Solve any equation using gradient descent

Gradient descent is an optimization algorithm designed to minimize a function. In other words, it estimates where a function outputs its lowest value.

This video demonstrates how to use gradient descent to approximate a solution for the unsolvable equation x^5 + x = 3.

We seek a cubic polynomial approximation (ax^3 + bx^2 + cx + d) to cosine on the interval [0, π].

[kushaagr]

You're narration is like from 90s american television. I liked it very much.

[kukikukic]

Bro i was home alone and wanted to watch some math shit, i instantly closed yt when i saw the intro😂

[Physics_HB]

The intro, the speaker's voice, and everything was beautiful in this video

[richardmarch]

While being just a machine learning tutorial, this has an unsettling vibe that I find very unique for a educational channel, and honestly much more captivating. Keep it up!

[DaMonster]

The fact that there’s no quintic formula was proved by Galois before dying in a duel at 20

[beautyofmath]

Beautiful and very well made video, I personally loved the old tv vibe to this, not to disregard the instructive yet nicely explained method of gradient descent. Subscribed

[sicko]

that voice kinda gives the vintage vibes of it, it's like you watching a very old vid. good shit homie

[Zacvh]

Bro this video is so fire. I get so annoyed by the voices in my actual school videos that they make you watch and this is a huge step up from that it actually makes this seem like it’s a top secret information like you’re debriefing the first nuclear tests or something

[Biggietothecheese]

Truly happy i found this gem of a chanel before it blows up

[stardreamix]

Amazing to hear a new algorithm to solve equations, even the non real ones! - Thanks for helping me understand!

[AhmAsaduzzaman]

AWESOME Video! Thanks! Trying to put some basic understanding on this: "We seek a cubic polynomial approximation (ax^3 + bx^2 + cx + d) to cosine on the interval [0, π]."
Let's say you want to represent the cosine function, which is a bit wavy and complex, with a much simpler formula—a cubic polynomial. This polynomial is a smooth curve described by the equation where a, b, c, and d are specific numbers (coefficients) that determine the shape of the curve.

Now, why would we want to do this?

Cosine is a trigonometric function that's fundamental in fields like physics and engineering, but it can be computationally intensive to calculate its values repeatedly.

A cubic polynomial, on the other hand, is much simpler to work with and can be computed very quickly.

So, we're on a mission to find the best possible cubic polynomial that behaves as much like the cosine function as possible on the interval from 0 to π (from the beginning to the peak of the cosine wave).

To find the perfect a, b, c, and d that make our cubic polynomial a doppelgänger for cosine, we use a method that involves a bit of mathematical magic called "least squares approximation".

This method finds the best fit by ensuring that, on average, the vertical distance between the cosine curve and our cubic polynomial is as small as possible. Imagine you could stretch out a bunch of tiny springs from the polynomial to the cosine curve—least squares find the polynomial that would stretch those springs the least.

Once we have our cleverly crafted polynomial, we can use it to estimate cosine values quickly and efficiently. The beauty of this approach is that our approximation will be incredibly close to the real deal, making it a nifty shortcut for complex calculations.

[elmoreglidingclub]

This is just awesome stuff. Makes me want to study math. To be patient and learn the fundamentals that gets me to being able to understand and solve these equations. Fascinating.

[akrammohamed]

I love this explanation, voice, simplicity.
Im guessing the voice is a text to speech trained on old 60s videos?

[cblpu]

Instead of squaring, you can raise to higher even powers(like 4, 6, ..etc), giving quicker convergence if i remember correctly.

[Daniel_Larson_Records]

There's something about the way you talk and edit the video together that actually makes it interesting. I can't put my finger on it. Maybe it's how novel it is? I don't know, but PLEASE make more videos like this. It's amazing, and I actually understood it completely (rare for someone so bad at math lol)

[olerask]

Nice video.
Next step is designing a suitable neural network (choose number of hidden layers, and nodes of each layer, as well as an activation function), and use gradient descent to “learn” the node values, such that the neural network f.ex. produces a regression function to a set of points.

[ycty]

fantastic video ur gonna blow up soon (this is a threat)

[ananthakrishnank]

Thank you for the video!! Took some time to grasp the second example.

No surprise. This gradient descent optimization is at the heart of machine learning.

[AhmAsaduzzaman]

Yes, solving the equation x^5 + x = y for x in terms of y is much more complex than solving quadratic equations because there is no general formula for polynomials of degree five or higher, due to the Abel-Ruffini theorem. This means that, in general, we can't express the solutions in terms of radicals as we can for quadratics, cubics, and quartics.

However, we can still find solutions numerically or graphically. Numerical methods such as Newton's method can be used to approximate the roots of this equation for specific values of y. If we're interested in a symbolic approach, we would typically use a computer algebra system (CAS) to manipulate the equation and find solutions.

[brickie]

A perfect video to watch at 2 am, especially the intro... now I will have a ton of time to think about this algorithm because there is no way i will sleep 😂 but seriously, very much interesting. I will delve deeper into it 👍