numerical-analysis

[viscid crag]

do we assume?

rounding to nearest

occurs by the machine?

[fathom rain]

I would assume round to even. But I guess depends on implementation?

and did you mean 1/y is less than machine epsilon or y less than machine epsilon?

[viscid crag]

y

do we round to nearest for y?

if i can figure that out question is ez, since 1/y is allowed to have more decimal digits than the mantissa

of the the two numbers being dividied

[fathom rain]

What do you mean by that? If it rounds to even or nearest for 1/y? Again, I would assume that depends on implementation. I think default 754 is to even

[viscid crag]

its a rounding method when u cant represent number w floating point precison

[fathom rain]

ok so you mean smallest possible number of the data type? ```
julia> floatmin(Float64)
2.2250738585072014e-308

julia> eps(Float64)
2.220446049250313e-16

[viscid crag]

oh wait i figured question out im dumb

[fathom rain]

julia> 1/(1E-400)
Inf

😛

[viscid crag]

xD the question was asking about cancellation error the whole time, sleep deprivation hits a mfer diff and make dem illiterate

[brave crypt]

would DFAs and regex conversion fit into here?

I just need to check if this NFA describes (ab)*(ba)*+a(a)*

I hand verified most of the cases but I'm concerned I missed something

[sharp peak]

@brave crypt I'm not completely familiar with that notation, but if + is an union then aren't you missing the "a" word?

[brave crypt]

Aw dammit, okay I have an older variation but I thought it didn't really make sense

and yes + is union, its notation from Miruoka

[sharp peak]

As a deterministic automaton it doesn't make sense because q0 has two paths with the same input, I think it might be missing an $\epsilon$ path between q3 and q2, but it's been a while since I studied this stuff

[pine jetty]

nanashi

[brave crypt]

Oh, do NFAs need to abide by that condition as well?

[sharp peak]

Sorry for not being of much help

No

[brave crypt]

thats what I was assuming from the start

oh

okay then this would be acceptable as an NFA conversion?

[sharp peak]

Do you not require to make explicit the empty string paths?

[brave crypt]

my prof wasn't really clear on that, and the book wasn't making it to clear if they wanted empty strings either

i see it in explicit calculations when they do some abstract stuff, but otherwise there is no other mention

[sharp peak]

I remember it being more or less a must because not every path was connected by empty strings, but I'd probably lie if I asserted anything, I don't think I'll be of much help

Sorry

[brave crypt]

alright, well thank you for reminding me of that bit about NFAs, I'll think I'll go with the second variation and see if there is much else to be done

[brave crypt]

aight i've got one more for checking over if anyone else is interested, I tried to use some minimizing techniques from JFLAP to get it down a bit more:
$((a^*b^a^)^b)^$

[pine jetty]

Manzareh

[brave crypt]

I spent some time today optimizing the second and the first regexes to see if it would be easier to check my work and all I was able to get down to was $(ab)^(ba)^ \cup a^*$ and $(a^b)^$

[pine jetty]

Manzareh

[brave crypt]

I used a regex equality algorithm (this one: https://bakkot.github.io/dfa-lib/regeq.html) and did some minor commutative madness but the first was the one I couldn't figure how to get any shorter

it appears with JFLAP the generation of NFAs with the regex to NFA converter tends to get less noisy (with all the empty states) as the regex gets shorter which mostly lines up with what I've read, however I still don't really find either of these solutions particularly satisfying

[hardy pier]

is there a way of figuring out what equation would describe a path traced out by an object moving through a vector field? (in my case the vector field would be very similar to a gravitational well thingy) https://media.discordapp.net/attachments/712782199732174888/889961685102764103/unknown.png

here's an equation I was able to come up with for such a vector field, idk if it's even correct, I'm assuming it is lol
[b-a] is the vector from a to b
R is the radius of the circle

[vivid raven]

Hi does anyone here know lambda calculus?

[wide spear]

Lambda calculus should go in #foundations probably

[vivid raven]

Got it, thanks.

[pine jetty]

EmilB

[pine jetty]

EmilB

[fathom shell]

Anyone good with secant method that can help to determine the radius of a cone with surface area $S=1200 m^2$ and height $ h=20m$? Im not sure what my function should be in this case

[pine jetty]

EmilB

[wide spear]

It looks like you need help with coding

#computing-software is the place for that

[merry tinsel]

indeed

thanks!

[green locust]

I don't know if this is the good channel to ask, I don't know if there's one (the CS discord doesn't seem to have one either)

I have this exercise, I have the intuitive idea but I really struggle with properly writing it down and formulating the argument

Let $s, k \in K<<A^>>$ be proper series. Show that $(s + t)^ = s^(ts^)^*$

[pine jetty]

Philémon

[green locust]

These are formal series

And a proper series is a series over a semiring $K$, so a function $s : A^* \to K$ such that $s(\epsilon) = 0_K$

[pine jetty]

Philémon

[wide spear]

You might have better luck in #advanced-analysis

[green locust]

Well, this is computer science though

That's why I went on this channel

[civic ledge]

hi

anyone familiar with taylor series here?

i wanted to know what kind of projects can be done under the said topic

[wide spear]

What sorts of projects are you interested in

#advanced-analysis might be a better fit depending on what your interests are

[civic ledge]

any

a mini project to be precise @wide spear

[wide spear]

What level of math are you at

[civic ledge]

school grad

i mean college level

or maybe eulers method

not necessarily taylors

Bisection method or central difference, forward or backward difference also..

You there @wide spear ?

[fathom rain]

how big is a "mini project"

[civic ledge]

not that big,

https://wp.nyu.edu/francescopaparella/2017/12/21/numericalmethods2017/

u could refer this

its more or less like an investigation or a research about a particular topic that could be used in engineering applications

[fathom rain]

If that is your teacher, maybe do something on finite volume methods? he seems to be into fluid mechnaics

[civic ledge]

nope he isn't

one professor suggested me the webiste

to refer*

[fathom rain]

ok

[civic ledge]

while doing the project

eulers method
not necessarily taylors
Bisection method or central difference, forward or backward difference also..

based on these topics

or related to computational

[fathom rain]

I have ideas related to my research, but I don't have much time to explain stuff and I don't know your background. But I can give you easy to digest references

[civic ledge]

tq

[fathom rain]

One idea would be to describe the spectral distribution of the Laplacian discretized with FDM https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative using the theory of generalized locally Toeplitz sequences

I am pretty sure your teacher has never heard of that 🙂

(it is kind of an advanced form of Fourier analysis)

but easy to learn the basics for somebody with like msc background

just tell me if it sounds interesting and i can give you references

[civic ledge]

yea sounds cool

but the problem is i dont really have a msc background

still UG

[fathom rain]

Other people should come with suggestions 🙂

[wide spear]

You can do perturbation analysis for a differential equation

This also touches upon Taylor series which you mentioned earlier

Say you have a differential equation $u''+\eps u=u^2$ or something which has a solution $u(t,\eps)$ and then you solve first for $\eps=0$ and then perform a Taylor series expansion of $u$ around $\eps=0$

[pine jetty]

Angetenar

[wide spear]

And then you can compare this with what you get numerically

[lusty spire]

Hi everyone. I was wondering if anyone's got any resource recommendations for transitioning from applied math graduate school into industry. Graduating in May and I'm starting the job search!

[wide spear]

What is your area of specialization

What do you want to do in industry

[lusty spire]

I do most work in algorithm optimization and numerical solutions to continuous problems. Mostly the former

[lusty spire]

What most interests me are jobs in computer systems (I'd love to work for Intel as an Algorithm Engineer for instance), or aerospace, or digital signal processing

[wide spear]

Ok if you want to work in aerospace, brush up finite element stuff

[fathom rain]

what country?

[lusty spire]

United States

[fathom rain]

aerospace is 99% fvm not aware of anybody using fem (except for structural computations)

[wide spear]

https://boards.greenhouse.io/boomsupersonic/jobs/4017480004 is a job posting for plane design

If you want to work at Intel on something like MKL then you should be familiar with how algorithms work in hardware contexts

[lusty spire]

This is great information, first off thank you

[prime kraken]

on signal processing, i can recommend petre stoica's spectral analysis of signals and yonina eldar's compressed sensing book (i forget the name)

[wide spear]

https://arxiv.org/abs/1906.06440

This is some stuff that Intel people are doing

[fathom rain]

I would say if you want to work in aerospace it is almost expected to have a PhD, at least from my experience

[wide spear]

If Intel interests you, then Nvidia and AMD are also doing similar things

[lusty spire]

I'd say working in that realm (Intel) is my top pick

[wide spear]

Nvidia is more GPU focused, do you have experience with cuda and stuff

[lusty spire]

I do not

[wide spear]

https://developer.download.nvidia.com/books/HTML/gpugems/gpugems_pref01.html

[lusty spire]

I have a class this semester where I can read up on basically anything I want, so if you have any recommendations for what I could read into for that area I'd be grateful!

[wide spear]

Also communication avoiding algorithms are a big deal at Intel and co

[lusty spire]

Okay, I'm going to look into that and their MKL. Thank you for the info!

[fathom rain]

maybe mixed precision computing can be of interest?

https://nhigham.com/2015/10/08/the-rise-of-mixed-precision-arithmetic/

[loud bough]

Any ideas on how to start this? Does it just mean we need k terms of the Taylor series to get a reasonable estimate of the root?

[raven dew]

As for how to start it - one method is to just write x_(n+1) - r in terms of x_n and r and expand f(x_n) and f'(x_n) as taylor polynomials with enough terms that you get something non-zero; stuff will start to cancel out and you'll be left with the quadratic convergence required

(I'd use r = 0 WLOG tbh)

[loud bough]

Thank you I will try this

[raven dew]

np

[hybrid parrot]

@past pebble you're solidly in qfin right? maybe you can help @tardy ember

hes a phd math looking into qfin industry

[past pebble]

what's the context? were posts deleted or on a different channel?

[hybrid parrot]

oh no I just thought to introduce @tardy ember to someone familiar with qfin industry

[tardy ember]

Hello!

[past pebble]

hello

i should say though that qfin is very varied and the term itself thus rather meaningless (not to mention that finance itself is all about detail so often you'll find that "good firms" exist in the same sense as "good universities", i.e. comparison makes more sense at a team/group level)

[tardy ember]

Interesting, that makes sense

[hybrid parrot]

some firms i more associate with papers than others though

😏

[past pebble]

well, e.g. AQR is very academic (lots of staff having PhDs) and publish plenty of whitepapers (probably not much in the peer reviewed space), but more from an econ perspective

and I don't think there's enough math to keep a math PhD entertained in the smart beta/factor investing land, though I'm ignorant of the deets

lopez de prado was there for a bit trying to bring ML to the space, but I guess it didn't work

[hybrid parrot]

lopez do prado, wherever he is now, has some neat results on hitting times and analytical optimal entry/exit

ive seen aqr names on a bunch of arxiv papers so it sounds like a place for phds yeah

[tardy ember]

Makes sense

For what it's worth I'm largely accepting that I'll probably enter into qfin (or really anything other than academia) on its terms rather than my own if that makes sense. Whether I find the work as interesting as my current stuff, it does still seem like somewhat interesting work, pays well, etc. Hence why I'd also consider e.g. trading.

[past pebble]

sure, but there's plenty of firms doing very different things is all, finance is all about fine detail and finding a niche and being good at it

people often think, and this is even more true on all the internet forums, that finance is all about stocks, etfs, or exchange traded products

[hybrid parrot]

uhh wheres that quant shop with lots of ex-msri peeps, some still publishing non-quant papers

always funny seeing a rando result in eg protein folding coming out of a quant

[past pebble]

i believe e.g. de shaw (the person) is doing protein folding himself nowadays, not finance

[wide spear]

DE Shaw has a research division

There's also Renaissance

Which was founded by Jim Simons

[past pebble]

and as for trading, there's a vast spectrum there too. one could just execute (for a PM/signal), or you could be taking views of the market (so being a PM, or being part of designing the strategy), or something in between or both; and again one might ask if a quant trader is someone guiding trading algorithms, or someone trading high touch illiquid derivatives

[tardy ember]

@past pebble sorry I was out getting food

So right now I'm looking at trading and/or QR internships mostly. I can't quite do AQR or SIG until summer 2023

Also don't know coding or stats so I don't know if research is at all accessible in the near future

Been looking at Jane Street, also today was told about De Shaw so maybe that too

[wide spear]

There's also Optiver and 2 sigma and 5 rings

Hudson river trading

[hybrid parrot]

Flow Traders

Ginkgo

Jump Trading or w/e its called

Point72

[tardy ember]

I'll keep these names in mind

[hollow hare]

how can ODEs like y'(x) = y(sin(x)) for y(0) = y0 be solved?

solved numerically

are there algorithms for such ODEs?

[pine jetty]

itsmeyash31

[dawn viper]

ig since |sin x| < |x|

you can numerically solve it

otherwise

[prime kraken]

isnt this separable

or am i missing smth

[random hornet]

They probably don't mean y\times sin(x) on the right, but rather that y depends on sin(x) or sth

The latter is not immediately sensible to me though

[prime kraken]

ah

[languid gate]

a closed form sol will exist in this case?

[wide spear]

No it won't

Why would a closed form solution exist

Anyways

I don't see what the issue is

You just solve it numerically

[hollow hare]

`but how do you solve it numerically? I don't think it is that straightforward

[wide spear]

Forward euler?

I can write up the details later but you just get a time marching procedure

[mint hemlock]

y'(0) = y(sin(0) = y(0)
y(delta t) = y(0) + y'(0)*delta t + O(delta t^2)
y'(delta t) = y(sin(delta t)) = y(delta t + O(delta t^3)) ~ y(delta t)
and so on

that could be wrong but that's my first instinct to check

[wide spear]

If sin(t) is not a point that you have explicitly you can just compute it with linear/polynomial interpolation

[dawn viper]

^ + linear/polynomial interpolation

[cinder blaze]

hi everyone, I'm trying to simulate a physic system in which I smoothly transport a particle by slowly changing the hamiltonian. I'm using the scipy package on python with eigh function to diagonalize. When I take the inner product between neighbors states (that should be nearly identical) I get negative signs in some cases, like in here:
product:(0.9966026884156858-0.05932605607477859j), psi1: (0.032943245023489+0j), psi2: (0.03339802473783493+0j)
product:(0.9966084381126055-0.0584749753051199j), psi1: (0.03339802473783493+0j), psi2: (0.03403795510530472+0j)
product:(0.9969452119066843-0.05402623734400029j), psi1: (0.03403795510530472+0j), psi2: (0.03485994413495064+0j)
product:(-0.9975294263339476+0.04629996657310793j), psi1: (0.03485994413495064+0j), psi2: (-0.035831790417432785+0j)
product:(0.9981950162523199-0.03622141578551043j), psi1: (-0.035831790417432785+0j), psi2: (-0.036901943385601044+0j)
product:(0.9987735107759269-0.024920471350837484j), psi1: (-0.036901943385601044+0j), psi2: (-0.03800549358129501+0j)
product:(0.9991503307680749-0.013362420884015513j), psi1: (-0.03800549358129501+0j), psi2: (-0.0390763133764267+0j)

psi1 and psi2 are the first coordinate of each eigenvector

any ideas?

[wide spear]

Coding questions go in #computing-software

[slow shale]

Hi everyone, I had a question about the nelder-mead algorithm. I know that its quite slow but I'm having trouble figuring out its order. I don't think its 0, 1, 2 since its quite slow so I think its 3 or greater?

[wide spear]

By order, do you mean runtime or error

[slow shale]

Sorry I should've clarified, I mean error

[wide spear]

Not very much is known about the convergence of Nelder-Mead

There are some results in very special cases but overall, the picture remains fuzzy

[slow shale]

Oh I see, thanks! Is this the same case for its runtime?

[wide spear]

I mean usually with runtime, you just multiple # of iterations * cost/iteration

And we can't say anything about # of iterations

[slow shale]

makes sense

thank you

[slow shale]

I guess I also have a follow up, unrelated question. I'm being asked what optimization methods might be useful when we are stuck in ravines where the condition number of the Hessian is large. The options are preconditioning, gradient descent, newton's method, and conjugate gradient.

I've ruled out gradient descent because the gradient changes way too rapidly in such situations for gradient descent to be efficient

And more generally I think first-order methods struggle in situations like this. But I'm not sure about the other three

[wide spear]

By ravines do you mean local minima

I think the best thing to do here is precondition

Precondition is a pretty vague thing but generally it refers to techniques that make the problem easier by reducing the condition number

One way you can do this is transform the function being optimized by stretching/squeezing to make the level sets look more like circles

So the condition number of the Hessian will decrease as the gradient won't change as rapidly

[slow shale]

Oh I see. I can select multiple answers here, so do you think the other methods could also be useful? I don't think newton's would be good here since second-order methods require the inversion of a hessian and if the hessian is very ill-conditioned then that might be tough

I'm not sure about conjugate gradient though

[wide spear]

I mean in general, you don't have any guarantee of convergence with any of the 3 if you have non-unique minima right

[slow shale]

yeah

[wide spear]

They don't have any ways of getting out of local minima

If you were doing ML I would suggest hand tuning hyperparameters in your SGD

Lol

[slow shale]

Hmm then yeah I think you're right preconditioning might be the best

thank you!

I actually just found this https://stats.stackexchange.com/questions/386091/why-newtons-method-is-not-sensitive-to-ill-conditioned-hessian

which seems to indicate that newton's could be good here haha

[wide spear]

That doesn't help Newton get out of local minima though

[lusty spire]

Would anyone be willing to try to explain aliasing in regards to interpolatory quadrature and the Discrete Fourier Transform? I've gone to my professor on the subject twice now and his explanations aren't getting through to me

[outer meteor]

any recommendations on introductory expositions on Hilbert transform?

[naive moat]

I mean, what are you looking for other than the definition?

[naive moat]

Unrelated question. In the paper

Kaprzyk, S. "An algebraic formulation of the simplex linear method for multiple integrals over the d-dimensional domain." Computer Physics Communications 183.1 (2012): 20-25.
Kaprzyk finds that integrals in N-D of the form

equal

The context is basically that the former integral is useful for piecewise rational interpolation for integration in n-dimensional spaces, tessellated by tetrahedra

**My question is: ** is there a deeper meaning to a formula like this? Like, what's the role in this special structure for multidimensional integrals? I get the Jacobian determinant comes about for a change of variables, but this formula seems to have so much structure (and is a ratio of determinants!)

In the paper, he derives the formula via a proof by induction

but I'm afraid the intuition escapes me, despite the fact that the highly structured form of the result seems like there should be a deeper meaning or some intuition

For example, since determinants are involved, I imagine a clever change of variables should yield a Jacobian determinant times some easily integrable function that produces the results, but I can't see how to do that easily (what change of variables? how to keep track of all the integration ranges? etc)

[orchid sequoia]

Looks like Cramer's rule

[sharp ore]

What is r^(d) ? You showed us a formula for r_0 and then one for r, how are we supposed to link the two together?

[naive moat]

Sorry, you're right. r^(d) is (1/d!) r_0^(d)

[sharp ore]

What about phi_d?

[naive moat]

It's the d-th integral of phi_0

Here's the additional pics to above:

[naive moat]

I mean, the part that blows my mind is that the resulting structure is independent of the integrand, so long as the function's argument is of the linear combination form

[wide spear]

It isn't though?

The right hand column of the determinant in the numerator is very dependent on the integrand

[naive moat]

Yes. The result isn't fully independent, I meant the structure of the result, the fact that there's a closed-form solution regardless. I edited the above comment to reflect that

[sharp ore]

so I didn't do it all the way but I think the trick here is just to develop the determinant along the last column of phi_d, so you get a sum of the integrals phi_d.
To get that sum from the r_0 I think you just need to make multiple changes of variables u_d = everything between the parenthesis of phi_0 and then u_i = ... and so on iteratively

Uh wait, I guess you only need to do the change of variables once for u_d and proceed by induction if you suppose the formula true for d-1 variables z1,...,z_(d-1)

I only did a cursory check with the first change of variable but you can already see the vandermonde determinant start to appear.

[naive moat]

Yeah, that's what the author does in the paper. I can see how one would reproduce the result algebraically; I suppose my question at its heart is, is there a geometric intuition behind the result? Maybe a path forward to deriving it without induction?

[sharp ore]

I'm not sure there's straightforward geometrical intuition. I guess both determinant and integral represent a volume so looking at what volume they represent could be something to explore, but that's not going to be something you think of when seeing the integral, it'd rather be an interpretation after the fact. I suggest trying to visualize all the volumes involved with simple examples, in geogebra for instance.
As for the "how do we get the intuition to find this formula", I think it's not about geometrical intuition but more of an analytical one: you just have to see that you can do a linear change of variables for each individual integral and see what you get, then t'recognize a pattern.

Maybe someone with better understanding of volumes than me will be able to give you a better answer though

[loud bough]

I'm probably missing something very obvious, but in the last line he claims that the fact that N'_k(r) = 0 implies that N'(r) = 0. I don't see how that follows
https://www.whitman.edu/documents/Academics/Mathematics/burton.pdf

[brittle atlas]

Here's a nice book https://bookstore.ams.org/gsm-29 it is intro grad level but I'm not sure you can really do much with Hilbert transform at a lower level

[outer meteor]

@brittle atlas appreciate it. intro grad level is great.

[brittle atlas]

no problem! beyond that, probably something like Stein Singular Integrals book or another book on singular integrals would have more stuff

if the Duoandikoetxea book isn't what you're looking for, you could try asking in the analysis channel also

[opal mason]

I'm stuck on trying to show that there is a householder matrix for any 2 vectors with the same length in C^M

okay so as I was typing this I think I just got an idea.

I'll type it anyways if it's a dud

[pine jetty]

dk

dk

[opal mason]

Anyone have any tips of how to approach this, or if my idea is correct?

[wide spear]

Are you using v for the householder matrix when you should be using u?

[opal mason]

I'm using u, but I wrote my algebra out in v (so I did divide out v^*v)

What I basically did was just expand out $$Fx = (I - 2 \frac{vv^*}{v^*v})x$$ and got that $Fx = \alpha y$ only when $$2\frac{v^*x}{v^*v} = 1$$

[pine jetty]

dk

[opal mason]

So I tried showing that can be reduced to something true, i.e. 0 = 0. I was able to reduce it to $${\bar \alpha}y^*x = \alpha x^*y$$

[pine jetty]

dk

[opal mason]

So I just need to show that for a given x, there exists a y vector such that above holds, which works when x and y are real multiples (which I'm thinking is technically different from the scaling done by $\alpha$ (i.e. since I assumed $x \neq \alpha y$), since $\alpha = e^{i\theta}$ is a complex number when at least $\theta \neq 0$

[pine jetty]

dk

[opal mason]

<@&286206848099549185>

[wide spear]

@opal mason I will take a closer look today

[pine jetty]

whzup

[green locust]

Hi ! Does anyone have experience with well quasi-orders ?
I have an exercise about decidability of certain problems that uses WQOs
I'm usually terrible at computability but I think I found something convincing, so if someone is up to talk about it and shatter my dreams, don't hesitate to ping me 😄

Maybe that's more of a CS thing but I'm trying nonetheless

[wide spear]

You might also try in #foundations for questions about computability

[green locust]

Thanks, I'll try it !

[fathom shell]

how can i compute least squares with cholesky factorization?

Can someone tell me what im doing wrong in this process?

im supposed to get this matrix returned by the function : [6., -2., 2.] but instead im getting this [1. 0. 0. 0.]

[wide spear]

Are you solving the normal equations?

The normal equations say that $X^TXx=X^Ty$

[pine jetty]

Marius von Hagen

[wide spear]

So you should replace rhand with np.dot(Ain.T,rhand)

(Also np.matmul(A,B) is preferred over np.dot(A,B) for matrix multiplication)

Also you should be multiplying A^TA, not AA^T

[fathom shell]

Thanks alot, it works

[wide spear]

@opal mason look at theorem 2.1.13 in http://www.cse.zju.edu.cn/eclass/attachments/2015-10/01-1446086008-145421.pdf

You should be able to adapt this proof to your problem

[opal mason]

I already submitted, but thank you! I'll take a look.

hmm link doesn't work for me

[wide spear]

It’s just very slow

[proud lion]

https://cdn.discordapp.com/attachments/326138783181438976/896052770577190982/unknown.png

this is for cryptography class so I hope it's the right channel

but I need help with figuring out the probabilities

don't really get it because isn't every deck going to be unique

so wouldn't the probability be 1/6 for every deck

[wide spear]

#probability-statistics

[proud lion]

ty

[fathom shell]

when would we want to find the pseudoinverse(Moore penrose) of a non square matrix ?

[wide spear]

To solve least squares

[fathom shell]

Why is it generally a bad idea to use the inverse to calculate least squares ? as in why do we use cholesky or LU instead of just doing inverse?

[wide spear]

The normal equations are ill conditioned

Calculating the inverse is always bad

Also because of floating point reasons

[fathom shell]

why cant we just use pseudo invese for ill conditioned? is it because it is slower than factorization?

[wide spear]

Calculating the pseudo inverse is very slow because the SVD is slow

[fathom shell]

i see, so in rough terms its all about optimization

[fathom rain]

https://arxiv.org/pdf/1201.6035.pdf

[wide spear]

"Computing the inverse requires more arithmetic operations than computing an LU factorization."

Especially when doing this for the normal equations like EmilB was doing you can do even better than LU with Cholesky

[fathom rain]

"We do not address the question of computational
efficiency, but we do note that there is evidence that using the inverse is
sometimes preferable from the performance perspective [3]."

🙂

just arguing against "always bad"

In general I agree 🙂

[wide spear]

Ok sure

If you follow [3] then they present 2 cases where A^-1 is faster than LU

Anyways

You should never solve the normal equations in the first case

[hearty finch]

Hi

I was wondering guys

How can I solve a system of equation with module operation, like for example

In fact, the algorithm does not need to find the solution, it needs to determine if there is a solution or not

I do not know if a regular Gaussian elimination would work and in case it does, I do not know how to perform the modification to it

[fathom shell]

"Interpolation is not often something that’s useful in machine learning, but rather something that we often use AI for. Feeding data to a computer and allowing it to make educated guesses for us, especially when it comes to millions of lines of data, is helpful in many different fields. "

Why is it not useful for machine learning? And i suppose by AI they mean neural networks, so why is interpolation good for NN?

[prime kraken]

with a deep enough network and enough data, the underlying structure can be inferred from the rough data

what's more, interpolating the input data incorrectly can make the network interpolate and extrapolate worse

interpolation and extrapolation don't generate any new information. they just exploit what is already there. the network already does this

[fathom shell]

so in laymans terms why is interpolation not good for ML , and more common in Deeplearning?

[prime kraken]

because at best it does nothing, and at worst it produces incorrect data

i'm not quite sure what you mean by machine learning there

[hearty finch]

mod is supposed to be 2

I am doing a XOR operation system of equations

[prime kraken]

interpolation is often an optimization task that can be approached in several ways. deep learning is one of them.

[hearty finch]

Would you please tell me a little bit more?

[fathom shell]

why would you use interpolation instead of just linear regression?

[prime kraken]

linear regression does exactly what the name implies. it assumes the data follows a linear relationship. this is simply not true for most data, and so if you interpolate in such a simple fashion, you make mistakes

the problem is if you take this data and then give it to the network as if it were the correct data (which it isn't)

you then teach your network that it should do this linear interpolation, when it could do much better if you didn't confuse it

[hearty finch]

So, the linear operations should be redifined to work in Z/2Z ?

Sorry, I think I do not understand. It is being a little bit confusing to me

If R is not considered a finite field, I do think it is not, then I do not have experience with Gaussian elimination over finite fields

Now I get it

There is a way to know if a solution exists without performing the gaussian elimination?

The concept is amazing, it blew my mind

May I ask why did you have to work with this?

Me too hahahaha

"Given N friends, and given M relations of the form A -> B where A and B are friends, find if there is a partition of at most size 2 such that each member of the partition have an odd number of friends inside the partition"

Is it working in O(n^3)?

Would you mind sharing it with me?

No problem, thanks in advance

[wide spear]

https://hal.archives-ouvertes.fr/file/index/docid/688254/filename/linalgff.pdf

[simple vigil]

I suppose this isn't directly related but maybe someone will benefit... looking for something related to numerical methods/ hpc internships e.g. the NVIDIA SWE and MathWorks that entertain undergrad applications. Anyone know of anything good/ worth looking into

https://nvidia.wd5.myworkdayjobs.com/en-US/UniversityJobs/job/US-CA-Santa-Clara/Software-Intern--Math-Libraries_JR1933622

stuff similar to this

[hearty finch]

It looks pretty interesting

Thanks for the sharing

[leaden breach]

@brave crypt

[wide spear]

This is not the correct place for this

You should ask about this in #advanced-number-theory

[fathom shell]

Can anyone help with interpreting Least squares(OLS) in matrix form, im so lost.

[wide spear]

What is your question

[fathom shell]

So i have this dataset right, where prmres is the dependent variable and the others are the independent. How would set this data up in matrices for least squares ?

[wide spear]

Ok so

Given $m$ data points ${y_i}$ measured at $m$ places ${x_i}$, the goal is to determine parameters ${\beta_i}$ so that $f(x,\beta)=\sum_{i=1}^n\beta_i\varphi_i(x)$ and ideally, $y_i=f(x_i,\beta)$ but we cannot have this because there are too many $y_i$ and $x_i$ for a linear relationship so we want to minimize the residual $r_i(\beta)=y_i-f(x_i,\beta)$

[pine jetty]

Marius von Hagen

[wide spear]

To convert this into the matrix form, we take $X_{ij}=\varphi_j(x_i)$ and $y$ as the vector of $y_i$ so we have $\beta=(X^TX)^{-1}X^Ty$

[pine jetty]

Marius von Hagen

[wide spear]

$\varphi_j$ is some function (possibly nonlinear) of $x$

[pine jetty]

Marius von Hagen

[fathom shell]

can you give an example?

[wide spear]

In your case, X will be the tall skinny vector formed out of oms, konk, and nypr, and y will be the vector pmres

[fathom shell]

i get that but idk how to represent X in matrix form

[wide spear]

$X=\begin{bmatrix}5.355&2&0\5.551&2&0\5.62&1&0\\vdots&\vdots&\vdots\end{bmatrix}$

[pine jetty]

Marius von Hagen

[fathom shell]

is X a coefficient matrix? @wide spear

[wide spear]

Coefficients of what?

[fathom shell]

ok so i have this python code to solve a least squares solution, it just uses np.linalg.lstq .

the arrays in the pic above corresponds to the this picture, where area is the independent variable

so what im asking if this is correct

[wide spear]

Sure

[fathom shell]

are you knowledgeable on Stata? Do you know where the least squares solution is given in Stata?

[wide spear]

No

[fathom shell]

alright thanks for the help man, i need to validate my python answers in some other program

[wide spear]

Actual coding questions go in #computing-software

[fathom shell]

actually i think it is wrong, i read up on it and they say its called a "Design Matrix" and the whole first column should be 1's

[prime kraken]

what else do you know about the model? that sounds like one of the variables is known to be scaled by 1

is this like a polynomial? are the quantities just scaled and added?

the model you have built there says that your data is of the form area = a bedrooms + b age + c price

and you're solving for a, b, and c

having an extra column of ones could be interpreted as either knowing something else about the variables, or maybe having an extra parameter to denote a base area that doesn't depend on the other variables

say area = a bedrooms + b age + c price + (1) d, and d is this base area

[naive moat]

Okay, so I seem to be needing to efficiently evaluate Vieta's formulas: https://en.wikipedia.org/wiki/Vieta's_formulas

Imagine I have the roots r1, r2, ... of the polynomials mentioned:

In particular, I have these roots, and I want to know if there's a quicker way of evaluating this (getting the rhs) than just straightforwardly

for k=1,2,3, ... say up to 7

[prime kraken]

you mean you want to compute both the LHS and the RHS? cuz the RHS is pretty simple

[naive moat]

I have these $r_{i_j}$

[naive moat]

imagine I don't know these a_{n-k}/a_n

[prime kraken]

right, so you want the LHS

[naive moat]

yes, and I'm hoping there may be a quicker way to evaluate it than just as given

based on these Vieta's formulas I would think there might be?

[prime kraken]

i don't think so

[naive moat]

it's likely gunna be the bottleneck to what I'm doing so I'm hoping for some way 😄

[prime kraken]

i would almost suggest you do the first few small polys with the sum and product definition

and then for large polynomials, actually find the coefficients and use the RHS

[naive moat]

ya I'll be doing it for like k=1,2

[naive moat]

but "actually find the coefficients" seems to be done with the LHS?

[prime kraken]

no, i mean build the poly

you know the roots, right=

[naive moat]

yea ok

[prime kraken]

so (x - x1)(x-x2)... is the poly

and the coefficients of the poly are what is used in the rhs

the computer does poly multiplication with convolutions, which should be rather efficient

[naive moat]

yep, that should be faster for the larger sum-prods I would think

ty!

[prime kraken]

you can compare it if youd like, but this way you save yourself using 2 loops and the cost is that of repeated convolutions

[naive moat]

so since it's on the computer, x is a number, it seems like I'd have to do it for like multiple x's and compute the coefficients via finite differences

[prime kraken]

what do you actually know about the polynomials

earlier you said you have the roots

if you have the roots, there's no need to do finite differences for anything here

[naive moat]

polynomial manip is weirdly far from my numerical experience

but I found this: https://stackoverflow.com/questions/33067755/how-to-efficiently-find-coefficients-of-a-polynomial-from-its-roots

so I think I'm good

[prime kraken]

that's literally what i told you to do

lol

[naive moat]

yes

[fathom shell]

its beta 0

[prime kraken]

if you say so 😛 the name beta 0 alone means nothing to me out of context lol

[fathom shell]

https://en.wikipedia.org/wiki/Design_matrix but yes my question was not the best, i shoulda asked how to format the linear regression model as a matrix and not just the data

[prime kraken]

if you can provide the data model, we can set up the model matrix

it sounds like the assumption is that the data behaves like y = a x1 + b x2 + c x3 + x4

[proud lion]

Can someone help me with this?

[prime kraken]

you can do this in two steps. first, find the general solution in terms of all the free parameters (if any). if there are free parameters, you can then solve an optimization problem to make the sum of the coefficients zero (or some other constraint) if it's not evident how to do so at a glance

[fathom shell]

can you only plot regression with more than 1 feature by using a 3d plot?

[opal mason]

This is more of a theoretical question to something involved applied computation math, but I can't find a suitable channel so I'll drop it here>

My question is on subdifferentials
https://www.stat.cmu.edu/~ryantibs/convexopt-F16/scribes/subgrad-scribed.pdf

So one of the examples they give in this PDF about subgradients is the max of 2 convex, differentiable functions. They claim for x such that f_1(x) = f_2(x), the subgradient at f(x) is a convex combination of the gradients of f_1 and f_2. Geometrically this is easy to understand, but how do I go about proving this? The case when f_1(x) /= f_2(x) is pretty much trivial, but I'm not sure how to do the equals case.
The example in consideration:

preferably I'd like a hint for both directions (showing that it's only a subgradient if it's a cvx combination of the gradients), but a proof sketch/actual proof will do as well

[prime kraken]

first, if they are equal, they have the same subgradient

next, a sketch in R that you can go over to generalize to R^n 😛

we have the convex combination x1 a + x2 (1 - a)

let x2 = x1 + c, for some constant c

then the combination is rewritten as x1 a + (x1 + c) (1 - a) = x1 a + x1 - a x1 + c - ca = x1 + c(1-a)

we know 0 <= a <= 1 due to the definition of a convex comb

this means the result ranges from x1 to x2, i.e. either [x1,x2] or [x2,x1]

this is pretty much the same definition of a subgradient itself, since that is a range of slopes, so to speak, for which the subtangents lie below the graph

i.e. if the subgradient is [y1,y2] and x1 and x2 are in this set, then [x1,x2] is a subset of [y1,y2]

and graphically it's the region between two lines

idk if that helps you, i just woke up lol

[opal mason]

so I'm kind of lost

So for the 2D case, $\xi = \lambda\nabla f_1({\bf x}) + (1-\lambda)\nabla f_2({\bf x})$
Then if I suppose $\nabla f_2({\bf x}) = \nabla f_1({\bf x}) + {\bf c}$, so $c$ is some vector here
and then aaaaaa
I get $\nabla f_1({\bf x}) + {\bf c}(1-\lambda)$

[prime kraken]

mhm

[pine jetty]

dk

[prime kraken]

parentheses

[opal mason]

wait so is this the sketch for showing a convex combination of the gradients is a subgradient?

or the other way around

[prime kraken]

as you said

that taking a cnv comb of subgradients is a subgradient

[opal mason]

you mean cnv comb of gradients?

It's really late here and my thought is very slow

I'm still failing to see where I get the subgradient from the last expression

[prime kraken]

oh i guess ur right, yeah

[opal mason]

how would I show that every subgradient at that point is a convex combination of gradients then? So starting with some subgradient, showing that it's a convex comb of grads

I feel like I'm missing something

[prime kraken]

hmm

probably backwards

start with the subgrad at x being an interval

and show that a subset can be expressed as a cnvx comb

i don't see a good way of showing the endpoints of the interval are the grads of f1 and f2 right now, tho

[vale imp]

Anyone know why centers for rbf neural networks are chosen using k means clustering? I'm going to use an rbf neural network to approximate some non linear function and I was thinking of using gradient descent to minimize the error and treat the centers as extra parameters just like the weights. Is there any reason why that approach would be inefficient compared to k means clustering?

[vale imp]

Would it lead to overfitting because the algorithm might now assign the centers numerical values that best fit the data, but may not be the actual point about which the data is centered?

[grim bobcat]

Kind of a simple problem but I have no idea with computational stuff. I have a simplical cone (Given 3 linearly independent vectors $v_1,v_2,v_3$ in $\mathbb R^3$, the simplical cone spanned by them is the set of all linear combinations with positive coefficients), and I wanna find out in an efficient manner if a point lies inside (Including on the boundary). The points I wanna know if are inside are specifically integer points if that helps. I've thought that, this cone is actually the intersection of the 3 half spaces given by the planes spanned by every 2 vectors from $v_i$, but i'm not sure how to find which part of the half-space I should even take (And therefore which orientation of the cross product I should take). I also thought there might be an easier way to find this out using recession directions but I haven't really fleshed that thought out

[pine jetty]

ShiN

[wide spear]

I'm heading out now but I'll answer if it is unanswered when I get back

[grim bobcat]

Thanks!

[prime kraken]

the straight up naive way is to set the vectors as columns of a matrix, invert the mat, and multiply by the coordinate vector

you'll get the coords in the basis of the v_i and then just check if they're all positive

this ignores the points being integers tho

[grim bobcat]

Oh nice, I didn't consider that since I was too focused on the half-space idea 😅

There might be a better way but at least this works haha

Thanks

[leaden breach]

Hi could someone help and point me in the right direction to solve this problem:

I want to write a python script that does the following:

1. Isolate a square root term on the LHS.
2. Square both sides of the equation.
3. Add and subtract terms accordingly to get 0 on LHS.
4. Isolate another square root term on the LHS. [ i.e. step 1) ]
5. Repeat step 2).
6. Repeat step 3).
7. Repeat step 1).
   ETC . . .

Would this be "solving equations iteratively in Python" ? (I have very little experience with code.)
(And I know I can just tell WolframAlpha to solve this for me, but it would be cool to this piece of a problem I am working on all by myself, in a way.)

[wide spear]

Another solution: truncate the cone at a point past your point of interest, then determine if the point of interest is inside this closed convex polyhedron

This is a classical problem that you will find many algorithms for

[gaunt crag]

is the cengage edition of sipsers introduction to the theory of computation fine

like does it have errors in it?

i bought the book for $20 new when the main published edition is for like $200

[random hornet]

Might be better asked in #book-recommendations ; and I think Cengage is the authorised publisher for both hardcover and paperbacks.

[opal mason]

More theoretical, but any ideas on how to show this is convex? (or counterexample?)

if i'm picturing it correctly geometrically, is this just a shifted quadratic, with lower bound of 1, which should be convex?
I've tried just a bunch of algebraic manipulation and I'm just not getting anywhere

[grim bobcat]

Has someone here worked with polyhedra in sage? there's something dumb that is driving me crazy and I can't find any documentation on

[wide spear]

[grim bobcat]

so when you ask for a vertex representation of a polytope it gives you a list that's like 'A vertex at (x,y,z), A vertex at (x_2,y_2,z_2)' etc.
All I want is that, but without the words, and in vector form. Just a list of the vectors making up the vertices

is that too much to as kfor

[wide spear]

#computing-software

[grim bobcat]

I guess

i'll reposti I just dk if that channel gets any traction

[hazy basin]

I have a homology theory with coefficients in GF(2) that I ultimately need to find representatives for generating sets of the homology groups. I'm guessing that this is a fairly standard problem (compute the homology groups) that there might be a python package for? A brief search shows that I might be able to find some routine for computing the Smith normal form instead?

*The boundary maps are sparse finite dimensional matrices

Is computing the Smith normal form the standard route? / Is there a package that can do that for a very sparse boundary maps with dimension ~10k x 10k ?

[wide spear]

If you don't get a response here you might also want to ask in #point-set-topology

I would ping max but max is no longer with us

[marble kestrel]

differentiate with respect to x twice

[prime kraken]

i think given that A is pos def, x is any vector in R^n

so it's convex

hmm

nvm they gave the cond that the result is >= 1

[marble kestrel]

the derivative will be $$A(x - e_1) + (x - e_1)^TA = A(x-e_1)+ (A(x-e_1))^T$$

the second derivative will be $A+ A^T= 2A$ which is the hessian, and is positive semi-definite so you’re function is convex.

[pine jetty]

c squared

[prime kraken]

but you can first look for the condition for a vector v, and then let v = x-e_1

you sure that works? they're asking for the convexity of the set, not the func

[marble kestrel]

oh whoops. misread

[maiden badger]

2x - y = 0
x^3 + x - y = 0

am exact ans is (1, 2)

if my guess is (2, 3) and after my first iteration calculations, my values are (1.5, 3.5)
there's must be a mistake in my calculation

[barren walrus]

Hi friends I have no idea where to ask this question but maybe you guys have an answer: Is it possible to calculate partial derivatives of an pde even though it has no analytical solution? Like for example the 3dim Navier Stokes Equations?

[opal mason]

Doesn't this only pertain to functions not sets

Turns out there was a very simple counterexample and I was too fixed on showing convexity

[barren walrus]

Which method are using to solve iteratively?

[marble kestrel]

i mean, technically functions and variables are sets. but anyway, i just misread ur question and the advice i gave shows that ur function is convex. not that the set in question is convex

yea i was working through it. i got it to almost work out but there were some terms that just didn’t want to disappear

[wide spear]

You'd probably want to use a sympletic solver right

Equations coming from physics usually conserve energy

[edgy sage]

Given an array $a$ of $n \leq 1000$ integers with $1 \leq a_i \leq 250$, compute the number of distinct subarrays of $a$.

I have a solution using rolling hash that seems to work, but what is the deterministic solution?

I feel it should use string algorithms

[pine jetty]

IlIIllIIIlllIIIIllll

[brave crypt]

hey all

i have a problem i wanna write an algorithm for

but im v confused on how to start

[wide spear]

Best to ask in the CS server, linked in #old-network

[gaunt crag]

what would happen if you defined a new type of automataon saying that each state needs to have >=1 outward arrows for each letter in the alphabet

would any new special properties arise

[loud bough]

https://math.stackexchange.com/questions/1470990/x-n1-x-n-fx-n-over-fx-0-find-c-and-s-such-that-e-n1-ce-n#

For this question on a modified Newton's method problem, I can't comment on stackexchange, but in the last line of Gordon's answer, isn't f(r) = 0, so he is dividing by zero? My own answer came out quite a bit differently, I got the same s value but C = f''(r)e0/[2f'(r)]

[vale imp]

Suppose I wanted to make a linear fit to a dataset with vector input and output, by minimizing the least square error. Then the least square error equation would be
$$E = \frac{1}{2}\sum_i(W\vec{x}^{(i)} - \vec{y}^{(i)})^2$$
And I have to find the weight matrix $W$ that would minimize this. Can't I treat this as a colelction of linear regression problems? Like the least squared error for the j'th component of the output would be
$$\frac{1}{2} \sum_i ( \vec{w}_j \cdot \vec{x}^{(i)} - y^{(i)}_j )^2$$
And each component equation is independent from the next. So can I treat these as simple linear regression problems and find the rows of the weight matrix from the one step formula of linear regression that used the pseudoinverse (I forgot the exact formula) ?

[pine jetty]

Abrar

[vale imp]

I think I can but I see other people use methods like gradient descent or particle swarm optimization to get to the minimum instead for this sort of problems with vector outputs so I'm confused.

[brave crypt]

Hi Guys, I have a scenario, I want to turn it into a mathematical model,

The scenario:
I have a car that makes a reservation for the network resources in advance (use it in the future), That is, when the car reaches the coverage of the Base station (as you can see in the attached Foto), for example, the car has been reserved resources for a while, however, this reserved time may change (update) before the car enters the coverage of Base station due to uncertainty in traffic and movement. Updating the reservation takes the following steps:

1. cancel partial (overbook): If the reserved time is more than the time car needs it right now.

2. Cancel all and new reservation (underbook): If the reserved time is less than the time car needs it right now.

Cancel Partial:
In the cancel partial there are different strategies, for example:
a) If 70% of the reservation is overbooked, should we cancel it all and reserve a new reservation, or cancel it partially why?
b) If 50% of the reservation is overbooked, should we cancel it all and reserve a new reservation, or cancel it partially why?
c) If 20% of the reservation is overbooked, should we cancel it all and reserve a new reservation, or cancel it partially why?

There are cancelation strategies and fees in our model:
Peak-On (i.e. Traffic flow is very high in this Base station coverage):

1. Cancellation of reservation before entering the car the base Station coverage. The cancellation fee is 5%.
2. Cancellation of the reservation after entering the car the base Station coverage. The cancellation fee is 20%.

Peak-Off (i.e. Traffic flow is very low in this Base station coverage):

1. Cancellation of reservation before entering the car the base Station coverage. The cancellation fee is 0%.
2. Cancellation of the reservation after entering the car the base Station coverage. The cancellation fee is 5%.

[wide spear]

I'm going to be honest, this question doesn't really fit here

[hidden marsh]

I have a question,

In class I've proved the following inequality

Where Ax=b and Ax^=b^

Now I want to come up with a proof that there exists a b thats not b^ s.t. the equality holds

but I'm not sure where to start

I can see that I can easily get the equality if I assume |A^-1| |b-b^|=|A^-1(b-b^)| and |A| |x|=|Ax|

but I'm sure theres a stronger assumption I can make about b

any help is appretiated

[orchid sequoia]

You can consider vectors that maximize the operator norms, but I don't remember the outline of the proof

[limpid wagon]

hi

can someone suggest some projects to be done with newton raphson method

a unique one infact

where you explain the project objective

methodology

problem statement

and algorithm

[vale imp]

Bump

[brave crypt]

https://financialdata.tiiny.site/ I pasted the data that I got using python. My problem is that the PCA gives me 49 eigenvalues and I have 93 variables. Isn't that strange ? I should have as much eigenvalues as variables no ?

[prime kraken]

you have the data arranged as a matrix, yeah?

[wide spear]

Data only has 49 rows

You'd have 93 variables if data had 93 rows

[wide spear]

No one is going to do your homework for you

[prime kraken]

unless...

jk jk. but...

[brave crypt]

Why rows

Isn’t it supposed to be related to columns ?

[fathom rain]

google rank

[prime kraken]

well, PCA deals with the eigenvalues of the gram matrix. you could build that either as A^TA or AA^T

[brave crypt]

A^TA

[prime kraken]

regardless, since the matrix is 49 x 93, apparently, then it can at most have rank 49

[brave crypt]

Matrix here is 93 93

[prime kraken]

yep

and 93-49 eigenvalues are 0

[brave crypt]

Why 94 -49 ?

[prime kraken]

as a lazy proof, you can use the SVD to prove this

if the original matrix is 49 x 93, let's call it A

then it has an SVD such that A = USV^H, where U is an orthonormal mat of size 49 x 49, S has the singular values along its main diagonal and is of size 49 x 93, and V is orthonormal and of size 93 x 93

then A^H A = V S^H U^H U S V^H, and sice U is orthonormal, U^H U = I

then S^H and S are diagonal and you get a diagonal matrix D = S^H S of size 93 x 93 with at most 49 nonzero entries

and A^H A = V D V^H, all matrices square of size 93 x 93, V orthonormal, and D a diagonal matrix, so this is the eigenvalue decomposition of A^H A

and it has the same rank as A, and the eigenvalues are the absolute value squared of the singular of A

the rank of A is <= min(number of rows, number of columns)

so the rank of A^H A follows the same property

[brave crypt]

Whats H ?

[prime kraken]

hermitian transpose

complex conjugate transpose

not needed here if the data is all real, so just replace it with a regular transpose

[brave crypt]

How you know 49

[prime kraken]

because of how diagonal matrices multiply

try this

take $\begin{bmatrix} a && 0 && 0 && 0 \\ 0 && b && 0 && 0 \end{bmatrix}$

[pine jetty]

Edd

[prime kraken]

call that S

and calculate S^T S to convince yourself

this is easy to generalize to arbitrary sized diagonal matrices

[limpid wagon]

hii

does someone know about NR method

newton raphson

and thomas algorithm

@prime kraken ?

@wide spear

[wide spear]

If you have a question, ask it

[limpid wagon]

okay i wanted to know the applications of it

newton raphson

and thomas algorithm

[wide spear]

We aren't here to do your project for you

[limpid wagon]

ik'

i wanted to know what kind of projects acan be done

obv i have to do

[prime kraken]

anything where one has to optimize something

[limpid wagon]

okay

do you have suggestions?

[wide spear]

We aren't here to design your project either

[brave crypt]

still don't get it

they are diagonal yes but i don't the link with non zero

[prime kraken]

dude write it on a piece of paper and do it

and here's a related proof, too

https://math.stackexchange.com/questions/48989/how-to-prove-textrankab-leq-min-textranka-textrankb

[brave crypt]

oh i get it

[brave crypt]

it's because S is 49,93

so maximum u have 49 non zero entry right

for D

"S has the singular values along its main diagonal and is of size 49 x 93"

[wide spear]

Someone could use a linear algebra refresher

[brave crypt]

that was my problem

i though that S matrix was diagonal

[brave crypt]

how do you know that ?

[prime kraken]

i literally showed it above

do you know what an SVD is?

[brave crypt]

that is has the same rank as a ?

[brave crypt]

no

[prime kraken]

yes

do you know what a symmetric matrix is

[brave crypt]

y

A^TA is symmetric

[prime kraken]

here's a quick proof

https://math.stackexchange.com/questions/349738/prove-operatornamerankata-operatornameranka-for-any-a-in-m-m-times-n

[brave crypt]

i get his proof

ut now i'm stuck at the end

how did he get dim = dim and rand = rand from the first equality xd

[wide spear]

#linear-algebra

[sand knoll]

The methods in question are PCA, EM algorithm, Neural networks and Naive Bayes

Obiviously for PCA there is a positive effect as we want to find underlying correlations. For the other techniques, I am less familiar with them.

[sand knoll]

Perhaps for neural networks we can consider correlation as a feature to work with, so it would not really matter.

[hidden marsh]

I have a question about matrix decomp
so I know how to decompose a matrix A = L*U
where L is a lower triangular matrix and U is an upper triangular matrix
Now let A = [4,-1,-1;-1,4,-1;-1,-1,4]
I was able to decompose it into LU but one of the question asks me to decompose it to Ltrans(L)
But I don't know any method of doing that
Any help would be appretiated

[wide spear]

See Cholesky decomposition

[fathom shell]

can anyone help with gaussian quadrature

[orchid sequoia]

[fathom shell]

i need to estimate this integral using the gaussian-legendre quadrature formula

And the question is "Would you consider the result of Gaussian quadrature in this case a good approximation? If no, why do you think, this integral is difficult to approximate? "

but idk what to make of it

[fathom shell]

so the polynomial : (1)/(x**2+0.0001) and im integrating with range [0,1] and degree 2 and i get the result : 23.82841

So that begs the question which i dont know the answer to: Why is this a bad estimation?

[fathom shell]

to get a result of 156 i would need the degree to be n = 100, that wouldnt make sense tho as the degree is the degree of the polynomial

[sage vapor]

where are you seeing a polynomial

[slow shale]

Hi, I was sort of confused about this question and wanted to see what others had to say

My thinking is that as the model parameter increases, the training error decreases but the test error will increase too due to overfitting

For this reason I was thinking that the first one is probably not correct but, the second one sort of makes sense to me as well

[hard escarp]

Hi. Just curious. Would this be the right channel for numerical analysis stuff?

[wide spear]

Yes

[near trench]

Could questions on automata theory work here? I don't quite understand how NFAs are made.

[wide spear]

Best to ask in the CS server

[untold crypt]

I have a question to do with LaGrange interpolations. better to ask here or #advanced-analysis ?

[wide spear]

Fine here

[untold crypt]

okay i think i figured it out but i was just having trouble reasoning it out this derivative. im still not sure if i did it correctly. Here is part of my work

im trying to find omega prime given omega.

this is the prompt:

[wide spear]

Ok that looks fine

[untold crypt]

okay i was mostly concerned with the making sure everything was accounted for properly in the product rule expansion.

[wide spear]

Coding questions go in #computing-software

[cosmic karma]

how do I build this block tridiagonal matrix in matlab? I was able to build T but not A

Update: I was able to build a block diagonal matrix will all matrix T's. I do not know how to include the identity matrix yet

[brave crypt]

Hi quick question: What norm should be used for the banach fix point theorem? I have a fix point iteration of the form $x^{(k+1)} = \Phi(x^{[k)}) = Ax^{(k)} + b$ where A is a matrix and b a vector, now i wanted to do:
$||\Phi(x) - \Phi(y) || = ||A(x-y) || \leq ||A|| \cdot ||x-y||$, i would like $||A||$ to be my contraction number but for some norms i get $||A|| \geq 1$

[pine jetty]

Gewisser Fler

[wide spear]

Whatever norm works

You'll get convergence in whatever norm you choose

And on fin dim vector spaces, all norms are equivalent

[fathom rain]

Easiest is to use Kronecker products. What you have there is 2d Laplace so just ```

n1=4;n2=4;

L1=toeplitz([-2 1 zeros(1,n1-2)],[-2 1 zeros(1,n1-2)])

L1 =

-2     1     0     0
 1    -2     1     0
 0     1    -2     1
 0     0     1    -2

L2=toeplitz([-2 1 zeros(1,n2-2)],[-2 1 zeros(1,n2-2)]);

L2D=kron(L1,eye(n2))+kron(eye(n1),L2);

Modified so you can have different `n` in each dimension

[cosmic karma]

This worked thank you!

[brave crypt]

I'm doing a PCA

I'm a little bit lost to know if I should remove outlier ?

and how to know if they are outlier

[prime kraken]

robust PCA is a big deal, since covariance matrices are susceptible to them

but sophisticated techniques are beyond your reach right now

you could try something simple

vectors whose norm differs too much from that of the others, for example, could be discarded

or entrywise computing the quartiles, too

[brave crypt]

@prime kraken i computed quartiles to find outlier

but idk if it's usefull to delete them

[prime kraken]

it is

[brave crypt]

Can I send you my notebook ?

[prime kraken]

no

[brave crypt]

ok x)

@prime kraken

for example for the column the mean is 36 and the max is 156

if he is considered as an outlier using quartiles, should i remove it ?

it's not a value that is crazy but it's also considered as an outlier by the strategy so idk

[prime kraken]

it also depends on what the data represents, so i can't comment

[brave crypt]

oh ok

[main urchin]

Given the error amount (higher_order-lower_order)*step_size, what is the proper way to accept or reject a step in adaptive Runge Kutta? And what is the formula to adjust the step_size for the next step? Everybody on the internet says something different...
I have the iterator like this:

const auto result     = method_type         ::apply   (problem_, step_size_);
const auto evaluation = error_evaluator_type::evaluate(problem_, step_size_, result);

if (evaluation.accept)
{
  problem_.value = result.value;
  problem_.time += step_size_;
  step_size_     = evaluation.next_step_size;
}
else
{
  step_size_     = evaluation.next_step_size;
  return operator++(); // Retry with the adapted step size.
}

now trying to implement error_evaluator_type::evaluate which returns a boolean saying whether to accept or reject the current step and the next step size, but stuck AF:

struct error_evaluator
{
  constexpr static auto absolute_tolerance = 1e-6;
  constexpr static auto relative_tolerance = 1e-3;

  template <typename problem_type, typename time_type, typename result_type>
  static error_evaluation<time_type> evaluate(
    const problem_type& problem  , 
    const time_type     step_size, 
    const result_type&  result   )
  {
    time_type error = NORM(result.error / (absolute_tolerance + relative_tolerance * MAX(problem.value, result.value)));
    return {error < time_type(1), step_size / error};
  }
};

Boost.Odeint uses a weird scheme that I have not seen in any book: https://www.boost.org/doc/libs/1_63_0/libs/numeric/odeint/doc/html/boost_numeric_odeint/odeint_in_detail/steppers.html#boost_numeric_odeint.odeint_in_detail.steppers.controlled_steppers
I just want the classic step size adaptation method

[wide spear]

Accept in error < threshold (decided by you)

Formula for step size: double if error is very small, half if error is large

Or whatever amounts you want

[sand knoll]

I want to make a comparison of the convergence rate of different matrices for conjugate gradient vs gaussian elimination. I have the expression $\frac{|| x_k- x^* ||}{||x^*||} \sim (1- \frac{1}{\sqrt{K_A})})^k$ for the convergence of the CG method. Im wondering if there is a similar expression to be obtained for the convergence rate of gaussian elimination involving condition numbers s.t I may compare use the condition numbers to evaluate which is the quickest method?

I believe $\frac{|| x_k- x^* ||}{||x^*||} \sim (1- \frac{1}{K_A)})^k$ holds for steepest descent, but does it work for A = LU?

here $x^$ represents the true solution of $Ax^ = b$ and $x_k$ is the solution of conjugate gradient at the k-th iteration.

[pine jetty]

Fredrikpiano

Fredrikpiano

Fredrikpiano

[wide spear]

The run time of Gaussian elimination doesn't depend on the condition number because it isn't an iterative method

[sand knoll]

well I believe the gaussian elimination in this context is LU decomposition

[wide spear]

You don't have a convergence rate either

LU decomposition isn't iterative

[sand knoll]

ahh, yeah that makes sense. But then the best comparison would be to state the time complexity for the different methods I guess

[wide spear]

Yes

[sand knoll]

right thanks

[hollow pagoda]

not sure if this is the correct place to ask but

how do I turn a max min objective function into a linear program?

[sand knoll]

@wide spear sorry for the ping. I kept finding different answers for the time complexity for the conjugate gradient. There is a paper titled "An Introduction to the Conjugate Gradient Method without the agonizing pain" which lists the time complexity as $O(m \sqrt{K_A})$ where m is the number of nonzero entries in the mtx. Then if the number of unknowns is equal to $1, N=1$, there would be nothing to gain, but as N increases ($N \geq 2$) the CG algorithm will always converge faster. Does this sound reasonable?

[pine jetty]

Fredrikpiano

[wide spear]

Yes, that is Shewchuk's document right?

What is N

[sand knoll]

yes, Shewchuck. N is just to denote the dimension i .e N = 10 generates a 10x10 matrix in the matlab program I am running to generate some matrices which I want to compare the convergence rate for. I choose the infinity norm for when I am computing the condition numbers (||A^-1|| ||A||_\infty), but I could have chosen other norm as well right?

[wide spear]

Usually the condition number refers to the ratio of the largest and smallest eigenvalue

Clearly there is nothing to be gained when N=1

But you haven't said what you're comparing CG to

[sand knoll]

Comparing it to LU decomposition with time complexity O(n^3)

[wide spear]

Ok

[sand knoll]

one can relate condition numbers to a mtx being singular as well right? If the condition number is very large for example

[wide spear]

Yes

[sand knoll]

I also have the following exercise Im working on from my teachers lecture notes

Im thinking I should show that A has one lin. independent column and work from there

[wide spear]

#linear-algebra

[sand knoll]

yep

thanks

[hollow pagoda]

if anyone could help or give me hints it would be much appreciated...

[wide spear]

What are you minimizing with respect to

What are you maximizing with respect to

[hollow pagoda]

in this case i am minimizing the i's

and maximizing the sum ai1 s1 ... + ain xn

[wide spear]

What do you mean by minimizing the is

The ai's?

[hollow pagoda]

yes, i think we are choosing an i

such that a_i will be minimized

and then we "maximize" by choosing a x

I think the first step would be to replace the entire

with "A"

and then try to translate this minimiztion equation into a set of constraints

but im not sure how

[wide spear]

Wait are you minimizing the sum of the a_i

What does minimize the a_i mean

You can rewrite that sum as Ax, yes

[hollow pagoda]

minimizing ai1 + ai2 ... + ain

and after that, maximizing ai1x1..ainxn

[wide spear]

Ok so first you minimize the 1-norm of A

And then maximize the 1-norm of Ax

[hollow pagoda]

i believe so

[wide spear]

Why do you want this as a linear program

What constraints do you have on A aside from Ax<=1

[hollow pagoda]

this is the only constraint

and this is an exercise, the rest of the numbers were fairly straightforward

[wide spear]

I see

Wait

We don't minimize abs(ai1)+abs(ai2)+... right

We minimize ai1+ai2+...?

Then there is no minimizer A

You can make the sum as small as you want

[hollow pagoda]

wait i might've been wrong

[wide spear]

Ok that makes more sense I think

[hollow pagoda]

so first i minimize the sum, and i pick the biggest "component"

[wide spear]

Yeah

I'll assume that A is fixed then

Then you have m linear programs to minimize a_i^Tx

Then you have one linear program to maximize c^Tx where x is a vector with entries summing to 1 and c is the vector of minimizers of the a_i^Tx

And you use the constraint when minimizing a_i^Tx

I think

[hollow pagoda]

ok so you're suggesting i use two linear programs?

[wide spear]

Yes

[hollow pagoda]

hmmm ok i'll try that

thanks!

[tall solar]

Can someone remind me about how random matrix theory is related to compressed sensing?

For example in the matrix completion problem you need to assume the entries are missing randomly. So you are basically dealing with Bernoulli random variables. Then you have a matrix of Bernoulli variables. Somehow properties about random matrices of 0 and 1s is relevant somehow. So I think random matrix theory comes in here. But I wanna know the core results that are relevant

[prime kraken]

you're interested in matrices that satisfy the restricted isometry property, then

[wide spear]

https://www.mathc.rwth-aachen.de/~rauhut/files/LinzRauhut.pdf

Is this what you want

[prime kraken]

looks good

[tall solar]

Thank you so much I think the tools from probability chapter is exactly what I was looking for

[tropic loom]

Would this channel fit for Automata theory?

[wide spear]

No, you should ask about automata theory in the CS server, linnked in #old-network

[tropic loom]

👍

[terse flower]

hi friends possible to compute cholesky factorization on non square matrix? 👀

^matlab

[wide spear]

#computing-software

[terse flower]

thank you ❤️

[marble kestrel]

anybody here know machine learning?

[wide spear]

You might have better luck asking in the ML server linked in #old-network but you can always try asking here

[marble kestrel]

alr, thanks

[marble kestrel]

just going to ask here in case:
alr, so i have some data:

X1    X2  Label
0.4  0.4  Red
0.8  0.8  Red
0.2  0.4  Blue
0.4  0.2  Blue
0.8  0.2  Blue
0.2  0.8  Blue

and i have the following kernel K(x,y) = y^T x / ||y|| ||x|| and i am tasked with mapping each data point in the original feature space to the new feature space representation implicit by K(x, z).

im kind of unsure what to do...

just to make sure im understanding, the original feature space is the set of ordered triplets that look like
(x1, x2, label) correct?

[viral tendon]

Anyone can help me with calculating norm of inner products?

I've got u,v in V
V is a space of inner products

| | u | | =3 ; | |u+v| |=4 | |u-v| |=6

how do I find v?

or | |v| |

[random hornet]

#linear-algebra is more appropriate I guess

[prime kraken]

use the associated inner product that induces the norm

for example, $\langle u, u \rangle = \Vert u \Vert^2 = 9$

[pine jetty]

Edd

[wide spear]

#linear-algebra indeed

[prime kraken]

indeed 😛

[viral tendon]

how does it work with | | u + v | | =4 -> | | u + v | | ^2 = 16?

does it give the option to break it?

[wide spear]

#linear-algebra

[prime kraken]

check what i wrote over in linalg

i explained it over there

[finite salmon]

i'm using RStudio and i'm trying to analyse a dataset. one of the variables in it is "product_brand" where each entry is a brand name e.g. "APPLE". i want to make a histogram of a different variable in the dataset but i only it to consider data from a specific brand, does anyone know what i would write?

[wide spear]

#computing-software

[finite salmon]

ah sorry

[hazy basin]

Sage, Magma, and Mathematica will compute smith normal forms for integer matrices

[woven shale]

Does heun's method give the same solutions as Euler's method? (Specifically the forward euler)

I'm a little confused on this, because Heun's is considered a 2nd order method, so I don't see how it would give the same answer as Euler's. But if it's just an improved Euler's method, shouldn't it give the same answer?

[wide spear]

No it doesn't give the same answer as Euler's method

[tribal pulsar]

How does the green part works?

I search around the internet and find "finite difference approximation" but how?

[prime kraken]

you can read about it on almost any google search about finite difference approx :p

https://en.m.wikipedia.org/wiki/Finite_difference_method

[fathom rain]

Intuitively think maybe like this: f'(x)=df/dx Here, df is a small difference in f and dx is a small difference in x. So, df=f(x1)-f(x2) where x1 and x2 are close, and then dx=x1-x2. f'(x)=df/dx is only true as the difference goes to zero and for finite differences you only have that f'(x) is approximated by df/dx

In your case you are interested in f'(x) for x=p_{n-1} and you have that p_{n-2} is a point close to p_{n-1}. Hence, just plug in your values to approximate the derivative f'(p_{n-1}), that is df=f(p_{n-1})-f(p_{n-2}) and dx=p_{n-1}-p_{n-2}

[tribal pulsar]

That makes sense, thank you!

[sleek sundial]

A question for seasoned data scientists. Let’s say we have a dataset and we suspect it has a lower-dimensional underlying structure. Obviously, we’d like to work with fewer dimensions. But all dimensionality reduction algorithms I know of presuppose the structure of the manifold: PCA thinks of the data as an ellipsoid; Isomap only works if you happened to correctly guess the topology you embed the data into, considering the usual guess is a flat plane, it isn’t very useful most of the time; t-SNE thinks of the data as a collection of globular clusters, and tries to fit anything into that framework; and so on. I’m curious, is there an algorithm that can learn the underlying manifold directly? As an atlas? Or, even better, the algorithm to extract not only the topology of the data space, but also its curvature in every point?

[wide spear]

UMAP?

https://arxiv.org/pdf/1802.03426.pdf

[sleek sundial]

Isn’t it just a slightly better version of t-SNE?

[wide spear]

Based on the paper, it seems like they aren't assuming anything about the underlying manifold

[civic ledge]

anyone knows about the redlich-kwong equation of state
i wanted to know if that eqn could be solved using numerical methods?

[fathom rain]

From Wikipedia: ```
This equation only implicitly gives Z as a function of pressure and temperature, but is easily solved numerically, originally by graphical interpolation, and now more easily by computer. Moreover, analytic solutions to cubic functions have been known for centuries and are even faster for computers.

[civic ledge]

ok can this be used for doing mini projects?

this equation

for finding something

[fathom rain]

No idea, I have never heard of this equation.

[civic ledge]

oohk

could u just go through it

and lmk if possible

[fathom rain]

ask your teacher

[toxic wagon]

hello! i have asked a question in #help-14 regarding finite difference approximation. im not sure how to tackle this problem.

[wide spear]

Can you repost your question here and close the question channel

[toxic wagon]

Hello! I currently have a problem with using finite difference approximation for solving the eigenvalue problem of a periodic boundary condition. Right now I only have the second-order forward and backward approximations for f'(x) and central approximation for f''(x). From there, I don't know where to proceed. As far as I know I should set up a banded symmetric matrix A?

The eigenvalue problem is as follows:
y''(x) = -lambda y
y(0) = y(1)
y'(0) = y'(1)

here it is @wide spear

from what I know about FDA, i'm supposed to have some equation that looks like
$$
\begin{bmatrix} 2 & 1 & 0 & 0 \ 1 & 2 & 1 & 0 \ 0 & 1 & 2 & 1 \ 0 & 0 & 1 & 2 \end{bmatrix}
\begin{bmatrix} y_1 \ y_2 \ y_3 \ y_4 \end{bmatrix} = -\lambda
\begin{bmatrix} y_1 \ y_2 \ y_3 \ y_4 \end{bmatrix}
$$

[wide spear]

Do you want to fix that matrix

[toxic wagon]

$$
\begin{bmatrix} 2 & 1 & 0 & 0 \ 1 & 2 & 1 & 0 \ 0 & 1 & 2 & 1 \ 0 & 0 & 1 & 2 \end{bmatrix}
\begin{bmatrix} y_1 \ y_2 \ y_3 \ y_4 \end{bmatrix} = -\lambda
\begin{bmatrix} y_1 \ y_2 \ y_3 \ y_4 \end{bmatrix}
$$

[pine jetty]

gristCollector

[toxic wagon]

there we go lol

thats what i believe the matrices look like for dirichlet boundaries

but im not sure how this is going to look like in periodic boundaries

[wide spear]

With periodic boundary conditions you'll get a 1 in the top right and bottom left corners of the matrix

The [1 2 1] wraps around, in a sense

Also don't some of those need to be negative

Shouldn't it be 1 -2 1

[toxic wagon]

oh right, because $y(0) = y(1)$, which means $y_1 = y_4$

[pine jetty]

gristCollector

[toxic wagon]

oh right, should it?

ah yeah it is whoops

it is supposed to be -2

but is that the only change to this matrix?

[wide spear]

Yes, that's the only change

[toxic wagon]

ohhhh

ok that makes a lot of sense actually

and now i just use the same methods as i use in dirichlet to calculate for the eigenvalues?

[wide spear]

Sure

[toxic wagon]

i see, thank you very much!

[fathom rain]

and eigenvalues of the circulant are f(t_j)=-2+2cos(t_j) for t_j=(j-1)2*pi/n j=1,...,n

[toxic wagon]

may i ask why this is the case? our lectures just passed over this tbh and i didnt understand how

[fathom rain]

Because circulant matrices form a matrix algebra that can be diagonalized by, for example, the Fourier matrix, and in that case the eigenvalues are given by sampling the symbol (function f here, which generates the matrices (the elements of the matrix are given by the Fourier coefficients of f)), by for example that grid t_j. So, all eigenvalues of all circulant matrices are given in the same way.

[toxic wagon]

i see

im still trying to understand the last bit about fourier coefficients but i think i get the gist of it

[fathom rain]

If you have Dirichle BC instead the grid is just t_j=j*pi/(n+1) instead (same symbol f)

f(t)=-2+2cos(t)=exp(-i*t)-2+exp(i*t) then Fourielr coefficients are f_{-1}=1 f_0=-2 and f_1=1 and that is the elements on the upper diagonal, main diagonal, and lower diagonal of the matrix (and in the circulant case for periodic BC you just add 1 oin top right, bottom left corners)

Dirichlet -> $T_n(f)$ and Periodic -> $C_n(f)$ and Neumann -> $T_n(f)+R_n$ where $T_n$ is a generated Toeplitz matrix $C_n$ is a circulant and $R_n$ is a low-rank matrix (only 1 in top left and bottom right corners)

[pine jetty]

Sven-Erik

[fathom rain]

(they all share the same symbol, eigenvalues are just given by different grids)

[wide spear]

-¿

[toxic wagon]

ok i think i understand it

[fathom rain]

In engineering the matrices that diagonalizes these matrices (when combining different BC) are usually denoted dct-* and dst-* (discrete cosine and discrete sine transformes where *=1,2,3...)

https://en.wikipedia.org/wiki/Discrete_sine_transform

[tall solar]

Anyone here know about proximal operators

[brave crypt]

any help with how to get started on this stochiastic calc problem

[wide spear]

#advanced-probability

[brave crypt]

i'll post it there

[obsidian oracle]

Hello everybody.
anyone has the demonstration or a paper that discusses the rate of convergence of stochastic gradient descent when the gradient is L-Lipschitz continues? I can't find it in any paper

[prime kraken]

https://www.stat.cmu.edu/~ryantibs/convexopt-F18/lectures/stochastic-gd.pdf

[obsidian oracle]

Thank you. it just say SGD does not improve when we further assume f has Lipschitz gradient. So I conclude that there is no proof yet under this condition.

[prime kraken]

there are several papers on related topics though, just look it up on google scholar

there's other stuff here https://www.lpsm.paris/pageperso/picard/gradsto_pointdevue.pdf

[obsidian oracle]

Thank you. This powerpoint contains a reference which has a demonstration under this condition.

[prime kraken]

👍

[neon snow]

Hey! I want to model a spiral transfer with a low thrust engine from an orbit of 2000 km to 40 000 km. Any advice on how to model this? I'm thinking express the equation of motions in cylindrical coordinates:
[
\frac{d^{2}r}{dt^2}-r\left(\frac{d\theta}{dt}\right)^2 = -G\frac{M}{r^2}
]
for the radial part, and for the tangential part:
[
r\frac{d^{2}\theta}{dt^2}+2\frac{dr}{dt}\frac{d\theta}{dt}=\frac{F}{m}
]
where (F) is the force of the engine then. I suppose I could just numerically solve this in MATLAB? How can I rewrite this system in a form that I can feed to a numerical method, like \texttt{ode45} in MATLAB?

[pine jetty]

Victor H

[wide spear]

To feed this into a pre-built solver, rewrite this as a system of 1st order equations

[neon snow]

Yeah, I'm struggling a bit with how haha, I totally forgot lol sorry.

How would I start off? If I remember correctly you usually express the second derivative by introducing an extra variable, and use that as an intermediate variable... I think

[wide spear]

Write $x=\frac{dr}{dt}$ and $y=\frac{d\theta}{dt}$ so your differential equations become [\frac{dx}{dt}-ry^2=-G\frac{m}{r^2}] and [r\frac{dy}{dt}+2xy=\frac{F}{m}] and [y=\frac{d\theta}{dt}] and [x=\frac{dr}{dt}] so you have [\frac{d}{dt}\begin{bmatrix}\theta\r\x\y\end{bmatrix}=\begin{bmatrix}y\x\ry^2-\frac{Gm}{r^2}\\frac1r\qty(\frac{F}{m}-2xy)\end{bmatrix}]

[pine jetty]

陆景和

[neon snow]

You're such a fking chad

Thanks mate, I really appreciate it

I'll try to fix this, and inevitably fail and probably ask again. But damn bro, thanks a whole bunch

[radiant vortex]

I did it like this. Can someone verify if its okay?

[orchid sequoia]

it seems ok ig

[wide spear]

#odes-and-pdes

[marble kestrel]

if anybody here knows anything about ML or something similar and is willing to help, id really appreciate it. pls ping me if you do.

so i have the following question i need help with. im pretty sure the weights matrix has dimension 2 (its a square 2 by 2 matrix) and the biases are also two dimensional.

im pretty confused on part (b) though

[wide spear]

Chain rule

[marble kestrel]

no, no, like. i dont know how all of these things are supposed to come together. i can do chain rule just fine, i just dont even know what to apply it to

to get from the input layer to the middle layer

[pine jetty]

c squared

c squared

c squared

[marble kestrel]

is this on the right track?

[wide spear]

Sure

[marble kestrel]

so then what? just calculate partial derivatives?

[wide spear]

Yeah

[marble kestrel]

sick. so happy i was doing that right. thanks

[wise yoke]

wait so like

optimisation of the svm hard margin thing

they say set b = -1 and seem to negate the t and x

i thought it would be optimised by setting to b = 1 as the constraint no?

confused why they selected the A matrix as each instance of -t^ix^i and not just postive

[fresh oxide]

guys, how would i go about finding eigenvector of a matrix with eigenvalue 1?
in code?

[mint hemlock]

The power method is sufficient assuming that 1 is a maximal eigenvalue, or the inverse power method if it’s a minimal eigenvalue.
But if this not the case, that’s okay, there are surely other iterative methods that are slipping my mind rn.
If you don’t care about time complexity, finding nonzero x that solves (A-I)x=0 would work

[fresh oxide]

i know

but power method wont get me as many marks 😂 precisely cuz of the time complexity

ill use both tho

thnx

im basically trying to find steady state vector for a markov chain

[mint hemlock]

Yeah, doing (A-I)x=0 is only funky because A-I is singular

hm yeah, do you have an expected time complexity?

[fresh oxide]

idk tbh 😂

[mint hemlock]

So given a method that has a region of absolute stability D (think about some methods’ stability regions here such as Euler’s), as you allow h->0, in order for your method to converge, does h\lambda_k need to be in this region D. In other words, can there be an eigenvalue with h\lambda_k outside of D without the method diverging (hint: this criterion is a part of what makes a method conditionally stable, think again about Euler’s method)

[fresh oxide]

just the power method will be considered comparitively inelegant to (A-I)x = 0 i guess

wait, u are saying power method is faster than finding eigenvector using (A-I)x = 0?

[mint hemlock]

Well it depends on if 1 is a maximal eigenvalue

[fresh oxide]

https://colab.research.google.com/drive/1IVTP88qBbXzb3rKERwcMHssLnhfBjadS#scrollTo=85498567

i have already implmented power method here i guess

[mint hemlock]

doing (A-I)x=0 would require Gaussian Elimination of A-I I would think

[fresh oxide]

yep

thats O(n^3)

[mint hemlock]

mhm

[fresh oxide]

power method for a large matrix is..

[mint hemlock]

Isn’t that just doing Ax for multiple x? so that’s n^2

then some number of iterations

[fresh oxide]

complexity of dot product is O(n^3)

right?

tho i guess if its a vector with matrix it could be O(n^2), not sure

[mint hemlock]

Nonetheless neither method is fast

Is there anything known about A?

or any property it possesses

[fresh oxide]

yeah its o(n^2)

with vector

and convergence is linear

[mint hemlock]

indeed

[fresh oxide]

idk what to make of that tbh

thats why these numerical methods are kind of tricky

how fast would a large matrix converge?

eh, i might just do the power method cuz im probably gonna run out of time

[mint hemlock]

yeah rate of convergence is |\lambda_1/\lambda_2|, so you would need to know how large 1 is in comparison to the next largest eigenvalue

[fresh oxide]

its a markov chain

so.. i think

wait lemme just use eig

these are the eignvals in this case

this is just a dummy version tho

actual matrix is something like this

[mint hemlock]

ah yeah, so power method should converge quite nicely as long as you have similar ratios to that

[fresh oxide]

each row adds up to one