Newton-Raphson method

Newton-Raphson method is a recursive algorithm for approximating the root of a differentiable function. We know simple formulas for finding the roots of linear and quadratic equations, and there are also more complicated formulae for cubic and quartic equations.  The Newton-Raphson method is a method for approximating the roots of polynomial equations of any order. In fact the method works for any equation, polynomial or not, as long as the function is differentiable in a desired interval.

The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line , and one computes the x-intercept of this tangent line . This x-intercept will typically be a better approximation to the function’s root than the original guess, and the method can be iterated. (Show in formulas below)

Suppose ƒ : [a, b] → R is a differentiable function defined on the interval [a, b] with values in the real numbers R. The formula for converging on the root can be easily derived. Suppose we have some current approximation xn. Then we can derive the formula for a better approximation, xn+1. The equation of the tangent line to the curve y = ƒ(x) at the point x=xn : y = f’(xn) (x- xn) + f(xn) where, ƒ’ denotes the derivative of the function ƒ. The x-intercept of this line (the value of x such that y=0) is then used as the next approximation to the root, xn+1.

In other words, setting y to zero and x to xn+1 gives 0 = f’(xn) (x- xn) + f(xn)
Solving for xn+1 gives, xn+1 = xn – f(xn) / f’(xn)

Proof of quadratic convergence for Newton’s iterative method:

According to Taylor’s theorem, any function f(x) which has a continuous second derivative can be represented by an expansion about a point that is close to a root of f(x). Suppose this root is  Then the expansion of f(α) about xn is

where ξn is in between xn and a. Since is the root, (1) becomes:

Generalizations - Newton Fractal: When dealing with complex functions, Newton’s method can be directly applied to find their zeroes. Each zero has a basin of attraction in the complex plane, the set of all starting values that cause the method to converge to that particular zero. These sets can be mapped as in the image shown. For many complex functions, the boundaries of the basins of attraction are fractals.

  • Newton fractal for three degree-3 roots P(z) = z^3 - 1, coloured by root reached. (A Julia set for the rational function associated to Newton’s method for f : z→z3−1. Coloring of Fatou set according to attractor, the roots of f)
  • Newton fractal for P(z) = z^3 +2z +2 . Points in the red basins do not reach a root.
  • Newton fractal for P(z) = z^5 - 3iz^3 -(5+2i)z^2, coloured by root reached, shaded by number of iterations required.

Image: Fractal de Newton

My handsome Link. I will go on a tangent here. Link is my 3yo lead dog. He is my heart and soul dog. He has been urban mushing with me since the beginning. We learned together. We crashed together. We got up together. He and I began the journey that led me to mushing groups, to meeting my mushing obsessed boyfriend, to holding mushing workshops for those in my area. What would I do without you Link? I can’t even imagine. We are just mushing each day to the fullest :) photo credit @piperpippa

meangreenlimabean asked:

Jenn, I can't stop thinking about that X-files ep where Scully got dressed and we saw nekkid Mulder in her bed as she left. I woke up at 2am last night with the most bizzare idea, follow me for a moment. WHAT IF that Marie girl from the fan fiction episode was the next prophet after Kevin? and WHAT IF Dean really does turn into a woman like Marie said? and WHAT IF we are left with a shot of very male Dean getting dressed and leaving nekkid Cas asleep in his bed. I'm freaking out!!!

While part of this is confusing me, seriously, it would be so fucking easy to make destiel explicitly canon without saliva ever being swapped. And this is definitely one of the ways.

One of things that annoys me the most if that people want to say Jensen won’t allow Destiel to happen. I hate that Jensen’s professionalism is disparaged like that. Jensen, in the end, will do what Carver and Co want because it’s his job and he is the absolute consummate professional. While he might change lines to make something sound more like Dean, he won’t rewrite scripts. He, in the end, does what Carver tells him. We saw this with the whole Shake It Up thing. He didn’t want to do it, but it was important enough that Carver insisted and Jensen complied. Because he’s a professional.

So that got way off tangent, but yeah. Destiel could easily be made explicitly canon without sex or even kissing. And that’s what people don’t seem to understand about us Destiel shippers. We’re happy to keep the sex in the pages of our fanfics. In the end, we just want Cas and Dean to be happy, and the truth is, they’d make each other happy. 

I want a family and dog for Sammy, and Cas and Dean for Cas and Dean. 

God has put this calling on my life and I have all these questions. Why? How? When? The answer I do know is that He is going to make it happen. Somehow, someway, He will make a way.

It’s kinda cool how He does that. He teases us. He puts this calling or desire in our hearts. We see it; we feel it. But we are left with our mouths full of questions. He dangles it in front of our eyes and all we can do is trust. Trust that God will put strength in out weak bodies to go out there and do it. Trust that God will put wisdom is our mind to fulfill this calling. Trust that God will provide whatever needs to be provided. Trust. Easier said than done right? You know when He has something planned for you it’s going to turn out beautiful. So what do you have to fear?

Sometimes we are presented with a problem which cannot be solved by simple algebraic means. For instance, if we needed to find the roots of polynomials , we would find that the tried and true techniques just wouldn’t work. However, we will see that calculus gives us a way of finding approximate solutions

Newton–Raphson method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function: f(x) = 0

Consider x1 to be the initial guess root of the function f(x) which is essentially a differential function. Now, to derive better approximation, a tangent line is drawn as shown in the figure. The equation of this tangent line is given by:

y = f’(x1) (x- x1) + f(x1)

As shown in the figure, f(x2) = 0 . Therefore, 0 = f’(x1) (x2- x1) + f(x1)
Solving, x2 = x1 – f(x1) / f’(x1). Repeating the above process for xn and xn+1 terms of the iteration process, we get the general iteration formula for Newton-Raphson Method as:

xn+1 = xn – f(xn)/f’(xn)

 Example: Consider f’(x) = 3x^2  -1. Lets determine the guess value.
 f(1) = 1 -1 -1 = -1 and f(2) = 8 – 2 -1 = 5

Therefore, the root lies in the interval [1, 2]. Since, f(3/2) = 0.875 >0. Consider x1= 1.5 as the initial guess root of the function f(x) = x3−x−1.
f(1.5) = 1.53 – 1.5 – 1 = 0.875
f’(1.5) = 3 * 1.52- 1 = 5.750

Using Newton’s iteration formula:
x2 = x1 – f(x1)/f’(x1) = 1.5 – 0.875/5.750 = 1.34782600. The iteration for x3, x4, …. is done similarly.

n                                        xn                                                     F(xn)

1                               1.34782608696                            0.100682173091

2                               1.32520039895                             0.002058361917

3                                1.32471817400                            0.000000924378

4                                1.32471795724                             0.000000000000

(Example from CodeWithC.com)

Steps to find root using Newton’s Method


4 golden circles

Loved this construction from Bùi Quang Tuån, via Alexander Bogomolny at Cut-the-Knot. He has the proof. I wanted to explore the relationship of this aesthetically pleasing construction. That is it is so phi-full is pretty sweet.

Two tangent pairs of overlapping congruent circles. If the overlapping circles pass through each other’s center, bam! Golden ratio all over the place.  On GeoGebraTube.

so at my old high school we had a regular substitute teacher the entire school affectionately referred to as “The Bike Sub“ because he would, without fail, drag his biking equipment in with him every class, and at the slightest provocation, spiral each and every class into an incoherent tangent on his biking hobby for the full hour of the timeslot

his secondary trademark was that he’d stick pennies to the walls of the classroom literally every time he subbed and refused to tell anyone how he did it

The last time I saw him was in my twelfth grade law class and he spent the entire class completely disregarding the syllabus and talking about aura reading and how to train yourself to see the auras of others (it involved a lot of deliberately unfocusing your eyes and assigning mystical value to the blurry outlines everything magically gained as a result)

public school is a wild ride, kids.

when ur in bed at night just know that somewhere, out there, Jimmy mcgill is probably on his pullout couch with a full cup of cucumber water and a shitload of random heavy liquor drinks mixed in crying into his pillow about how much his big brother screwed him over and how he thought his brother was proud of him for becoming a lawyer when in reality it’s the exact opposite and im sorry i lost track of what I was saying

Some stuffs I wanna share with my followers in high school

- Don’t destroy yourself over grades. The only reason you should work that hard in high school is so that you can hopefully land a scholarship or two. You may think “it’s fine, I can still graduate with mediocre grades” but I promise you— you’re gonna wish you worked just a bit harder when you watch your crippling debt pile up. There is no shame in hard work.
- Get off your phone for a sec. Get off the computer. Sit in silence and ask yourself what you want to spend your life doing. Have dreams, have goals, remember to just think sometimes. I know it’s stressful but it’s a bad habit to have a problem for every solution. Give yourself a chance and ask yourself the big questions throughout high school. Try to pave a path for yourself.
- If you don’t have friends, if bullying is brutal, if all your friends have graduated, fucking change schools or look into alternative schooling programs your high school has to offer, if you have the opportunity. I wish I did this. All my friends graduated and I was stuck with my grade my senior year— and my grade was full of dickbags. I’m really not sure why I stayed. I had a horrible senior year— so if you’re in a similar situation, just trust me when I tell you to fucking leave. In retrospect, you may realize you had nothing to miss there anyway.
- Eat. Fucking. Breakfast. Pack snacks, too. You guys are going through a scary amount of puberty and you gotta eat. High school wakes you guys up way too early and it really doesn’t make sense— they’re taking away the sleep you need, but just make sure they’re not taking your nutrients.
- Limit your intoxication. There will always be opportunities to get fucked up wherever you go, and you’ll find cooler people to do it with. Learn to cope with pain while sober. You have more important shit to be getting done. Get your fucking act together and focus on what matters. Take better care of your body. Get your shit done. There is a time and a place to turn up— do not lose control over your life. It can happen fucking easily. REALLY easily.
- Don’t argue with teachers in class. Teachers will sometimes call you out for not working diligently and it’s fucking rude and humiliating. Rather than talking back, just ask “Can I talk to you about it after class?” It’s easier to get their respect that way. If you act like a shitstain you’re gonna be treated like one.
- Report people. Report teachers. Report anyone who’s doing something fucked. I don’t get why none of you utilize this. One thing that’s funny in high school is that you can get that little shit in your chemistry class /grounded/. You guys can fucking ground each other, that’s hilarious.
- Write notes on your body if you wanna cheat on a test. Teachers only look for phones these days.
- Don’t let yourself get patronized by adults. Their authoritative personalities don’t actually exist in real life— they’re just fucking rude sometimes. If your teacher actually disrespects you in a way that they wouldn’t disrespect an adult, get your parents involved. This is really only if you need a serious change to be made— an example could be unfair dress code policies or teachers who steal your phone without a warning or whatever.
- Don’t just skip class. Go to class, get attendance, and get a pass to the nurse. Say you’re gonna throw up and they won’t say no.
- Don’t resent high school for teaching you algebra instead of teaching you how to take out a loan. That’s not high school’s job. That’s your parent’s job or the bank’s job. Or you can honestly just google it. You know you can.
Here’s what you SHOULD try to get out of high school:
- A decent work ethic.
- Some understanding of social norms.
- Interest in certain fields of study. Dreams, goals.
You may think to yourself: Wow, this list is so fucking ableist! Not everyone can—!
Listen… When I was in high school, I gave up on shit easily because I “couldn’t” do it. I had some medical reasons for shit too, we all do. But looking back at it, I think… Hey, I’m sure I coulda done that. It would’ve just taken me longer to get good at it. I would’ve struggled way more than the other kids, but I wasn’t completely hopeless.
I want all of you to know that, too. There’s a future for you all. None of you are hopeless. There is reward in hard work. There is a better life for you all outside of high school. The pain will end. You’ll end up somewhere YOU want to be if you work for it. Don’t be intimidated by the idea of earning what you want— you can all do it. So yah.

saint patrick’s day

this is your yearly reminder that saint patrick’s day is a celebration of the genocide of celtic paganism, its religious texts, and its practitioners. please dont celebrate it and especially dont wear green for it because that has to do with the orange and green wars which are a totally different topic. wear black in mourning and in support for the “snakes”. educate people and help stop the celebration of genocide


AU in which Spock’s blind as a bat and Jim forgets he’s a touch-telepath. MORE UNDER THE CUT.

I was going off on a tangent the other day about how much love I have for Spock wearing ZQ’s thick glasses, and thayes gave me this adorable scenario of Spock losing them and Jim using it as an excuse to hold hands. SO THIS ONE’S FOR YOU, TAYLOR.

And please if anyone has a Spock-with-glasses headcanon send them to me because they are my only source of sustenance.  

Keep reading

pick up lines the signs would probably use

aries: i bet my tongue is stronger than yours, wanna find out?

taurus: are your parents bakers? cause you’re a cutie pie.

gemini: your face is a work of art. we should frame it with my legs.

cancer: did you just fart? cause you blew me away.

leo: are you an orphanage? cause i wanna give you kids.

virgo: i wish i was your derivative so i could lie tangent to your curves.

libra: i’m not asian, but i’ll eat your cat.

scorpio: nice shoes.. wanna fuck?

sagittarius: do you work at papa john’s? cause i’d like a pizza that ass.

capricorn: do you have eleven protons because you’re sodium fine.

aquarius: are you from china? cause i’m china get in your pants.

pisces: roses are red, violets are fine, if i’ll be the 6, will you be the 9?

Everyone needs to stop what they are doing and go watch The Great Sperm Race, which is a not only a truly fantastic documentary about sperm, it’s narrated by RICHARD ARMITAGE.  Listen to me. Listen to my wisdom. 

  • Richard Armitage says all the naughty words you giggled at when you were seven.  Sex. Semen. Vagina. Orgasm. 
  • The entire fertilization operation is dramatized by dressing up grown men and women in white suits and having them run desperately through mountain ranges that symbolize the vaginal walls.
  • “The cervix is, quite simply, sperm hell.” 
  • They turn The Gherkin in London into a giant testicle.  Richard Armitage says the words “giant testicle.” 
  • You get to watch TV-appropriate sex between two average-looking white people while hearing Richard Armitage say phrases like “performance between the sheets.” 
  • At one point they go into a tangent about sex workers and LAPDANCES. 

In short, this documentary is a thing of beauty and you must watch it immediately. It’s called THE GREAT SPERM RACE. Narrated by RICHARD ARMITAGE. How could you go wrong?????