Загрузка видео...

Не удалось загрузить видео

Возникла проблема при загрузке этого видео. Это может быть связано с временными проблемами сети или видео может быть недоступно.

На главную

How to sketch a polynomial function graph. This is a 10th degree polynomial.

Math Magician

4,950 subscribers

21,894 просмотров • 1 месяц назад •via X (Twitter)

Anya Rossi• Live Now

Private livecam show

Комментарии: 2

Фото профиля Manu

Manu1 месяц назад

@davidobrador M'ha fet pensar en l'activitat que ens vas fer a l'Arabat0!

Фото профиля Ado_rabbit

Ado_rabbit1 месяц назад

这个不是高中学过的穿针引线法吗？

Похожие видео

This video, created by my dear coauthor Mahdi E Kahou for our teaching and papers, shows how overparameterized neural networks produce smooth function approximations even in the context of the Runge phenomenon. Some background. Imagine you want to approximate the Runge function using polynomial interpolation at equally spaced points. It is well known that, despite targeting an infinitely differentiable function, such a polynomial approximation produces oscillatory behavior that worsens with the degree of the polynomial. In other words, higher-degree polynomial approximations might not improve accuracy. Instead, approximate the Runge function with a neural network (here, two layers are just to make the example concrete; nothing fundamental depends on it). As you increase the number of parameters well above the 11 training points (in our example, a two-layer neural network with 128 nodes each), you nicely converge to the target, without wild oscillations. Yes, this has much to do with double descent and benign overparameterization, but the main punchline of this post is that neural networks are really very different types of animals than polynomial approximations. And yes, Chebyshev nodes and splines exist, and in this case, they will prevent the oscillations. But that's not the point. Chebyshev nodes and splines still confront Faber’s theorem, which states that for any system of polynomial interpolation nodes, there exists a continuous function whose sequence of interpolating polynomials diverges as the number of nodes grows to infinity. Faber’s theorem does not apply to neural networks because they are not polynomials. The notebook, if you want to check the details, is here: Stay tuned for more on this 👀

This video, created by my dear coauthor Mahdi E Kahou for our teaching and papers, shows how overparameterized neural networks produce smooth function approximations even in the context of the Runge phenomenon. Some background. Imagine you want to approximate the Runge function using polynomial interpolation at equally spaced points. It is well known that, despite targeting an infinitely differentiable function, such a polynomial approximation produces oscillatory behavior that worsens with the degree of the polynomial. In other words, higher-degree polynomial approximations might not improve accuracy. Instead, approximate the Runge function with a neural network (here, two layers are just to make the example concrete; nothing fundamental depends on it). As you increase the number of parameters well above the 11 training points (in our example, a two-layer neural network with 128 nodes each), you nicely converge to the target, without wild oscillations. Yes, this has much to do with double descent and benign overparameterization, but the main punchline of this post is that neural networks are really very different types of animals than polynomial approximations. And yes, Chebyshev nodes and splines exist, and in this case, they will prevent the oscillations. But that's not the point. Chebyshev nodes and splines still confront Faber’s theorem, which states that for any system of polynomial interpolation nodes, there exists a continuous function whose sequence of interpolating polynomials diverges as the number of nodes grows to infinity. Faber’s theorem does not apply to neural networks because they are not polynomials. The notebook, if you want to check the details, is here: Stay tuned for more on this 👀

Jesús Fernández-Villaverde

47,506 просмотров • 5 месяцев назад

still on understanding zkSNARKs, how do we get from polynomial to proof?

still on understanding zkSNARKs, how do we get from polynomial to proof?

Nyakio

12,096 просмотров • 5 месяцев назад

You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

Mathelirium

15,566 просмотров • 8 месяцев назад

Polynomial division can look intimidating, but we can make this more accessible by drawing rectangles!

Polynomial division can look intimidating, but we can make this more accessible by drawing rectangles!

Howie Hua

36,278 просмотров • 3 лет назад

Zero Fee Perps are now LIVE on Polynomial Trade. 0% maker & taker fees on ETH and BTC perpetuals with our trading points program ending October 31st. Same execution speed. Same deep liquidity. Zero trading fees. Welcome to Polynomial "Uptoberfest."

Zero Fee Perps are now LIVE on Polynomial Trade. 0% maker & taker fees on ETH and BTC perpetuals with our trading points program ending October 31st. Same execution speed. Same deep liquidity. Zero trading fees. Welcome to Polynomial "Uptoberfest."

Polynomial

30,313 просмотров • 1 год назад

Harmonics of Roots of Complex Polynomials polynomial equation by Simone Conradi

Harmonics of Roots of Complex Polynomials polynomial equation by Simone Conradi

Mathelirium

18,529 просмотров • 1 год назад

The evolving lemniscates of a time-choreographed complex polynomial. *DM me if you would like to see the math framework...bots are killing me.

The evolving lemniscates of a time-choreographed complex polynomial. *DM me if you would like to see the math framework...bots are killing me.

Mathelirium

18,318 просмотров • 1 год назад

Polynomial Superchain is live 🟢✨ Stake now at This is a significant milestone in our mission to hyperscale on-chain derivatives trading, and we invite everyone to join us. Details on staking and our points program below 👇

Polynomial Superchain is live 🟢✨ Stake now at This is a significant milestone in our mission to hyperscale on-chain derivatives trading, and we invite everyone to join us. Details on staking and our points program below 👇

Openstock

120,686 просмотров • 2 лет назад

The roots of a polynomial aren't just numbers, but a moving geometric field that reacts like a resonant surface in the complex plane. See the description for the animation below

The roots of a polynomial aren't just numbers, but a moving geometric field that reacts like a resonant surface in the complex plane. See the description for the animation below

Mathelirium

23,908 просмотров • 5 месяцев назад

This is how you turn a matrix into a graph:

This is how you turn a matrix into a graph:

Tivadar Danka

68,555 просмотров • 3 лет назад

One of the first polynomial root plots I made 2 years ago after learning the idea from Simone Conradi

One of the first polynomial root plots I made 2 years ago after learning the idea from Simone Conradi

Mathelirium

15,640 просмотров • 5 месяцев назад

$5 billion in volume processed. Over a year live on our chain. Today we're launching Polynomial Fusion - something that doesn't exist anywhere else in crypto. The world's first hybrid orderbook system. Here's why this changes everything 🧵

$5 billion in volume processed. Over a year live on our chain. Today we're launching Polynomial Fusion - something that doesn't exist anywhere else in crypto. The world's first hybrid orderbook system. Here's why this changes everything 🧵

Openstock

63,393 просмотров • 1 год назад

#UEFN Scene Graph trick: Teleporting a player with SetGlobalTransform instead of TeleportTo skips the loading screen!!! Today's v38.00 update added a function for retrieving a `fort_character`s Entity which allows for this trick to function!

#UEFN Scene Graph trick: Teleporting a player with SetGlobalTransform instead of TeleportTo skips the loading screen!!! Today's v38.00 update added a function for retrieving a `fort_character`s Entity which allows for this trick to function!

krisp

12,542 просмотров • 1 год назад

This is obviously a sketch but it's terrifying how real it is.

This is obviously a sketch but it's terrifying how real it is.

Ian Miles Cheong

109,167 просмотров • 3 лет назад

Introducing stock perpetuals on Polynomial ✨ NVDA, TSLA, HOOD, PLTR, GOOG, COIN live now. 100x leverage. Cross-margin them with crypto, indices, commodities, and forex.

Introducing stock perpetuals on Polynomial ✨ NVDA, TSLA, HOOD, PLTR, GOOG, COIN live now. 100x leverage. Cross-margin them with crypto, indices, commodities, and forex.

Polynomial

10,328 просмотров • 10 месяцев назад