Trapezoidal Rule
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Trapezoidal Rule
In numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule)
is an approximate technique for calculating the definite integral
The trapezoidal rule is one of a family of formulas for numerical integration called Newton
Cotes formulas, of which the midpoint rule is similar to the trapezoid rule.
Simpson's rule is another member of the same family, and in general has faster convergence
than the trapezoidal rule for functions which are twice continuously differentiable, though not
in al specific cases.
However for various classes of rougher functions (ones with weaker smoothness conditions),
the trapezoidal rule has faster convergence in general than Simpson's rule.
Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions
are integrated over their periods, which can be analyzed in various ways.
For nonperiodic functions, however, methods with unequally spaced points such as Gaussian
quadrature and ClenshawCurtis quadrature are general y far more accurate;
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ClenshawCurtis quadrature can be viewed as a change of variables to express arbitrary
integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied
accurately.
Uniform Grid
For a domain discretized into "N" equal y spaced panels, or "N+1" grid points (1, 2, ..., N+1),
where the grid spacing is "h=(ba)/N", the approximation to the integral becomes
Nonuniform Grid
When the grid spacing is nonuniform, one can use the formula
Error analysis
The error of the composite trapezoidal rule is the difference between the value of the integral
and the numerical result:
It follows that if the integrand is concave up (and thus has a positive second derivative), then
the error is negative and the trapezoidal rule overestimates the true value.
This can also be seen from the geometric picture: the trapezoids include all of the area under
the curve and extend over it. Similarly, a concavedown function yields an underestimate
because area is unaccounted for under the curve, but none is counted above.
If the interval of the integral being approximated includes an inflection point, then the error is
harder to identify.
Even though the trapezoidal formula is considered to be less efficient in approximating definite
integral, it is found surprisingly efficient in some cases of periodic functions.
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We may observe that the approximation using the Trapezoidal rule for 20sinx dx wil give
the same value as the integral = 0. But the trapezoidal approximation done for 0sinx dx wil
yield the difference consistent with the error bounds for the rule.
In the first cast the Trapezoidal rule is applied to the function which is integrated over its ful
period. In this case, the graph will have one portion concave down in the interval (0,) and
another equal part of the graph is concave up in the interval (,2). This phenomenon
cancels the errors which occur when the approximation is done using Trapezoidal rule.
But in the second case, the integration is done over half period, and hence the error occurred
remain yielding not so accurate approximation.
When compared to Simpson's rule, the trapezoidal rule is less efficient in approximating a
definite integral. But the trapezoidal partitions done on the graph provides a clear visual
explanation to the concept applied. The formula is also
easy to memorize and can be applied with ease.
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