EE 103, Winter ’09, Prof SEJ:
HW 1 Sol.
Page 1 of 12
Applied Numerical Computing
Instructor: Prof. S. E. Jacobsen
HW1 Solution
Students: Distributed HW solutions are a component of the course and should be
fully understood.
SEJ
Problem 1)
To find the roots of the equation
0
1
1
.
62
2
=
+
−
x
x
, using fourdigit
arithmetic we compute:
b
2
−
4
ac
=
fl
((
−
62.1)
2
)
−
4
=
fl
(3856.41)
−
4
=
3856
−
4
=
3852
=
fl
(62.0645)
=
62.06
therefore if we use fourdigit arithmetic and the quadratic formula, the two roots will be:
fl
(
r
1
)
=
−
b
+
b
2
−
4
ac
2
a
=
(62.1
+
62.06)
2
=
62.08
fl
(
r
2
)
=
−
b
−
b
2
−
4
ac
2
a
=
(62.1
−
62.06)
2
=
0.02
But the roots of the equation,
0
1
1
.
62
2
=
+
−
x
x
(to seven decimal places) are
0838928
.
62
1
=
r
and
0161072
.
0
2
=
r
. If we round these numbers to four digits, we will
have
08
.
62
1
=
r
and
01611
.
0
2
=
r
. What we found instead are
fl
(
r
1
)
=
62.08 and
fl
(
r
2
)
=
0.02
.
So
the
relative
errors
are



)
(

1
1
1
r
r
r
f
−
=
0.0
and



)
(

2
2
2
r
r
r
f
−
=
24
.
0
01611
.
0
00389
.
0
≈
The percentage error for the second root is approximately 24%. This error is very large
and clearly unacceptable. The reason for this large error is
subtractive cancellation,
which occurs when two close numbers are subtracted (in this case b=62.1 and
06
.
62
4
2
≈
−
ac
b
). while at least one of them is subject to error (here we have error due
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