Exercises 40—43 refer to another algorithm, known as Horner’s rule, for finding the value of a polynomial.
Algorithm 11.3.4 Homer’s Rule
[This algorithn computes the value of a polynomial
a
[
n
]
x
n
+
a
[
n
−
1
]
x
n
−
1
+
⋯
+
a
[
2
]
x
2
+
a
[
1
]
x
+
a
[
0
]
by nesting successive additions and multiplications as indicated in the following parenthesization:
(
(
⋯
(
(
a
[
n
]
x
+
a
[
n
−
1
]
)
x
+
a
[
n
−
2
]
)
x
+
⋯
+
a
[
2
]
)
x
+
a
[
1
]
)
x
+
a
[
0
]
.
At each stage, starting with
a
[
n
]
, the current value of polyval is multiplied by x and the next lower coefficient of the polynomial is added to it.] Input: n[a nonnegative integer],
a
[
0
]
,
a
[
1
]
,
a
[
2
]
,
…
,
a
[
n
]
[an array of real numbers], x [a real number] Algorithm Body:
p
o
l
y
v
a
l
:
=
a
[
n
]
for
i
:
=
1
to n
p
o
l
y
v
a
l
:
=
p
o
l
y
v
a
l
⋅
x
+
a
[
n
−
i
]
next i
[
A
t
t
h
i
s
p
o
i
n
t
p
o
l
y
v
a
l
=
a
[
n
]
x
n
+
a
[
n
−
1
]
x
n
−
1
+
⋯
+
a
[
2
]
x
2
+
a
[
1
]
x
+
a
[
0
]
.
]
Output: polyval [a real number]
42. Let
t
n
=
the number of additions and multiplications that are performed when Algorithm 11.3.4 is executed for a polynomial of degree n. Express
t
n
, as a function of n.