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Algebra Homework 3 Solutions
Problem 5: Use fourdigit rounding arithmetic and the formulas to find the most
accurate
approximations to the roots of the following quadratic equations. Compute the
relative error.
a)
b) 1.002x^{2} + 11.01x+ 0.01265 = 0.
Solution : The quadratic formula states that the roots of ax^{2} + bx + c = 0 are
a) The roots of are approximately
x_{1} = 92.24457962731231,
x_{2} = 0.00542037268770.
We use four digit rounding arithmetic to find approximations to the roots. We
find the
first root:
which has the following relative error:
has the following relative error:
We obtained a very large relative error, since the
calculation for
involved the subtraction
of nearly equal numbers . In order to get a more accurate approximation to
, we need
to use an alternate quadratic formula , namely
Using four digit rounding arithmetic, we obtain:
which has the following relative error:
b) The roots of 1.002x^{2} + 11.01x+ 0.01265 = 0 are approximately
x_{1} = −0.00114907565991,
x_{2} = −10.98687487643590.
We use fourdigit rounding arithmetic to find approximations to the roots.
If we use the generic quadratic formula for the calculation of
, we will encounter the
subtraction of nearly equal numbers (you may check). Therefore, we use the
alternate
quadratic formula to find
;
which has the following relative error:
We find the second root using the generic quadratic formula:
which has the following relative error:
Similar Problem
The roots of 1.002x^{2} − 11.01x+ 0.01265 = 0 are approximately
x_{1} = 10.98687487643590,
x_{2} = 0.00114907565991.
We use fourdigit rounding arithmetic to find approximations to the roots. We
find the
first root:
which has the following relative error:
If we use the generic quadratic formula for the
calculation of
, we will encounter the
subtraction of nearly equal numbers . Therefore, we use the alternate quadratic
formula
to find
:
which has the following relative error:
Problem 6: Suppose that fl(y) is a kdigit rounding approximation to y. Show that
(Hint: If
then
If
then
)
Solution : We have to look at two cases separately.
Case
:
since
so
since
by assumption
Case :
Note that
and
For example,
since so
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