Estimating Maximum Error (Multivariable)

In summary, the conversation is about finding the maximum possible error in a computed value using the formula T = x(e^y + e^-y), given that x and y have maximum possible errors of 0.1 and 0.02 respectively. The solution involves using the second derivative of the function and the equation |E| <= (1/2)*M*(|x-xo| + |y-yo|)^2, where M is the largest possible value of the second derivative at the given point. However, there seems to be a discrepancy between the calculated error of 0.036 and the book's answer of 0.31.
  • #1
d-rock
14
0
Estimating Maximum Error, Odd Question? (Please Help Today!)

Homework Statement


Suppose that T is to be found from the formula T = x(e^y + e^-y), where x and y are found to be 2 and ln(2) with maximum possible errors of |dx| = 0.1 and |dy| = 0.02. Estimate the maximum possible error in the computed value of T.


Homework Equations


|E| <= (1/2) * M * (|x - xo| + |y - yo|) ^ 2


The Attempt at a Solution


dT/dx = (e^y + e^-y) and dT/dy = x(e^y - e^-y)

d2T/dx2 = 0, d2T/dy2 = x(e^y + e^-y) @ (2, Ln(2)) = 2(2 + 1/2) = 5

and lastly d2T/dxy = (e^y - e^-y) @ (2, Ln(2)) = 2 - 1/2 = 1.5


So I believe the M maximum is 5, for the second derivative with respect to y.

Then, I used the equation:

|E| <= (1/2) * 5 * (|dx| + |dy|)^2 = 2.5 * (0.1 + 0.02)^2 = 0.036


So my answer was 0.036. But the book's answer is 0.31. So I am unsure what went wrong or if I overlooked something...?
 
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  • #2
I'm not exctaly sure what you doing? Is that a quadratic apporimation to the error

Why not use a linear approximation to the function at the point by finding the gradient of the function there? (which you already have, but then use it to estimate the error...)

dependent on the function, for small dx & dy, generally the linear term will be largest (unless its an extremum) with higher order terms decreasing in magnitude.
 
  • #3
I don't know what you mean?

The error formula in my book (which is surrounded by Linearization, etc.) has

|E| <= (1/2)*M*(|x-xo| + |y-yo|)^2

where M is the largest possible value of the second derivatives at the point given. Then I just assumed that (x - xo) and (y-yo) are the dx and dy values they gave me, but I'm guessing it's wrong.

So, what linear approximation?
 

Related to Estimating Maximum Error (Multivariable)

What is maximum error and why is it important in multivariable estimation?

Maximum error refers to the largest possible difference between an estimated value and the true value of a variable. In multivariable estimation, it is important because it helps us understand the accuracy and reliability of our estimates, which is crucial for making informed decisions based on the data.

How is maximum error calculated in multivariable estimation?

Maximum error is calculated by finding the absolute value of the difference between the estimated value and the true value, and then dividing it by the true value. This value is then multiplied by 100 to get a percentage, which represents the maximum error.

What factors can affect the maximum error in multivariable estimation?

The maximum error in multivariable estimation can be affected by a variety of factors, such as the size and quality of the dataset, the complexity of the model used for estimation, and the assumptions made about the data. Other external factors, such as human error and measurement errors, can also contribute to the maximum error.

How can we minimize the maximum error in multivariable estimation?

There are several strategies that can help minimize the maximum error in multivariable estimation, such as using a larger and more diverse dataset, selecting a simpler and more appropriate model for estimation, and validating the assumptions made about the data. It is also important to carefully consider and address any potential sources of error in the estimation process.

What are some limitations of maximum error in multivariable estimation?

Maximum error is a useful measure of accuracy in multivariable estimation, but it is not without limitations. For example, it only gives us information about the largest potential error and does not tell us anything about the distribution of errors. Additionally, it assumes that all errors are equally important, which may not always be the case. It is important to consider these limitations when interpreting maximum error in multivariable estimation.

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