All Questions
Tagged with floating-point numerical-methods
163
questions
4
votes
2
answers
110
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 ...
- 10.5k
1
vote
2
answers
64
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 ...
- 125
1
vote
0
answers
49
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} \...
2
votes
1
answer
73
views
Numerically stable way to compute ugly double fraction
I am looking for a numerically stable version of this (ugly) equation
$$
s^2=\frac{1}{\frac{1}{\beta_1}+\frac{1}{\beta_2}W}
$$
where
$$
\beta_1 = c_1-c_2m+(m-c_2)b\\
\beta_2 = \frac{1}{2}\left((a-m)^2-...
- 272
0
votes
1
answer
59
views
Proof of `TWOSUM` implementation in "double-double" arithmetic
"double-double" / "compensated" arithmetic uses unevaluated sums of floating point numbers to obtain higher precision.
One of the basic algorithms is ...
- 5,707
1
vote
0
answers
159
views
Show that $x+1$ is not backward stable
Suppose we use $\oplus$ to compute $x+1$, given $x \in \mathbb{C}$. $\widetilde{f(x)} = \mathop{\text{fl}}(x) \oplus 1$. This algorithm is stable but not backward stable. The reason is that for $x \...
- 2,783
1
vote
2
answers
173
views
Another way to compute the epsilon machine
Why the next program computes the machine precision? I mean, it can be proved that the variable $u$ will give us the epsilon machine. But I don't know the reason of this.
Let
$a = \frac{4}{3}$
$b = a −...
- 153
0
votes
2
answers
99
views
Tricks in the floating point operations for better numerical results
I'm attempting to comprehend a passage from the book "Computational Modeling and Visualization of Physical Systems with Python" which I may be mentally fatigued to grasp. Here's the issue: ...
- 15
2
votes
1
answer
182
views
Is there a stable algorithm for every well-conditioned problem?
Reading these notes on condition numbers and stability, the summary states:
If the problem is well-conditioned then there is a stable way to solve it.
If the problem is ill-conditioned then there is ...
0
votes
0
answers
60
views
Secant method optimization - initial guesses with floating point precision?
Say I want to find the root of $f(x) = e^{-x} - 5$, and assume I start with initial guesses $x_0 = -3$ and $x_1 = 3$.
I define my update function as $x_i = x_{i-1} - f(x_{i-1}) * \frac{x_{i-1} - x_{i-...
- 3,588
1
vote
1
answer
172
views
Does using smaller floating-point numbers decrease rounding errors?
I started learning about floating point by reading "What Every Computer Scientist Should know About Floating-Point Arithmetic" by David Goldberg. On page 4 he presents a proof for the ...
0
votes
1
answer
60
views
Representation of rounding error in floating point arithmetic. [duplicate]
It is well known that in a Floating point number system:
$$
\mathbb{F}:=\{\pm \beta^{e}(\frac{d_1}{\beta}+\dots +\frac{d_t}{\beta^t}): d_i \in \{0,\dots,\beta-1\},d_1\neq 0, e_{\min}\leq e \leq e_{\...
- 1,356
3
votes
1
answer
109
views
How to compute this "smooth max operator"?
I was seeking for an alternate way to activate each neuron of a neural network non-linearly. Eventually, I came up with the following binary operation:
$$
x \lor y = \log (\exp x + \exp y)
$$
With $-\...
- 2,049
0
votes
0
answers
44
views
Is converting between roots and coefficients of a polynomial numerically stable?
Assume we're on a computer using 32-bit floats (or something similar), and I'm converting back and forth between the $n$ coefficients of a polynomial and the corresponding $n$ roots of the polynomial. ...
- 2,230
0
votes
1
answer
53
views
storing decimal number into computer with finite mantissa
I am learning about numerical methods and the following link caught my attention:
https://www.iro.umontreal.ca/~mignotte/IFT2425/Disasters.html
So from what I understand 0.1 is not exactly ...
- 109