Oscar Veliz
Oscar Veliz
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exp(x) explained
The algorithms behind the exponential function in the computer revealed. Lesson also includes discussion of the pow(x,y) function, historical digital computing of the exponential function, modern versions of the algorithm, and a high-precision approach to finding any number of digits for e.
Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters:
0:00 MATLAB
0:16 Euler's Number
0:38 Computing e^x
0:52 Computing pow(x,y)
1:19 pow(x,y) source code
1:56 Computing e^x continued
3:21 exp(x) source code
4:03 Long Story Short
4:29 High-precison e
4:54 e-spigot
5:58 Why does e-spigot work?
6:22 Mathemaniac
6:52 Oscar's Notes
7:28 Outro
Recommended Viewing:
@mathemaniac 's exp video ua-cam.com/video/u1taDXNzFto/v-deo.html
Computing π: Machin-like formula ua-cam.com/video/M_fTdDx8IlY/v-deo.html
Reference links:
math.h github.com/openbsd/src/blob/master/include/math.h
exp(x) source code netlib.org/fdlibm/e_exp.c
pow(x,y) source code netlib.org/fdlibm/e_pow.c
"Algorithms for Digital Computers" by Hastings et. al. press.princeton.edu/books/paperback/9780691626949/approximations-for-digital-computers
"The Mathematical-Function Computation Handbook" by Beebe link.springer.com/book/10.1007/978-3-319-64110-2
"A Spigot Algorithm for the Digits of π" by Rabinowitz & Wagon www.tandfonline.com/doi/abs/10.1080/00029890.1995.11990560?journalCode=uamm20
Background music "Drifting at 432 Hz" by @UnicornHeads
#NumericalAnalysis #SoME3 #exp
Переглядів: 8 193

Відео

Bairstow's Method
Переглядів 7 тис.Рік тому
Bairstow's Method for finding the roots of polynomials including complex roots. Discussion of method derivation, relation to synthetic division of two variables, stopping condition, selection of initial values, fractals, and historical context. Submission for Summer of Math Exposition 2 contest by @3blue1brown. Example code hosted on GitHub github.com/osveliz/numerical-veliz Chapters: 00:00 Int...
Graeffe's Method
Переглядів 3,6 тис.2 роки тому
Graeffe's Root-Squaring Method (also called Graeffe-Dandelin-Lobachevskiĭ or Dandelin-Lobachesky-Graeffe method) for finding roots of polynomials. The method solves for all of the roots of a polynomial by only using the coefficients and does not require derivatives nor an interation function. This lesson provides a history of the method, motivates "why" the method works, and walks through an e...
Generalized Bisection Method for Systems of Nonlinear Equations
Переглядів 2 тис.2 роки тому
Generalization of the Bisection Method for solving systems of equations. This lesson explains the algorithm for a 2 dimension example based on Harvey-Stenger's approach using bisecting triangles. It includes a visualization of the method in action on an example nonlinear system. Other methods for solving in 3 dimensions and for larger systems are also discussed as well as hybrid approaches. Exa...
Generalized False Position & Alternative Secant Methods
Переглядів 7372 роки тому
False Position Method for Nonlinear Systems (aka Generalized Regula Falsi) along with two Alternative Secant Methods. Includes discussion of history and primary sources along with numeric examples and visualizations. Example code hosted on GitHub github.com/osveliz/numerical-veliz Chapters: 0:00 Scaffolding 0:25 Korganoff 1:02 Robinson 1:32 Some History 1:50 Robinson Continued 2:51 Robinson ver...
Global Newton's Method - It Always Converges
Переглядів 6 тис.2 роки тому
Globally convergent modification of Newton's Method that uses backtracking whenever a test point would not cause the function iterations to shrink in absolute value based on the Armijo's Search. Lesson also covers fractals using Global Newton Method as well as solving systems of nonlinear equations. Example code hosted on GitHub github.com/osveliz/numerical-veliz Chapters: 0:00 Intro 0:30 Liter...
Halley's Method for Systems of Nonlinear Equations
Переглядів 2,9 тис.2 роки тому
Halley's Method for Solving Systems of Nonlinear Equations. Submission for The Summer of Math Exposition. Lesson includes motivation & explanation of notation, description of the method, numerical example, discussion of order, and comparison with the Method of Tangent Hyperbolas. Example code hosted on GitHub github.com/osveliz/numerical-veliz Chapters: 0:00 Wikipedia 0:44 Intro 0:54 Recommende...
Broyden's Method
Переглядів 11 тис.3 роки тому
Broyden's Method for solving systems of nonlinear equations. Lesson covers motivation, history, examples, discussion, and order of this Quasi-Newton Method. It also explains the "Good" and "Bad", as well as the third version of the method. Example code hosted on GitHub github.com/osveliz/numerical-veliz Chapters: 0:00 Intro 0:22 Newton's Method According to Broyden 1:08 Nonlinear System Example...
Approximating the Jacobian: Finite Difference Method for Systems of Nonlinear Equations
Переглядів 6 тис.3 роки тому
Generalized Finite Difference Method for Simultaneous Nonlinear Systems by approximating the Jacobian using the limit of partial derivatives with the forward finite difference. Example code on GitHub www.github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:13 Prerequisites 0:32 Refresher 0:43 What is the Jacobian? 2:06 Approximating the Jacobian 3:00 Finite Differences 3:21 Note on Notation...
Steffensen's Method for Systems of Nonlinear Equations
Переглядів 3,1 тис.3 роки тому
Generalized Steffensen's Method for Simultaneous Nonlinear Systems originally credited to J. F. Traub. Video shows how to solve nonlinear systems by approximating the Jacobian. Example code on GitHub www.github.com/osveliz/numerical-veliz Chapters 0:00 Prerequisites 0:20 Intro 0:40 Traub 1:24 Soleymani et al 1:58 Explaining Notation 2:32 1D Example 3:06 Two Methods - Same Method 3:20 System of ...
Secant Method for Systems of Nonlinear Equations
Переглядів 5 тис.3 роки тому
Generalized Secant Method for Simultaneous Nonlinear Systems originally credited to Wolfe and Bittner. Lesson shows how to solve nonlinear systems without the Jacobian, nor the need to approximate it, in a straightforward and visual manner. Example code on GitHub www.github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:15 Prerequisites 0:25 Secant Method Recap 0:45 Literature 1:00 Secant Me...
Newton's Method for Systems of Nonlinear Equations
Переглядів 16 тис.3 роки тому
Generalized Newton's method for systems of nonlinear equations. Lesson goes over numerically solving multivariable nonlinear equations step-by-step with visual examples and explanation of the Jacobian, the backslash operator, and the inverse Jacobian. Example code in MATLAB / GNU Octave on GitHub: github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:12 Prerequisites 0:32 Background 0:58 Set...
Video Mistakes II: The Sequel
Переглядів 4053 роки тому
This video corrects mistakes in my videos on Taylor Series Origin, Ternary Search, Dichotomous Search, Fixed Point Iteration for Fixed Point Iteration System of Equations with Banach, and Wegstein's Method. Thanks to commenters who pointed these errors out. If you find other mistakes feel free to comment or post in the GitHub Issues Forum for the code repository www.github.com/osveliz/numerical...
Brent's Minimization Method
Переглядів 8 тис.3 роки тому
Hybrid minimization algorithm combining Golden-section Search and Successive Parabolic Interpolation (Jarratt's Method) that is guaranteed to locate minima with superlinear convergence order. Example code github.com/osveliz/numerical-veliz Chapters: 0:00 Intro 0:16 Scaffolding 0:31 Motivation 1:17 Parabolic Interpolation Review 1:48 Renaming Variables 2:40 Brent's Method Algorithm 3:19 SPI Beha...
Successive Parabolic Interpolation - Jarratt's Method
Переглядів 4 тис.3 роки тому
Optimization method for finding extrema of functions using three points to create a parabola that is then used to find the next approximation to the solution. This lesson visualizes the behavior of the method with numeric examples as well as its convergence through fractals. Based off the paper "An iterative method for locating turning points" by P. Jarratt. Example code github.com/osveliz/nume...
Sublinear Convergence #MegaFavNumbers
Переглядів 8923 роки тому
Sublinear Convergence #MegaFavNumbers
Golden-section Search
Переглядів 19 тис.3 роки тому
Golden-section Search
Fibonacci Search
Переглядів 30 тис.4 роки тому
Fibonacci Search
Dichotomous Search
Переглядів 11 тис.4 роки тому
Dichotomous Search
Ternary Search
Переглядів 10 тис.4 роки тому
Ternary Search
Computing π: Machin-like formula
Переглядів 2,9 тис.4 роки тому
Computing π: Machin-like formula
Origin of Taylor Series
Переглядів 13 тис.4 роки тому
Origin of Taylor Series
What is Order of Convergence?
Переглядів 37 тис.4 роки тому
What is Order of Convergence?
Video Mistakes and How to Fix Them
Переглядів 1 тис.4 роки тому
Video Mistakes and How to Fix Them
Subscriber Milestone - 5 Ways to Help the Channel
Переглядів 2974 роки тому
Subscriber Milestone - 5 Ways to Help the Channel
Householder's Method
Переглядів 8 тис.4 роки тому
Householder's Method
Halley's Method
Переглядів 10 тис.4 роки тому
Halley's Method
Laguerre's Method
Переглядів 7 тис.4 роки тому
Laguerre's Method
Aberth-Ehrlich Method
Переглядів 2,7 тис.5 років тому
Aberth-Ehrlich Method
Durand-Kerner Method
Переглядів 4 тис.5 років тому
Durand-Kerner Method

КОМЕНТАРІ

  • @irajgharaee7057
    @irajgharaee7057 7 днів тому

    I was 9 when you uploaded this video and now I almost got my bachelor in CS

  • @gsUPD
    @gsUPD 27 днів тому

    GOAT

  • @user-df9zz5qv6l
    @user-df9zz5qv6l 28 днів тому

    Please explain, I have a negative discriminant. At the same time, the equation well clearly has a root On which it depends and how to fix it?

  • @mominarehman3924
    @mominarehman3924 Місяць тому

    how to find norm square of matrix In Jn kindly guide me

  • @TJBEATSAMV
    @TJBEATSAMV Місяць тому

    Very well explained , thanks !!

  • @walterdiaz2003
    @walterdiaz2003 Місяць тому

    @5:00 the reality is that none of them discovered differential calculus. The person who discovered it was Madhava from India in 1300. It's well know that intelectual property was subtracted from Kerala and these dudes managed to understand it 200 or 300 years later. All credits go to India.

  • @onlinemathtutor2361
    @onlinemathtutor2361 Місяць тому

    Hey,Is all this convergence or divergence totally depend upon intial guess ?

    • @OscarVeliz
      @OscarVeliz Місяць тому

      Both the initial and g(x).

  • @matiassantacruz5487
    @matiassantacruz5487 Місяць тому

    Absolutely amazing video, thank you!

  • @michatarnowski580
    @michatarnowski580 Місяць тому

    I would really like to see a proof of why Horner scheme gives the quotient. I understand why it gives the remainder of division, but it giving the quotient looks like magic. I saw some proofs by solving equations for polynomial coefficients, but I wonder if there's a quick and simple argument, at least for low degrees like 2 and 3.

  • @user__unknown
    @user__unknown Місяць тому

    Thanks for such a wonderful video, it made the topic easily understandable.

  • @guillaumeleclerc3346
    @guillaumeleclerc3346 Місяць тому

    Thanks!

    • @OscarVeliz
      @OscarVeliz Місяць тому

      Thank you sooo much @guillaumeleclerc3346 for my first ever Super Thanks!!

    • @guillaumeleclerc3346
      @guillaumeleclerc3346 Місяць тому

      @@OscarVeliz you deserve it for sure, this channel replaced reference textbook for me when I have to implement solvers 🙂

  • @obigwejesse2004
    @obigwejesse2004 Місяць тому

    How did you get the 1+root5/2 from

  • @SempatikBalkc
    @SempatikBalkc Місяць тому

    so succesfull thank you

  • @karolleszynski1963
    @karolleszynski1963 Місяць тому

    Best stuff i could find on yt. Well done

  • @saidberk6599
    @saidberk6599 2 місяці тому

    you have no Idea how many people you are helping from diffenrent places of the world. thanks a lot.

  • @gecaprathamesh6583
    @gecaprathamesh6583 2 місяці тому

    Thank you! Was able to understand the method in one go! :)

  • @habibullahamjad2037
    @habibullahamjad2037 2 місяці тому

    I want to a video of how regula falsi is mathematically same like regula falsi can be converted to secant method so they are mathematically same you why we call a secant method a contribution how these two are different

  • @moontiger6393
    @moontiger6393 2 місяці тому

    Can't believe how few views this has for just have fantastic it is, needed to find a simple and fast method for computing minima today, and this video explained exactly what was needed in a mere few minutes. Otherwise it would have taken much longer for me to be sure of what I was doing, thank you so much!

  • @taquiansari4833
    @taquiansari4833 2 місяці тому

    extremely helpful .. tysm

  • @piupolino2618
    @piupolino2618 2 місяці тому

    I find it ironic that you need the root to find out whether it'll converge at the root. Great video, very helpful!

    • @OscarVeliz
      @OscarVeliz 2 місяці тому

      I discuss this irony in my followup video ua-cam.com/video/FyCviw2ZA2o/v-deo.html

  • @yorailevi6747
    @yorailevi6747 2 місяці тому

    Most results on google for this method link to this video… is it not well known or not used much?

    • @OscarVeliz
      @OscarVeliz 2 місяці тому

      The method is not well known which leads to not being used much. Seems like every numerical methods textbook covers fixed point iteration, but then stops there without going over Wegstein or Steffensen.

  • @ballapranavachaitanya3975
    @ballapranavachaitanya3975 3 місяці тому

    thanks bro very helpful

  • @nikolajlapkovskij
    @nikolajlapkovskij 3 місяці тому

    Absolutely fantastic explanation. Even after 12 years. Huge thank you and much much much appreciated!!!

  • @Dicergrey
    @Dicergrey 3 місяці тому

    You elucidate the method effortlessly in such simple words. This channel is pretty helpful in helping me, a beginner easily grasp the steps of these methods, found it jus now.

  • @hendohenderson1159
    @hendohenderson1159 3 місяці тому

    You sure that’s not 10^50 factorial at 0:59 😂

  • @reikofz93
    @reikofz93 3 місяці тому

    Oscar, I wish your videos were more popular. We give millions of views to other UA-camrs that deal with content that is as captivating as useless. On the other hand, you deal with actual problems that can be encountered in school or at work. You provide simple and effective explanations, resources, and code. This is a very pragmatic and scientist oriented approach. No fricking fireworks and smoke, and this penalizes you. I hope that real life rewards you as you deserve. Keep up the good work!!

    • @OscarVeliz
      @OscarVeliz 3 місяці тому

      Thank you for this comment. It means a lot.

  • @pauselab5569
    @pauselab5569 3 місяці тому

    one of the best iteration theorem is banach's fixed point theorem which is great for all the continued fraction and continued root because they all satisfy the conditions for the theorem

  • @goop69_
    @goop69_ 3 місяці тому

    useless video, no coding

    • @OscarVeliz
      @OscarVeliz 3 місяці тому

      Code is provided in the GitHub repo for the channel. Link in description.

  • @alexandertaffe227
    @alexandertaffe227 4 місяці тому

    I finally figured out what I was doing wrong after I found this video. Thank you!!

  • @wolfix20021
    @wolfix20021 4 місяці тому

    Thank you so much! just within 4 minutes you open gates of the secant method. Much appreciated!

  • @karim8913
    @karim8913 4 місяці тому

    yeah

  • @koreedeiqbal9141
    @koreedeiqbal9141 4 місяці тому

    Please explain Which software is used for this video making ❤

    • @OscarVeliz
      @OscarVeliz 4 місяці тому

      PowerPoint and Microsoft Mathematics 😁

  • @user-gb5ee9yi8k
    @user-gb5ee9yi8k 4 місяці тому

    unable to find if there exist any numerical iterations base on Laurent series or complex Fourier seris

  • @user-gb5ee9yi8k
    @user-gb5ee9yi8k 4 місяці тому

    there are too-scarce sources discussing error terms and error analaysis mid-terms and finals ever propose these forgettable error problems

  • @user-gb5ee9yi8k
    @user-gb5ee9yi8k 4 місяці тому

    there are too-scarce sources discussing error terms and error analaysis mid-terms and finals ever propose these forgettable error problems

  • @wolfix20021
    @wolfix20021 4 місяці тому

    Short yet quite adequate, Thanks a lot!!!!

  • @fireknuckles2678
    @fireknuckles2678 4 місяці тому

    How will you know the fixed point n that the aitken accelerated you to

  • @irisvera5490
    @irisvera5490 4 місяці тому

    The root of what value??? someone plz explain 2:53

    • @OscarVeliz
      @OscarVeliz 4 місяці тому

      I have a follow up video answering this and other commonly asked questions ua-cam.com/video/FyCviw2ZA2o/v-deo.html

  • @theyammi437
    @theyammi437 4 місяці тому

    Simply put and straight to the point. If only Textbooks were this way.

  • @jacquesbautista-plante5497
    @jacquesbautista-plante5497 4 місяці тому

    my teacher explained it terribly, thank you so much for clearing up this muck!

  • @atharvasinnarkar4285
    @atharvasinnarkar4285 4 місяці тому

    dude this covers 4 lectures from trefethen bau ....covered in just 7 mins...amazing stuff!

  • @hlnebula2828
    @hlnebula2828 4 місяці тому

    Where do the a values come from 2:50

    • @OscarVeliz
      @OscarVeliz 4 місяці тому

      They come from Hastings Jr's paper (link in description). If I recall correctly, they were determined through interpolation.

  • @olbluelips
    @olbluelips 4 місяці тому

    Wonderful! I needed a way to approximate implicit relations robustly! I'll need to change it so that instead of finding a minimum/maximum, it will find the complex solution, but this is really useful thanks for the upload

    • @OscarVeliz
      @OscarVeliz 4 місяці тому

      Instead of dividing by 2, try dividing by 2i

    • @olbluelips
      @olbluelips 4 місяці тому

      @eliz Finally got it implemented! It was harder than i expected, because i didn't understand that even when globally convergent, newton's method is really sensitive to the initial guess. I had it converging on erroneous solutions for a bit because i didn't understand how to pick a good initial

  • @oussamaelbazzim3578
    @oussamaelbazzim3578 5 місяців тому

    Its 2024 and this is still one banger of a video.Great explanation as usual.Many thanks !

  • @holyshit922
    @holyshit922 5 місяців тому

    They switch to the methods which do not guarantee convergence , strange approach

  •  5 місяців тому

    Great video. If you ever feel like getting back to this topic, I'd love to see your take on TOMS 748, which is considered(?) superior to Brent's method (albeit, some doubt has been cast from a very thorough review by Gregory W. Chicares).

  • @jkgan4952
    @jkgan4952 5 місяців тому

    Great Video! Thanks so much!!!

  • @gaaraofddarkness
    @gaaraofddarkness 5 місяців тому

    2:45 how did you arrive at the expression of alpha as the ratio of error ratios?

    • @OscarVeliz
      @OscarVeliz 4 місяці тому

      Check out my video on order of convergence ua-cam.com/video/JTinepDn1dI/v-deo.html

  • @stephon9449
    @stephon9449 5 місяців тому

    thanks!

  • @Rachotilko
    @Rachotilko 6 місяців тому

    There is a variant of this method for simultaneous solving of all roots, possibly with multiplicities > 1.