Last Updated：2025/12/03
(mathematics) A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n.
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Lagrange's interpolation formula
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Source Word
Lagrange's interpolation formula
Noun
uncountable
(mathematics)
A
formula
which
when
given
a
set
of
n
points
(x_i,y_i),
gives
back
the
unique
polynomial
of
degree
(at
most)
n
−
1
in
one
variable
which
describes
a
function
passing
through
those
points.
The
formula
is
a
sum
of
products,
like
so:
∑ᵢⁿy_i∏_(j
ne
i)x-x_j/x_i-x_j.
When
x=x_i
then
all
terms
in
the
sum
other
than
the
iᵗʰ
contain
a
factor
x-x_i
in
the
numerator,
which
becomes
equal
to
zero,
thus
all
terms
in
the
sum
other
than
the
iᵗʰ
vanish,
and
the
iᵗʰ
term
has
factors
x_i-x_j
both
in
the
numerator
and
denominator,
which
simplify
to
yield
1,
thus
the
polynomial
should
return
y_i
as
the
function
of
x_i
for
any
i
in
the
set
1,...,n.
Japanese Meaning
ラグランジュの補間公式：数学における公式で、与えられた n 個の点 (x_i, y_i) を通る一意の、最高次数 n − 1 以下の多項式（補間多項式）を構成する。公式は、各点に対応する補間基底多項式の加重和として表され、各補間基底多項式は対象の点で 1 となり他の点で 0 となるように設計されている。
Sense（1）
(mathematics)
A
formula
which
when
given
a
set
of
n
points
(x_i,y_i),
gives
back
the
unique
polynomial
of
degree
(at
most)
n
−
1
in
one
variable
which
describes
a
function
passing
through
those
points.
The
formula
is
a
sum
of
products,
like
so:
∑ᵢⁿy_i∏_(j
ne
i)x-x_j/x_i-x_j.
When
x=x_i
then
all
terms
in
the
sum
other
than
the
iᵗʰ
contain
a
factor
x-x_i
in
the
numerator,
which
becomes
equal
to
zero,
thus
all
terms
in
the
sum
other
than
the
iᵗʰ
vanish,
and
the
iᵗʰ
term
has
factors
x_i-x_j
both
in
the
numerator
and
denominator,
which
simplify
to
yield
1,
thus
the
polynomial
should
return
y_i
as
the
function
of
x_i
for
any
i
in
the
set
1,...,n.
