“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.
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