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-\section{Errors in Numerical
-Computations}\label{errors-in-numerical-computations}
-
-In any numerical method, \textbf{error} is inevitable. Understanding
-\textbf{what kinds of errors occur} and \textbf{why} is essential to
-building reliable and accurate computations.
-
-We mainly classify errors into two major types: - Truncation Error -
-Round-off Error
-
-\subsection{What is Error?}\label{what-is-error}
-
-Let's remind ourselves what error is: \[
-\text{Error} = \text{True Value} - \text{Approximate Value}
-\] However, often the \textbf{true value} is unknown, so we focus on
-\textbf{reducing} and \textbf{analyzing} different types of errors
-instead of eliminating them completely. This can be done by using
-relative error when using iterative methods and is calculated as
-follows: \[
-\text{Relative Error} = \frac{\text{Best} - \text{Second to best}}{Best}
-\]
-
-\subsection{Truncation Error}\label{truncation-error}
-
-Truncation error occurs \textbf{when an infinite process is approximated
-by a finite process}.\\
-In simple terms, it happens \textbf{when you cut off or ``truncate''
-part of the computation}. An example of this could be using a finite
-number of terms from a Taylor Series expansion to approximate a
-function.
-
-Approximating $e^x$ by the first few terms of its Taylor series:
-
-$e^x \approx 1 + x +\frac{x^2}{2!} + \frac{x^3}{3!}$
-
-The error comes from \textbf{neglecting} all the higher order terms
-($\frac{x^4}{4!}, \frac{x^5}{5!}$), \ldots).
-
-Truncation error occurs when using numerical methods such as
-approximating and calculating derivatives and integrals. A
-representation of the truncation error is show in the figure below.
-Using our numerical methods we are left if some degree of error.
-
-\begin{figure}
-\centering
-\includegraphics{figures/truncationError.png}
-\caption{Representation of truncation error under a curve}
-\end{figure}
-
-In order to reduce truncation error there are a few things we can do: -
-Include more terms (higher-order methods) - Decrease step sizes (e.g.,
-smaller \(\Delta x\) in approximations) - Use better approximation
-algorithms.
-
-\subsection{Round-off Error}\label{round-off-error}
-
-Round-off error is caused by \textbf{the limited precision} with which
-computers represent numbers. Since computers cannot store an infinite
-number of digits, \textbf{they round off} after a certain number of
-decimal or binary places. For example, instead of representing pi with
-infinite decimal places it may be rounded off to approximately 16 digits
-depending on number of bits and the representation of the bits.
-
-In other words, round-off error happens because of how computers store
-numbers. For a double-floating point, the number is stored using
-64-bits. The more bits we use, the more precise of a number we can
-store. However, it makes it costs us more memory making it more
-computational expensive.
-
-While individual round-off errors may seem negligible, their effects can
-\textbf{accumulate over repeated computations}, leading to significant
-inaccuracies. This is particularly problematic in operations such as
-\textbf{subtracting two nearly equal numbers}, where \textbf{loss of
-significance} can occur, severely reducing numerical precision and
-amplifying the impact of round-off error.
-
-\subsubsection{How to Reduce Round-off
-Error:}\label{how-to-reduce-round-off-error}
-
-To reduce round-off error, use higher-precision data types when storing
-numerical values. Additionally, code and algorithms should be structured
-to \textbf{avoid subtracting nearly equal numbers}, a common source of
-significant error. Finally, employing \textbf{numerically stable
-algorithms} is essential for minimizing the accumulation of round-off
-errors during computation.
-
-\subsection{Total Error}\label{total-error}
-
-Truncation and round-off error are inversely proportional, meaning that
-if we decrease one, the other increases. If we want to minimize total
-error we must find the optimal point between step size and error.
-
-\begin{figure}
-\centering
-\includegraphics{figures/totalError.png}
-\caption{Total Error}
-\end{figure}