71 lines
No EOL
2.4 KiB
Markdown
71 lines
No EOL
2.4 KiB
Markdown
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# Error Analysis
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In computational applications there are two main sources of error that we need to worry about:
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1. roundoff (or rounding) errors
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2. approximation errors
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**Rounding errors** originate from the necessity of having to represent numbers in a computer using a finite number of digits. **Approximation errors**, instead, refer to those errors that are intrinsic of the way we numerically solve a specific problem. In particular, these would be present even in the ideal case where we were able to carry out the computation without rounding errors.
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A simple example of approximation error can be seen in the case where we estimate the value of the function
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$$
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e^x = \sum_{n=0}^{\infty} {x^n\over n!} = 1 + x + {x^2\over 2} + {x^3\over 3!} + \ldots
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\
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\approx
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\
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\sum_{n=0}^{n_{\rm max}} {x^n\over n!}\,,
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$$
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by evaluating the series only up to a finite order $n_{\rm max}<\infty$. In this case, the approximation comes with a *truncation error* of ${\rm O}(x^{n_{\rm max}+1})$.
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:::i "Info"
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aasdasdasd
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:::
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> [!NOTE]
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> Let $f(x)$ and $g(x)$ be two functions defined on some unbounded subset $D$ of the positive real numbers, and $g(x)$ be non-zero in a neighborhood of $a\in D$ (often $a=0$). We write
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$$
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f(x)={\rm O}(g(x))
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\quad
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\text{as}
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\quad
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x\to a
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$$
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> if there exist positive numbers $\delta$ and $M$ such that for all $x$ with $0 < |x-a| < \delta$,
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$$
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|f(x)| \leq M |g(x)|\,.
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$$
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> Equivalently, we may say that $f(x)={\rm O}(g(x))$ if
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$$
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\underset{x\to a}{\lim \sup}{|f(x)|\over |g(x)|}<\infty\,.
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$$
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> If, instead, it holds that
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$$
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\underset{x\to a}{\lim}{f(x)\over g(x)}=1\,,
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$$
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>we write $f(x) \sim g(x)$ as $x\to a$.
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In this chapter, we present a general discussion on *roundoff errors*, not tied to a specific numerical problem we want to solve. In later chapters, we discuss several different methods to solve specific problems and study their approximation errors in detail.
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<font size= "6"> Exercises </font>
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1. Consider the function $f(x)=e^x$ in the interval $x\in [0,1]$. Write a program that calculates the corresponding approximating series:
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$$
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g_N(x)=\sum_{n=0}^N {x^n\over n!}\,.
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$$
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**(a)** Verify that the absolute error $\Delta=|f(x)-g_N(x)|$ scales approximately with $x^{N+1}/(N+1)!$ for $N=1,2,3,4$.
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**(b)** The error $\Delta$, in the given interval in $x$, differs from $x^{N+1}/(N+1)!$ . Why is that and for which values of $x$? |