# Questions tagged [error-estimation]

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### Is there some kind of expected error margin for my Monte Carlo algorithm?

My Monte Carlo algorithm starts by placing some circles in the plane with potential overlaps. I then place a large circle somewhere and compute the overlapping area of this larger circle with the ...
1 vote
60 views

### Regula falsi with error in x-axis

I want the regula falsi to get x +- .0001 so that f(x) = 0. But all the implemetations I see get x so that f(x) +- .0001 = 0 which doesn't make much sense. (f(x) = x^3). How do I stop the regula ...
1 vote
104 views

### Adaptive step size constrained to a limited number of iterations

I'm solving a differential equation on the form $\ddot x = f( \dot x, x)$ on a microchip within a limited (real world) time frame, hence I want to use an adaptive step size to get as good of a result ...
122 views

554 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 ...
1 vote
67 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)
352 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?
534 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 \...
85 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 ...
39 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 ...
302 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 ...
1 vote
212 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 ...
53 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 ...
46 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 ...
189 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)) ...
455 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 ...
1 vote
85 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 $f_a$ ...
2k 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 ...
1 vote
92 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 ...
1k 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: $$e^x =\sum_{k=0}^\infty \frac{x^k}{k!}$$ ...
637 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 ...
67 views

154 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 ...
74 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 ...
135 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,...
266 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 ...
1 vote
79 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 ...
1 vote