Questions tagged [error-estimation]

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161 views

Is order of matrix multiplication affecting numerical accuracy of the result?

I have to multiply three matrices of floats: A (100x8000), B (8000x27) and C (27x1). Is ...
39 views

What does it mean when saying “we want $\Lambda$ to be $\tilde{O}(1)$ as a function of $M$”?

What does it mean when saying "we want $\Lambda$ to be $\tilde{O}(1)$ as a function of $M$"? (it appears on the top of page 12 of this paper)
75 views

What does $\tilde{O}_P(N^\alpha)$ mean?

What does $\tilde{O}_P(N^\alpha)$ mean? It appears in an estimation error mention in this paper, in the second paragraph on page 3. What does big O subscript P mean in a probability context?
171 views

Proof that (x-y)(x+y) is more accurate than x²-y²

I was carrying on my reading of What Every Computer Scientist Should Know About Floating-Point Arithmetic but got stuck on the proof of Theorem 2 (page 34). At some point it says: \begin{align} (x \...
49 views

Proof that a guard digit bound the error of subtraction

I was reading What Every Computer Scientist Should Know About Floating-Point Arithmetic, which is extremely interesting. But I have some troubles understanding the proof of Theorem 9 (page 33). First ...
26 views

Rigorous error bounds for eigenvalue solvers

I computed the first four eigenvalues of a quite large ($2^{24}\times 2^{24}$) but very sparse matrix. I used pythons in-build function sparse.linalg.eigsh to compute them. I need a validation that ...
97 views

What is the state of the algorithmic art for floating point arithmetic on complex numbers?

Most modern compilers and processors implement the IEEE 754 binary formats for floating point numbers. IEEE 754 guarantees that the addition, subtraction, multiplication, division, and square root ...
55 views

Addition errors in IEEE754 floating point representation

So in class, we were talking about the idea of floating point precision in IEEE754 format, and how, when some numbers are added, precision is lost. My professor then gave the following example of a ...
45 views

Compile-time error control vs. interval arithmetic?

I use interval arithmetic for reliable computing. Now, a procedure coded in a good implementation of interval arithmetic takes perhaps about eight times as much as the same procedure carried out ...
35 views

Q-Learning Error Bounds

I have searched a lot for this, but apparently there is no result on calculating any bound on the error $||Q-Q^*||$ when I stop Q-learning after say $N$ iterations ($Q$ is the vector of Q-values at ...
114 views

Computing the error bound of floating-point expression

How should I compute the maximum absolute and relative error of the following IEEE-754 floating-point expression? a.y + (x - a.x) * ((b.y - a.y) / (b.x - a.x)) ...
92 views

What happens to a file after too many copy-pasting?

Let's say we're talking about a 1 GB video file. It's copy-pasted from hard disk D1 to the hard disk D2, then from D2 to D3, and so on, all using Windows. If we continue this process for like 1 ...
64 views

Machine error in computer arithmetic

I'm wondering if if is possible to have a function $f$ such that there exists $x,y$ such that we have $f_t(x) > f_t(y)$ where $f_t$ denotes the true value of $f$ and $f_a(x)<f_a(y)$ where $f_a$ ...
1k views

How to estimate floating-point precision of function?

Let's say I have a function that consists solely of floating-point operations where the last operation rounds the computed value to a predefined number of digits. And I feed this function with a range ...
88 views

Which implementation for the Maclaurin Series for the cosine function is better?

First, sorry if this post is off-topic. I consider it too analytic for stack overflow. In Numerical Analysis subject I must explain which one is better (has less error). The recursive ...
400 views

Avoiding overflows while computing $e^x$ by Taylor series

I'm coding a program to calculate the value of $e^x$ by using the Taylor expansion, that is: $$e^x =\sum_{k=0}^\infty \frac{x^k}{k!}$$ ...
219 views

The stability of log(1+x)

I am trying to understand why the formula $$\frac{\log(1+x)}{(1+x)-1} \times x,$$ which simply reduces down to $\log(1+x)$, is considered as more stable to compute than $\log(1+x)$. In my head it ...
53 views

99 views

How likely is it that a computer miscalculates 1+1? [closed]

Of course, normally a fully-functional computer will calculate 1+1=2. However, the physics governing the behavior of a chip is quantum mechanical. So in principle there is a certain probability that ...
66 views

Learning parameters of noise and filter coefficients from data where data and noise both have Gaussian distributions

Assume $X$ and $N$ are two sets of vectors (observations) from a normal distribution, where $X$ represents clean data and $N$ represents noise; and $A$ a projection matrix of a filter. The scenario is ...
88 views

Representing Computations on Transcendental Numbers

Consider the set of transcendental numbers that are not compressible to a finite base-2 representation. How can I compute multiples (more generally, any algebraic computation) of one of these numbers,...
260 views

Monetary computations theory (manual/textbook)

My problem is due to the fact that I am manipulating a set of amounts that span over some intervals of time (start date/end date) and that are rounded to cents. I have to multiply each of them by some ...
65 views

Why is the precision of floating point numbers worse for smaller numbers?

Why is the machine error/epsilon higher between a pair of two lower numbers than a pair of two high numbers? For example, between the two smallest numbers possible in 5 bit mantissa and the two ...
36 views

Error estimates of piecewise-linear curve approximations

In order to plot a curve a set of points is usually calculated based on some formula. The function FPLOT in MATLAB also supports plotting with some error tolerance. Its help says the following about ...
I have this problem: Consider the problem of calculating the integral $$y_n =\int_{0}^{1} \dfrac{x^n}{x+10} \mathrm{d}x$$ for a positive integer $n$. Observe that y_n + 10y_{n-1} = \...