h =
abs
(
f
(
x
))
f
∏∏
(
x
)
10
where I have used the absolute value abs() to ensure that the root is real. I also divide the final result
by 10 to ensure that h is small enough that nder1() doesn't terminate too early.
nder1() can terminate in one of two ways: if the error keeps decreasing, nder1() will run until the amat[]
matrix is filled with values. However, if at some step the error increases, nder1() will terminate early.
The variable safe specifies the magnitude of the error increase that causes termination. Refer to the
code listing for details.
From my testing with a few functions, nder1() nearly always terminates by exceeding the safe limit.
[6.27] Find Bernoulli numbers and polynomials
[Note: since this tip was written, Bhuvanesh Bhatt has also written a function to find both Bernoulli
numbers and polynomials. Bhuvanesh' function is smaller and handles complex arguments. You can
get it at his site (see the "More Resources - Web sites" section for the URL), and at ticalc.org. Also, M.
Daveluy has written a Bernoulli number function, which can be found at ti-cas.org.]
Bernoulli numbers are generated from the Benoulli polynomials evaluated at zero. Bernoulli
polynomials are defined by the generating function
te
xt
e
t
−1
=
✟
n=0
∞
B
n
(
x
)
t
n
n!
Bernoulli polynomials can also be defined recursively by
[1]
B
0
(
x
)
= 1
[2]
d
dx
B
n
(
x
)
= nB
n−1
(
x
)
for [3]
¶
0
1
B
n
(
x
)
dx = 0
n m 1
The first few Bernoulli polynomials are
B
0
(
x
)
= 1B
3
(
x
)
=
2x
3
−3x
2
+x
2
B
1
(
x
)
=
2x−1
2
B
4
(
x
)
=
30x
4
−60x
3
+30x
2
−1
30
B
2
(
x
)
=
6x2−6x+1
6
B
5
(
x
)
=
6x
5
−15x
4
+10x
3
−x
6
The nth Bernoulli number is denoted as B
n
. The Bernoulli numbers can be defined by the generating
function
t
e
t
−1
=
✟
n=0
∞
B
n
t
n
n!
6 - 39
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