線型近似

一般の関数(緑線)を一次関数(赤線)で近似する.

数学における線型近似(せんけいきんじ、: linear approximation)とは、一般の関数一次関数を用いて(より正確に言えばアフィン写像を用いて)近似することである。

例えば、2回微分可能な一変数関数 f は、テイラーの定理n = 1 の場合により、

と表せる。R2は剰余項である。線型近似は剰余項を落とした

となる。この近似は xa に十分近い場合に成り立つ。この式の右辺はちょうど元の f のグラフの (a, f(a)) における接線の表式となっており、そのことから、接線近似とも呼ばれる。

をaにおけるfの標準線型近似といい、x=a をセンターという。

線型近似は多変数関数に用いることもでき、この場合は導関数の代わりに関数行列が用いられる。例えば、微分可能な実関数 f(x, y) は、(a, b) に十分近い (x, y) においては次のように近似できる。

右辺は z = f(x, y) のグラフの (a, b) における接平面の表式となっている。

さらに一般に、バナッハ空間においては

と表される。ここで Df(a) は f の a におけるフレシェ微分である。

線型近似を用いて の近似値を求めてみよう。

  1. という関数を考える。この関数について f(25) を求めればよい。
  2. 微分すると である。
  3. 線型近似により となる。
  4. 小数に直すとおよそ2.926であるが、これは確かに真の値2.924…に近い。

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