Absolute Error of Sum is not Greater than Sum of Absolute Errors
Theorem
Let $X_1, X_2, \ldots, X_n$ be approximations to a collection of (true) values $x_1, x_2, \ldots, x_n$ respectively.
Let $\Delta X_i$ be the absolute error of $X_i$ in $x_i$ for $i \in \set {1, 2, \ldots, n}$.
Then:
- $\ds \map \Delta {\sum_{i \mathop = 1}^n X_i} \le \sum_{i \mathop = 1}^n \Delta X_i$
That is, the absolute error of a sum of approximations is not greater than the sum of the absolute errors of those approximations.
Proof
The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
- $\ds \map \Delta {\sum_{i \mathop = 1}^n X_i} \le \sum_{i \mathop = 1}^n \Delta X_i$
for appropriately defined $X_1, X_2, \ldots, X_n$ and $x_1, x_2, \ldots, x_n$.
Basis for the Induction
$\map P 2$ is the case:
- $\map \Delta {X_1 + X_2} \le \Delta X_1 + \Delta X_2$
We have:
\(\ds \Delta X_1 + \Delta X_2\) | \(=\) | \(\ds \paren {X_1 - x_1} + \paren {X_2 - x_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {X_1 + X_2} - \paren {x_1 + x_2}\) |
This needs considerable tedious hard slog to complete it. In particular: Until it is truly specified what the "approximation" is, we can't make sense of this. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Thus $\map P 2$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $\ds \map \Delta {\sum_{i \mathop = 1}^k X_i} \le \sum_{i \mathop = 1}^k \Delta X_i$
from which it is to be shown that:
- $\ds \map \Delta {\sum_{i \mathop = 1}^{k + 1} X_i} \le \sum_{i \mathop = 1}^{k + 1} \Delta X_i$
Induction Step
This is the induction step:
This theorem requires a proof. In particular: I understand this intuitively, but because of the loose way we have defined errors and approximations, I can't make a sensible proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall n \in \N: n \ge 2: \ds \map \Delta {\sum_{i \mathop = 1}^n X_i} \le \sum_{i \mathop = 1}^n \Delta X_i$
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.5$ Absolute and Relative Errors: $3.5.4$