Last Updated：2025/12/05
Sentence
関数 f が x について凸（すなわち ∂^2f/∂x^2 > 0）である場合、ルジャンドル変換を行うことで x を共役変数 p = ∂f/∂x に置き換え、同じ情報を p の関数として表す新しい関数が得られる。
Quizzes for review
For a function f that is concave up in x (so ∂2f/∂x2 > 0), performing a Legendre transformation replaces x with its conjugate variable p = ∂f/∂x, yielding a new function that encodes the same information in terms of p.
See correct answer
For a function f that is concave up in x (so ∂2f/∂x2 > 0), performing a Legendre transformation replaces x with its conjugate variable p = ∂f/∂x, yielding a new function that encodes the same information in terms of p.
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Related words
Legendre transformation
Noun
(mathematics)
Given
a
function
f(x,y,z,...)
which
is
concave
up
with
respect
to
x
(i.e.,
its
second
derivative
with
respect
to
x
is
greater
than
zero),
an
involutive
procedure
for
replacing
x
with
another
variable,
say
p=∂f/∂x
thus
yielding
another
function,
say
F=F(p,y,z,...).
This
new
function
contains
all
of
the
information
of
the
original
f
encoded,
as
it
were,
within
it
so
that
∂F/∂p=x
and
applying
a
similar
transformation
to
F
yields
the
original
f.
The
formula
is:
F(p,y,z,...)=p·x(p)-f(x(p),y,z,...)
where
x
must
be
expressed
as
a
function
of
p.
(Note:
The
concave
upwardness
means
that
∂f/∂x
is
monotonically
increasing,
which
means
that
p
as
a
function
of
x
is
invertible,
so
x
should
be
expressible
as
a
function
of
p.)
Japanese Meaning
数学におけるルジャンドル変換とは、ある関数 f(x,y,z,...) が変数 x に関して上に凸である（すなわち、x に関する二階微分が正である）場合に、x をその偏微分 p = ∂f/∂x により新たな変数 p と置き換え、F(p,y,z,...) = p·x(p) - f(x(p),y,z,...) という形で定義される新しい関数に変換する手法です。この変換により、元の関数 f の全ての情報が F に含まれることになります。
Related Words
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