“spline”的概念、定义、翻译、参考文献-科学参考

spline

Mathematics

Interpolating with polynomials of high degree, as is necessary with Lagrangian interpolation for many data points, can be computationally difficult, and the interpolating polynomial can fluctuate significantly between data points. Instead, it may be better to interpolate using different low-degree polynomials between the different data points. Given data points (x₀,y₀),…(x_n,y_n), where a = x₀ < ⋯ < x_n = b, a cubic spline is a function f(x) satisfying
- • f(x) is defined by some cubic polynomial p_i(x) on each interval [x_i,x_i+1];
- • f(x_i) = y_i at each data point;
- • f(x) has a continuous (see continuous function) second derivative.
Higher-degree splines can be similarly defined.

Statistics

A set of polynomials, one for each sub-interval, that give an approximation to the function f(x), defined on some interval a ≤ x ≤ b, where a=x₀ < x₁ <⋯< x_n=b is a subdivision of the interval a ≤ x ≤ b. The polynomials are all of the same degree, d, and are chosen so that the values of the polynomials and their first (d − 1) derivatives are continuous at the intermediate points of subdivision and the values of the polynomials agree with the value of f(x) at each of the points.

Computer

In its simplest form a spline function (of degree n), s(x), is a piecewise polynomial on [x₁,x_N] that is (n– 1) times continuously differentiable, i.e.
$\begin{array}{l} s (x) \equiv polynomial of degree n \\ x_{i} \leq x \leq x_{i + 1}, i = 1, 2, \dots, N - 1 \end{array}$
These polynomial ‘pieces’ are all matched up at points (called knots):
$x_{1} < x_{2} < \dots < x_{N}$
in the interior of the range, so that the resulting function s(x) is smooth. The idea can be extended to functions of more than one variable. Cubic splines—spline curves of degree 3—provide a useful means of approximating data to moderate accuracy. Splines are often the underlying approximations used in variational methods. See also B-spline.

单词	spline
释义	spline Mathematics Interpolating with polynomials of high degree, as is necessary with Lagrangian interpolation for many data points, can be computationally difficult, and the interpolating polynomial can fluctuate significantly between data points. Instead, it may be better to interpolate using different low-degree polynomials between the different data points. Given data points (x₀,y₀),…(x_n,y_n), where a = x₀ < ⋯ < x_n = b, a cubic spline is a function f(x) satisfying • f(x) is defined by some cubic polynomial p_i(x) on each interval [x_i,x_i+1]; • f(x_i) = y_i at each data point; • f(x) has a continuous (see continuous function) second derivative. Higher-degree splines can be similarly defined. Statistics A set of polynomials, one for each sub-interval, that give an approximation to the function f(x), defined on some interval a ≤ x ≤ b, where a=x₀ < x₁ <⋯< x_n=b is a subdivision of the interval a ≤ x ≤ b. The polynomials are all of the same degree, d, and are chosen so that the values of the polynomials and their first (d − 1) derivatives are continuous at the intermediate points of subdivision and the values of the polynomials agree with the value of f(x) at each of the points. Computer In its simplest form a spline function (of degree n), s(x), is a piecewise polynomial on [x₁,x_N] that is (n– 1) times continuously differentiable, i.e. $\begin{array}{l} s (x) \equiv polynomial of degree n \\ x_{i} \leq x \leq x_{i + 1}, i = 1, 2, \dots, N - 1 \end{array}$ These polynomial ‘pieces’ are all matched up at points (called knots): $x_{1} < x_{2} < \dots < x_{N}$ in the interior of the range, so that the resulting function s(x) is smooth. The idea can be extended to functions of more than one variable. Cubic splines—spline curves of degree 3—provide a useful means of approximating data to moderate accuracy. Splines are often the underlying approximations used in variational methods. See also B-spline.
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