6 ms·

Gaussian integration is cool

constantcrying 1y ago
A good introduction to the basics. What is also worth pointing out and which was somewhat glanced over is the close connection between the weight function and the polynomials. For different weight functions you get different classes of orthogonal polynomials. Orthogonal has to be understood in relation to the scalar product given by integrating with respect to the weight function as well. Interestingly Gauss-Hermite integrates on the entire real line, so from -infinity to infinity. So the choice of weight function also influences the choice of integration domain.
- creata 1y ago
  Sorry if this is a stupid question, but is there a direct link between the choice of weight function and the applications of the polynomial? Like, is it possible to infer that Chebyshev polynomials would be useful in approximation theory using only the fact that they're orthogonal wrt the Wigner semicircle (U_n) or arcsine (T_n) distribution?
  - constantcrying 1y ago
    Yes. Precisely that they are orthogonal means that they are suitable. If you are familiar with the Fourier series, the same principle can be applied to approximating with polynomials. In both cases the crucial point is that you can form an orthogonal subspace, onto which you can project the function to be approximated. For polynomials it is this: https://en.m.wikipedia.org/wiki/Polynomial_chaos https://en.m.wikipedia.org/wiki/Polynomial_chaos
    - sfpotter 1y ago
      What you're saying isn't wrong, per se, but... There are polynomials that aren't orthogonal that are suitable for numerics: both the Bernstein basis and the monomial basis are used very often and neither are orthogonal. (Well, you could pick a weight function that makes them orthogonal, but...!) The fact of their orthogonality is crucial, but when you work with Chebyshev polynomials, it is very unlikely you are doing an orthogonal (L2) projection! Instead, you would normally use Chebyshev interpolation: 1) interpolate at either the Type-I or Type-II Chebyshev nodes, 2) use the DCT to compute the Chebyshev series coefficients. The fact that you can do this is related to the weight function, but it isn't an L2 procedure. Like I mentioned in my other post, the Chebyshev weight function is maybe more of an artifact of the Chebyshev polynomials' intimate relation to the Fourier series. I am also not totally sure what polynomial chaos has to do with any of this. PC is a term of art in uncertainty quantification, and this is all just basic numerical analysis. If you have a series in orthgonal polynomials, if you want to call it something fancy, you might call it a Fourier series, but usually there is no fancy term...
      - constantcrying 1y ago
        I think you are very confused. My post is about approximation. Obviously other areas use other polynomials or the same polynomials for other reasons. In this case it is about the principle of approximation by orthogonal projection, which is quite common in different fields of mathematics. Here you create an approximation of a target by projecting it onto an orthogonal subspace. This is what the Fourier series is about, an orthogonal projection. Choosing e.g. the Chebychev Polynomials instead of the complex exponential gives you an Approximation onto the orthogonal space of e.g. Chebychev polynomials. The same principle applies e.g. when you are computing an SVD for a low rank approximation. That is another case of orthogonal projection. >Instead, you would normally use Chebyshev interpolation What you do not understand is that this is the same thing. The distinction you describe does not exist, these are the same things, just different perspectives. That they are the same easily follows from the uniqueness of polynomials, which are fully determined by their interpolation points. These aren't distinct ideas, there is a greater principle behind them and that you are using some other algorithm to compute the Approximation does not matter at all. >I am also not totally sure what polynomial chaos has to do with any of this. It is the exact same thing. Projection onto an orthogonal subspace of polynomials. Just that you choose the polynomials with regard to a random variable. So you get an approximation with good statistical properties.
        sfpotter 1y ago
        No, I am not confused. :-) I am just trying to help you understand some basics of numerical analysis. > What you do not understand is that this is the same thing. It is not the same thing. You can express an analytic function f(x) in a convergent (on [-1, 1]) Chebyshev series: f(x) = \sum_{n=0}^\infty a_n T_n(x). You can then truncate it keeping N+1 terms, giving a degree N polynomial. Call it f_N. Alternatively, you can interpolate f at at N+1 Chebyshev nodes and use a DCT to compute the corresponding Chebyshev series coefficients. Call the resulting polynomial p_N. In general, f_N and p_N are not the same polynomial. Furthermore, computing the coefficients of f_N is much more expensive than computing the coefficients of p_N. For f_N, you need to evaluate N+1 integral which may be quite expensive indeed if you want to get digits. For p_N, you simply evaluate f at N+1 nodes, compute a DCT in O(N log N) time, and the result is the coefficients of p_N up to rounding error. In practice, people do not compute the coefficients of f_N, they compute the coefficients of p_N. Nevertheless, f_N and p_N are essentially as good as each other when it comes to approximation.
  - sfpotter 1y ago
    Chebyshev polynomials are useful in approximation theory because they're the minimax polynomials. The remainder of polynomial interpolation can be given in terms of the nodal polynomial, which is the polynomial with the interpolation nodes as zeros. Minimizing the maximum error then leads to the Chebyshev polynomials. This is a basic fact in numerical analysis that has tons of derivations online and in books. The weight function shows the Chebyshev polynomials' relation to the Fourier series . But they are not what you would usually think of as a good candidate for L2 approximation on the interval. Normally you'd use Legendre polynomials, since they have w = 1, but they are a much less convenient basis than Chebyshev for numerics.
    - creata 1y ago
      True, and there are plenty of other reasons Chebyshev polynomials are convenient, too. But I guess what I was asking was: is there some kind of abstract argument why the semicircle distribution would be appropriate in this context? For example, you have abstract arguments like the central limit theorem that explain (in some loose sense) why the normal distribution is everywhere. I guess the semicircle might more-or-less be the only way to get something where interpolation uses the DFT (by projecting points evenly spaced on the complex unit circle onto [-1, 1]), but I dunno, that motivation feels too many steps removed.
      - sfpotter 1y ago
        If there is, I'm not aware of it. Orthogonal polynomials come up in random matrix theory. Maybe there's something there? But your last paragraph is exactly it... it is a "basic" fact but the consequences are profound.
        efavdb 1y ago
        Could you guys recommend an easy book on this topic?
        sfpotter 1y ago
        Which topic?
seanhunter 1y ago
I thought when I first saw the title that it was going to be about the Gaussian integral[1] which has to be one of the coolest results in all of maths. That is, the integral from - to + infinity of e^(-x^2) dx = sqrt(pi). I remember being given this as an exercise and just being totally shocked by how beautiful it was as a result (when I eventually managed to work out how to evaluate it). [1] https://mathworld.wolfram.com/GaussianIntegral.html https://mathworld.wolfram.com/GaussianIntegral.html
- deleted 1y ago
  [deleted]
- niklasbuschmann 1y ago
  Gaussian integrals are also pretty much the basis of quantum field theory in the path integral formalism, where Isserlis's theorem is the analog to Wick's theorem in the operator formalism.
  - srean 1y ago
    Indeed. It's the gateway drug to Laplace's method (Laplace approximation), mean field theory, perturbation theory, ... QFT. https://en.m.wikipedia.org/wiki/Laplace%27s_method https://en.m.wikipedia.org/wiki/Laplace%27s_method
    - chermi 1y ago
      Maybe I should just read the wiki you linked, but I guess I'm confused on how this is different than steepest descent? I'm a physicist by training so maybe we just call it something different?
      - dawnofdusk 1y ago
        They are the same for physicists who analytically continue everything. Steepest descent is technically for integrals in the complex plane, and Laplace's method is for integrating over the real line.
extrabajs 1y ago
What is Fig. 1 showing? Is it the value of the integral compared with two approximations? Would it not be more interesting to show the error of the approximations instead? Asking for a friend who isn’t computing a lot of integrals.
- mturmon 1y ago
  Fig 1 could use a rethink. It uses log scale, but the dynamic range of the y-axis is tiny, so the log transform isn't doing anything. It would be better shown as a table with 3 numbers. Or, maybe two columns, one for integral value and one for error, as you suggest.
  - rd11235 1y ago
    Yeah - my guess is this was just a very roundabout solution for setting axis limits. (For some reason, plt.bar was used instead of plt.plot, so the y axis would start at 0 by default, making all results look the same. But when the log scale is applied, the lower y limit becomes the data’s minimum. So, because the dynamic range is so low, the end result is visually identical to having just set y limits using the original linear scale). Anyhow for anyone interested the values for those 3 points are 2.0000 (exact), 1.9671 (trapezoid), and 1.9998 (gaussian). The relatives errors are 1.6% vs. 0.01%.
  - beansbeansbeans 1y ago
    [dead]
tim-kt 1y ago
No, the integral cannot be "expressed" as a sum over weights and function evaluations (with a "="), it can be approximated with this idea. If you fix any n+1 nodes, interpolate your function, and integrate your polynomial, you will get this sum where the weights are integrals over (Lagrange) basis polynomials. By construction, you can compute the integral of polynomials up to degree n exactly. Now, if you choose the nodes in a particular way (namely, as the zeros of some polynomials), you can increase this to up to 2n+1. What I'm getting at is that the Gaussian integration is not estimating the integrals of polynomials of degree 2n+1, but it's evaluating them exactly.
- Onavo 1y ago
  Convolution?
  - tim-kt 1y ago
    What about it?
- beansbeansbeans 1y ago
  You're totally correct, my writeup in that section definitely was not clear. I've updated the blog, hopefully it's better. I've also given you a shoutout in the end of the post in my edit log, if that's cool with you
  - tim-kt 1y ago
    Sure, thanks. I also saw this sentence, which was not there in the last version (I think): > That is to say, with n nodes, gaussian integration will approximate your function's integral with a higher order polynomial than a basic technique would - resulting in more accuracy. This is not really the case, Gaussian integration is still just interpolation on n nodes, but the way of choosing the nodes increases the integration's exactness degree to 2n-1. It's actually more interesting that Gaussian integration does not require any more work in terms of interpolation, but we just choose our nodes better. (Actually, computing the nodes is sometimes more work, but we can do that once and use them forever.)
drmikeando 1y ago
My mind was exploded by this somewhat similar technique https://en.wikipedia.org/wiki/Tanh-sinh_quadrature https://en.wikipedia.org/wiki/Tanh-sinh_quadrature - it uses a similar transformation of domain, but uses some properties of optimal quadrature for infinite sequences in the complex plane to produce quickly convergent integrals for many simple and pathological cases.
curtisszmania 1y ago
[dead]
actinium226 1y ago
While I love seeing mathy articles on HN, the lead image in this article is terrible. At first glance I see we're talking about numerical integration so I assuming the red part is this method that is being discussed and that it's much better than the other two. Then I look at the axis, which the caption notes is log scale, and see that it goes from 1.995 to 2. Uh-oh, is someone trying to inflate performance by cutting off the origin? Big no-no, but then wait a minute, this is ground truth compared to two approximations. So actually the one on the right is better. But the middle one is still accurate to within 0.25%. And why is it log scale? Anyway point is, there's lots of room for improvement! In particular it's probably not worth putting ground truth as a separate bar, just plot the error of the two methods, then you don't need to cut off the original. And ditch the log scale.
- beansbeansbeans 1y ago
  [dead]
- beansbeansbeans 1y ago
  Yeah - I totally agree. Some others on the thread pointed it out as well. I ended up replacing it with a table additionally having the error percentages. Thanks for the fair critique! I've also mentioned your username in the edit logs at the end of the blog if you don't mind.
algorithmsRcool 1y ago
Did anyone else's antivirus complain about an exploit on this page? ---EDIT--- I'm about 98% sure this blog has a browser hijack embedded in it targeted at windows+MSEDGE browsers that attempted to launch a malicious powershell script to covertly screen record the target machine
- beansbeansbeans 1y ago
  That's a major claim. The only thing different in this blog post from my others is that I've embedded an executable python notebook in an iframe. It's a marimo notebook that runs code using WASM in your browser. That project is open source too, with no exploit as far as I know. The code for my blog is here : https://github.com/RohanGautam/rohangautam.github.io https://github.com/RohanGautam/rohangautam.github.io If you could point to anything specific to support that claim, would be nice.
  - algorithmsRcool 1y ago
    I would be happy to be wrong on this one. But I've gotten two pretty convincing threat notifications when visiting the page from the Sentinel One antivirus platform saying that my msedge process had been compromised by a known exploit. I'll try to get more details. I should note, I do not believe the site is malicious, but i am worried about 3rd party compromise of the site without the owner's knowledge
    - beansbeansbeans 1y ago
      I see, that's strange. Yeah, feel free to share the details/logs with me - you can open an issue on my blog's repo with the relevant details and system info. However, I'm still suspecting it's something specific to your antivirus not knowing what to do with WASM code(which is used on this page). I found something similar on Reddit: https://www.reddit.com/r/eaglercraft/s/heVtPy60lG https://www.reddit.com/r/eaglercraft/s/heVtPy60lG. I wonder if that's the issue.
- BLanen 1y ago
  You should not be using antivirus browser plugins anyway.
Malipeddi 1y ago
This is a neat and accessible introduction to the topic. Nice work! I want to mention nesting quadratures like Gauss-Kronod like Clenshaw-Curtis/Fejér. The nesting means that the quadrature points for lower order are included in the points required for higher orders. This allows you to increase your order of accuracy if required with fewer total evaluations.

A good introduction to the basics. What is also worth pointing out and which was somewhat glanced over is the close connection between the weight function and the polynomials. For different weight functions you get different classes of orthogonal polynomials. Orthogonal has to be understood in relation to the scalar product given by integrating with respect to the weight function as well. Interestingly Gauss-Hermite integrates on the entire real line, so from -infinity to infinity. So the choice of weight function also influences the choice of integration domain.

creata 1y ago
Sorry if this is a stupid question, but is there a direct link between the choice of weight function and the applications of the polynomial? Like, is it possible to infer that Chebyshev polynomials would be useful in approximation theory using only the fact that they're orthogonal wrt the Wigner semicircle (U_n) or arcsine (T_n) distribution?
- constantcrying 1y ago
  Yes. Precisely that they are orthogonal means that they are suitable. If you are familiar with the Fourier series, the same principle can be applied to approximating with polynomials. In both cases the crucial point is that you can form an orthogonal subspace, onto which you can project the function to be approximated. For polynomials it is this: https://en.m.wikipedia.org/wiki/Polynomial_chaos https://en.m.wikipedia.org/wiki/Polynomial_chaos
  - sfpotter 1y ago
    What you're saying isn't wrong, per se, but... There are polynomials that aren't orthogonal that are suitable for numerics: both the Bernstein basis and the monomial basis are used very often and neither are orthogonal. (Well, you could pick a weight function that makes them orthogonal, but...!) The fact of their orthogonality is crucial, but when you work with Chebyshev polynomials, it is very unlikely you are doing an orthogonal (L2) projection! Instead, you would normally use Chebyshev interpolation: 1) interpolate at either the Type-I or Type-II Chebyshev nodes, 2) use the DCT to compute the Chebyshev series coefficients. The fact that you can do this is related to the weight function, but it isn't an L2 procedure. Like I mentioned in my other post, the Chebyshev weight function is maybe more of an artifact of the Chebyshev polynomials' intimate relation to the Fourier series. I am also not totally sure what polynomial chaos has to do with any of this. PC is a term of art in uncertainty quantification, and this is all just basic numerical analysis. If you have a series in orthgonal polynomials, if you want to call it something fancy, you might call it a Fourier series, but usually there is no fancy term...
    - constantcrying 1y ago
      I think you are very confused. My post is about approximation. Obviously other areas use other polynomials or the same polynomials for other reasons. In this case it is about the principle of approximation by orthogonal projection, which is quite common in different fields of mathematics. Here you create an approximation of a target by projecting it onto an orthogonal subspace. This is what the Fourier series is about, an orthogonal projection. Choosing e.g. the Chebychev Polynomials instead of the complex exponential gives you an Approximation onto the orthogonal space of e.g. Chebychev polynomials. The same principle applies e.g. when you are computing an SVD for a low rank approximation. That is another case of orthogonal projection. >Instead, you would normally use Chebyshev interpolation What you do not understand is that this is the same thing. The distinction you describe does not exist, these are the same things, just different perspectives. That they are the same easily follows from the uniqueness of polynomials, which are fully determined by their interpolation points. These aren't distinct ideas, there is a greater principle behind them and that you are using some other algorithm to compute the Approximation does not matter at all. >I am also not totally sure what polynomial chaos has to do with any of this. It is the exact same thing. Projection onto an orthogonal subspace of polynomials. Just that you choose the polynomials with regard to a random variable. So you get an approximation with good statistical properties.
      - sfpotter 1y ago
        No, I am not confused. :-) I am just trying to help you understand some basics of numerical analysis. > What you do not understand is that this is the same thing. It is not the same thing. You can express an analytic function f(x) in a convergent (on [-1, 1]) Chebyshev series: f(x) = \sum_{n=0}^\infty a_n T_n(x). You can then truncate it keeping N+1 terms, giving a degree N polynomial. Call it f_N. Alternatively, you can interpolate f at at N+1 Chebyshev nodes and use a DCT to compute the corresponding Chebyshev series coefficients. Call the resulting polynomial p_N. In general, f_N and p_N are not the same polynomial. Furthermore, computing the coefficients of f_N is much more expensive than computing the coefficients of p_N. For f_N, you need to evaluate N+1 integral which may be quite expensive indeed if you want to get digits. For p_N, you simply evaluate f at N+1 nodes, compute a DCT in O(N log N) time, and the result is the coefficients of p_N up to rounding error. In practice, people do not compute the coefficients of f_N, they compute the coefficients of p_N. Nevertheless, f_N and p_N are essentially as good as each other when it comes to approximation.
- sfpotter 1y ago
  Chebyshev polynomials are useful in approximation theory because they're the minimax polynomials. The remainder of polynomial interpolation can be given in terms of the nodal polynomial, which is the polynomial with the interpolation nodes as zeros. Minimizing the maximum error then leads to the Chebyshev polynomials. This is a basic fact in numerical analysis that has tons of derivations online and in books. The weight function shows the Chebyshev polynomials' relation to the Fourier series . But they are not what you would usually think of as a good candidate for L2 approximation on the interval. Normally you'd use Legendre polynomials, since they have w = 1, but they are a much less convenient basis than Chebyshev for numerics.
  - creata 1y ago
    True, and there are plenty of other reasons Chebyshev polynomials are convenient, too. But I guess what I was asking was: is there some kind of abstract argument why the semicircle distribution would be appropriate in this context? For example, you have abstract arguments like the central limit theorem that explain (in some loose sense) why the normal distribution is everywhere. I guess the semicircle might more-or-less be the only way to get something where interpolation uses the DFT (by projecting points evenly spaced on the complex unit circle onto [-1, 1]), but I dunno, that motivation feels too many steps removed.
    - sfpotter 1y ago
      If there is, I'm not aware of it. Orthogonal polynomials come up in random matrix theory. Maybe there's something there? But your last paragraph is exactly it... it is a "basic" fact but the consequences are profound.
      - efavdb 1y ago
        Could you guys recommend an easy book on this topic?
        sfpotter 1y ago
        Which topic?

seanhunter 1y ago

I thought when I first saw the title that it was going to be about the Gaussian integral[1] which has to be one of the coolest results in all of maths. That is, the integral from - to + infinity of e^(-x^2) dx = sqrt(pi). I remember being given this as an exercise and just being totally shocked by how beautiful it was as a result (when I eventually managed to work out how to evaluate it). [1] https://mathworld.wolfram.com/GaussianIntegral.html https://mathworld.wolfram.com/GaussianIntegral.html

deleted 1y ago
[deleted]
niklasbuschmann 1y ago
Gaussian integrals are also pretty much the basis of quantum field theory in the path integral formalism, where Isserlis's theorem is the analog to Wick's theorem in the operator formalism.
- srean 1y ago
  Indeed. It's the gateway drug to Laplace's method (Laplace approximation), mean field theory, perturbation theory, ... QFT. https://en.m.wikipedia.org/wiki/Laplace%27s_method https://en.m.wikipedia.org/wiki/Laplace%27s_method
  - chermi 1y ago
    Maybe I should just read the wiki you linked, but I guess I'm confused on how this is different than steepest descent? I'm a physicist by training so maybe we just call it something different?
    - dawnofdusk 1y ago
      They are the same for physicists who analytically continue everything. Steepest descent is technically for integrals in the complex plane, and Laplace's method is for integrating over the real line.

extrabajs 1y ago

What is Fig. 1 showing? Is it the value of the integral compared with two approximations? Would it not be more interesting to show the error of the approximations instead? Asking for a friend who isn’t computing a lot of integrals.

mturmon 1y ago
Fig 1 could use a rethink. It uses log scale, but the dynamic range of the y-axis is tiny, so the log transform isn't doing anything. It would be better shown as a table with 3 numbers. Or, maybe two columns, one for integral value and one for error, as you suggest.
- rd11235 1y ago
  Yeah - my guess is this was just a very roundabout solution for setting axis limits. (For some reason, plt.bar was used instead of plt.plot, so the y axis would start at 0 by default, making all results look the same. But when the log scale is applied, the lower y limit becomes the data’s minimum. So, because the dynamic range is so low, the end result is visually identical to having just set y limits using the original linear scale). Anyhow for anyone interested the values for those 3 points are 2.0000 (exact), 1.9671 (trapezoid), and 1.9998 (gaussian). The relatives errors are 1.6% vs. 0.01%.
- beansbeansbeans 1y ago
  [dead]

tim-kt 1y ago

No, the integral cannot be "expressed" as a sum over weights and function evaluations (with a "="), it can be approximated with this idea. If you fix any n+1 nodes, interpolate your function, and integrate your polynomial, you will get this sum where the weights are integrals over (Lagrange) basis polynomials. By construction, you can compute the integral of polynomials up to degree n exactly. Now, if you choose the nodes in a particular way (namely, as the zeros of some polynomials), you can increase this to up to 2n+1. What I'm getting at is that the Gaussian integration is not estimating the integrals of polynomials of degree 2n+1, but it's evaluating them exactly.

Onavo 1y ago
Convolution?
- tim-kt 1y ago
  What about it?
beansbeansbeans 1y ago
You're totally correct, my writeup in that section definitely was not clear. I've updated the blog, hopefully it's better. I've also given you a shoutout in the end of the post in my edit log, if that's cool with you
- tim-kt 1y ago
  Sure, thanks. I also saw this sentence, which was not there in the last version (I think): > That is to say, with n nodes, gaussian integration will approximate your function's integral with a higher order polynomial than a basic technique would - resulting in more accuracy. This is not really the case, Gaussian integration is still just interpolation on n nodes, but the way of choosing the nodes increases the integration's exactness degree to 2n-1. It's actually more interesting that Gaussian integration does not require any more work in terms of interpolation, but we just choose our nodes better. (Actually, computing the nodes is sometimes more work, but we can do that once and use them forever.)

drmikeando 1y ago

My mind was exploded by this somewhat similar technique https://en.wikipedia.org/wiki/Tanh-sinh_quadrature https://en.wikipedia.org/wiki/Tanh-sinh_quadrature - it uses a similar transformation of domain, but uses some properties of optimal quadrature for infinite sequences in the complex plane to produce quickly convergent integrals for many simple and pathological cases.

curtisszmania 1y ago

[dead]

actinium226 1y ago

While I love seeing mathy articles on HN, the lead image in this article is terrible. At first glance I see we're talking about numerical integration so I assuming the red part is this method that is being discussed and that it's much better than the other two. Then I look at the axis, which the caption notes is log scale, and see that it goes from 1.995 to 2. Uh-oh, is someone trying to inflate performance by cutting off the origin? Big no-no, but then wait a minute, this is ground truth compared to two approximations. So actually the one on the right is better. But the middle one is still accurate to within 0.25%. And why is it log scale? Anyway point is, there's lots of room for improvement! In particular it's probably not worth putting ground truth as a separate bar, just plot the error of the two methods, then you don't need to cut off the original. And ditch the log scale.

beansbeansbeans 1y ago
[dead]
beansbeansbeans 1y ago
Yeah - I totally agree. Some others on the thread pointed it out as well. I ended up replacing it with a table additionally having the error percentages. Thanks for the fair critique! I've also mentioned your username in the edit logs at the end of the blog if you don't mind.

algorithmsRcool 1y ago

Did anyone else's antivirus complain about an exploit on this page? ---EDIT--- I'm about 98% sure this blog has a browser hijack embedded in it targeted at windows+MSEDGE browsers that attempted to launch a malicious powershell script to covertly screen record the target machine

beansbeansbeans 1y ago
That's a major claim. The only thing different in this blog post from my others is that I've embedded an executable python notebook in an iframe. It's a marimo notebook that runs code using WASM in your browser. That project is open source too, with no exploit as far as I know. The code for my blog is here : https://github.com/RohanGautam/rohangautam.github.io https://github.com/RohanGautam/rohangautam.github.io If you could point to anything specific to support that claim, would be nice.
- algorithmsRcool 1y ago
  I would be happy to be wrong on this one. But I've gotten two pretty convincing threat notifications when visiting the page from the Sentinel One antivirus platform saying that my msedge process had been compromised by a known exploit. I'll try to get more details. I should note, I do not believe the site is malicious, but i am worried about 3rd party compromise of the site without the owner's knowledge
  - beansbeansbeans 1y ago
    I see, that's strange. Yeah, feel free to share the details/logs with me - you can open an issue on my blog's repo with the relevant details and system info. However, I'm still suspecting it's something specific to your antivirus not knowing what to do with WASM code(which is used on this page). I found something similar on Reddit: https://www.reddit.com/r/eaglercraft/s/heVtPy60lG https://www.reddit.com/r/eaglercraft/s/heVtPy60lG. I wonder if that's the issue.
BLanen 1y ago
You should not be using antivirus browser plugins anyway.

Malipeddi 1y ago

This is a neat and accessible introduction to the topic. Nice work! I want to mention nesting quadratures like Gauss-Kronod like Clenshaw-Curtis/Fejér. The nesting means that the quadrature points for lower order are included in the points required for higher orders. This allows you to increase your order of accuracy if required with fewer total evaluations.