HP HP-32S - Round-Off Error and Underflow

Round-Off Error and "Underflow"
Round-off Error. The limited (12-digit) precision of the calculator
can cause errors due to rounding off, which adversely affect the itera
tive solutions of SOLVE and integration. For example,
[(Ixl + 1) + 1015]2 - 1030 = 0
has no roots because fix) is always greater than zero. However, given
initial guesses of 1 and 2, SOLVE returns the answer 1.0000 due to
round-off error.
Round-off error can also cause SOLVE to fail to find a root. The
equation
\x2 - 71 = 0
has a root at \J7 . However, no 12-digit number exactly equals \ff, so
the calculator can never make the function equal to zero. Further
more, the function never changes sign. SOLVE returns the message
NO ROOT FND. However, the final estimate of x (press(T)to see it) is
the best possible 12-digit approximation of the root when the routine
quits.
"Underflow." Underflow occurs when the magnitude of a number is
smaller than the calculator can represent, so it substitutes zero. This
can affect SOLVE results. For example, consider the equation
J_
x2
whose root is infinite in value. Because of underflow, SOLVE returns a
very large value as a root. (The calculator cannot represent infinity,
anyway.)
272 C: More About Solving an Equation