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Thread: Numerical Analysis - Relative Error

  1. #1
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    Numerical Analysis - Relative Error

    Hi, I am working on the following problem and need some assistance:

    For what range of values of x will the computed value of f(x) given below have large relative error? Assume 0 $\displaystyle \le$ x $\displaystyle \le$ 100

    f(x) = 10 - $\displaystyle \sqrt{(100-x)}$

    I know that catostrophic error can occur when subtracting two numbers of similar magnitude, so in our case would we present the most relative error when x is equal to 0 and 100? Or do I need a wider range of numbers?

    Thanks
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  2. #2
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    Re: Numerical Analysis - Relative Error

    Quote Originally Posted by jnava View Post
    Hi, I am working on the following problem and need some assistance:

    For what range of values of x will the computed value of f(x) given below have large relative error? Assume 0 $\displaystyle \le$ x $\displaystyle \le$ 100

    f(x) = 10 - $\displaystyle \sqrt{(100-x)}$

    I know that catastrophic error can occur when subtracting two numbers of similar magnitude, so in our case would we present the most relative error when x is equal to 0 and 100? Or do I need a wider range of numbers?

    Thanks
    Assume some reasonable relative error for the square root operation. See what happens for x1 which is close to 100, like 100 minus a few times the relative error times 100. Then see what happens with x2 = 100 - x1, which is a number close to zero.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Numerical Analysis - Relative Error

    Quote Originally Posted by jnava View Post
    Hi, I am working on the following problem and need some assistance:

    For what range of values of x will the computed value of f(x) given below have large relative error? Assume 0 $\displaystyle \le$ x $\displaystyle \le$ 100

    f(x) = 10 - $\displaystyle \sqrt{(100-x)}$

    I know that catostrophic error can occur when subtracting two numbers of similar magnitude, so in our case would we present the most relative error when x is equal to 0 and 100? Or do I need a wider range of numbers?

    Thanks
    With a simple step You obtain...

    $\displaystyle f(x)= 10- \sqrt{100-x} = \frac{x}{10+\sqrt{100-x}}$ (1)

    ... and any criticity disappears. In particular for 'small' values of x is $\displaystyle f(x) \sim \frac{x}{20}$...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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