Questions tagged [floating-point]
Mathematical questions concerning floating point numbers, a finite approximation of the real numbers used in computing.
474 questions
-1 votes
1 answer
98 views
How do we pick how many decimals to use when solving problems? [closed]
I was working on an exercise from a chemistry textbook which was about an application of the osmotic pressure formula. When I checked the solution the answer give was $62.05$ while I was getting as a ...
- 133
0 votes
0 answers
42 views
Stability of specific floating point operation algorithm
Problem: We want to determine whether the following algorithm is stable or not. Data is $x_{1},x_{2} \in \mathbb{C}$, Solution is $x_{1}(x_{2}+1)$, computed as $\text{fl}(x_{1}) \otimes (\text{fl}(x_{...
- 1,191
0 votes
0 answers
54 views
2 votes
1 answer
80 views
What is the stochastic error introduced by a converting a random (real) number to a floating point?
What is the variance of the relative error introduced by a converting a random real number to a float? I was able to develop a formula which, empirically, is off by a factor of 2. Let $\mathcal F \...
- 6,261
4 votes
2 answers
197 views
Why does this modified Newton's method fail to converge for N > 1 iterations?
I am investigating the behavior of an approximation of 1/sqrt(x) which uses a similar method to the more famous Quake III "Fast Inverse Square Root". Quake's FISR uses an approximation of ...
- 153
1 vote
2 answers
208 views
Accurately Converting Between Binary Floating Point and Decimal Character String.
This must have been studied before, and has been one of the confusion I've been having. Suppose we want to convert a astronomically big floating point number (so big that it's floating point ...
- 355
1 vote
0 answers
69 views
Necessary condition for sum of floating point numbers x + y to be x
In my numerical analysis text book there is this problem Assume a machine with base $\beta$, and $p$ mantissa digits, and that $u$ is the round-off error of a particular number. The number $∣x∣=(0....
- 147
0 votes
1 answer
103 views
Forward Stability of $f(x) = x+1$
I was reading: https://www.cs.usask.ca/~spiteri/CMPT898/notes/numericalStability.pdf Where I read that the algorithm $\tilde{f}(x)=fl(x)\bigoplus1$ for the problem $f(x)=x+1$, is stable but not ...
4 votes
2 answers
178 views
pow and its relative error
Investigating the floating-point implementation of the $\operatorname{pow}(x,b)=x^b$ with $x,b\in\Bbb R$ in some library implementations, I found that some pow ...
7 votes
0 answers
182 views
Algebraic Structures involving 𝙽𝚊𝙽 (absorbing element).
IEEE 754 floating point numbers contain the concept of 𝙽𝚊𝙽 (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return ...
- 12.3k
3 votes
0 answers
75 views
Solve $10^{10^z} = 10^{10^x}+10^{10^y}$ for $z$ with floating point accuracy
In the following equation $$10^{10^z} = 10^{10^x}+10^{10^y}$$ I want to find an algorithm that computes $z$ in a floating point accurate manner given any values of $x$ and $y$ (e.g. $x=y=2000$). The ...
2 votes
2 answers
123 views
How to transform this expression to a numerically stable form?
I have this function $$f(x, t)=\frac{\left(1+x\right)^{1-t}-1}{1-t}$$ Where $x \ge 0$ and $t \ge 0$. I want to use it in neural network, and thus need it to be differentiable. While it has a ...
- 135
1 vote
0 answers
124 views
Proof that $\epsilon_{mach} \leq \frac{1}{2} b^{1-n}$
I have a question about the proof of the following statement: For each set of machine numbers $F(b, n, E_{min}, E_{max})$ with $E_{min} < E_{max}$ the following inequality holds: $\epsilon_{mach} \...
- 107
1 vote
0 answers
184 views
Why does TI-84 show scientific notation for zeros sometimes but not others?
When graphing a function and then going through the process to calculate the zeroes (left bound, right bound, guess), is there a reason that sometimes it shows y = 0, but there are other times when it ...
- 481