HP 15c User Manual
Page 99
Section 4: Using Matrix Operations
99
10
10
0
3
3
1
3333333333
.
0
1
LU
,
which is nonsingular. The singular matrix B can't be distinguished from the nonsingular
matrix
1
9999999999
.
3
3
D
since they both have identical calculated LU decompositions.
On the other hand, the matrix
LU
A
10
3
1
10
0
3
3
1
0
1
9999999999
.
1
3
3
is nonsingular. Using 10-digit accuracy, matrix A is decomposed as
0
0
3
3
1
3333333333
.
0
1
LU
.
This would incorrectly indicate that matrix A is singular. The nonsingular matrix A can't be
distinguished from the singular matrix
9999999999
.
9999999999
.
3
3
C
since they both have identical calculated LU decompositions.
When you use the HP-15C to calculate an inverse or to solve a system of equations, you
should understand that some singular and nearly singular matrices have the same calculated
LU decomposition. For this reason, the HP-15C always calculates a result by ensuring that all
decomposed matrices never have zero pivots. It does this by perturbing the pivots, if
necessary, by an amount that is usually smaller than the rounding error in the calculations.
This enables you to invert matrices and solve systems of equations without being interrupted
by zero pivots. This is very important in applications such as calculating eigenvectors using
the method of inverse iteration (refer to page 130).
The effect of rounding errors and possible intentional perturbations is to cause the calculated
decomposition to have all nonzero pivots and to correspond to a nonsingular matrix A + ΔA
usually identical to or negligibly different from the original matrix A. Specifically, unless
every element in some column of A has absolute value less than 10
−89
, the column sum norm
||ΔA||
C
will be negligible (to 10 significant digits) compared with ||A||
C
.
The HP-15C calculates the determinant of a square matrix as the signed product of the
(possibly perturbed) calculated pivots. The calculated determinant is the determinant of the