“Simpson’s rule”的概念、定义、翻译、参考文献-科学参考

Simpson’s rule

Mathematics

A rule to find approximations for the definite integral

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

Divide [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, where n is even. Denote f(x_i) by f_i, and let P_i be the point (x_i, f_i). Take an arc of the parabola through the points P₀, P₁, and P₂, an arc of a parabola through P₂, P₃, and P₄, similarly, and so on. This is why n is necessarily even. The resulting Simpson’s rule gives

$\frac{1}{3} h (f_{0} + 4 f_{1} + 2 f_{2} + 4 f_{3} + 2 f_{4} + \dots + 2 f_{n - 2} + 4 f_{n - 1} + f_{n})$

as an approximation to the integral. In general, Simpson’s rule is much more accurate than the trapezium rule, and the error is bounded by

$\frac{{(b - a)}^{5}}{2880 n^{4}} max_{a \leq x \leq b} | f'''' (x) | .$

The rule is named after the English mathematician Thomas Simpson (1710–61), though the rule is due to Newton, as Simpson himself acknowledged.

Statistics

A method for finding an approximate value for an integral, which is usually more accurate than the trapezium rule. Suppose we wish to find an approximate value for . The interval a ≤ x ≤ b is divided into an even number, 2n, of sub-intervals, each of length h=(b − a)/(2n), at points a=x₀, x₁, x₂,…, x_2n−1, x_2n=b, where x_r=(a +rh) for r=0, 1, 2,…, 2n − 1, 2n. Writing y_r=f(x_r), the integral is approximated by $Simpson’s rule$
Simpson’s rule gives the exact answer when f is a polynomial of order 3 or less. In general, the error in using Simpson’s Rule is approximately proportional to 1/n⁴, so that if the number of sub-intervals is doubled then the error is reduced by a factor of 16. Simpson’s rule is based on fitting a quadratic graph in each adjacent pair of sub-intervals such that the graph passes through the points (x_2r, y_2r), (x_2r+1, y_2r+1) and (x_2r+2, y_2r+2).

Chemical Engineering

A numerical integration method used to obtain the approximate value to a definite integral. It is applied to odd numbers of data and is given by:
$A = \frac{S}{3} (F + L + 4 E + 2 O)$

where S is the width of the intervals between data, F and L are the first and last ordinate, E is the sum of the even-numbered ordinates, and O is the sum of the remaining odd-numbered ordinates.

Computer

The approximation
$\int_{x_{i}}^{x_{i} + 2} f (x) d x \approx \frac{1}{3} h (f (x_{i}) + 4 f (x_{i + 1}) + f (x_{i + 2}))$
where h = x_i+1 − x_i. It is used in numerical integration.

单词	Simpson’s rule
释义	Simpson’s rule Mathematics A rule to find approximations for the definite integral $\int_{a}^{b} f (x) d x .$ Divide [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, where n is even. Denote f(x_i) by f_i, and let P_i be the point (x_i, f_i). Take an arc of the parabola through the points P₀, P₁, and P₂, an arc of a parabola through P₂, P₃, and P₄, similarly, and so on. This is why n is necessarily even. The resulting Simpson’s rule gives $\frac{1}{3} h (f_{0} + 4 f_{1} + 2 f_{2} + 4 f_{3} + 2 f_{4} + \dots + 2 f_{n - 2} + 4 f_{n - 1} + f_{n})$ as an approximation to the integral. In general, Simpson’s rule is much more accurate than the trapezium rule, and the error is bounded by $\frac{{(b - a)}^{5}}{2880 n^{4}} max_{a \leq x \leq b} \| f'''' (x) \| .$ The rule is named after the English mathematician Thomas Simpson (1710–61), though the rule is due to Newton, as Simpson himself acknowledged. Statistics A method for finding an approximate value for an integral, which is usually more accurate than the trapezium rule. Suppose we wish to find an approximate value for . The interval a ≤ x ≤ b is divided into an even number, 2n, of sub-intervals, each of length h=(b − a)/(2n), at points a=x₀, x₁, x₂,…, x_2n−1, x_2n=b, where x_r=(a +rh) for r=0, 1, 2,…, 2n − 1, 2n. Writing y_r=f(x_r), the integral is approximated by $Simpson’s rule$ Simpson’s rule gives the exact answer when f is a polynomial of order 3 or less. In general, the error in using Simpson’s Rule is approximately proportional to 1/n⁴, so that if the number of sub-intervals is doubled then the error is reduced by a factor of 16. Simpson’s rule is based on fitting a quadratic graph in each adjacent pair of sub-intervals such that the graph passes through the points (x_2r, y_2r), (x_2r+1, y_2r+1) and (x_2r+2, y_2r+2). Chemical Engineering A numerical integration method used to obtain the approximate value to a definite integral. It is applied to odd numbers of data and is given by: $A = \frac{S}{3} (F + L + 4 E + 2 O)$ where S is the width of the intervals between data, F and L are the first and last ordinate, E is the sum of the even-numbered ordinates, and O is the sum of the remaining odd-numbered ordinates. Computer The approximation $\int_{x_{i}}^{x_{i} + 2} f (x) d x \approx \frac{1}{3} h (f (x_{i}) + 4 f (x_{i + 1}) + f (x_{i + 2}))$ where h = x_i+1 − x_i. It is used in numerical integration.
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