (see
“Determining the Accuracy of a Matrix Solution” below.
4. Compute the residual to test your results.
5. Resolve the system of linear equations using LSQ.
14-16 Matrices and Linear Algebra