numerical-analysis

[wide spear]

Technically when working on numerical methods everything you do is discretized and you are once again in fin dim vec space land

But a lot of the intution from functional analysis is very valuable

[solid igloo]

I mean, should I pay close attention to the different topologies that happen in rudin's functional analysis? Or should I focus on normed spaces or Banach spaces, to assume almost all theorems hold, without having to remember so many topologies.
Each of his theorems have different hypotheses that make it much difficult to track. Maybe I just have to study harder and it will be worth it? I heard there is even functional analysis on graphs that would be cool

[sinful idol]

Sobolev spaces

[pine jetty]

Evil Bean

[wary scarab]

on second thought does it relate to convergence of the fixed point?

[delicate falcon]

yo

[heady mesa]

The one on the left probably won't converge.

Whereas the one on the right does.

$ln(1)=0$ so $\frac{1}{\ln(x)}$ is going to have a vertical asymptote in this domain.

Why is this important?
At an asymptote the derivative is going to be massive and is going to get bigger.
(I'd suggest actually differentiating and observing the derivative of the function on the right in this domain).

Whereas the expression on the right has a nice derivative and is a smooth function on the domain.

I'd suggest looking up the 1D Fixed point theorem, one of the conditions is that near the root you need: the function to have an absolute derivative less than 1.

[pine jetty]

Max

[idle cosmos]

Hello, I have a question about the rayleigh quotient itération. Whats the theorical convergence of the method if x0 (the initial vector we choose) and v (the eigenvector to wich we converge) if |x0-v| > 1 ?

[wide spear]

Have you looked at the convergence proof in Demmel

[kind copper]

Nah but I can ask him he’s my professor rn 😭

[wide spear]

Oh nice

Is this 221

[kind copper]

Yeah lmao how you know

[wide spear]

I too took 221 with jim

[kind copper]

LMAO

[wide spear]

Anyways you should read the proof in the book before asking him

[unborn cedar]

i had this qn i had to answr
Let ℝd be equipped with the Euclidean norm, let D⊂ℝd be a closed set, let g: D→D be a contraction with Lipschitz constant L=1/3, let x0∈D and assume that D⊂B2(x0). By the contraction mapping principle, there exists a fixed point x*∈D of g, and the sequence (xk)k generated by the iteration

x_{k+1}=g(x_k)\quad\text{for}\ k\in\mathbb{N}

converges to x*.

What is the smallest number N∈ℕ such that for any dimension d≥1, and for any D, g and x0 as characterised above, we have

x_k\in B_{10^{-8}}(x^*)\quad\forall,k\ge N?

Hint: Consider the concrete example D=[0,2], g(x)=x/3 and x_0=2 and find the index N_1 in that particular case; this shows that, in general, the smallest possible N working for all cases is larger N_1.

To then find an N that works for all cases of D,g as in the statement of the question, you have to write down a short mathematical argument. You manage to make N=N_1, you have identified N.

the answer is N=18

ok nice the latex didnt convert

this is the written explanation for why n=18 but tbh i dont really understand lol...like the whole shabang. if anyone could help me out id be forever grateful 🙏

[edgy zinc]

What does "$O(1)$ numbers" mean?

My understanding is that if $f=O(g)$, then there exists $C>0$ s.t. $\qty| \frac{f(x)}{g(x)} | \leq C$

So if you say a constant $A$ is $O(1)$, you're saying that $|A/1|=|A|\leq C$, which doesn't seem to be particularly meaningful, as it is true of every number

[pine jetty]

Douglas

[wide spear]

Order of magnitude

Technically an abuse of notation but quite common

[edgy zinc]

Meaning what exactly?

[wide spear]

A and B are close to 1

Closer to 1 than to 10 or 0.1

[edgy zinc]

okie dokie

[wide spear]

Which part don’t you understand?

[edgy zinc]

related question... given that A=O(1) means A is close to 1, I presume it means that A=[integral up to some point in the interval]/[integral over entire interval], rather than just the numerator?

cos if the total integral is 4π eg and A is just the integral across some fraction of the interval, then A might be 11 or something, in which case A is certainly closer to 10 than 1

[edgy zinc]

this is in the context of the trapezium rule btw

[wide spear]

Scaling an integrand by a constant doesn't change any of the theory

[edgy zinc]

what's the significance of specifying O(1) then, as opposed to O(10), or any other?

[molten knot]

f=O(1) means f<=C*1 for some c>0

O(•) implicitly means some constant times the general order of convergence

so we pull out all constant terms into that C

[wide spear]

Machine epsilon is the distance between 1 and the next floating point number

[edgy zinc]

right but what does eps have to do with O(1)?

[wide spear]

eps changes when you get further away from 1

[barren brook]

does anyone know how we can compute an error bound for Romberg integration?

[wide spear]

Start with the error bound for the trapezoidal rule, then see how the error behaves under Richardson extrapolation

[barren brook]

Thank you. I will try that

[edgy zinc]

How are we getting 5.27?

[wide spear]

What is S_i defined as

[edgy zinc]

It's defined as in 5.21

[wide spear]

Ok

Taylor expand f(b) and f((a+b)/2) around f(a)

[brave crypt]

When doing quadrature by decomposing domain into piecewise polynomial approximations(lagrange), is there anything we can say about the error depending where you are at in the interval?

To restate:
For lagrange polynomial interpolation the error increases at the bounds. What can we say when we do a piecewise polynomial interpolation? What about in case of quadrature?

[wide spear]

Do you mean that the error is large at the endpoints of the interval?

[brave crypt]

Yea not bounds

[wide spear]

This is dependent a lot on grid spacing

[brave crypt]

Well assuming equidistant

[wide spear]

For the interpolation points used for the polynomial approximation

Ok well equidistant points are bad

[brave crypt]

Yes but it doesnt matter for quadrature

[wide spear]

For quadrature, you can do better with non equidistant points

At any rate, this is the error for polynomial interpolation

For interpolation points x_i

The product on the right hand side will vary in your interval, and for equidistant points, is minimized at x=x_i and x at the center of the interval

[brave crypt]

Yea I get this but I guess my question was more about quadrature because it was covered in class recently

We discussed newton cotes formulas

And we also discussed a possible way to minimize errors was by breaking the integration interval into subintervals and applying polynomial interpolations there

[wide spear]

Yes newton cotes is unstable for high degree unless you are very careful

Ok sure

[brave crypt]

Yea we talked about that a little. I don’t know the extended story though.

I guess my question was more like can we say anything about where error is highest or lowest when doing this subinterval polynomial interpolation

As it relates to quadrature tho

[wide spear]

Well, precisely because equidistant point based polynomial interpolation is useless at high degree

[brave crypt]

Wym by careful then?

Like weird interpolating points?

[wide spear]

You don't want your interpolation weights to come from a polynomial interpolation

You want to do a least squares method to get the weights instead

[hasty ferry]

How can one show: A matrix ( A \in M_n(\mathbb{R}) ) is invertible if and only if there exists a permutation matrix ( P \in \prod_n ) such that all principal minors ( (P \cdot A)_m ) of ( P \cdot A ) are invertible for ( m = 1, \dots, n ).

[pine jetty]

Sentinel

[wide spear]

Hmmm #linear-algebra

[hasty ferry]

its actaully from a numerical analysis lecture

thats why i posted it here

but it may not fit here u are righjt

[wide spear]

I understand but there is a linear algebra fact

Yes

[hasty ferry]

would you like to help 🙂

[wide spear]

I regret to inform you that I do not want to think about permutation matrices at the moment

[hasty ferry]

Thanks though i ve posted it in linear algebra with my intended solution

which doesnt work fully

[brave crypt]

I dont know either but my guess is something like invertible principle minors things relates to gaussian elimination way of finding determinants

And you want to show thats non zero

Id try that atleast

[wicked pecan]

Yeah isn’t that cus of evenly spaced points?

Just staying I read none of the messages above or below that message I just saw it and instantly replied

[wide spear]

[orchid lagoon]

I have a question about subspace iteration. If we run this algorithm on symmetrical matrices to calculate the 10 largest eigenvalues. We simulate matrices with size ranging from 10 to n. I simulated my algorithm and for each bigger matrix n, I saved the amount of iterations until convergence.

The relation between amount of runs & size of matrix was linear. But I fail to understand why...

[wide spear]

By subspace iteration, do you mean arnoldi iteration?

[orchid lagoon]

I'm not sure what arnoldi iteration is. I use this algorithm:

Iteration matrix Q, QR decomposition in second step

I really appreciate your reply

[wide spear]

Oh

QR iteration

[raven panther]

Hey
im trying to read more about butcher tablaues, implicit RK and diagonally implicit RK methods, where do i look at?
my ultimate goal is implementing a parallel DIRK solve

[wide spear]

What do you mean

Like resources?

For learning?

[raven panther]

Yes

im pretty much vanilla in numerical analysis

just learned about runge-kutta a couple of days back

[wide spear]

This should be good

I'm not entirely sure what you mean by a parallel DIRK solve

Like do you have a pde that you want to use DIRK to time step for

Or are you trying to do a parallel in time thing

[raven panther]

yk normal parallel RK?

parallel in time yes

you can set it up for implicit and diagonally implicit RK as well

[wide spear]

[orchid lagoon]

If I have a sparse matrix and I assign it as sparse(B) in matlab. Then when doing gram schmidt on vectors, do I have to do this step? vector1 = sparse(vector1 ./ norm(vector1));

Mister angetenar, does this statement ring any bells?

Hmmm, operations in gram schmidt might make elements nearly zero instead of zero, which would make a vector not sparse anymore

hmmmmmmmmm

[wide spear]

Gram Schmidt is generally not preferred for a numerical algorithm

Are you doing a QR factorization?

[orchid lagoon]

Yessir, and Gram Schmidt is obligatory

[wide spear]

What do you mean obligatory

[orchid lagoon]

EUh if I don't do it I won't pass

[wide spear]

Ok

[orchid lagoon]

Is it correct that that algorithm can cause 0 elements to become non zero?

[wide spear]

Gram schmidt will not preserve sparsity

https://en.wikipedia.org/wiki/Gram–Schmidt_process#Numerical_stability

Check the modified gram-schmidt presented here

[orchid lagoon]

So my matrix becomes non sparse, but would zero elements become non zero?

ok I will check it out, thanks angetenar!

[edgy zinc]

Why are the nodes indexed by n,i rather than i only?

[wide spear]

Just notation

I wouldn't think too much about it

[edgy zinc]

So when he refers to "two groups of parameters", they are the nodes and the approximation weights, not n and i?

[wide spear]

The author may be trying to emphasize that the x_n,i will vary with different numbers of quadrature points

[wide spear]

Paramters isn't the best term, I would say degrees of freedom

But yes, you can vary the qudrature nodes and the quadrature weights at each node

[edgy zinc]

right ok

[fathom rain]

I would say it is more common to use x_{i,n} but it just means x_{n,i}=h*i+a=(b-a)i/n+a, i=0...n

(or some other grid)

[edgy zinc]

ye i know what the nodes represent

just wondering wot was up with the notation

[fathom rain]

for a specific grid type, the node points are usually defined by n and the point indices i, not more complicated

and if you use multiple grid types you need one or more extra indices to define which

[edgy zinc]

I am not seeing the thing that is supposed to be easy to see (5.48)

[wide spear]

Have you done the integration by parts k+1 times

[edgy zinc]

This is as far as I got before thinking I had gone wrong somewhere

there should be alternating signs in the margin of the D/I table

[wide spear]

You can ignore the n!/(2n)!, that won't affect the integral

[edgy zinc]

yeah i included the n! (and not the (2n)!) so as to avoid a 1/facotrial thing

[wide spear]

Hmmmm

[edgy zinc]

it just makes the table look a little bit tidier i think

[wide spear]

Yeah sure

Ok let's see how I would do this

First, recognize that if n is even, then phi_n only contains even degree terms

Similarly if n is odd, then phi_n contains only odd degree terms

So if n is even and k is odd or vice versa, the integral will be 0 right

Integral of an odd function on a symmetric interval

[edgy zinc]

hmmmmm

this does make sense

that is a nice argumenr

[wide spear]

Now, we consider when both n and k are even

Hmmmmm

Ok really what you should do is show that the integral of phi_n(x) from -1 to 1 is 0

There are much better presentations of Legendre polynomials that actually discuss their properties

(Namely being an orthogonal basis)

(In which case 5.48 follows immediately)

[edgy zinc]

how exactly does this argument work in this context?

we have an inner product $(f,g)=\int_{-1} ^1 f(x) g(x) \dd{x}$

then what?

[pine jetty]

Douglas

[edgy zinc]

(phi_i, phi_j)=0, i≠j, is what we're saying?

[wide spear]

Yes

[edgy zinc]

and how does that get you (x^k, phi_n)=0

[wide spear]

If k<n, then you can write x^k as a sum of phi_i for i<=k and split the integral up

And each piece of the integral will be 0

[edgy zinc]

hmmmm yes

you are right this is a much nicer argument than doing the integral by tabulation

ty for the help

[wide spear]

<@&268886789983436800>

[unkempt moth]

gone

[bold marsh]

Anyone here knows about b-spline?

[grave spoke]

What about it?

[bold marsh]

I just want to understand it, for example what are the knots and control points in a spline?

Also how do I define a B-spline?

[grave spoke]

Sorry, I'm a bit preoccupied rn, but I can try to answer later if no one does

[bold marsh]

np, tysm

[alpine mirage]

so basically you first choose some arbitrary knots, which are points on the x-axis

this choice defines a set of functions constructed with a recursive relation, where each function (B-spline) is nonzero only on some limited interval

and then you choose control points, which are something like coefficients in linear combination of vectors

but in multiple dimensions

so you take some average of these B-splines (basis splines) weighted by control points

or if you're familiar with the notion of partition of unity, it's a very similar idea with basis splines as functions

[bold marsh]

What is this term? I looked it up and it says blending functions that are used to blend the control points

[alpine mirage]

yea, maybe it's better to view it as a weighted average of control points

B-splines look something like this

if you enforce them to sum up to 1, you get something like a "smoothly changing weight" for the weighted average of control points

[bold marsh]

A partition of unity

I see

[alpine mirage]

yea exactly

[bold marsh]

Another question, how do I calculate the points on the b-spline curve?

I read that there's an Algorithm called DeBoor-Cox

But how to apply that Algorithm?

I found the Cox-de boor formula

[bold marsh]

tysm, for explaining b-spline

[alpine mirage]

sorry i had a call
yeah you apply the Cox-de boor formula which is just computing the values recursively from the splines of lower degree

[bold marsh]

I forgot to mention that I'm making a presentation about b-spline

[azure socket]

I have an amazing video for you to watch then

https://www.youtube.com/watch?v=jvPPXbo87ds&ab_channel=FreyaHolmér

you caaan skip straight to the B-Splines chapter (also what I did originally) but you will likely end up watching the whole video. especially if youre presenting on b-splines, like I had to

shes really good at telling the story and the visuals just keep you entranced

[coarse root]

Just realised this might fit better in here, oh well

[wide spear]

Care to copy over what is going on

[coarse root]

I'm using a modified Newton's method for solving a 2x2 non-linear system. I've substituted the Jacobian's elements with the following expressions:
[
\begin{cases}
\frac{\partial f_1}{\partial K_1}\approx\frac{f_1(K_1+s,,K_2)-f_1(K_1,,K_2)}{s}\
\frac{\partial f_1}{\partial K_2}\approx\frac{f_1(K_1,,K_2+s)-f_1(K_1,,K_2)}{s}\
\frac{\partial f_2}{\partial K_1}\approx\frac{f_2(K_1+s,,K_2)-f_2(K_1,,K_2)}{s}\
\frac{\partial f_2}{\partial K_2}\approx\frac{f_2(K_1,,K_2+s)-f_2(K_1,,K_2)}{s}
\end{cases}
]

[wide spear]

What system are you solving

[pine jetty]

Eruc

[wide spear]

Also what are you plotting

[coarse root]

The truncation error (Y) plotted against different values for the small supplement s in the expressions (X).

The truncation error is the total of the entire problem (it's a combination of different methods feeding into each other) but the only thing being varied here is the value of s

[wide spear]

Is this after a single iteration step

Or after it's converged?

[coarse root]

After 6 iterations, sufficiently converged for the problem

But so it seems there's a certain value of s that minimises the truncation error, and then it grows, and then starts approaching the same minimum asymptotically. What's going on with that? Where can I look more into this?

[wide spear]

Hmmmm

It's hard to say what is going on without knowing more about what f is, but this kind of reminds me of what happens when you vary the parameter in successive over relaxation

[coarse root]

To be more specific, the system being solved is to optimise two constants of a second order non-linear ODE so that the solution to the ODE fullfils two separate criteria

So the entire problem involves errors from ODE-solving (RK4), polynomial interpolation, Newton-Rhapson, and then this error dependent on s

[wide spear]

Are the two lines two different error measures

[coarse root]

They're the errors of the two different constants being optimised

[wide spear]

What does f look like

[coarse root]

I don't really have a expression for them, but f1 gives the distance between two points on the ODE curve that have two specific properties. The constants are being optimised so that f1 = 0 (i.e. the same point is supposed to have both properties)

And f2 is something similar but less complicated

[wide spear]

Do you have any method of directly computing the error in the derivative calculation of f

[violet gull]

Hey guys, what's the link between the fourier transform and the fourier series? If there is one at all. I kinda what they each are on their own, not sure how they relate though. Thanks 🙂

[wide spear]

Fourier series are the Fourier transform on a periodic interval

Ask further questions in #advanced-analysis

[midnight plover]

im using gauss seidel iteration (+ overrelaxation) to remove the divergence from a velocity field when simulation the incompressible uniform euler fluid equations, but its slow for high resolutions because i need a bunch of iterations. when iteratively reducing the divergence im subtracting it from the velocity field at every step.

i want to use a multigrid method but im confused on whether im supposed to be projecting and restricting the actual velocities of each cell (thus seemingly destroying local information when interpolating back onto the fine grid) or if im supposed to keep a separate error array, and after solving for it at each level interpolate it onto the fine grid by subtracting it from the actual velocity array (thus keeping local velocity differences but subtracting an interpolated divergence term)

[wide spear]

Have you considered a numerical method that won't introduce divergence at all

[midnight plover]

like what

[wide spear]

Compatible finite elements or a Lagrangian approach a la Chorin's vortex method

[sinful idol]

Projection methods avoid having to deal with divergence too

[edgy zinc]

wouldn't it make more sense to use the notation $P_N (n)$?

N is a parameter that we control, whereas n is entirely random

[pine jetty]

Douglas

[wide spear]

I wouldn’t think too much about it

Sometimes there is nothing deep about notation

[edgy zinc]

maybe

but it feels weird

like you wouldnt diffeentiate wrt an index

i mean you could

but it would be a little weird

[wide spear]

Lol yes

<@&268886789983436800>

[edgy zinc]

im confused

why are we pinging mods

[wide spear]

Ratking behavior is not channel appropriate

[edgy zinc]

oh

[wide spear]

Rafiking

At any rate, there is far worse notation that you will come across I imagine

[edgy zinc]

Why is $\sigma$ not equal to $p_1 p_2/N$?

This is what you get if you compare coefficients of 5.67 and 5.68

[pine jetty]

Douglas

[wide spear]

#advanced-probability

[edgy zinc]

ye sorry these are from my numerical methods lecture notes

so just defaulted to this channel

but u r correct

prob makes more sense here

[kind copper]

Does anyone know why for sparse matrices randomized truncated SVD (like Halko et al.) performs empirically poorly something like Arnoldi iteration (for example ARPACK)? Specifically for PCA

[brave crypt]

In my numerical analysis course. We get some theory questions too. Like what are the draw backs of newton ralphson theory. The topics we have are.

• Error calculation
• taylor theorem
• Newton raphson method
• secant method
• bisection method
• interpolation (linear quadratic)
• newton dividend diff interpolation
• lagrangian
• trapeziod
• Simpsons 1/3 rule

How do i study the theories. I never studied theory for maths before and our teacher does not tell us

Can ppl here tell me?

[normal mauve]

Anyone has good recommendations for introductory books on numerical optimisation?

[vapid plume]

I would say just read books, there are many good books on numerical methods/analysis. I recommend James F. Epperson's book, it is very good at the introductory level. You can always search on the web for other recommendations or ask in the books channel.

[brave crypt]

okay thank you very much ❤️

[sinful idol]

A good classic is "An Introduction to Numerical Analysis" by Atkinson.

[sinful idol]

Nocedal should be the one of the most commonly used.

[brave crypt]

can i send link here?

we are following a book at uni and that book is uploaded in internet by the ppl i think

Numerical Methods with Applications by Authors: Autar K Kaw | Co-Author: Egwu E Kalu, Duc Nguyen

[sinful idol]

You can also check out convex optimization books, usually they include a significant amount of numerical algorithms inside.

[brave crypt]

https://nm.mathforcollege.com/textbook-numerical-methods-with-applications/

chapter 9 is optimization

[sinful idol]

Sounds like a very applied book.

[brave crypt]

ikr but we get question in exam like:

1. in which of the following function does root jumping occure while useing the newton raphson methods? analyse graphically.
   a. y = x^2
   b. y = sinx
   c. y = cosx

this is maybe explanation of the draw back of newton.

idk

[sinful idol]

Um... this is an applied question.

[brave crypt]

but idk what this is

its not like thenormal maths

i saw

[sinful idol]

The maths i know is proof based maths

[wide spear]

What is wrong with that question

[sinful idol]

Nothing wrong, but its something where you can just draw and guess.

No proof needed.

But not a theoretical question (in my opinion).

[minor osprey]

It seems fine as introductory if you aren’t really coming from a math background. Seems like the book is geared towards engineering majors

[real hull]

hello, does anyone have any references for error analysis/estimation for various spline interpolation methods? (cubic spline, spline with tension, thin plate spline), or does anyone know any methods and can explain?

[wide spear]

Taylor series

[sinful idol]

Most approximation theory textbooks should cover rigorous error estimates for splines.

The techniques used depend on the function spaces. For instance, taylor series would work for C^k functions but probably not so helpful for H^k functions.

[woeful schooner]

Hello everyone i need a reference for the research im doing on weddle’s method so if anyone could help me i would appreciate it alot

[fathom rain]

Seems to be an obscure Newton-Cotes method, never heard of it 🙂

so I assume compare it with with the standard ones?

[fathom rain]

https://scholar.google.com/scholar?q=thomas+weddle&hl=sv&as_sdt=0%2C5&as_ylo=1800&as_yhi=1900

[marble socket]

I want to choose $n+1$ pairwise distinct points $x_0, \dots, x_n \in [0,1]$ to extremize the value of the integral
$$L = \int_0^1 |w(x)| , dx, \qquad w(x) = \prod_{k=0}^n (x - x_k).$$
I know that $L$ is minimized when we take the $x_k$ to be the zeros of the Chebyshev polynomial of the second kind $U_{n+1}(x)$. But what choice of the $x_k$ maximizes $L$?

[pine jetty]

Eduardo León

[marble socket]

Errr, never mind. I just realized I can solve this using calculus.

Bruh.

[wide spear]

Lol

[teal panther]

LR = A where L and R are up and down triangle matrizes, prove that $\frac{|| \left|L\right|\left|R\right| ||\infty}{||A||\infty} \leq \kappa_\infty(L)$

[pine jetty]

Jemaseda

[teal panther]

can someone help me with this? i have no idea what im doing

[wide spear]

What have you tried

[teal panther]

Well not much pure algebra playing and the < things to approximate or smtn

[wide spear]

Well you can prove it for diagonal matrices at least right

[molten knot]

any suggestions on how to handle (d)? I tried the hint but I keep getting cross terms that I dont know to handle

the derived scheme with $f=0$ is
$$\frac{1}{2}(W_2^{m+1}+W_2^{m},\chi)=\frac{1}{\tau}(W_1^{m+1}-W_1^{m},\chi)$$
$$a(\frac{1}{2}(W_{1}^{m+1}+W_{1}^{m}),\chi)=\frac{1}{\tau}(W_2^{m+1}-W_2^{m},\chi).$$

[pine jetty]

ansh

[molten knot]

W^(m+1/2)=(W^(m+1)+W^(m))/2 btw

[wide spear]

Well what do you get after multiplying the system

[molten knot]

so if i test eq1 with $W_1^{m+1/2}$ and eq2 with $\triangle_h W_2^{m+1/2}$ the new system is
$$(W_1^{m+1/2},W_2^{m+1/2})=\frac{1}{2\tau}(|W_1^{m+1}|^2-|W_1^{m}|^2)$$
$$(\triangle W_1^{m+1/2}, \triangle W_2^{m+1/2})=-\frac{1}{2\tau}(|\nabla W_2^{m+1}|^2-|\nabla W_2^{m}|^2)$$

[pine jetty]

ansh

[molten knot]

(with some rearranging and integrating by parts)

but how do you show stability from here?

oh theres a typo it should be grad(W2) instead of grad(W1) in the second equation RHS

[alpine mirage]

is there some standard method of computing step in multivariate newton method up to quadratic term in taylor expansion

$$ 0 = f(x) + Df(x) h + \frac 1 2 D^2 f(x) (h, h) $$

how to get h is my question

[pine jetty]

Bwoonoid, a crank

[wide spear]

Newton's method for root finding?

I've literally never seen anyone include the quadratic term before lol

[alpine mirage]

the exercise probably aims to show that including it is not needed

theoretically it could make sense however, but idk how to calculate the step

[edgy zinc]

Where is A+B=1 coming from? Is that imposed just to make A and B "nice" numbers?

[wide spear]

If you are integrating a constant function, you need A+B=1 for the quadrature to be correct

This is a common thing that you'll see in numerical quadrature

[edgy zinc]

what's quadrature?

[wide spear]

Numerical integration

[edgy zinc]

uh

but you're not integrating a constant function here

you're integrating f

[wide spear]

You are integrating an arbitrary function f yes

[edgy zinc]

so how does

If you are integrating a constant function, you need A+B=1 for the quadrature to be correct
help us with integrating arbitrary f?

[wide spear]

It's a basic check on accuracy for the easiest functions to integrate

If you cannot numerically integrate a constant, then you have no hope of doing anything more sophisticated

[tall cypress]

Hi guys, can I get some help? Suppose $$A = \begin{bmatrix} A_{11} & A_{21}^T \ A_{21} & A_{22}\end{bmatrix}$$ is symmetric PD. I want to prove that $|A_{21}A_{11}^{-1}|2 \leq \kappa{2}(A)^{1/2}$, where $\kappa_2$ is the condition number under the operator 2-norm. The hint is considering $A$'s Cholesky factorization, but I can't seem to figure it out...

[pine jetty]

{Wilhelm}

[wide spear]

Well, what is the cholesky factorization of A

[tall cypress]

We don't assume anything about its factorization, only since it is PSD it can be done. The hint suggests using the block Cholesky factorization $$A = \begin{bmatrix}R_{11}^T & 0 \ R_{12}^T & R_{22}^T \end{bmatrix} \begin{bmatrix}R_{11} & R_{12} \ 0 & R_{22}\end{bmatrix}$$

[pine jetty]

{Wilhelm}

[tall cypress]

Where $A_{11} = R_{11}^TR_{11}$ is its Cholesky factorization itself

[pine jetty]

{Wilhelm}

[tall cypress]

I managed to bound it by $\kappa_2(A)$, but not its square root

[pine jetty]

{Wilhelm}

[tall cypress]

I really appreciate any insights :)

[wide spear]

What did you manage to bound

2-norm or 2-norm squared

[tall cypress]

2 norm

This is part (c) of an exercise, and this is what I have proved so far, first this factorization: $$\begin{bmatrix} I & 0 \ -A_{21} A_{11}^{-1} & I \end{bmatrix} \begin{bmatrix}
A_{11} & A_{21}^\tp \ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} I & -
A_{11}^{-1}A_{21}^\tp \ 0 & I \end{bmatrix} = \begin{bmatrix} A_{11} & 0 \ 0 & S
\end{bmatrix}.$$

[pine jetty]

{Wilhelm}
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tall cypress]

$\lVert A_{ij} \rVert_2 \le \lVert A \rVert_2, \quad i,j \in {1,2}$

[pine jetty]

{Wilhelm}

[tall cypress]

and $\kappa_2(S) \le \kappa_2(A)$

[pine jetty]

{Wilhelm}

[tall cypress]

Interestingly, the Schur complement $S$ simplifies to $R_{22}^TR_{22}$, and of course it should be enough to show that $|A_{21}A_{11}^{-1}|_2^2 \leq \kappa_2(S)$, which might be where the exercise is leading to

[pine jetty]

{Wilhelm}

[tall cypress]

Oh, I think the root part comes in by the fact that $\kappa_2(A) = \kappa_2(R^TR) = \kappa_2^2(R)$

[pine jetty]

{Wilhelm}

[tall cypress]

OHHHHH SOLVED IT :)))))

[tall cypress]

you use cholesky to show $A_{21}A_{11}^{-1} = R_{12}^TR_{11}^{-T}$, then the whole norm is less than $|R_{12}|2|R{11}^{-1}|_2$, and then you use the bound on block matrix norms above

[pine jetty]

{Wilhelm}

[tall cypress]

Ok, if anyone can give me a hint on this following problem that would be great:

Suppose that you have a solution to the following least squares problem,
$$
\min \left| \begin{bmatrix} A \ \textbf{c}^T \end{bmatrix} \textbf{x} - \begin{bmatrix} \textbf{b} \ d \end{bmatrix} \right|_2
$$

You can make no assumptions on the method that was used to solve it. How can you leverage the solution above to efficiently solve $\min |A\textbf{x} - \textbf{b}|_2$? The method should probably include the use of the Sherman-Morrison formula https://en.wikipedia.org/wiki/Sherman–Morrison_formula. Thanks!

[pine jetty]

{Wilhelm}

[violet nymph]

yo i think i desparately need some help
on numerical analysis
https://ustspace.s3-ap-southeast-1.amazonaws.com/upload/lbTzVGpf53mxcwyg286fCz60MufPq36f/FinalExam2019.pdf
I need some help with Q2, Q4 and Q5b

[wide spear]

To clarify, this isn't your exam right

You are practicing from a past exam?

[fathom rain]

we wait 120mins and answer 😄

[violet nymph]

yes

correct

this is a pastpaper from 2019

My progress so far with Q2

wait no the linear system is solved wrong

well to be honest now i am not too sure if my previous step is even correct
i somehow got a = (h-1)/(h+1) c

pls help

[violet nymph]

well that works too but this is a pastpaper lol

the year is written

[violet nymph]

Pls help with Q4 and Q5(b)

[violet nymph]

My current progress with Q4

But somehow the error term become zero

Idk what’s wrong for now, pls help

i will be back in around 3 hours, i need a quick nap

[barren star]

I've recently started a project and I am trying to implement numerical methods using python.
https://github.com/TheKidKy/numerical_analysis

[heady mesa]

Nice, although I would suggest making sure you maximise the use of libraries if possible as they will use code, most likely written in C which is WAY faster than Python.

Python is generally considered slow, if you like the syntax but want more speed without losing the Syntax I'd recommend learning Julia

[barren star]

should I switch to C?

[heady mesa]

Personally I would switch to Julia, but that has its own problems (slow compile times) but after that it is as fast as C but has a python feel

C is probably the fastest all around, ie. in compile time and run time

But python is the most convenient to code and there are lots of libraries

For instance, say you decide to use Python that would be fine just make sure to do as much code as possible in libraries such as Pandas, Numpy ext. which are written in C

Also Python has some really nice numerical analysis libraries such as Mpmath which I'm not sure the others have

[violet nymph]

hello guys i am back again

https://ustspace.s3-ap-southeast-1.amazonaws.com/upload/lbTzVGpf53mxcwyg286fCz60MufPq36f/FinalExam2019.pdf

anyone have any idea on how to do Q5b?
FYI THIS IS NOT AN ONGOING EXAM, THIS IS A PASTPAPER FOR PRACTICE

[violet nymph]

to confirm this is not an ongoing exam, i am trying to do it since yesterday so....

[wide spear]

What have you tried

[violet nymph]

well fun fact, i just found the proof in a textbook

[wide spear]

Lol nice

[violet nymph]

yah

it was the step 1 and 2 of proofing error bound theorem

cool

[molten knot]

taylor series expansion y around a and compute the difference

[neon remnant]

hi

[wide spear]

Hello

[lofty widget]

suggest any playlist for this topic man ..i have exams coming in 3months

[heady mesa]

Look at your notes and watch videos by topic rather than one set course...

this is the stage of maths where you need to be learning how to learn from notes and text as opposed to videos... as rarely does specialized content have video tutorials

[flat plank]

This is how i decided to solve this.

As i'm supposed to pick 3 random ODE methods and either secant/bisection method for task b i have 6 possible solutions.

import numpy as np
from solvers import solver

# RHS of ODE
def rhs(t, y, m, k, g):
    v = y[1]
    dydt = v
    dvdt = -k/m * v**2 - g/m
    return np.array([dydt, dvdt])

def nonlinear_function(H, m, k, g, T, target_position, method):
    initial_conditions = np.array([0.0, H])
    
    # Wrapper
    def rhs_with_params(t, y):
        return rhs(t, y, m, k, g)

    t0 = 0.0
    dt = 0.01
    t, y = solver(rhs_with_params, initial_conditions, t0, dt, T, method)
    
    # Final position at t = T
    final_position = y[-1][0]
    
    # Return the difference from the target position
    return final_position - target_position

# Parameters
m = 1.3
g = 9.81
k = 0.05
T = 1.5
target_position = 10.0

# Differential methods
methods = ["Heun", "Ralston", "Runge-Kutta"]

# Results storage
results = []

for method in methods:
    print(f"\nUsing Differential Solver: {method}")

    def F(H):
        return nonlinear_function(H, m, k, g, T, target_position, method)
    
    # Bisection
    root_bisection = bisection(F, 0, 20)
    print(f"Bisection root: {root_bisection}")
    results.append((method, "Bisection", root_bisection))

    # Secant
    root_secant = secant(F, 0, 20)
    print(f"Secant root: {root_secant}")
    results.append((method, "Secant", root_secant))

print("\nSummary of Results:")
for method, solver, root in results:
    print(f"{method} with {solver}: Root = {root}")```

bisection() and secant() come from a worksheet we used in a lab and they're known to work

[wide spear]

I don’t know what your question is

But

Coding questions go in #computing-software

[heady mesa]

Fairs but it's numerical analysis

If you think there is a simple mistake it could be worth asking an AI chat bot, they good at picking up on simple mistakes

[wide spear]

Well they also didn’t ask a question

[heady mesa]

If you are just asking if it's right then look at the accuracy of your answer, are they accurate? If not how can you make it more accurate

[wide spear]

At least not that I can see

[heady mesa]

True 😭😂

[heady mesa]

/ what could you change to make your answer more accurate, if it doesn't make it more accurate then it's likely you've made a mistake

If you can't spot the mistake after a bit of investigation then just redo the whole thing again lmao honestly has helped me spot so many mistakes in the past

[wide spear]

!da2a

[copper abyss]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[heady mesa]

Such a mathematician thing to say lmao

[idle light]

Dear all, I having trouble with the discontinuous Garlerkin method (DG). My question is the following , by applying the DG method could I deduce a upwind/downwind discretization scheme for my PDE (a PDE to the convection diffusion type) ?

[final sleet]

Is it true that accuracy in floating point numbers cannot be accurately calculated for precise precision and accuracy: https://en.wikipedia.org/wiki/Floating-point_arithmetic

[still forum]

Yes, you can't rely on floating point numbers to be exactly accurate, try doing $0.1+0.2$ in python or something. Even if you write doing 0.3 on a computer, its really being stored as a slightly different number close to 0.3

[pine jetty]

jamiecjx

[still forum]

Floating point numbers generally only satisfy the standard arithmetic operations up to a known error, so for example, given floating point numbers $x,y \in \bb{F}$, we have $x \oplus y = (x+y)(1+\delta)$ where $|\delta| < \frac{\eps_M}{2}$ and a $\eps_M$ is a constant

[pine jetty]

jamiecjx
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[still forum]

said constant is usually about 2 * 10^{-15} in double precision, so whilst you can't calculate precisely with floating point, you can rigorously bound computations (with things like interval arithmetic)

[versed bison]

also a related topic https://en.wikipedia.org/wiki/Machine_epsilon

[wide spear]

I mean not really "related"

This is just what it is

[dusky pivot]

Can someone please suggest me a good book/resource to study Runge Kutta methods, very funadmentally and in great detail? Would be great if it also mentions any suggestions on programming the different models of RK, like RK 7, 8 or higher in Python.
But, primarily I want to understand the method, the derivation of the formula, and it's application in real life problems.

[signal fiber]

Apparently the golden ratio pops out of nowhere when doing natural cubic spline interpolation! Basically knots (nodes) at phi^2k makes the interpolation neither converge nor diverge. Is this well-known?

[sterile holly]

I'm also open to using another library for this task.
I solved my first numerical eigenvalue problems yesterday lol.

Can someone please suggest me a good

[sterile holly]

I was helped :)

[edgy zinc]

please would someone explain the highlighted bit?

Also...

1. "The case mu=1 is called "1-sublinear convergence". Don't you identify the nature of the convergence by the order q (q-convergence), so wouldn't it make more sense to say "the case q=1"?
2. Why does he talk about mu=0 when the inequality mu satisfies prohibits that case? And what does it mean to lie between q=1 and q>1?

[vagrant jackal]

if the approximation is correct to two decimal places then the errror (x_k - x*) is on the order of 10^(-3) meaning -log(x_k - x*) is somewhere between 2 and 3. similarly if they agree at 5 decimal places the expression would be 5--6

[edgy zinc]

ok

[edgy zinc]

what about this

[vagrant jackal]

if you say q = 1 then people would assume you are referring to the linear case unless otherwise specified. both special cases start with "1-" to refer to the order q

[obtuse vault]

Maybe you should look into why that is

That’s the beauty of math after all

[edgy zinc]

so is it the case that if q=1 then 0≤mu≤1 (i.e. closed interval, not open)?

possibly one for probability, but what does it mean to choose the nodes?

the nodes are the data points, i.e. a random sample from a population, so how is there any choice in their spacing?

[wide spear]

There are a few variants of the interpolation problem

One is scattered data interpolation where you have no choice over where the data is located

But in other cases, you might have a function that is very expensive to compute so you fit a polynomial approximation to evaluate it more efficiently

In which case you do have a choice over the interpolation nodes

[ionic veldt]

Maybe someone here can help me with a question about numerical integration.

Specifically, the remainder theorem for the "midpoint method". I highlighted it in red.

I can't find a proof of this theorem anywhere that does not assume the second derivative of the function is continuous. The theorem's formulation seems to state only that the second derivative must exist, i.e., continuity is not guaranteed.

I have found only these proofs so far:

https://math.stackexchange.com/a/4327333/861268
https://www.macmillanlearning.com/studentresources/highschool/mathematics/rogawskiapet2e/additional_proofs/error_bounds_proof_for_numerical_integration.pdf

But they both assume continuity of second derivative.

I'm exhausted from searching. It feels like this is some completely unimportant detail.

P.S.: Maybe I should've posted it in calculus chat because I've found this theorem in calculus book? Here: https://openstax.org/books/calculus-volume-2/pages/3-6-numerical-integration

[heady mesa]

Essentially you are approximating the curve using a straight line

Or in other words the first order Taylor series of the function.

Then integrating the straight line.

The error for a first order Taylor series is dependent on the second derivative.

I'll let you put it together more formally but that's the gist.

[grave spoke]

It says "If M is the maximum value of |f''(x)| over [a, b], ..."

That's an additional condition that replaces the requirement of continuity of the second derivative

The maximum value might not exist in general if f'' is not continuous on [a, b]

[ionic veldt]

That's what's done in this proof https://math.stackexchange.com/a/4327333/861268 At first, he integrates the Taylor's approximation of the function for one interval and then summing it for n intervals. But I've no idea how this can be done without assuming that f''(xi) (which is used in remainder for Taylor's approximation) is continuous on interval.

[ionic veldt]

So, am I right in understanding, based on what has been said, that the theorem states that whether the second derivative is continuous or discontinuous, we should have an M value for this formula to make sense?

[grave spoke]

Yeah, the requirement that the second derivative is continuous is usually only required in the theorems to ensure that the second derivative is bounded

[ionic veldt]

And how, then, can it be proven? I mean, I’ve kept the proofs above in my main post, so is it correct to assume in the proof that it's continuous? The reason why I'm so curious about that is: what if in some practical case/problem I'll come across a case where f'' is discontinuous?

[grave spoke]

Well, check in which parts of the proof do you need continuity of the second derivative

The intermediate value property still holds for derivatives even though they are not continuous, for example

https://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)

[ionic veldt]

So this part of the proof (1st one, where he uses mean value theorem for integrals) can be performed by using Darboux's theorem without assuming the continuity, right?

[grave spoke]

Instead of using the mean value theorem (which is possible, but might require monotonicity of f''), you can just bound f'' by M

[edgy zinc]

He refers to 4.8 as the absolute error. How are we justifying the absorption of O(e^2) into the other error terms?

Do we know that e/h is bigger than e^2, so we can ignore O(e^2) altogether?

[vagrant jackal]

epsilon and h are both small quantities (much less than 1), so ε² is guaranteed to be much smaller than ε and ε/h will be much larger than ε (even if h were larger than 1 it would not be much smaller, certainly not as small as ε²)

[edgy zinc]

but how do you know that e is not approx equal to h, in which case e/h≈1?

[vagrant jackal]

ε and h are both much less than 1 so that would be a much larger error

[edgy zinc]

right ok i was thinking about things the complete wrong way around

yes you only care about the largest term, and (small)^2 is smaller than (small)/(small)

[edgy zinc]

where are the highlighted terms coming from?

like i understand that there will be some error

but how do you know that those terms give the correct error

[vagrant jackal]

taylor expand f centered at x

[ionic veldt]

But this will do it for one interval as I understand it. Will not the fact that the value of M might be different for different intervals to be obstacle to sum this error n times (for n different intervals I mean)?

[grave spoke]

M is the maximum value of |f''(x)| over [a, b]

[edgy zinc]

It should say "order" where it says "rate".

Is smoothness actually a necessary condition? If you had non-vanishing first derivative and f was only twice differentiable, wouldn't you still have quadratic convergence?

[grave spoke]

Continuous differentiability and a Lipschitz condition will be sufficient

[edgy zinc]

How do we get (9)?

I was using a slightly different formula for Simpson's, i.e. with $n-1/2$ and $n-1$ in the exponents, so I got the following.

$$\frac{h}{6} \frac{6-3h+h^2 -\cdots}{h+h^2/2 + \cdots} = \frac{1-h/2+h^2/6 +\cdots}{1+h/2+\cdots}$$

Using $(1+y)^{-1} \approx 1-y$, we find that the above expression is equal to $$ (1-h/2+\cdots) (1-h/2+h^2/6+\cdots)$$

But this gives $I_S = (e-1) (1+O(h))$ I think, and that isn't right.

Where have I gone wrong?

[pine jetty]

Douglas

[wide spear]

Should just be a taylor series?

[edgy zinc]

Which bit are you referring to? I've used Taylor series to get 1+O(h)

[wide spear]

When it says "expanding in h"

Wait huh

What is your analogous expression to 8

[edgy zinc]

Same but with $$1+4e^{-h/2}+e^{-h}$$

[pine jetty]

Douglas

[edgy zinc]

Just a sign change

[wide spear]

Are you sure nothing else changes

You should have different bounds on the sum as well

[pine jetty]

Douglas

Douglas

[edgy zinc]

@wide spear ^

actually i think i can see the mistake now

i've evaluated those sums incorrectly bc of the bounds being different

i will fix when i return to my desk

[edgy zinc]

turns out you still get (8) once you fix the sums

all good now

[wide spear]

Nice

[edgy zinc]

Not entirely "all good" as it turns out...

I've got a Taylor series with a quadratic term

The coefficient on the cubic term seems to be incorrect in my working, but I cba to find the arithmetical mistake

[wide spear]

You need to include a few more terms in both the numerator and denominator I think

[edgy zinc]

how do you know how many terms to include

[wide spear]

Well you want something to be O(h^4)

So you should at the very least include to h^5

[edgy zinc]

yeah but if i didnt have the worked solution available then i wouldnt know to go to h^5

hmmmm

ok so what wolfram says agrees with the answer once you multiply by h/6

[edgy zinc]

i think this is the main problem im having

i assume from the worked solution that the fact that the dots start after quadratic means thats how many terms he's going up to

[clever wave]

Could someone please give a (very ELIUndergrad) reason for why we care about (and how to account for) numerical stability in algorithms? I'm trying to write a basic matrix library for fun and every source I look at warns about naive algorithms leading to instability and I'm unsure of how to interpret any of this.

For context, we finished a first course in linear algebra at uni a few weeks ago and today I got the idea to write a basic toy library to handle basic matrix maths,but got bit scared by mentions of stability and such

[wide spear]

Do you know what numerical stability means

[clever wave]

not really

AFAIK it has to do with small deviations in input data leading to large/undesirable changes in the output

But that's about all I know

[wide spear]

Are you familiar with floating point arithmetic

[clever wave]

Yes

[wide spear]

Ok so

Let's take an algorithm that isn't numerically stable

The computation of the jordan decomposition

Consider the matrix $\begin{bmatrix}1&1\x&1\end{bmatrix}$

[pine jetty]

Angetenar

[clever wave]

okay

[wide spear]

If $x=0$, then the jordan form is the matrix, or $\begin{bmatrix}1&1\0&1\end{bmatrix}$

[pine jetty]

Angetenar

[wide spear]

If $x\neq0$, then the jordan form is $\begin{bmatrix}1+\sqrt{x}&0\0&1-\sqrt{x}\end{bmatrix}$

[pine jetty]

Angetenar

[wide spear]

So any perturbation to x can dramatically change the output

[clever wave]

ahhhh yeah I see

[wide spear]

This interacts poorly with floating point arithmetic

[clever wave]

Okay so if we get any floating point inaccuracies then our resulting matrix could be completely wrong

[wide spear]

Something like 1/3 can't be represented exactly

So floating point errors build up

[clever wave]

yeah

[wide spear]

And then you can get things that are very wrong

If you are using things that are not numerically stable

[clever wave]

that makes sense, so if JNF is unstable like this due to the way it works overall, is it just avoided in general? or are there tricks to still compute it

[wide spear]

JNF is numerically unsalvageable

The Schur decomposition is the preferred numerical alternative

[clever wave]

ahh I see, that makes sense

Thank you :)

[manic charm]

can u do this with another representation that is not floating point, something like fixed point?

does the problem persist in that representation?

[wide spear]

I mean the jordan form is not stable

Regardless of what arithmetic you use

Doing something like fixed point eliminates one source of perturbations

But doing something like 9*(1/9) in fixed point still wouldn't equal 1 would it

[manic charm]

it would not

[wide spear]

So that doesn't really help

[edgy zinc]

In the fourth picture, shouldn't it say "quadratic convergence order", as $q=2$ whereas $\mu=f''(x^)/2f'(x^)$

[pine jetty]

Douglas

[edgy zinc]

mu is the rate and depends on f'' and f'

but q is always 2 in the non-degenerate case

oh actually

hmmm

"q-convergence rate"

that seems like a weird way of phrasing things if q is the order

[dull coral]

Newton’s method is a method for root finding. Modify Newton’s method in one
dimension as a method for finding minimum/minimizer of a function. Write down
such an algorithm and using the convergence theorem proved in class, state the
differentiability requirements on the function whose minimum/minimizer is being
sought.
someone please explain me this question.

[wide spear]

What don't you understand

[heady mesa]

Where do minimums/maximums occur @dull coral

Hence what equation are you solving?

Input this into NR

What are the requirements for NR to converge?

Hence what are the requirements for this function that we are minimizing / maximizing

[final sleet]

How do you explain the nature of unary and binary from a numerical analysis standpoint

[heady mesa]

It all boils down to how those numbers are stored and operated on within a computer. Lots of work already done on binary because that is how computers stored floats anyways

[torpid pike]

If I'm answering a problem and the problem boils down to "find m where m is the largest value s.t. m! < 0.9999*10^15" is stating "17! < max, 18! > max" sufficient?

I realized that i interpreted the question wrong

it's actually "find m where m is the largest value s.t. for 0 < k < m, k!(m-k)! < 0.9999*10^15"

nvm m! > k!(m-k!) for all k < m?

someone kill me

[raven dew]

For 0 < k < m yes (binomial coefficients), though that doesn't seem too relevant to what you previously said

[torpid pike]

Nah I figured it out

[raven dew]

Nice

[azure yew]

hello, i would like to have some help

[pine jetty]

Result:

2

[azure yew]

Assuming delta t = epsilon, and using milstein formula, the mark scheme is wrong

i just dont get it

why is there epsilon next to the 2nd term

[wide spear]

Do you mean delta

[azure yew]

sorry, delta yes

well, i got 1/4 ..... and then (....) - delta

how can delta go above the fraction

can it?

[wide spear]

Is this the provided solution

[azure yew]

yes

[wide spear]

What is your solution

[azure yew]

also if u look at deltaW_k , you can write it as W_k+1 - Wk correct?

[heady mesa]

<@&268886789983436800>

[unkempt moth]

thanks

[rose meteor]

So recently I was looking into ways to approximate different numerical series. Like series that apply to natural numbers. And I derived this formula for approximating the factorial function (basically the gamma function):
where
b = floor(x)
d = x - b

and this function does converge to x! for all real numbers as k grows to infinity. What i dont understand though is that if i replace b = floor(x) with c = ceiling(x) the function still converges to x! except much much quicker. And this is strange to me because I designed this formula with b = floor(x) without even thinking about ceiling(x) until much later. This also doesnt make sense to me because d is almost always negative. Anyways heres the graph of both and if anybody could help me that would be great.

Also heres the Desmos graph:

[late dirge]

https://www.youtube.com/watch?v=sct27JMTDKw

[wide spear]

Ummmmmm

[wide spear]

What is the role of k?

[heady mesa]

Isn't d just the fractional part? But when we change to c we are finding d=the negative fractional part

so perhaps the speed of convergence is due to the convexity, does have you tested a mix of values and their range of convergence?

[heady mesa]

(ie. when using c we are approaching the limit from below but when we are using b we are approaching the limit from above, which we would expect to have different rates of convergence)

[rose meteor]

As k approaches infinity the equation converges to x! for all real numbers

[rose meteor]

Yeah i have tried testing a lot of things https://www.desmos.com/calculator/setly6ocqm

And i think i understand why they look different but i dont understand why ceil is much more exact

i may be able to get floor to look like that tho if i tweak it enough

Also this method does converge extremely quickly whith k values of 12 having an error of 0.001

[cunning moat]

is this the correct channel for a question about a problem from Finite Elements Method?

[cunning moat]

or more exactly the theory behind the problem

[heady mesa]

Yes

[heady mesa]

How are you calculating this error? As x get's larger we would also expect the error between $\Gamma(x)$ and your approximation to get bigger?

[pine jetty]

Max

[heady mesa]

Whenever studying something it's great to find your own way, but perhaps do some research on what other methods people have done and compare. It may be that actually this is quite slow. (Like newton raphson method converges quadratically so in 12 iterations we get a root that is roughly 4096 times as accurate as the first root...)

Also for error, doing some sort of integral might be better? Ie. $$\int_a^b\lvert \Gamma(x)-ApproxOfGamma(x)\rvert dx$$, where $a,b$ is the interval of interest.

[pine jetty]

Max

[cunning moat]

ok so, when we were solving the Poisson equation generally using 2D FEM with linear polynomials, we derived an element-wise computation formula for the local stiffness matrix components that doesnt really depend on the shape function since it uses the transformation to the standart triangle
Similarly when we were on the theory of L2-projection for 2D FEM using linear polynomials a formula was given to us for the local mass matrix components something of the sort $$\frac{|\tau|}{12}*
\begin{matrix}
2 & 1 & 1 \
1 & 2 & 1 \
1 & 1 & 2
\end{matrix}$$
Both of which are firmulas for triangulation mesh
(Im not familiar how the discord bot for TeXiT works so idk if that will render and im a newbie at using LaTeX as a whole)
My question is, why would those formulas not work for solving the wave equation (where its requried to compute both the mass and stiffness matrix) when both of them are independent on the shape functions but only on the type of the mesh (to be triangulation) and on the triangle you are in currently?

[pine jetty]

HelODeh

[rose meteor]

Ok wait this is really smart thanks for the help

however

one very strange thing i have noticed

[rose meteor]

Sorry it took a little bit to test this

but ive noticed that for large values being apporximated

like say the approximation of 216.513!

taking the approximation of like 1.513!

and then scaling it up ith 2.513 * 3.513 * 4.513... and so on until 216.513

there is actually less error in doing it that way then just straight up taking the approximation of 216.513!

which is really starange to me because i dont understand how that could happen unless the error for larger numbers was not proportional to theyre size

[wide spear]

Do you have a question

[wary oriole]

yes wait a min

    n = a.shape[0]

    for k in range(0, n - 1):
        if a[k, k] == 0:  # stop if pivot is zero
            raise LinAlgError("Singular matrix")
        for j in range(k + 1, n):
            for i in range(k + 1, n):
                m_ik = a[i, k] / a[k, k]  # compute multipliers for current column
                # a[i, k] = m_ik
                a[i, j] = a[i, j] - m_ik * a[k, j]  # apply transformation to remaining submatrix```

so this is my implementation of lu decomp without pivoting. somehow it's not correct. i based my on above algo

this is step by step

my result is

 [ 4 -4 -6]
 [ 4 -2 -1]]```
which is the supposed U. but I can't do L at the same time

so this algo according to bhte book, you are not suppose to store L, or U, instead change U in place in A as upper triangluar including diagonal, L as lower traangluar portion of A, but not including diagonal as it's unit diagonal. so my result is correct in U, but that A[3, 2] should be 0.5 according the hand-calculation.

 [ 4 -4 -6]
 [ 4 0.5 -1]]

this is intended.

how to correct it?

[wary oriole]

oh I got it. np.array has dtype. if you do

  a[i, k] = m_ik```
on a integer matrix, with m_ik = 0.5, numpy will make it 0. Just declare dtype=float

[gray tundra]

here is the iterations for newton raphson method. They are computing relative error in each iterations and i was wondering why is it that eps = (x(k+1)-x(k))/x(k)

my problem with this formula is that i think that the denominator should be the closest value to the real value which is x(k+1)

[outer jacinth]

can any help me learn plu and lu compostion

anyone *

[wide spear]

[outer jacinth]

im like wondering how to compute both because my teacher isnt very clear

[wide spear]

Here is an outline

[outer jacinth]

coding it for me inst the issue i think its more connceputally

[wide spear]

Well, do you understand gaussian elimination?

[outer jacinth]

yes kind of essientially isnt the point of it getting it to row echelon form without any swaps or zeros

[wide spear]

Do you understand how gaussian elimination helps us solve systems of linear equations

[granite aurora]

@wide spear you familiar with taylor series and how they help integrate differential equations?

[wide spear]

If you have a question just ask

If I know I might answer

[granite aurora]

i have a problem that states the following:
derive and explain the method of Taylor series for integrating a differential equation
y' = f(x,y) , y(x0) = y0
Then asks what is the main issue of using this method to integrate such equations

[wide spear]

This is more appropriate for #odes-and-pdes

[granite aurora]

it involves ODE but its a numerical method that approximates the integral thats why i thought to ask here first but thanks ill try there as well

[outer jacinth]

i mean not entirely

all ik is u get a answer but im really bad at thinking of things abetrially with math 18

linear algerbra *

[wide spear]

Ok you should solve a few linear systems with gaussian elimination

By hand

[outer jacinth]

i mean ik how you just row reduce the matrix

right isnt that about it

[outer jacinth]

is there any good textbooks or videos i can watch to learn this stuff

[wide spear]

Any intro linear algebra textbook should cover this

Friedberg Insel Spence is what I used

[outer jacinth]

okay thanks

what edition

[wide spear]

I don't remember

[vast nexus]

Hi, how would you solve problems like this?

[wide spear]

Numerically?

[vast nexus]

Yes

[outer jacinth]

is there any good youtube videos on this stuff

im debating if its even possible to get a 60 % in this class

[vast nexus]

I didn't take good notes in the first couple lectures and I am worried because I can't find resources on our problems 😭

[fathom rain]

plot and visually pick an interval

[vast nexus]

my professor did not use a graphing calculator in class while solving this but i don't remember his approach

do you know a numerical approach?

[fathom rain]

test a few values and see where there sign of the functions changes. you can use newton's method to find the roots

[vagrant jackal]

my guess is that it intends you to evaluate at a bunch of points and note where the sign changes, then go "by IVT there is a solution here"

[fathom rain]

like on the first one, you have different signs for x=0 and x=1

which should tell you something

[vast nexus]

that's true, so [0, 1] would be a solution?
since f(0) < 0 < f(1)

[fathom rain]

yes, like cloud said above, ivt: https://en.wikipedia.org/wiki/Intermediate_value_theorem

[fathom rain]

on b you have 3 roots = 3 sign changes = 3 intervals

[brave crypt]

Why is discrete 1-d laplacian poorly conditioned

Like its supposed to be O(N^2) but i dont see it immediately, do I have to go thru what row reduction or is there a quciker way

Where N is 1/h

[wide spear]

What is supposed to be O(N^2)?

[fathom rain]

it has a zero of order 2, since the spectral symbol is 4sin^2(t/2)

the smallest eigenvalue (dirichlet) is 4sin^2(pi/(2(n+1))) and the largest is 4sin^2(npi/(2(n+1)))

[brave crypt]

Matrix condition number

[brave crypt]

Are u supposed to get this with the greens function thing?

Im not exactly sure what a spectral symbol is sorry

[fathom rain]

spectral symbol describes the eigenvalue distribution. it will also describe haw fast the smallest eigenvalue goes to zero

[brave crypt]

Ok that sounds important

[fathom rain]

i just derive it from the matrix

w8 i can get you a reference

[brave crypt]

I would appreciate this thanks!

[fathom rain]

it is describe for example in here https://link.springer.com/book/10.1007/978-3-319-53679-8

[brave crypt]

Wow this is cool

);

I lost internet but I have mobile data

Does this book introduce spectral symbols too?

In class we are going to go over spectral methods in more depth

But that doesnt show up until page 300 of the textbook

[fathom rain]

yes and for the laplacian you have it for example in remark 10.2

[fathom rain]

it is a rather obscure concept for most people. but very powerful

[brave crypt]

Yeah it sounds obscure but it seems like it that would be a lot of information

Its just linear algebra though?

[fathom rain]

yes

[brave crypt]

Like spectral symbol is probably only relevant for those high dimensional operators

Yeah my book doesnt have anything about spectral symbol

[fathom rain]

it comes from operator theory, i use it for highly efficient and accurate eigenvalue computations

[brave crypt]

Imma see if trefethens book has it

[fathom rain]

probably not

[brave crypt]

Ok cool

[fathom rain]

if you look at any book about pseudospectra or structured matrices you will find it

[brave crypt]

Yeah I dont know if I am ready to formally learn operator theory as I havent taken a class on functional analysis yet

[wary oriole]

anyone doing FEM has experience with NGSolve? If I want to create a 3by3 squares (like tic tac toe) geometries in 2D do I have to create nice Rectangles? I feel like there is a easier way to do this?

[crisp lake]

I'm having a brainfart or something & can't figure out how/why Clenshaw summation works.

[wide spear]

The quadrature technique?

[crisp lake]

No, the efficient method for evaluating finite sums of functions satisfying 3-term recurrences.

[wide spear]

Ah ok

Well

What don't you understand

[crisp lake]

It’s described in this paper if you can grasp it & somehow explain it to me: https://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071856-0/S0025-5718-1955-0071856-0.pdf

[wide spear]

Is there a specific part that you don't understand

[crisp lake]

The proof of why the method is correct. I guess I don't follow the penultimate line «Then application of the recurrence relation (6) shows that…» and the final equation.

[wide spear]

Ok I would first try to understand why this works for just chebyshev polynomials as shown in equations 1-3

In particular, pick some A_n and compute the sum directly and using the reversed recurrence relation by hand

[molten knot]

the way I do it is set the endpoints and draw lines between them

and then set an index inside and out to denote the boundary

there might be an easier way tho

[crisp lake]

I just tried that & must have miscalculated or something because the results didn't line up.

[wide spear]

Well, let's see the calculation

[crisp lake]

$2\cdot T_0(x)+3\cdot T_1(x)+5\cdot T_2(x)+7\cdot T_3(x)$ was what I just tried.

[pine jetty]

Nadia

[crisp lake]

I actually did it on Microsoft whiteboard on my iPad Mini, so in principle, there should be some way to capture it.

[wide spear]

Screenshot?

Latex would be nicer to look at but alas

[crisp lake]

Here:

[wide spear]

Don't use the backwards recurrence relation in (6) for now

Use (2)

[crisp lake]

I'll give it another shot.

[crisp lake]

I got it wrong again.

[wide spear]

I'm just going to do this with 3 terms

So

$f(x)=A_0+A_1T_1(x)+A_2T_2(x)$

[pine jetty]

Angetenar

[wide spear]

The reverse recurrence gives $B_4=B_3=0$, $B_2=A_2$, $B_1=A_1+(4x-2)B_2=A_1+(4x-2)A_2$, and $B_0=A_0-B_2+(4x-2)B_1=A_0-A_2+(4x-2)A_1+(4x-2)^2A_2$ right?

[pine jetty]

Angetenar

[wide spear]

So in theory, $f(x)=B_0-(2x-1)B_1=A_0-A_2+(4x-2)A_1+(4x-2)^2A_2-(2x-1)A_1-(2x-1)(4x-2)A_2$

[pine jetty]

Angetenar

[wide spear]

And using $2x-1=\cos\theta$ and grouping like terms, this becomes $A_0+A_1\cos\theta+A_2(2\cos^2\theta-1)$

[pine jetty]

Angetenar

[wide spear]

Which is correct

[crisp lake]

I'm getting wrong answers repeating this in Maxima.

[wide spear]

Do you understand the computation I did at least

[crisp lake]

Yes, something is going seriously awry in my hand calculations.

[wide spear]

Hopefully you figure that out

Now as to the intuition as to why this works

As we perform the reverse recursion down to n=0, we use the recurrence relation that defines the T_n right

So like if we look at the A2 part

We start with A2, then (4x-2)A2, then (4x-2)^2A2-A2

Which recursively builds up T2

[crisp lake]

Still unsure why it's not working.

[wide spear]

[crisp lake]

There is the maxima file.

[wide spear]

I don't know maxima

[crisp lake]

What's done there is trivial enough that it shouldn't be an issue.

[wide spear]

I think the only issue might be the expansions of FA vs FB

[crisp lake]

I'm not sure what you mean.

[crisp lake]

The Chebyshev polynomials are shifted i.e. $T^*_n(x)=T_n(2x-1)$.

[pine jetty]

Nadia

[crisp lake]

The updated maxima file:

[wide spear]

Is it correct?

[crisp lake]

The two match now. The problem was that the example in Clenshaw's paper used the recurrence coefficients for $T^*_n(x)=T_n(2x-1)$ and I was using the recurrence coefficients for $T_n(x)$. When I used a consistent set of recurrence coefficients for a consistent orthogonal polynomial basis it all worked.

[pine jetty]

Nadia

[wide spear]

Ok nice

Did the example I worked by hand help to provide some intuition about why clenshaw works?

[crisp lake]

I should probably do a similar exercise instead of using a concrete orthogonal polynomial basis.

[wide spear]

Sure

But doing some calculations by hand should provide some insight into why it works

[crisp lake]

I think there should be an orthogonal polynomial product formula like $\phi_{m}(x) \phi_{n}(x)=\sum_{k=0}^{mn}\alpha_{k}\phi_k(x)$ out there somewhere but am not finding generalisations of what I see for e.g. Chebyshev polynomials.

[pine jetty]

Nadia

[heady mesa]

If you want to come up with one for Chebyshev polynomials then use: $T_n(\cos \theta)=\cos(n\theta)$

[pine jetty]

Max

[heady mesa]

I think the result you are looking for follows pretty quickly when you use product of sum, ie.

[crisp lake]

I have the one for Chebyshev polynomials and am hoping for a more general version for orthogonal polynomials.

[heady mesa]

I doubt it exists

[crisp lake]

I'm willing to take restrictions on the class of orthogonal polynomials.

[heady mesa]

Things in general are really tricky, Chebyshev polynomials have some super nice identities and follow nice rules, generalizing them to even really nice and well known orthogonal polynomials such as Laguerre polynomials is usually unfruitful

[wide spear]

Yeah I'm not aware of any such product formulas even for the Legendre polynomials

[crisp lake]

I think Legendre have them.

Maybe it's Sturm-Liouville eigenfunctions, maybe it's some kind of umbral calculus. It's tougher to use the things as a basis without being able to take products.

[wide spear]

I'm not sure I agree

I very rarely need to multiply elements of a basis

Unless I'm taking an inner product of two functions in which case I don't want to explicitly multiply them either

[crisp lake]

For my purposes I do need multiplication.

[wide spear]

What are you trying to do?

[crisp lake]

These guys are representing probability densities with splines that are in plain monomial form, so the alternative strategy I'm thinking about is being able to represent the splines' polynomial pieces in a basis that facilitates taking expectations using the modelled density.

One of the tasks they do is computing a centile histogram.

[wide spear]

By splines you mean a bunch of polynomials of degree n on intervals with some continuity/differentiability condition at the endpoints right

If you're working with splines, sort of the natural basis to work with are the Bernstein polynomials

[crisp lake]

I can't actually tell how they're computing the spline to represent the probability density. I might be missing how their code is getting invoked because it's mostly the centile histogram root-finding.

[crisp lake]

Not M-splines?

[wide spear]

Well now it depends on what types of splines are being used

[crisp lake]

(Basically B-splines renormalised to integrate to 1.)

[wide spear]

Spline is very overloaded

Anyways are chebyshev polynomials not satisfactory for some reason

[crisp lake]

Maybe ignoring the orthogonal polynomials for a moment, they're trying to do Descartes' rule of signs or Vincent's and need to do $(cx+d)^{\deg p}p\left(\frac{ax+b}{cx+d}\right)$ efficiently somehow. Or at minimum a translation $p(x+a)$.

[pine jetty]

Nadia

[crisp lake]

Quantile histograms for distributions with densities given by splines, basically.

[wide spear]

Is direct evaluation too slow?

[crisp lake]

I'm not sure if it's a performance hot path or anything, but they (and I) seem to be adrift from much grounding in literature/theory/etc.

[crisp lake]

Still looking around for more quantile algorithms for distributions whose densities are represented by splines.

I'm pretty sure that their spline density functions are empirically determined but I have no idea how, so there's some weirdness about dynamically updating the quantile histograms as new samples are acquired.

[crisp lake]

Schumaker (2007) has Chebyshev splines and gives Budan-Fourier theorems for polynomials in the Chebyshev basis.

[brave crypt]

Hi is there anyone familiar with nonlinear parabolic pde?

[wide spear]

(Your question)

[crisp lake]

This can't be right. Somehow 1D Newton with bracketing interval & fallback to bisection has nowhere obvious to look it up.

[wide spear]

What do you mean look it up

[crisp lake]

When I look up Newton's method I don't see more stuff about how it's plugged into frameworks to stay within bracketing intervals, detect multiple root issues etc.

[wide spear]

I mean you can find an implementation of newton's method and see how they do it

But there's really not that much to it

[molten knot]

aren’t multiple roots just handled by sampling lots of initial conditions

[crisp lake]

I believe that Newton deals with n-fold roots by an integer factor in the denominator of the update.

[meager dome]

(I´m not sure if this questions goes here or in the cs channel!)

Hey! I have a doubt about some methods we have been using for proving some elementary bounds in a introductory numerical analysis class.

For example, we made bounds for the relative error of calculating a^3b with a, b numbers of the machine that were like 3\eps + O(\eps^2); in the proof we use constantly that x* = x(1 + \delta) (being x* the rounding of x as a number of the machine), and we use it operation by operation, i.e:

a^3b => ( ((a^2)* a)* b )* = ( ((a^2)(1 + \delta_1) a)* b )* = ( ((a^3(1 + \delta_1)(1 + \delta_2) b )* = a^3b(1 + \delta_1)(1 + \delta_2)(1 + \delta_3)
And for this we get the bound, knowing that |\delta_i| \leq \eps.

It was said in the class that one could "replace the operation in the machine with the real operation times a factor (1 + \delta)" with every normal operation except with substracting, and that this was related to the known catastrophic cancellation that happens when you substract similar numbers.

I get that with catastrophic cancellation in mind you have the fact that there is no good bound for substracting numbers in a machine, but I do not get how to precisely formalize the idea that if you want to multiply, sum or divide machine numbers you have a good bound (and you can operate like before) but with the substracting you can´t.

Any advice or resource would be helpful.

[wide spear]

Have you seen the proof of the sterbenz lemma

[meager dome]

No, thx for mentioning it. It seems useful for the doubts I have.

[minor vigil]

guys, anyone knows if there exists an efficient way to compute the full QR decomposition?

more specifically, if there exists clever way to obtain an orthogonal basis of Nu(A^T)

the reason for my question is because you can always obtain an orthogonal basis for the orthogonal complement of a single vector using house holder transformations

[wide spear]

There exist efficient QR algorithms, yes

[minor vigil]

$Rx= e_1 \Rightarrow R^2x = Re_1 \Rightarrow x = Re_1$

[pine jetty]

jeca tatu

[minor vigil]

where R is the householder reflection matrix

[wide spear]

One method is indeed based on householder reflections

[minor vigil]

and because R is orthogonal and R_1 = x, then all the other columns form a basis for the orthogonal complement of x

ok, but the full QR requires obtaining a basis for Nu(A^T) right?

I dont know if theres a clever way to do that without much effort

[wide spear]

How efficient of an algorithm do you want

[minor vigil]

something like to obtain an orthogonal basis for span{v1,...,vk} and get the complement for almost free

I dont know if thats possible

[wide spear]

Practically speaking, all QR decomposition algorithms will be O(n^3) unless your matrix is structured in some way

[minor vigil]

hmm

I just find curious how you can obtain the orth. complement of a single vector in a easy way

and I suspect that this trick could work for a collection of more vectors

[wide spear]

What does easy mean to you

[minor vigil]

I dont know, really

I just wanna know if it is at least something that can be visualized by some operations after performing householder to the set

Ill try to investigate it a little bit more

in the case of 1 vector the result is trivially by construction of the reflection of the first vector always to a multiple of e1

but for the subsequent operations you kinda look to the submatrices and things gets more complicated

[weak gyro]

hi can someone help me pls

I'm confused how is f'(0) = B and f'(2) = b

[weak gyro]

pls help <@&286206848099549185>

[thorny pulsar]

what is the best introductory book on numerical analysis (preferably at the graduate level) that is a good mix of theory and application?

[wide spear]

Are you looking for numerical linear algebra or numerical differential equations or numerical optimization or something else

[thorny pulsar]

honestly a bit of everything, nla, diff eq, integration differentiation, rootfinding, polynomial interpolation

if you have multiple book reccs I wouldn't mind

[wide spear]

There is an introductory textbook to numerical analysis, Burden and Faires, but I would say that it is at the undergraduate level

For grad level books, they are generally specialized into one of the specific areas that I mentioned

E.g. Demmel for numerical linear algebra and Iserles for numerical differential equations

[thorny pulsar]

okay thanks, I'll check them out

[next garden]

Does anyone know a good introductory text on the Diffuse element method / Diffuse Approximation ?

[fathom rain]

this seems to be the first hit and mostly cited https://link.springer.com/article/10.1007/BF00364252 any reason to use this? (probably pretty obscure method?)

[fathom rain]

sounds a bit like RBF:s?

[wide spear]

RBFs + moving least squares it seems to me

[next garden]

I am trying to find a robust way of computing the principal curvatures and principal directions of a smooth manifold defined as an SDF

So I am trying to find robust numerical ways of getting the partial derivatives

[next garden]

This is the original paper. But papers are usually much harder to understand, I was hopping for a textbook chapter or course notes. Those tend to be much more digestible.

Alternatively the paper i am currently reading suggests using lagange differentiation

[fathom rain]

no idea, never heard of this before

[next garden]

fair enough, thank you for the help still

[wide spear]

What techniques have you tried and deemed unsuitable?

[next garden]

regular finite differences (centered)

it's too unstable

too sensitive to the h parameter (the spacing)

[wide spear]

Hmmm did you try regular finite element

Or a voronoi method

Or a triangularization/spline based method

[fathom rain]

iga

[undone nebula]

I read your messages, are you trying to find real roots of a polynomial? I have an implementation on my GitHub which uses continued fractions if you need, it’s in TypeScript

It doesn’t have arbitrary precision or exact arithmetic though but you can just makes some changes if you need numerical stability

I went down a super deep rabbit hole on this too

[next garden]

I don't have a mesh so direct FEM won;t work. I am actually tring to triangluate the isosurface with a new method, so any method that relies on a mesh is redundant.

What is a voronoi mehtod? That is new to me? Can you offer an exmaple/resource?

[wide spear]

Generate a voronoi triangulation of your surface, then use that triangulation to do things

[next garden]

How do you generate a voronoi triangulation of an implicit?

[wide spear]

In what form is the SDF given to you?

Is it a function that you can evaluate wherever you want?

[next garden]

Yes, it;s a funciton pointer

so just some f(x,y,z)

that returns a scalar

[wide spear]

https://www.sciencedirect.com/science/article/pii/S001044852100169X

https://arxiv.org/abs/2407.19837

[green folio]

Bit of a long shot, but does anyone know how compute peak indexing for a sample that exhibits multiple bravais lattice structures in GSAS-II (or any other refinement software)

[wide spear]

Oh goodness

Ummm

I think that you'll have better luck in the physics server

discord.gg/physics

[versed marsh]

Let $-\pi \leq \theta \leq \pi$. The sequence $S_n = { sin(\theta)}$. This is an example of a bounded sequence but not a Cauchy Sequence?

[pine jetty]

Ashp116

[versed marsh]

The sequence $S_n = { 1/xsin(x)}$ is a Cauchy Sequence? If so, it is not monotonic right?

[pine jetty]

Ashp116

[wide spear]

#real-complex-analysis

[next garden]

Question In this document, section 2.2 talks about a technique called lagrange differentiation. I have not heard about this concept before (although it seems to come from a generalization of lagrange polynomials to higher dimensions).

I was hoping someone here would have so pointers to other resources. The mentioned paragraph doesn;t seem to have a reference to anything.

[fathom rain]

lagrange interpolation and then take derivatives of those

[next garden]

I don't think it is quite that. If you look at the coefficients. There is no parameter. It entirely yields constants.

So at minimum it has to be evaluating the parameters somehow.

It's also multiplying multiple basis together.

[wide spear]

What about the paragraph is unclear

[next garden]

This is not directly a lagrange polynomial. It is similar, but note how the numerator does not have a difference, it's just a constant. (Similar to the others)

And then look at the application of the polynomials. There is no paramater, the direct derivative of a lagrange polynomial would have a parameter. So at most this is taking the derivative of a LP and then evaluating its derivative at a fixed parameter

that's the only way I can think of where you would get a constant

[wide spear]

Huh

What parameter would it have

[next garden]

Not sure, I am tempted to say it is x=0 because then it would explain why the numberator is only the constants

however nrmally LPs start at 0 on the first knot

which is not what's happening here

that's why I am confused

[wide spear]

Are you confusing Lagrange polynomials with splines

[next garden]

no

[wide spear]

Lagrange interpolation does not have any parameters

[next garden]

?
https://en.wikipedia.org/wiki/Lagrange_polynomial

Any interpolation will have a parameter, because your input set you are interpolating over is just a point set

to get in betweens you need to blend them

for that you need a parameter

[wide spear]

No, you weight the basis according to the values at the interpolation points and add them

Which one of those is a parameter

[next garden]

The top

(x-x_0)

that x is a free variable

it's the parameter

[wide spear]

The xi are the interpolation points

[next garden]

yes

look at the numerator

[wide spear]

Ok that’s not how we use the term parameter

[next garden]

That's how it was used in the classes I took

parametric curves are called parametric curves because they take a parameter

[wide spear]

Ok whatever

[next garden]

the parameter being the free variable that allows you to interpolate

So the reason why I am confused with the text is precisely because that free variable/parameter is missing

so it should have been evaluated somehwere

[wide spear]

If I understand the text correctly, that is over a stencil with a fixed grid spacing delta x

[next garden]

but the text doesn;t say to what

[wide spear]

It gives the derivative values the grid points

[next garden]

Yes, but if you were to evaluate any of the lagrange basis polynomials at a given grid point, you would have a difference in the numerator

so that $b_{ik}$ should instead be something like $b_{ik} - b_{im}$

[pine jetty]

Makogan

[next garden]

like if oyu just grab the definition of a lagrange basis polynomial and evaluate it to a given point in space you get differences in the numerator

[wide spear]

Yes, the b_ik are differences

After 10c, for example, it says that b_i1=-delta x, bi2=0, and bi3=delta x

[next garden]

Yes, but that's only for the specific example given. I assume that you can express them as relative offsets from the pint at which you are taking hte derivative

[river thicket]

does anyone know that if i have

minimize $_\beta \vert \vert A \beta - y \vert \vert^2$

[pine jetty]

Jonathan

[river thicket]

and A is a compex matrix

if the solution is:

$\beta = (A^A)^{-1}A^ y$

where A^* is the conjugate transpose matrix of A ?

[pine jetty]

Jonathan

[wide spear]

Yes

[neat haven]

$\frac{dP(r)}{dr} = -\frac{G}{r^2}\left[\rho(r) + \frac{P(r)}{c^2}\right] \left[M(r) + 4\pi r^3 \frac{P(r)}{c^2}\right] \left[1 - \frac{2GM(r)}{r c^2}\right]^{-1},$

[pine jetty]

Farm

[neat haven]

Hello, I'm trying to solve the Tolman–Oppenheimer–Volkoff equation for modeling a neutron star using a polytropic equation of state of the form

What numerical method is recommended for integrating these equations? I've heard that Runge-Kutta methods are commonly used, but I'm not sure how to deal with the stiffness that might arise in the equations, especially near the center.

Sorry if this isn't the correct channel, but it is one of the most active.

-using a polytropic equation of state of the form
$P = K \rho^{\gamma},$

[pine jetty]

Farm

[wide spear]

Have you tried anything yet

If you haven't, start with RK4

If you have stiffness issues, then come back and we can discuss more

[neat haven]

I started with addressing the singularity at $r = 0$, then I tried to derive a Taylor series expansion for $M (r) and P (r) near R = 0$ and used the approximations for a small, non-zero radius. But really that's all.

I eventually added a standard fourth-order Runge-Kutta method to integrate the TOV equations. However, I've encountered some problems in the results that suggest there might be errors introduced when I used the method.

[pine jetty]

Farm

[wide spear]

Hmmmm

What happens if instead of integrating from 0 to r_1 you integrate from eps to r_1

Prescribing an initial condition for P(eps)

What problems are there in the results?

[violet sundial]

I'm trying to figure out how to get the orientation of a face-transitive polyhedron given a rotation matrix. Specifically, I'm thinking dice (beyond just 6-sided). If I label each face in some starting orientation and calculate normals for each face, I could rotate the normals once the polyhedron itself is rotated and find one which is pointing roughly "up", but that gets to be a lot of work as the number of faces increases. Is there some other method I'm missing?

[wide spear]

Are you only working in R^3?

Is your polyhedron convex?

What do you mean by a lot of work?

If you have N faces, you won't be able to do better than O(N) I think

Which your technique already is