This is not the document you are looking for? Use the search form below to find more!

Report home > Education

Trapezoidal Rule

0.00 (0 votes)
Document Description
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 all 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 non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate;
File Details
  • Added: June, 15th 2012
  • Reads: 455
  • Downloads: 5
  • File size: 179.87kb
  • Pages: 4
  • Tags: absolute value equations, central tendency, formula for circumference, probability worksheets, trapezoidal rule
  • content preview
Submitter
Embed Code:

Add New Comment




Related Documents

Absolute Value Equations

by: tutorvistateam_team, 4 pages

Before we start solving "Absolute Value Equations", we must understand actually what do we mean by "Absolute Value “. By "Absolute Value” , we mean the positive value of any number and it ...

Central Tendency

by: tutorvistateam_team, 4 pages

n statistics, the term central tendency relates to the way in which quantitative data tend to cluster around some value. A measure of central tendency is any of a number of ways of specifying this ...

Formula for Circumference

by: tutorvistateam_team, 4 pages

There are several figures in the geometry one of them is circle that has a combination of arch that makes the circle. A circle is a figure that has the radius that is equal for whole figure of circle ...

Probability Worksheets

by: tutorvistateam_team, 4 pages

To know the probability, first we have to understand the some terms that are closely related to the probability and with the help of all terms we can easily solve the probability worksheets. ...

Trapezoid Rule

by: vistateam123, 4 pages

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 ...

Simpson's Rule

by: tutorvistateam, 4 pages

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation: Simpson's rule also ...

Rule Booklet

by: hsemak, 41 pages

Rule Booklet for Techniche 2010

DIVERSITY-BASED INTERESTINGNESS MEASURES FOR ASSOCIATION RULE MINING

by: tetsuo, 8 pages

Association rule interestingness measures are used to help select and rank association rule patterns. Diversity-based measures have been used to determine the relative interestingness of ...

Rule Engine: Drools .Net

by: terttu, 15 pages

Rule Engine: Drools .NetAlbert Guojunyuo@gmail.com1IntroductionRule engineOverview of DroolsAdvantages of using rule engineDemo

Prove Product Rule

by: nishagoyal, 3 pages

Prove Product Rule We use product rule for calculating derivative of those functions, which are in multiply form, like (sin x.ln x), (x .ln x), (x .cos x) etc. Product rule tells that if two ...

Content Preview
Trapezoidal Rule
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 non-periodic functions, however, methods with unequally spaced points such as Gaussian
quadrature and Clenshaw-Curtis quadrature are general y far more accurate;
Know More About :- Partial Fraction Decomposition


Math.Tutorvista.com
Page No. :- 1/4

Clenshaw-Curtis 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=(b-a)/N", the approximation to the integral becomes

Non-uniform Grid
When the grid spacing is non-uniform, 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 concave-down 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.
Learn More :- Rate of Change


Math.Tutorvista.com
Page No. :- 2/4

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.


Math.Tutorvista.com
Page No. :- 4/4

ThankYouForWatching
Presentation



Document Outline

  • ﾿

Download
Trapezoidal Rule

 

 

Your download will begin in a moment.
If it doesn't, click here to try again.

Share Trapezoidal Rule to:

Insert your wordpress URL:

example:

http://myblog.wordpress.com/
or
http://myblog.com/

Share Trapezoidal Rule as:

From:

To:

Share Trapezoidal Rule.

Enter two words as shown below. If you cannot read the words, click the refresh icon.

loading

Share Trapezoidal Rule as:

Copy html code above and paste to your web page.

loading