Лінійна інтерполяція

Для заданих двох червоних точок, синя лінія — лінійний інтерполянт між ними, і значення y в x можна знайти через лінійну інтерполяцію

Лінійна інтерполяція — це інтерполяція функції f алгебраїчним двочленом P₁(x) = kx + c у точках x₀ та x₁, які належать відрізку [a, b].

Геометрична інтерпретація

З геометричної точки зору це означає заміну функції $f$ прямою,яка проходить через точки $(x_{0},f(x_{0}))$ та $(x_{1},f(x_{1}))$ .

Рівняння такої прямої має вигляд:

{\frac {y-f(x_{0})}{f(x_{1})-f(x_{0})}}={\frac {x-x_{0}}{x_{1}-x_{0}}}

звідси для $x\in [x_{0},x_{1}]$ маємо:

f(x)\approx y=P_{1}(x)=f(x_{0})+{\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{0})

Це і є формула лінійної інтерполяції, причому

f(x)=P_{1}(x)+R_{1}(x)\quad ,

де $R_{1}(x)$ — похибка формули, яка обчислюється за формулою:

R_{1}(x)={\frac {f''(\psi )}{2}}(x-x_{0})(x-x_{1}),\quad \psi \in [x_{0},x_{1}]

Для неї є справедливою наступна оцінка:

|R_{1}(x)|\leqslant {\frac {M_{2}}{2}}\max |(x-x_{0})(x-x_{1})|={\frac {M_{2}h^{2}}{8}},\quad M_{2}=\max _{[a,b]}|f''(x)|,\quad h=x_{1}-x_{0}.

Див. також

Білінійна інтерполяція

Джерела

Бахвалов Н. С. Численные методы. — М: Лаборатория Базовых Знаний, 2007. — 636 с. — ISBN 978-5-94774-815-4. (рос.)

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