“trapezium rule”的概念、定义、翻译、参考文献-科学参考

trapezium rule

Mathematics

An approximate value can be found for the definite integral

$\int_{a}^{b} f (x) d x,$

using the values of f(x) at equally spaced values of x between a and b. Divide the interval [a, b] into n equal subintervals of length h by the partition

$a = x_{0} < x_{1} < x_{2} < \dots < x_{n - 1} < x_{n} = b,$

where x_i + 1 − x_i = h = (b − a)/n. Denote f(x_i) by f_i, and let P_i be the point (x_i, f_i). If the line segment P_iP_i+1 is used as an approximation to the curve y = f(x) between x_i and x_i + 1, the area under that part of the curve is approximately the area of the trapezium shown in the figure, which equals $\frac{1}{2} h (f_{i} + f_{i} {_{+}}_{1})$ . By adding up the areas of all the trapezia, the trapezium rule gives

A single trapezium from the trapezium rule

$\frac{1}{2} h (f_{0} + 2 f_{1} + 2 f_{2} + \dots + 2 f_{n} {_{-}}_{1} + f_{n})$
as an approximation to the value of the integral. The error between this approximation and the definite integral is bounded above by

$\frac{{(b - a)}^{3}}{12 n^{2}} max_{a \leq x \leq b} | f^{″} (x) | .$

Simpson’s rule is significantly more accurate.

Statistics

A method of finding an approximate value for an integral, based on finding the sum of the areas of trapezia. Suppose we wish to find an approximate value for . The interval a≤x≤b is divided up into n sub-intervals, each of length h=(b−a)/n, and the integral is approximated by $trapezium rule$
where y_r=f(a+rh). This is the sum of the areas of the individual trapezia, one of which is shown in the diagram. The error in using the trapezium rule is approximately proportional to 1/n², so that if the number of sub-intervals is doubled, the error is reduced by a factor of 4. A more accurate method for approximating an integral is Simpson’s rule.
Trapezium rule. A method of approximate integration. A slight underestimate (as shown) will often be cancelled by a similar slight overestimate from another trapezium. Using narrower intervals will improve accuracy.

Chemical Engineering

A form of numerical integration used to find the approximate area under a curve by dividing it into parts of trapezium-shaped sections that form columns of equal width with bases that lie on the horizontal axis. Also known as the trapezoidal rule, it is calculated from:
$\int_{b}^{a} f (x) \approx (b - a) \frac{f (a) + f (b)}{2}$

Computer

The approximation
$\begin{array}{l} \int_{x_{i}}^{x_{i + 1}} f (x) d x ≃ 1 / 2 h (f (x_{i}) + f (x_{i + 1})), \\ h = x_{i + 1} - x_{i} \end{array}$
used as the basis for an extrapolation method in numerical integration.

单词	trapezium rule
释义	trapezium rule Mathematics An approximate value can be found for the definite integral $\int_{a}^{b} f (x) d x,$ using the values of f(x) at equally spaced values of x between a and b. Divide the interval [a, b] into n equal subintervals of length h by the partition $a = x_{0} < x_{1} < x_{2} < \dots < x_{n - 1} < x_{n} = b,$ where x_i + 1 − x_i = h = (b − a)/n. Denote f(x_i) by f_i, and let P_i be the point (x_i, f_i). If the line segment P_iP_i+1 is used as an approximation to the curve y = f(x) between x_i and x_i + 1, the area under that part of the curve is approximately the area of the trapezium shown in the figure, which equals $\frac{1}{2} h (f_{i} + f_{i} {_{+}}_{1})$ . By adding up the areas of all the trapezia, the trapezium rule gives A single trapezium from the trapezium rule $\frac{1}{2} h (f_{0} + 2 f_{1} + 2 f_{2} + \dots + 2 f_{n} {_{-}}_{1} + f_{n})$ as an approximation to the value of the integral. The error between this approximation and the definite integral is bounded above by $\frac{{(b - a)}^{3}}{12 n^{2}} max_{a \leq x \leq b} \| f^{″} (x) \| .$ Simpson’s rule is significantly more accurate. Statistics A method of finding an approximate value for an integral, based on finding the sum of the areas of trapezia. Suppose we wish to find an approximate value for . The interval a≤x≤b is divided up into n sub-intervals, each of length h=(b−a)/n, and the integral is approximated by $trapezium rule$ where y_r=f(a+rh). This is the sum of the areas of the individual trapezia, one of which is shown in the diagram. The error in using the trapezium rule is approximately proportional to 1/n², so that if the number of sub-intervals is doubled, the error is reduced by a factor of 4. A more accurate method for approximating an integral is Simpson’s rule. Trapezium rule. A method of approximate integration. A slight underestimate (as shown) will often be cancelled by a similar slight overestimate from another trapezium. Using narrower intervals will improve accuracy. Chemical Engineering A form of numerical integration used to find the approximate area under a curve by dividing it into parts of trapezium-shaped sections that form columns of equal width with bases that lie on the horizontal axis. Also known as the trapezoidal rule, it is calculated from: $\int_{b}^{a} f (x) \approx (b - a) \frac{f (a) + f (b)}{2}$ Computer The approximation $\begin{array}{l} \int_{x_{i}}^{x_{i + 1}} f (x) d x ≃ 1 / 2 h (f (x_{i}) + f (x_{i + 1})), \\ h = x_{i + 1} - x_{i} \end{array}$ used as the basis for an extrapolation method in numerical integration.
随便看	active methods active microwave instrument active mode active network active optics active pool active prominence active reconnaissance active region active remote sensing active sensor Active Server Pages active site active star active transition((of a clock signal)) active transport active vision active volcano active voltage active volt-amperes active widget ActiveX active zone activism activist research