HP HP-15C - Page 99

Section 4: Using Matrix Operations 97
Some matrices can't be factored into the L [/form. For example,
A —
0 1
1 2
for any pair of lower- and upper-triangular matrices L and U.
However, if rows are interchanged in the matrix to be factored, an
L U decomposition can always be constructed. Row interchanges in
the matrix A can be represented by the matrix product PA for some
square matrix P. Allowing for row interchanges, the LU
decomposition can be represented by the equation PA = LU. So for
the above example,
PA-
0 1
1 0
1 0
0 1
= LU.
Row interchanges can also reduce rounding errors that can occur
during the calculation of the decomposition.
The HP-15C uses the Doolittle method with extended-precision
arithmetic to construct the LU decomposition. It generates the
decomposition entirely within the result matrix. The LU
decomposition is stored in the form
U
It is not necessary to save the diagonal elements of L since they are
always equal to 1. The row interchanges are also recorded in the
same matrix in a coded form not visible to you. The decomposition
is flagged in the process, and its descriptor includes two dashes
when displayed.
When you calculate a determinant or solve a system of equations,
the L U decomposition is automatically saved. It may be useful to
use the decomposed form of a matrix as input to a subsequent
calculation. If so, it is essential that you not destroy the
information about row interchanges stored in the matrix; don't
modify the matrix in which the decomposition is stored.