減縮積分與數值自鎖

05-19

減縮積分與數值自鎖

來自專欄力學知識

一個曲線連續性的定義為：A curve can be said to have $C^n$ continuity if ${displaystyle displaystyle {frac {d^{n}s}{dt^{n}}}}$ is continuous of value throughout the curve.

對於一個多元函數 $g(x,y)$ ，如果 $frac{partial ^n g}{partial x^n}$ 是連續的，那麼就可以稱這個函數在 $x$ 方向具備 $C^n$ 連續性。

An $n$ -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree $2n ? 1$ or less by a suitable choice of the points $x_i$ and weights $w_i$ for $i = 1, ..., n$ . The domain of integration for such a rule is conventionally taken as $[?1, 1]$ , so the rule is stated as

${displaystyle int _{-1}^{1}f(x),dx=sum _{i=1}^{n}w_{i}f(x_{i}).}$

Gaussian quadrature as above will only produce good results if the function $f(x)$ is well approximated by a polynomial function within the range $[?1, 1]$ .

考慮積分 $int_{-1}^{1}f(x)dx$ ,採用Gauss積分來對其進行計算。若函數 $f(x)$ 具備 $C^n$ 連續性，則必須布置 $n$ 個Gauss積分點才能對