# Newton's multiple roots iterative process.

#### Saviour Muscat

Joined Sep 19, 2014
159
Hello,

I am trying to do my math home assignment(attached question2) and I am stuck what is Newton's multiple roots method. I did some found out on the internet and can't locate what is it referring? Newton's Raphson? Newton's modified?
Thanks,
Saviour

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#### ZCochran98

Joined Jul 24, 2018
165
The Newton-Raphson method is the same as regular Newton's method (it's the complete name, to be precise), so it's probably referring to the Modified Newton-Raphson method (like what was mentioned by Papabravo just before I hit "Post reply.")

#### KeithWalker

Joined Jul 10, 2017
1,912
Hello,

I am trying to do my math home assignment(attached question2) and I am stuck what is Newton's multiple roots method. I did some found out on the internet and can't locate what is it referring? Newton's Raphson? Newton's modified?
Thanks,
Saviour
This explains it much better than any youtube video.
Regards,
Keith

https://aaronschlegel.me/newtons-method-equation-roots.html

#### Papabravo

Joined Feb 24, 2006
16,775
I wasn't necessarily recommending the video. It just happened to be on the hit list.

#### Saviour Muscat

Joined Sep 19, 2014
159
Thank you all for your help!

If I understood well, is it referring to Newton's modified as attached?

Thank you
Saviour

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#### ZCochran98

Joined Jul 24, 2018
165
Thank you all for your help!

If I understood well, is it referring to Newton's modified as attached?

Thank you
Saviour

The one in the PDF is the "regular" Newton-Raphson method, though the "modified" one is based on it. The "modified" Newton-Raphson is a little more complicated:
$x_{i+1} = x_i - \frac{f(x_i)f'(x_i)}{(f'(x_i))^2-f(x_i)f''(x_i)}$
Source - a good video explaining it.

Hope this helps!

#### Saviour Muscat

Joined Sep 19, 2014
159
Thank you
ZCochran98

I will let you know when I work the whole solution.

#### Saviour Muscat

Joined Sep 19, 2014
159
Again Hello,

I did a tentative solution(attached) of question 2a,b and c.
1. On first page I found two roots in vicinity of, x=(2.5,3.5) and x=(-0.5,-1.5) then I did six iterations for 2.5 ( root=3) and eleven iterations for -0.5(root=-1).
2. Second page I did algebraic long division which I proved there are multiple roots.
3. Third page I used modified Newton's Raphson method which have shown that converges more faster to find the x=3 root.
I skipped some calculations because the pdf will be too long, and I hope that my solution is correct.
Please guide me accordingly if there are any mistakes.

Thank you,
Saviour Muscat

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#### ZCochran98

Joined Jul 24, 2018
165
The roots are correct. Newton's method is pretty tedious, isn't it? Even finding one root takes a decent number of iterations.

#### Saviour Muscat

Joined Sep 19, 2014
159
Thank you very much,
Saviour Muscat

#### Saviour Muscat

Joined Sep 19, 2014
159
Hello again,
The previous formula was not the exact one that I need to use given by
ZCochran98

please see fig1 attached the good one as given by instructor, I need to know what is g(x) stated in fig2 so I could be able to work out the given question.
Thanks,
Saviour

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Last edited:

#### Papabravo

Joined Feb 24, 2006
16,775
g(x) is the remainder after factoring out the root with multiplicity m

#### MrAl

Joined Jun 17, 2014
8,474
There is also the Lin-Bairstow method which i have used since the 1980's and also in a program to solve for the locus of roots of a transfer function. It looks like Wikipedia has an entry that uses a matrix to solve.
What they may not show is the reduction to n-1 or n-2 roots once one root or a pair of complex roots is found, but that just uses synthetic division.

#### bobokat

Joined Jul 16, 2020
2
How many of you know the background of Newton's inventions and theories? We all know he was a great scientist, but there are many other interesting facts in his life. You know about the laws of gravity, a multi-root iterative process, but did you know that throughout his life Newton believed that he descended from the Scottish nobles, but modern researchers have established that his ancestors were peasants.

#### MrAl

Joined Jun 17, 2014
8,474
How many of you know the background of Newton's inventions and theories? We all know he was a great scientist, but there are many other interesting facts in his life. You know about the laws of gravity, a multi-root iterative process, but did you know that throughout his life Newton believed that he descended from the Scottish nobles, but modern researchers have established that his ancestors were peasants.
Hi,

He also tried to establish a new written/speaking language that would be more logically precise and i think less redundant. He wanted it to take the place of the language currently being used in his geographical location. The way i understand it, it would be sort of like a set of logical statements that you would input to a logical reasoning program, but somewhat relaxed from that extreme, just as logical as possible without the pitfalls of a language like English where there are different rules for spelling in different situations even when the pronunciation is similar.

I almost wish it worked and the world adopted it because there are some rules i just can not see being a good idea. For example, "advertise" turns into "advertisement" which is clearly "advertise" plus the verb suffix "ment" which turns it into a noun, but the pronunciation is not 'adverise' + 'ment' it is "advertiz" + "ment".
To me that is just an unecessary complication that just makes it harder for foreigners to learn the language, and other complications like just keeping track of that difference. We have enough new words coming into the language over time without needed to resort to stupid changes in pronunciations like that. It also detracts from the meaning of the word where it sounds like it might have nothing to do with "advertising" anymore to new people or even people who are just not aware of that modification in the pronunciation.

Also, who else is in favor of respelling "pronunciation" to Popey's version "pronunskiation" (har har)

#### xox

Joined Sep 8, 2017
579
did you know that throughout his life Newton believed that he descended from the Scottish nobles, but modern researchers have established that his ancestors were peasants.

Newton's childhood home at Woolsthorpe Manor had been in the possession of his family for several generations prior so most likely would have been classified as yoemen, not peasants.

Not that it matters anyway. Our own family as well has held various titles throughout the ages, and yet we've had our fair share of paupers too. It all boils down to the individual really. Perceived clout, personal qualities, wealth, etc. All of these things go into what "class" a person and/or their family falls into. At the end of the day they're really nothing more than flimsy labels, IMO...