Newton's Method | Lecture 14 | Numerical Methods for Engineers

00:10:21
https://www.youtube.com/watch?v=cf_NK7NlWrs

Summary

TLDRNewtons metode er en rask og effektiv numerisk metode for å finne røtter av funksjoner, utviklet av Isaac Newton. Metoden bruker kalkulus for å tegne tangenten til en funksjon ved et gitt punkt og finner hvor denne tangenten krysser x-aksen for å estimere neste verdi. Formelen for metoden er x_{n+1} = x_n - f(x_n) / f'(x_n). For å illustrere metoden, ble det brukt et eksempel for å finne kvadratroten av 2, som viser hvordan man kan konvergere raskt mot det riktige svaret ved å iterere. En god initial gjetning er avgjørende for å oppnå rask konvergens.

Takeaways

  • 🔍 Newtons metode er en rask rotfunnmetode.
  • 📈 Den bruker kalkulus for å finne tangenter.
  • 🧮 Formelen er x_{n+1} = x_n - f(x_n) / f'(x_n).
  • ⚡ Rask konvergens med god initial gjetning.
  • 📏 Eksempelet viser hvordan finne kvadratroten av 2.
  • 📝 Krever en god startverdi for effektivitet.
  • 📊 Kan mislykkes hvis startverdien er for langt unna roten.
  • 🔗 Metoden er oppkalt etter Isaac Newton.
  • 📉 Tangentlinjen gir en lineær tilnærming til kurven.
  • 💡 Brukes ofte i numeriske beregninger.

Timeline

  • 00:00:00 - 00:05:00

    Newton's metode, oppkalt etter Isaac Newton, er en rask metode for å finne røtter av funksjoner ved hjelp av kalkulus. Metoden innebærer å tegne tangenten til kurven ved et punkt x_n, og deretter finne skjæringspunktet med x-aksen for å bestemme neste estimat x_n+1. Formelen for denne metoden er x_n+1 = x_n - f(x_n) / f'(x_n), hvor f' er den deriverte av funksjonen. En god startverdi nær roten er avgjørende for rask konvergens.

  • 00:05:00 - 00:10:21

    I et eksempel for å finne kvadratroten av 2, defineres funksjonen f(x) = x^2 - 2, med den deriverte f'(x) = 2x. Ved å bruke Newtons metode, starter vi med en initialverdi x_0 = 1 og beregner iterasjoner for å nærme oss kvadratroten av 2. Etter tre iterasjoner oppnår vi en verdi nær 1.41421, som viser den raske konvergensen til metoden.

Mind Map

Video Q&A

  • Hva er Newtons metode?

    En numerisk metode for å finne røtter av funksjoner ved å bruke kalkulus.

  • Hvordan fungerer Newtons metode?

    Den bruker tangenten til kurven ved et punkt for å estimere neste verdi.

  • Hva er formelen for Newtons metode?

    x_{n+1} = x_n - f(x_n) / f'(x_n)

  • Hvor raskt konvergerer Newtons metode?

    Den konvergerer veldig raskt hvis man har et godt startpunkt.

  • Hva er et eksempel på bruk av Newtons metode?

    Finne kvadratroten av 2 ved å iterere med funksjonen f(x) = x^2 - 2.

  • Hva er kravene for å bruke Newtons metode?

    Man trenger en god initial gjetning nær roten.

  • Hvem oppfant Newtons metode?

    Metoden er oppkalt etter Isaac Newton, som også oppfant kalkulus.

  • Hva er den viktigste fordelen med Newtons metode?

    Den har den raskeste konvergensen blant rotfunnmetoder.

  • Kan Newtons metode mislykkes?

    Ja, hvis den initiale gjetningen er for langt fra roten.

  • Hva er den matematiske betydningen av Newtons metode?

    Den gir en måte å approximere løsninger på ikke-lineære ligninger.

View more video summaries

Get instant access to free YouTube video summaries powered by AI!
Subtitles
en
Auto Scroll:
  • 00:00:09
    The next numerical method for root finding that I want to discuss is
  • 00:00:13
    called Newton's method, named after  Isaac Newton. Newton invented the
  • 00:00:20
    calculus, so the method fittingly uses calculus. This is a very fast method,
  • 00:00:28
    in fact the fastest method, and very  important method for solving problems
  • 00:00:33
    that require speed. Let's have a look at how to derive this method.
  • 00:00:41
    This is our x, y-axis. We're trying to find the root of y equals f of x. Let's draw the function
  • 00:00:50
    here. Say this is y equals f of  x. The root is the intersection
  • 00:01:01
    with the x-axis. This is an iteration  method, so let's say that we
  • 00:01:07
    already iterated n times, so we're at the value  of x_n. Let's say this is x_n. How do you find
  • 00:01:18
    x_n plus 1? Well, in Newton's  method, you draw the tangent line to
  • 00:01:24
    the curve at x equals x_n. If I can try to sketch the tangent here, it should look
  • 00:01:31
    something like that. That I would say is the tangent line, you find the tangent
  • 00:01:39
    line from calculus. The slope of this line is equal to the derivative of f evaluated
  • 00:01:52
    at x_n, Newton's calculus. Then the point of intersection with the x-axis
  • 00:02:02
    here will become our next guess for the root, x_n plus 1. We want to calculate a
  • 00:02:13
    formula then for x_n plus  1, so we need to write down
  • 00:02:17
    the equation for this line. What's the  equation for the line? We know the point
  • 00:02:25
    x_n or y_n here, so let me just call this  one y_n. We know y_n is equal to f of x_n.
  • 00:02:40
    We have the equation for the line then is Delta y. Y minus y_n is
  • 00:02:47
    equal to the slope, our f-prime of x_n times  Delta x, times x minus x_n. Then we're trying
  • 00:03:04
    to find x_n plus 1, so we want to find x equal to x_n plus 1 and then the value of y when x
  • 00:03:16
    equals n plus 1 is equal  to 0. Furthermore, we know
  • 00:03:24
    y_n equals f of x_n. That gives us the equation, y is zero, so 0 minus f of x_n is
  • 00:03:41
    equal to f-prime of x_n, x is x_n plus 1,  times x_n plus 1 minus x_n. That gives us an
  • 00:03:59
    equation for x_n plus 1, so we can solve that and we get x_n plus 1 is equal to x_n minus
  • 00:04:11
    f of x_n over f-prime of x_n with a little bit of algebra. This is the
  • 00:04:20
    equation for Newton's
  • 00:04:22
    method of root finding. It requires an initial guess, x-naught. X-naught should be
  • 00:04:29
    close to the root, otherwise you can run away from the root. But if you can guess a root
  • 00:04:36
    close enough to the root, you can get convergence to the root very quickly. Let's try an example.
  • 00:04:44
    Again, we're going to find square root of two,
  • 00:04:51
    and the value is 1.41421. We're going to iterate  here, we have our equation. F of x is our equation
  • 00:05:14
    for square root of 2, so it's x squared minus 2.  Our equation for f of x, the root of this equation
  • 00:05:28
    will be square root of 2. We need  the derivative, so f prime of x
  • 00:05:34
    is equal to 2x. Then Newton's iteration will be x_n plus 1 equals x_n minus
  • 00:05:50
    x_n squared minus 2 over  2x_n. This is what you would
  • 00:05:58
    program in a computer, this would be your correction factor to x_n. But by a hand
  • 00:06:04
    calculation, it's easier to combine them. If  I put them under 2x_n, we'll have 2x_n squared
  • 00:06:12
    minus x_n squared, which is x_n squared  plus 2 over 2x_n for a hand calculation.
  • 00:06:27
    Let's try this. We can choose the value of x naught. Let's take x naught equal to 1.
  • 00:06:37
    Then our value of x_1 from our formula here would be 1 plus 2, which is 3
  • 00:06:48
    divided by 2. So 1 plus 2 over 2 is three-halves, which is 1.5.
  • 00:06:59
    Our value of x_2 then would be n equals  1, so x_1 squared. Three-halves squared is
  • 00:07:09
    nine-quarters plus 2 divided by 2 times three-halves divided by 3, so this is
  • 00:07:23
    17 divided by 12. 17 divided by 12 is 1.416, bar there. Remember square root
  • 00:07:39
    of 2 is 1.41421, so we're ready, getting fairly close. Let's try x_3, what
  • 00:07:48
    does x_3 give us? X_3 then will be x_2 squared,  so 17 over 12 squared plus 2 divided by 2
  • 00:08:03
    times 17 over 12, which is 17 over 6. This is  a rational number, all the approximations here
  • 00:08:13
    will be rational numbers. Square root of 2 is an irrational number. This one you can
  • 00:08:19
    use a calculator, I can't do that in my head; 577 over 408. If you do this in
  • 00:08:30
    decimal is 1.41426. I suggest this  digit is wrong. You see that after
  • 00:08:49
    three iterations, we're already getting pretty close to the square root of 2,
  • 00:08:55
    converging very fast. This is our fastest method, fastest convergence. Let me summarize Newton's
  • 00:09:11
    method. Presumably invented by Newton,  but at least uses the calculus which was
  • 00:09:20
    invented by Newton. The idea then is that you start with some initial value,
  • 00:09:27
    and then you use the calculus to find the slope
  • 00:09:31
    of the tangent line to the curve and  where it intersects the x-axis becomes
  • 00:09:36
    your next value. You're basically approximating your curve by a
  • 00:09:42
    straight line. As you get closer and closer to the root, the straight line
  • 00:09:48
    approximation becomes more and more accurate; close to the root. When you do that you
  • 00:09:54
    get your formula, which is x_n plus  1 equals x_n minus f of x_n over
  • 00:10:00
    f prime of x_n. The only thing  you need is a good initial
  • 00:10:04
    guess for the root. Once you have a good initial guess, convergence is very rapid.
  • 00:10:11
    I'm Jeff Chesnoff. Thanks for watching and I'll see you in the next video.
Tags
  • Newtons metode
  • rotfunn
  • kalkulus
  • iterasjon
  • konvergens
  • kvadratrot
  • funksjon
  • tangentlinje
  • matematikk
  • numerisk metode