The Vandermonde Matrix and Polynomial Interpolation

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  • Опубликовано: 23 дек 2024

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  • @augustusnero5592
    @augustusnero5592 2 года назад +21

    This was the best explanation I've seen on the Vandermonde determinant.

    • @DrWillWood
      @DrWillWood  2 года назад

      Thank you very much!

    • @TESLA_13_DR
      @TESLA_13_DR Год назад

      @@DrWillWood noo, this is the best explanation video I have ever seen on yt, bravo!

    • @neolord50pro77
      @neolord50pro77 Год назад

      @@DrWillWoodit’s the best explanation in the universe

  • @nymiantoft5907
    @nymiantoft5907 3 года назад +27

    That was so elegant

  • @pianoforte17xx48
    @pianoforte17xx48 3 года назад +3

    Finally the determinant vandermonde after searching literally everywhere with no hope of understanding it. You are a legend

  • @1495978707
    @1495978707 3 года назад +7

    Cool vid! 3:30 I would like to note that, while for a proof, inversion of the matrix is probably the best way to go about this, practically speaking, inversion of the matrix is usually the worst way to go about solving a matrix equation. The idea is that if a matrix is invertible, A is unique, not that it exists at all. But if A isn’t unique, that means the interpolating polynomial isn’t unique. Anyhow, this matrix depends only on choice of node location, not value at the node, whether the equation has a unique solution depends only on whether the matrix is invertible, not on the y values. And that’s why it’s useful to do the proof this way.

    • @slavinojunepri7648
      @slavinojunepri7648 4 месяца назад

      Your comment is absolutely correct. The matrix equation can never results in my multiple solutions for A because of the polynomial interpolation uniqueness. Therefore, the polynomial doesn't exist if the Vandermonde determinant is zero.
      Using the Gauss-Jordan method would be best to solve for A than having to invert the Vandermonde matrix. Matrix inversion requires the use of cofactors, a very computationally expensive and wasteful process. I cannot recall another way for doing so that would have an acceptable time complexity, say polynomial for instance.

  • @robmarks6800
    @robmarks6800 3 года назад +9

    Lovely! Would like to see a continuation into the fourier transform matrix!

  • @af9466
    @af9466 8 месяцев назад

    Thanks for the video, this was a very interesting demonstration of an application of Vandermonde matrix I'd never heard of before, and all the steps were clear.

  • @callmedeno
    @callmedeno Год назад

    Wow that was peak clarity, you are gifted at this

    • @DrWillWood
      @DrWillWood  Год назад

      Thank you! Appreciate that! glad it was useful

  • @dmitrypolozkov1335
    @dmitrypolozkov1335 3 года назад +1

    Thank you! Great video, keep new uploads on!! Greetings from Russia❤️🧸

  • @pnachtwey
    @pnachtwey Год назад +2

    Yes, this is long winded. I just write the 4 equations and solve for the unknowns. This is simple if the points are equally space. Sometime one must use unequal intervals like t01 for the time between x0 and x1. I get equations in terms of time intervals. I make the substitutions for the time intervals so they aren't calculated over and over again.

  • @mb59621
    @mb59621 5 месяцев назад

    For representation of the general formula of vandermonde determinant , a double pi notation could be easier to understand , like a nested for loop.
    The left subscript is the lower limit, and right subscript is the upper limit here ..
    j=1π(n-1) {i=0π(j-1)} (xj - xi) is the correct representation then .

  • @yaniv242
    @yaniv242 Год назад

    at 7:15 when you substract a cols, i cant seem to understand why the second row after it only has x0 in the power of 1, shouldnt the action of sub x0^2 on the above col affect the whole col? sorry that part got me confused a bit

    • @jurrich
      @jurrich Год назад

      Note the blue text: we are subtracting "x0 * the previous column", for all rows. First we look at the effect that has on the first row, namely that it turns every element after the first into a zero. Then we look at what it does for the second row, so we start with [1, x1, x1². x1³, ...] and then apply same operation: the 1 stays the same; the next column gets "x0 * previous column = x0 * 1 = x0" subtracted, yielding x1 - x0; the next column gets x0 * x1 subtracted, yielding x1² - x0x1; the next column gets x0 * x1² subtracted, yielding x1³ - (x0 * x1²), and so on. The next row [1, x2, x2², x2³, ...] gets the same treatment, turning into [1, x2 - x0, x2² - x0x2, x³ - x0x2², ...], and so on all the way down the matrix.

  • @kafkayash2265
    @kafkayash2265 3 года назад +1

    Would really like to see an explanation regarding sylvester matrix too. Your videos are really great!

  • @henrik3141
    @henrik3141 Год назад

    Nice video. Just a small error at 0:53 to write that P_n = .... You also missed the chance of proving the "Key fact" by the vandermont determinant.

  • @BSK_666
    @BSK_666 Год назад

    Very clear, thank you so much doctor..

  • @redroth57
    @redroth57 3 года назад +1

    Awesome video

  • @AJ-et3vf
    @AJ-et3vf 2 года назад

    Awesome video! Thank you!

  • @General12th
    @General12th 3 года назад

    This is a nice little video!

  • @drioko
    @drioko Год назад +4

    what a long and complicated explanation. i don’t understand this and this video is making me panic

  • @DAIQRY
    @DAIQRY 2 года назад

    Amazing video!

  • @kafuu1
    @kafuu1 Год назад

    Nice work!

  • @martinepstein9826
    @martinepstein9826 3 года назад

    Great video. The Vandermonde determinant is really cool but I don't think it was necessary to show invertibility. If we know that Lagrange interpolation or whatever works for every choice of vector y that means the Vandermonde matrix is surjective, hence invertible.

  • @rauldurand
    @rauldurand 2 года назад

    Very cool! What software do you use for the animation?

    • @DrWillWood
      @DrWillWood  2 года назад

      thanks! I use Apple Keynote for the animations

  • @surajchess3114
    @surajchess3114 3 года назад

    But it doesn't say if a node is repeated then polynomial doesn't exist or multiple polynomials exist... so how to find the polynomial if a node is repeated? I can repeat the method for non repeating nodes and then multiply by (x-xr)^m when xr is repeated node and m is no of repetitions. Is this correct?

    • @derendohoda3891
      @derendohoda3891 2 года назад

      If a node is repeated then you have two output values for a single input which is not a function and so not a polynomial. So what you are doing is not interpolation. You are looking for polynomial regression, probably.

  • @djridoo
    @djridoo 3 года назад +1

    Very cool ^^

  • @uamdbro
    @uamdbro 3 года назад

    Nice, I always saw the existence proof given by explicitly constructing a polynomial (using Lagrange polynomials) and then using Taylor polynomials or something in actual practical applications.
    Though now that I am thinking about it, if all we want is an existence proof, doesn't that follow immediately from uniqueness? Seeing as the Vandermonde matrix (once you have fixed the x-values) represents a linear map on a finite dimensional vector space, where injectivity is equivalent to surjectivity?

    • @nikita_x44
      @nikita_x44 Год назад

      Proof of uniqueness assumes existence. if it exists, then exists at most one.

  • @area51xi
    @area51xi 10 месяцев назад

    You went from stating a rule regarding rows but then the example used columns which is very confusing.

  • @xizar0rg
    @xizar0rg 3 года назад +2

    The framework is essentially the same as what's in linear algebra textbooks (slightly more handwaving on the arithmetic than what I remember) from back in the day but presented more succinctly. (got tired of having to code it anew in fortran every time I needed it... damned kids and their libraries for everything.)

  • @bentationfunkiloglio
    @bentationfunkiloglio 3 года назад

    Love it

  • @plushiie_
    @plushiie_ 2 года назад

    omg you saved me!

  • @the_nuwarrior
    @the_nuwarrior 3 года назад

    its quite similar to the DFT matrix

  • @astricsll
    @astricsll Месяц назад

    I'm 14 and i sont understand this am i stupid

  • @richardfrederick1885
    @richardfrederick1885 Год назад

    The audio was terrible!!

  • @joaoheleno3700
    @joaoheleno3700 Год назад

    You're beautiful