Ill-Conditioned Matrices And The Condition Number - HP -15C Advanced Functions Handbook

98 Section 4: Using Matrix Operations
To calculate the determinant of a matrix, A for example, the
HP-15C uses the equation A = P^LU, which allows for row
interchanges. The determinant is then just (-l)
r
times the product
of the diagonal elements of U, where r is the number of row
interchanges. The HP-15C calculates this product with the correct
sign after decomposing the matrix. If the matrix is already
decomposed, the calculator just computes the signed product.
It's easier to invert an upper- or lower-triangular matrix than a
general square matrix. The HP-15C calculates the inverse of a
matrix, A for example, using the relationship
It does this by first decomposing matrix A, inverting both L and U,
calculating their product U^Lf
1
, and then interchanging the
columns of the result. This is all done within the result matrix —
which could be A itself. If A is already in decomposed form, the
decomposition step is skipped. Using this method, the HP-15C can
invert a matrix without using additional storage registers.
Solving a system of equations, such as solving AX — B for X, is
easier with an upper- or lower-triangular system matrix A than
with a general square matrix A. Using PA — LU, the equivalent
problem is solving LUX = PB for X. The rows of B are
interchanged in the same way that the rows of the matrix A were
during decomposition. The HP-15C solves LY = PB for Y (forward
substitution) and then UX = Y for X (backward substitution). The
L C7 form is preserved so that you can find the solutions for several
matrices B without reentering the system matrix.
The LU decomposition is an important intermediate step for
calculating determinants, inverting matrices, and solving linear
systems. The L U decomposition can be used in lieu of the original
matrix as input to these calculations.
Ill-Conditioned Matrices
and the Condition Number
In order to discuss errors in matrix calculations, it's useful to define
a measure of distance between two matrices. One measure of the