Questions tagged [error-estimation]

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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)
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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?
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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 \...
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1answer
45 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 ...
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1answer
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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 ...
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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 ...
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1answer
40 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 ...
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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 ...
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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 ...
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1answer
79 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)) ...
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1answer
87 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 ...
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1answer
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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 $f_a$ ...
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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 ...
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1answer
86 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 ...
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1answer
339 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: $$ e^x =\sum_{k=0}^\infty \frac{x^k}{k!} $$ ...
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190 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 ...
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Fitting a low-degree polynomial to a function on a finite 1d grid - a combinatorial problem?

I need to fit a low-degree polynomial $p$ (with $\text{deg}(p) \leq k$) to a function $f$ evaluated on the grid $\{0, 1, ... n-1\}$, so as to minimize the $L_\infty$ norm, i.e. minimize $\text{max}_{0 ...
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709 views

Floating Point Systems - Rounding Error in Taylor series

I have a question about rounding error. I have created a function to approximate exp(x) by summing the terms of its taylor series until the sum stops changing. (That is, 1 + x + x^2/2! ...). This ...
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Numerical Stability of Halley's Recurrence for Integer $n^{\mathrm{th}}$-Root

tl;dr? See last paragraph. If I use the initial value $2^{\left(\big\lfloor\lfloor\log_2 x \rfloor/n\big\rfloor + 1\right)}$ with Halley's recurrence in the compact form $ x_{k+1} = \frac{x_k\Big[A\...
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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 ...
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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 ...
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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,...
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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 ...
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1answer
63 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 ...
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1answer
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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 ...
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Is the per-vertex error over a PageRank iteration monotonically decreasing?

It seems to me that taken over the entire graph, the norm of the error vector must be monotonically decreasing, otherwise we couldn't guarantee that PageRank would ever converge. However, is the same ...
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Error accumulation in a numerical integration

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} = \...