Are there similar approaches of using sampling to compute the exact integral for non-polynomial functions?

asheng2

Would Gaussian Quadrature give us a better approximation than the trapezoid rule if we use less than n points or don't know the structure of f(x) beforehand?

dshernan

Is the "special" set of points just some points before/after each of the critical points? or inflection points? Or something else?

dl123

How to determine these special set of points and their weights?

jonasjiang

Is gauss quadrature also useful when we want to interpolate points?

richardnnn

If it's a polynomial of order n, and we can take n points, why can't we just solve for the coefficients and do the integral as polynomial integral is super trivial to implement?

ant123

How do we determine the set of points and their weights? Is that something we just play around with?

mangopi

Earlier in the video you mentioned that a lot of the functions/data sets we'll be working with are difficult to integrate, and I'm assuming if we're just given a data set then it may be difficult to represent using a function. Is the Gauss Quadrature useful when dealing with non polynomial data sets?

Coyote

Are these points easy to determine with a computer (or at least easier than trying to integrate the original function)?

MrRockefeller

How general is the sampling rule here? Like for any n degree of polynomials, any coefficient in each term? And does the domain have to be symmetric to y axis?

ddkim

How difficult is it to compute these points and weights, and how do we know that such points always exist?

corgo

How can we determine which points to sample in code? It's very intuitive when we are manually selecting points, but how can we write a program that "looks" and "finds" these determining points?

Murrowow

While this sounds nice and seems like it would make things a lot easier, I’m not fully understanding how exactly we obtain these special weights. Is there a special formula or heuristic that enables this?

Concurrensee

Is this any specialized data structure or hardware to do such kind of tests?

anj

How would you determine the special set of points?

jefftan

If these special weights are expensive to compute (for example if you need to compute square roots and division operations), might it be more computationally efficient to use a larger number of inexact sampling points that are easier to compute

TejasFX

Can we apply this or a similar idea to functions of another type? i.e log, fractional powers, etc.

gfkang

I am also confused as to how we would obtain these special set of points/weights? Isn't this a huge abstraction, it seems much more complicated than what's shown.

spidey

How do we determine these special points and weights? Is there a specific formula per degree or is it based on coefficients or etc.?

anag

Is this something we can use as long as we assume the data follows some polynomial curve that we can fit to (if we wanted a good approximate integral of a wonky curve)?

jefftan

If these special weights are expensive to compute (for example if you need to compute square roots and division operations), might it be more computationally efficient to use a larger number of inexact sampling points that are easier to compute

Are there similar approaches of using sampling to compute the exact integral for non-polynomial functions?

Would Gaussian Quadrature give us a better approximation than the trapezoid rule if we use less than n points or don't know the structure of f(x) beforehand?

Is the "special" set of points just some points before/after each of the critical points? or inflection points? Or something else?

How to determine these special set of points and their weights?

Is gauss quadrature also useful when we want to interpolate points?

If it's a polynomial of order n, and we can take n points, why can't we just solve for the coefficients and do the integral as polynomial integral is super trivial to implement?

How do we determine the set of points and their weights? Is that something we just play around with?

Earlier in the video you mentioned that a lot of the functions/data sets we'll be working with are difficult to integrate, and I'm assuming if we're just given a data set then it may be difficult to represent using a function. Is the Gauss Quadrature useful when dealing with non polynomial data sets?

Are these points easy to determine with a computer (or at least easier than trying to integrate the original function)?

How general is the sampling rule here? Like for any n degree of polynomials, any coefficient in each term? And does the domain have to be symmetric to y axis?

How difficult is it to compute these points and weights, and how do we know that such points always exist?

How can we determine which points to sample in code? It's very intuitive when we are manually selecting points, but how can we write a program that "looks" and "finds" these determining points?

While this sounds nice and seems like it would make things a lot easier, I’m not fully understanding how exactly we obtain these special weights. Is there a special formula or heuristic that enables this?

Is this any specialized data structure or hardware to do such kind of tests?

How would you determine the special set of points?

If these special weights are expensive to compute (for example if you need to compute square roots and division operations), might it be more computationally efficient to use a larger number of inexact sampling points that are easier to compute

Can we apply this or a similar idea to functions of another type? i.e log, fractional powers, etc.

I am also confused as to how we would obtain these special set of points/weights? Isn't this a huge abstraction, it seems much more complicated than what's shown.

How do we determine these special points and weights? Is there a specific formula per degree or is it based on coefficients or etc.?

Is this something we can use as long as we assume the data follows some polynomial curve that we can fit to (if we wanted a good approximate integral of a wonky curve)?

If these special weights are expensive to compute (for example if you need to compute square roots and division operations), might it be more computationally efficient to use a larger number of inexact sampling points that are easier to compute