“Gregory–Newton forward difference formula”的概念、定义、翻译、参考文献-科学参考

Gregory–Newton forward difference formula

Mathematics

Let x₀, x₁, x₂,…, x_n be equally spaced values, so that x_i = x₀ + ih, for i = 1, 2,…, n. Suppose that the values f₀, f₁, f₂,…, f_n are known, where f_i = f(x_i), for some function f. The Gregory–Newton forward difference formula is a formula involving finite differences that gives an approximation for f(x), where x = x₀ + θ‎h, and 0<θ‎<1. It states that

$f (x) \approx f_{0} + θ Δ f_{0} + \frac{θ (θ - 1)}{2!} Δ^{2} f_{0} + \frac{θ (θ - 1) (θ - 2)}{3!} Δ^{3} f_{0} + \dots,$

the series terminating at some stage. The approximation f(x) ≈ f₀ + θ‎Δ‎ f₀ gives the result of linear interpolation. Terminating the series after one more term provides an example of quadratic interpolation.

单词	Gregory–Newton forward difference formula
释义	Gregory–Newton forward difference formula Mathematics Let x₀, x₁, x₂,…, x_n be equally spaced values, so that x_i = x₀ + ih, for i = 1, 2,…, n. Suppose that the values f₀, f₁, f₂,…, f_n are known, where f_i = f(x_i), for some function f. The Gregory–Newton forward difference formula is a formula involving finite differences that gives an approximation for f(x), where x = x₀ + θ‎h, and 0<θ‎<1. It states that $f (x) \approx f_{0} + θ Δ f_{0} + \frac{θ (θ - 1)}{2!} Δ^{2} f_{0} + \frac{θ (θ - 1) (θ - 2)}{3!} Δ^{3} f_{0} + \dots,$ the series terminating at some stage. The approximation f(x) ≈ f₀ + θ‎Δ‎ f₀ gives the result of linear interpolation. Terminating the series after one more term provides an example of quadratic interpolation.
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