That is wrong.jm01's case just means there is no root to find in the vicinity of your first approximation.
And I guess because you didn't mention the word "mathematics" in YOUR argument I guess then you don't have a clue about what you're sayingAs you've not mentioned tangents anywhere in your reasoning, I can only assume you don't understand how Newton's method works.
Why don't you get a decent CAS and actually have a look at your *example*?And I guess because you didn't mention the word "mathematics" in YOUR argument I guess then you don't have a clue about what you're saying
The signs of P(z1) and P"(z1) need to be the same for a good 1st approximation using Newton's Method for finding roots of a polynomial.
Suppose we have a parabola, say.
The tangent drawn from say x=3 will cut the x axis further away from the root than the tangent drawn from say x=4. Is this the "reasoning" that you desire?![]()
The example you provided is flawed.And I guess because you didn't mention the word "mathematics" in YOUR argument I guess then you don't have a clue about what you're saying
The signs of P(z1) and P"(z1) need to be the same for a good 1st approximation using Newton's Method for finding roots of a polynomial.
Suppose we have a parabola, say.
The tangent drawn from say x=3 will cut the x axis further away from the root than the tangent drawn from say x=4. Is this the "reasoning" that you desire?![]()
Maybe I messed something up because I did it mentally.The example you provided is flawed.
The root of that polynomial is approximately 2.214014591. The reason that 2.3 gives a closer estimate than 2.1 is that it lies closer to the value of the root. It is not because it lies above the x-axis.Maybe I messed something up because I did it mentally.
But consider something like say
Why would x=2.1 have been a worse choice as a first approximation using Newton's Method of estimating polynomial roots over x=2.3?
If you start from the 1st approx x=2.1 a better estimate is not reached.The original question said "Using Newton's Method, under what conditions will a better estimate NOT be reached?"
In your example, whether I start from 2.1 or 2.3, a better estimate is reached with the next step.
lol. You just contradicted yourself in the same sentence.The root of that polynomial is approximately 2.214014591. The reason that 2.3 gives a closer estimate than 2.1 is that it lies closer to the value of the root. It is not because it lies above the x-axis.
Um...the distance between 2.10 to 2.21 is 0.11 whereas the distance between 2.21 and 2.30 is 0.09. I'd hardly call that a self-contradiction. We're talking on a very small scale here since both estimates are close to the actual root and clearly 2.3 is closer to the real root than 2.1 which means that after applying Newton's method with the initial approximation of 2.3, this gives a better estimate of the root.You said that the actual root was x=2.21... And you said that x=2.3 is closer to the root than x=2.1? They're pretty much about the same (I think)...
No one is perfect.I haven't looked into that parabola example yet. I'll do it tomorrow or something.
But it's not like it came to me in a dream or anything. It's published in one of the books of the perhaps one of the most prolific HSC Maths authors, ie. Jim Coroneos' 3U book. So it can't really be wrong.
Yeah I know what you mean but I believe that Coroneos had his booked checked by alot of people before publication. Plus most content in the books have been published since something like 1961... that's a long time compared to many other books... So the chance of any error is pretty slim.No one is perfect.
That's like saying Conquering Chemistry/Jacaranda Chemistry can't have anything wrong in it. It can. (Not to mention that I've found some errors in Jacaranda Chemistry during my HSC)
Well, the first edition of Excel Physics was published quite a long time ago. Not as long ago as 1961 but still, since it has errors in it, there's Coroneos might have some too.Yeah I know what you mean but I believe that Coroneos had his booked checked by alot of people before publication. Plus most content in the books have been published since something like 1961... that's a long time compared to many other books... So the chance of any error is pretty slim.
=\ spell the guy's name correctly: Coroneos.Coroneous'
I never really encountered a question that involved the closeness of a root. I shall have a look later but nevertheless the signs of P(z1) and P"(z1) do matter, which is what I said initially.So I am afraid that even the esteemed Coroneous is agreeing with what everyone else here has been saying