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Questions tagged [interval-arithmetic]

Interval arithmetic is the arithmetic of quantities that lie within specified ranges (i.e., intervals) instead of having definite known values.

Interval arithmetic is the arithmetic of quantities that lie within specified ranges (i.e., intervals) instead of having definite known values.

This concept is suitable for a variety of purposes. The most common use is to keep track of and handle rounding errors directly during the calculation and of uncertainties in the knowledge of the exact values of physical and technical parameters. The latter often arise from measurement errors and tolerances for components or due to limits on computational accuracy. Interval arithmetic also helps find reliable and guaranteed solutions to equations and optimization problems.

203 questions
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Why do you get zero by squaring [-1, 1] in interval arithmetic?

In the Wikipedia page on Interval Arithmetic, the example for the Dependency problem is that $f(x) = x^2 + x$ on the interval $\left[-1, 1\right]$ is $\left[-1, 2\right]$. I don't understand why this is the case. Looking at the…
Cedric Martens
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How many prime numbers in a given interval?

Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ? Example: [2, 10] --> 4 prime numbers {2, 3, 5, 7} [4, 12] --> 3 prime numbers {5, 7,…
alienflow
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Cartesian Product of Interval

I know what is a cartesian product of sets, for example, $M= \{1,2\} , N = \{a,b\} $ $M \times N = {(1,a), (1,b), (2,a) , (2,b)}$ but what is the Cartesian product of two intervals? say, $[-5,1] \times [-2,3]$
Leonardo
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Correctness of multiplication in ball arithmetic

We define $B(c,r) = \{x \in \mathbb{R} \ : \ |x - c| \le r\}$ with $r \ge 0$. Given two balls $b_1 = B(c_1, r_1)$ and $b_2 = B(c_2, r_2)$, multiplication is defined as follows: $$ b_1 * b_2 = B(c_1 c_2, r_1 r_2 + |c_1| r_2 + |c_2|r_1). $$ I've taken…
IOS_DEV
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Intersection of the Domain of the Logarithm of the Product of a Sequence of Sine Functions and $[0,1]$

Good afternoon. I have a couple of questions regarding a problem on the 2010 AMC 12A/AHSME. it's more on the reasoning of something I was using to try and solve the problem (turned out not to be fruitful). Here's the question. Let…
Eleven-Eleven
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Estimating $\tan(\cos(\sin1))$

The precise question through which I came across this particular estimation is as follows: If $x=\alpha$ is the maximum value of $x$ for which $ \left \lfloor{\sin^{-1}(\cos^{-1}(\tan^{-1}x))} \right \rfloor = 1$, then $\alpha$ lies in the…
Ankit Saha
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Show that a subset $I$ of $\textbf{R}$ is an interval if and only if whenever $a,b\in I$ and $0\leq t\leq 1$ then $(1-t)a + tb\in I$.

Show that a subset $I$ of $\textbf{R}$ is an interval if and only if whenever $a,b\in I$ and $0\leq t\leq 1$ then $(1-t)a + tb\in I$. MY ATTEMPT We say that a subset $I$ of $\textbf{R}$ is an interval iff $a \leq c \leq b$ and $a,b\in I$ implies…
user0102
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Notation Convention for integer in a certain range

I am wondering what notation I should use when writing that some variable is an integer within some range. What is the most common way to do this? Here are some ideas I have but I'm not sure what the usual convention is. For example, an integer on…
Quaz
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Let $g$ be a function defined on the interval $[0,2]$ and $x \le g(x) \le x^2-x+1$

Let $g$ be a function defined on the interval $[0,2]$ and $x \le g(x) \le x^2-x+1$ for $0 \le x \le 2$. Then $(A)$ $g$ must be a polynomial. $(B)$ $g$ must be continuous at $x = 1$. $(C)$ $g$ must be continuous at $x = 0$ and $x = 2$. $(D)$ $g$ must…
Deepak Gupta
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Is $[0,\infty )$ a closed interval or a half-closed interval?

Is $[0,\infty)$ a closed interval or a half-closed interval? My confusion is whether infinity is included or not.
Kantura
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A quadratic polynomial proof

Let $P(x)$ be a quadratic polynomial for which: $|P(x)| \leq1$ for all $x$ in the set $\{-1, 0, 1\}$. Prove that $|P(x)| \leq \frac{5}{4}$ for all $x$ in the interval $[-1, 1]$. How might I go about showing this?
Anonymous
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Why if f(x) = 2x - x on [-1, 1] results in [-1, 2] in Interval Arithmetic?

From the Section 3.1 of this paper The Dependency Problem. The main downside of interval arithmetic is that the computed bounds may be extremely pessimistic. As an example, consider the simple operation ← 2 −, evaluated on the range ∈ [−1, 1].…
Cedric Martens
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Differentiable formula that computes rotations in interval arithmetic (bounding box of a family of rotated rectangles)

I want to know how to perform a rotation in 2D interval arithmetic. That amounts to computing the tightest interval containing $$ x\ \mathrm{cos}(\varphi) + y\ \mathrm{sin}(\varphi), \tag{1}$$ where $x$, $y$ and $\varphi$ are all (real number)…
Will
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Definition of an Interval

I want to define both real intervals and integer intervals, and I want to avoid duplication of effort. As such, I put together the following definitions. I wanted to share and get feedback/critiques from the community. Are there any problems with my…
Ryan Pierce Williams
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What kind of algebra-like structure is interval arithmetic?

What kind of algebra-like structure is Interval Arithmetic (properties are defined in chapter 4)? Basically, interval addition and multiplication are both commutative and associative. Additive and multiplicative identity elements exist. But, the…
Denis Mazzucato
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