Numerical analysis (composite numerical integration)

Click For Summary

Discussion Overview

The discussion revolves around the application of the composite trapezoidal rule for numerical integration, specifically focusing on error estimation for the integrals of functions such as tan(x) and exp(x)sin(x). Participants explore how to determine bounds for the error in these integrations, including the necessary derivatives and values to substitute into the error formula.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about how to obtain a bound for the error of the integral of tan(x) from 0 to π/4 using the composite trapezoidal rule, referencing the error term formula.
  • Another participant points out that tan(π/2) is infinite, suggesting that the integration cannot be performed for that limit.
  • After clarifying the limits, a participant proposes substituting sec²(x) for the second derivative of tan(x) to find the maximum error bound.
  • There is a discussion about finding the maximum value of the second derivative, with one participant suggesting that it is 4 at π/4.
  • Another participant challenges the method of finding the maximum by stating that the derivative of the second derivative has no real roots, indicating that the maximum must occur at the boundary points.
  • One participant asks about the error bound for the integral of exp(x)sin(x) from 1 to 3, noting the second derivative and seeking guidance on the appropriate substitution for error estimation.
  • A later reply suggests that since exp(x) is monotonic and grows quickly, the maximum value of the second derivative should be evaluated at x = 3.

Areas of Agreement / Disagreement

Participants express differing views on how to determine the maximum of the second derivative for tan(x), with some suggesting boundary evaluation while others propose finding critical points. The discussion regarding the error bound for exp(x)sin(x) also indicates a lack of consensus on the substitution values for the second derivative.

Contextual Notes

There are unresolved aspects regarding the assumptions made in determining the maximum values of the second derivatives and the implications of boundary conditions versus critical points in the context of error estimation.

willbe
Messages
7
Reaction score
0
using composite trapezoidal rule with n=4 how can i get a bound for the error of I=integration tan(x) from x=0 to x=pi/2

i know that the term of error in comp trapezoidal rule is (b-a)/12 h^2 f''(eita)
i got the second derevative of tanx to be 2sec^2 x tanx then i don't know with what value exactly i need to substitute in this function to get the bound.
 
Physics news on Phys.org
Since tan(pi/2) is infinite, you cannot perform this integration.
 
sorry it is to x=pi/4
should i substitute by sec^2x by (1+tan^2x)
 
willbe said:
sorry it is to x=pi/4
should i substitute by sec^2x by (1+tan^2x)

You must have 0 < \eta < \pi/4

If you want a worst case error, differentiate again and find the value of \eta that maximizes f''.
 
ok i did and the second derivative is 2sec^2 x tanx now is the maximum value of this function is 4 as sec^2(x)=tan^2(x) + 1 ?
 
Taking the derivative of f'' and setting it to zero gives an immaginary root. So, there is no point of maximum.
The value of f'' at \pi/4 is 1, so we take it as the worst case.
h = \pi/4/4 = \pi/16
Then your error will be less then \frac{\pi/4}{12}\cdot \frac{\pi^2}{16^2}.
 
why we set the value of f'' to zero?
we set it to a value that gives us the maximum of the function so we substitute in 2sec^2(x)tan(x) which is equal to 2(1+tan^2(x))tan(x) by PI/4 so we get 2(1+1) which is 4.
right??
 
willbe said:
why we set the value of f'' to zero?
we set it to a value that gives us the maximum of the function so we substitute in 2sec^2(x)tan(x) which is equal to 2(1+tan^2(x))tan(x) by PI/4 so we get 2(1+1) which is 4.
right??

I have set the value of f''' to zero. The maximum of f'' happens when its derivative is zero.
Since f''' has no real roots the maximum of f'' must be at one of its boundary points 0 or \pi/4. The value of the funtion at 0 is 0 and at \pi/4 is 4 (and not 1 as I wrote previously). So in the expression for the error, you replace f''(\eta) by 4, getting \frac{\pi/4}{12}\cdot \frac{\pi^2}{16^2}\cdot 4
 
ok what about f(x)=exp x sinx using composite trapezoidal rule what is the bound of error for integration from 1 to 3

the second derevative will be 2exp x cos x then to get the bound of error which value should i substitute with??
 
  • #10
willbe said:
ok what about f(x)=exp x sinx using composite trapezoidal rule what is the bound of error for integration from 1 to 3

the second derevative will be 2exp x cos x then to get the bound of error which value should i substitute with??

Since exp x is monotone and grows very fast, the maximum value of f'' is at x = 3. This is the value to substitute in the expression.
 

Similar threads

Error of integration in numerical methods
  • · Replies 7 ·
Replies
7
Views
2K
Range of a function involving trigonometric functions
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Proof for interpolating polynomial and error approximation
  • · Replies 6 ·
Replies
6
Views
2K
Numerical Analysis - Richardson Extrapolation on Riemann Sum
  • · Replies 1 ·
Replies
1
Views
1K
Comp Sci Orbit of the Earth - numerical methods leapfrog
  • · Replies 3 ·
Replies
3
Views
3K
Simpson's Rule and Numerical Integration
  • · Replies 11 ·
Replies
11
Views
4K
Matlab Function for Composite Simpson
  • · Replies 3 ·
Replies
3
Views
2K
Numerical aperture of a Keplerian telescope
  • · Replies 1 ·
Replies
1
Views
3K
Mathematica Numerical integration over a Green's function
  • · Replies 13 ·
Replies
13
Views
3K
Numerical Analysis: Composite Trapezoidal Sum Rule
  • · Replies 5 ·
Replies
5
Views
4K