In many practical problems in numerical differentiation of a function
f
(
x
)
f(x)
that is known, observed, measured, or found experimentally to limited accuracy, the computing error is often much more significant than the truncating error. In numerical differentiation of the n-point Lagrangian interpolation polynomial, i.e.,
f
(
k
)
(
x
)
∼
Σ
i
=
1
n
L
i
n
(
k
)
(
x
)
f
(
x
i
)
{f^{(k)}}(x) \sim \Sigma _{i = 1}^nL_i^{n(k)}(x)f({x_i})
, a criterion for optimal stability is minimization of
Σ
i
=
1
n
|
L
i
n
(
k
)
(
x
)
|
\Sigma _{i = 1}^n|L_i^{n(k)}(x)|
. Let
L
≡
L
(
n
,
k
,
x
1
,
…
,
x
n
;
x
or
x
0
)
=
Σ
i
=
1
n
|
L
i
n
(
k
)
(
x
or
x
0
)
|
L \equiv L(n,k,{x_1}, \ldots ,{x_n};x\;{\text {or}}\;{x_0}) = \Sigma _{i = 1}^n|L_i^{n(k)}(x\;{\text {or}}\;{x_0})|
. For
x
i
{x_i}
and fixed
x
=
x
0
x = {x_0}
in
[
−
1
,
1
]
[ - 1,1]
, one problem is to find the n
x
i
{x_i}
’s to give
L
0
≡
L
0
(
n
,
k
,
x
0
)
=
min
L
{L_0} \equiv {L_0}(n,k,{x_0}) = \min L
. When the truncation error is negligible for any
x
0
{x_0}
within
[
−
1
,
1
]
[ - 1,1]
, a second problem is to find
x
0
=
x
∗
{x_0} = {x^\ast }
to obtain
L
∗
≡
L
∗
(
n
,
k
)
=
min
L
0
=
min
min
L
{L^\ast } \equiv {L^\ast }(n,k) = \min {L_0} = \min \min L
. A third much simpler problem, for
x
i
{x_i}
equally spaced,
x
1
=
−
1
,
x
n
=
1
{x_1} = - 1,{x_n} = 1
, is to find
x
¯
\bar x
to give
L
¯
≡
L
¯
(
n
,
k
)
=
min
L
\bar L \equiv \bar L(n,k) = \min L
. For lower values of n, some results were obtained on
L
0
{L_0}
and
L
∗
{L^\ast }
when
k
=
1
k = 1
, and on
L
¯
\bar L
when
k
=
1
k = 1
and 2 by direct calculation from available tables of
L
i
n
(
k
)
(
x
)
L_i^{n(k)}(x)
. The relation of
L
0
,
L
∗
{L_0},{L^\ast }
and
L
¯
\bar L
to equally spaced points, Chebyshev points, Chebyshev polynomials
T
m
(
x
)
{T_m}(x)
for
m
⩽
n
−
1
m \leqslant n - 1
, minimax solutions, and central difference formulas, considering also larger values of n, is indicated sketchily.