bimbo36

 

No it is a much much much wider subject than that.

 

Most mathematics is linear, and the first approach to any non linear maths is to try to linearise it (=find a linear approximation).

 

You really need to find out and understand what linear maths is.

I do not mean study all its ins and outs, that would take years,

just find out enough to recognise what is linear and what is not and to appreciate the principle consequences of that distinction.

 

The following polynomical falls into the ambit of linear maths

 

y = ax⁶ + bx⁵ + cx⁴ + dx³ + ex² + fx

 

because it is a linear combination of basis polynomials x⁶, x⁵, x⁴, x³, x² x.

 

this polynomial does not (is non linear)

 

y = x + xy

 

Note the word is basis not base. Detail is important in mathematics.

does this one have a graph like this ??

does the word basis has something to do with the propogation with an interval ??

linear maths ?? ok .. i never thought something which had a linear polynomial would end up so complicated ...

in my understanding ... i was classyfying all these to ..

linear polynomial ... a polynomial of degree one ..

non linear polynomials ... polynomials greater than degree one .. quadratic ... cubic ... and quartic .... ( and being able to solve them through factoring and indetities )

in my mind ,

the advanced maths meant ... functions .. differentiation ...intergrations ...

isnt all this .. related to graphs ??? the linear ... the non linear plolynomials??? the complex numbers ?? the differentiation ?? integration ???

sorry for going off topic too much ... but these are all the things running through my head right now ....

 

"Solving" means finding the "roots" ...

 

... a "root" (or "zero") is where the function is equal to zero:

 

 

 

How do we solve polynomials? That depends on the Degree!

 

 

The first step in solving a polynomial is to find its degree.

 

The Degree of a Polynomial with one variable is ...

 

... the largest exponent of that variable.

polynomial

 

When we know the degree we can also give the polynomial a name:

Degree Name Example

0 Constant 7

1 Linear 4x+3

2 Quadratic x2−3x+2

3 Cubic 2x3−5x2

4 Quartic x4+3x−2

a polynomial with degree one ? isnt that what is called by linear algebra ???

i am only familiar with the degree and coefficients in a polynomial ...

no , i am not familiar with the term base ...

i found this book online ...

a graduate introduction to numerical methods ...

let me see if i can focus on simple polynomials , to try to and solve a few questions .. or atleast try to understand them properly ...

in it ...

 

Polynomials and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1 Polynomials, Their Bases, and Their Roots . . . . . . . . . . . . . . . . . . . . 44
2.1.1 Change of Polynomial Bases . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.1.2 Operations on Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Examples of Polynomial Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.1 Shifted Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.2 The Newton Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.3 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.4 Other Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2.5 The Clenshaw Algorithm for Evaluating Polynomials
Expressed in Orthogonal Bases . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.6 Lagrange Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.7 Bernstein–B´ezier Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3 Condition Number for the Evaluation of Polynomials . . . . . . . . . . . 63

currently i am only familiar with .. polynomials , their degrees and their coefficients ...

i have not come across a lot of books like these ones ...

let me try to read its simple parts ... and see if it makes any sense to me ...

changes of polynomial bases ... ??

i have to understand that change a bit ...

i will spend my free time in understanding the polynomials ... so that sometime in the future i can try one or two examples like that mentioned above .....

thanks ...

 

First, and I cannot stress the importance of this enough.

 

You have exact solutions in this case.

 

Numerical methods are all about the real world.

In the real world your equation is likely to be presented something like

 

 

1.0057X² - 11.993725689X - 28.000000007 = 0

 

and you are not sure about the last few digits of any of the coefficients.

 

This is where you will use numerical methods to get not an exact answer but an answer that is good enough.

 

Back to the example, please note that this forum allows the use of superscript and subscript (you will find them on the second row of the toolbar) to better present mathematics.

 

Your equation is

 

1X² - 12X- 28 = 0

 

rearrange

 

12X = X² -28

 

X = (X² -28) / 12

 

Now look back at the formula I presented in post 2

 

We have a formula for X, but it is in terms of X.

 

If we take a 'guess' for X and insert it into this formula we can calculate another value of X.

 

If we take this new value of X and insert it into the formula we can calculate yet another value for X

 

and so on.

 

The hope is that each 'iteration' (=approximation) will get better than the last and the calculations will converge on the correct answer.

 

That is why this is called the recursiv.e or iterative method.

 

I have shown the results of doing this in the spreadsheet.

 

Have you any experience with spreadsheets?

thanks a lot ... this has to be the best explanation i have ever read so far ...

all of this makes a bit more sense right now ...

even the books i read were very confusing , so was some websites i went through ...

none of them had proper simple examples ... my text book was called computer oriented numerical method ... it had computer programs dealing with polynomial numerical methods ... but it was lacking good example questions ... so i had a very hard time understanding all those....

i have little experience using excel sheet ... i can do table , autosum and little things like that in it .. at this point i dont know if it is capable of doing advanced maths on its tables ....

anyway i have to re read everything when i get more free time ...

 

Using the generalized quadratic formula for polynomials of degree 2.

 

[latex]ax+bx+c[/latex]

[latex]x^2-12x-28[/latex]

 

Sub the values for the variables into the generalized quadratic formula

 

 

[latex]\frac{12\pm \sqrt{-12^{2}-4(1)(-28)}}{2(1)}[/latex]

[latex]\frac{12\pm 16}{2}[/latex]

 

Finding the roots.

 

[latex]\frac{28}{2}=14[/latex]

[latex]\frac{-4}{2}=-2[/latex]

 

No it is useless if the quadratic has complex roots like this one X^2-5x+25.

you mean numerical method is useless if the quadratic has complex roots ???

let me start with the very basics itself ...

 

"Solving" means finding the "roots" ...

... a "root" (or "zero") is where the function is equal to zero:



How do we solve polynomials? That depends on the Degree!


The first step in solving a polynomial is to find its degree.

The Degree of a Polynomial with one variable is ...

... the largest exponent of that variable.
polynomial

When we know the degree we can also give the polynomial a name:
Degree Name Example
0 Constant 7
1 Linear 4x+3
2 Quadratic x2−3x+2
3 Cubic 2x3−5x2
4 Quartic x4+3x−2

this is therefore a quadratic equation ...

i actually took this example from youtube , to make sure ... this problem has a proper solution ...

anyway , this is how you usually solve a polynomial .. right ?

https://www.youtube.com/watch?v=g7_llQnLepA

x = 14 and x= -2

is the answer ...

how else would i solve this ??

where would i apply the numerical methods on this one ??

 

Numerical Methods and errors

Interpolation

Numerical Differentiation

Numerical Integration

Solution of Algebraic and Transcendental Equations

Numerical Solution of a system of Linear Equations

Numerical Solution of Ordinary differential equations

Curve fitting

Numerical Solution of problems associated with Partial Differential Equations

we had that much amount of stuffs to study ..

there are many stuffs i dont understand about all those ...

i am trying to find the simplest from it to work with examples ...

i think ...

 

Solution of Algebraic and Transcendental Equations

Numerical Solution of a system of Linear Equations

 

i have no idea , which one is simplest among those two to look for an example ...

does those two only deal with polynomials of degree one ??

even if it is only dealing with polynomials of degree one , how does an example look like .. ?? and what am i trying to solve ? what is this interval they all speak of ... and what increments or dicrements in these problems ? the co eficients ? the degree of the polynomial ?

i have lot of doubts regarding polynomials and numerical method .. can someone help me understand some basics of it ... ?

how do i dissassemble this sofa of polynomial properly ?

and how does numerical method help ? can i have an example of a simple problem where i can utilize numerical method ???

what do i do here ?

sweet .. thanks for all the suggestions ...

i am a user at library.nu .. ( the site has loads of ebooks ..) so please suggest a nice book if you remember any ...

even though i can understand bit of complex numbers .. mostly the basic operation like addition, subtraction, divison ...

i dont know what else is there ...

but i cant understand any part of "complex geometry"

i have three months to study everything in the "heading "

where should i start ..

my whole life is depending on this crap ..

seriously ..

help ...

d/dx sin = cosx

d/dx cosx = - sin x

d/dx tanx = sec²x

then chain rule ...

composite two functions and composite three functions ...

d/dx f(x)og(x)

Df(x)og(x) * Dg(x)

------------------------------------------------------------------------------

d/dx f(x)og(x)oh(x)

Df(x)og(x)oh(x) * Dg(x)oh(x) *D h(x)

i have sort of figured out chain rule for two composite functions .. and thank god it works ..

sin x xrise 4 xrise 3

so this is three composite functions ..

i cant get the answer the way i am working with ..

i wish i had more examples to work out, of functions including only "two composite functions"

i will deal with three , later ...

derivative (outer function ) (whole terms) * derivative (inner function)

?

thanks

need a short break and a bit of revisions ..

be right back ..

so you throw this package in the box ?

functions ..

 

what is it ?

 

 

a function has y

 

a function has x

 

a function has f(x)

 

an equal sign ..

 

then numbers ..

 

roots ..

 

trigonomteric stuffs .. like sin cos .. theta ? sinx ?

 

 

fractions ...

 

sometimes the x is with numbers like multiplied ..

 

then some are with minus signs ..

 

some have ln .. or logarithm in it ..

 

 

 

 

the equations .. the quantites ..

 

are those like ..

 

y= 3apples+four and a half mangoes ?

 

find dy /dx ?

 

yea ..?

 

dy/dx of 3 apples + dy/dx of four and a half mangoes

 

 

 

 

try to find the change in y ..

 

when the quantities of apples and mangoes changes ?

 

 

 

I think you're getting confused for nothing.

 

Do you know what the derivative of sinx is?

Do you know what the derivative of sqrt(X) is?

 

The first couple of posts were about trying to understand what he question asks. I doubt that they're asking for an infinite series or sum according to what I see from the poster. Do correct me if I'm wrong.

 

Now, to the point. You don't need a graph. You don't need to explain this formula in words. Derivate with chain rule.

 

Just so we know your level (and hence how to helpyou) can you tell us what grade this is for, and if you've done any sort of derivatives and chain rule exercises?

if differentiation was about finding slopes ..

between points ..

how many points are we dealing with here ?

i think the derivative of sinx is cosx ..

sqrt of x = x^1/2

which is 1/2*x^1/2-1

isnt it ?

This may help:

 

http://www.sciencefo...post__p__409619

 

In this case you have several functions inside each other. sin(x) inside a square root, inside a square root with another sin x, etc.

i am writing that down .. and learning it ..

very nice tutorial mr cap

dude i was thinking about putting it in a graph .. one by one

in a transparent graph ..

one top of another ..

to help clarify the use of f(x) is saying we have a function that takes x as the input....so f(4) is the same as putting a 4 for every place you see an x on the right hand side of the equality.

wtf is this shit then ?

this function takes x as input ?

the whole sugaroot of sin1 + sugar root of sin1 + sugar root of sin 1 ..

 

is different from sugar root of sin1+sugar root of sin1+sugar root of sin 1

if i had one graph that was like ..

f(x)= sugar root of sinx

which could take x as input ...

so f(1) would be like ...

f(1)= sugar root of sin1

find dy/dx

silly but if you have .. it would be really nice if you could reply

how do i graph one ?

with f (1)

f(2)

f(3)

then find the slope ?

which is same as y = ^

i hope ..

anyway ..

help

ok .. cool

was working my ass off ..lol .. and not reaching anywhere ... exhausted to the mathz ...

now , since its like a sum .. i should use chain rule ? yes ?

could you show me how to ?

Suppose you want to calculate the derivative of a curve at some particular point. Draw a tangent to the curve at that point. The slope of the tangent gives you the derivative. This is geometrical approach.

 

The algebraical approach is little different. Let y =f(x) which means is a function of x. A very small change in y is denoted by dy and a very small change in x is denoted by dx. The derivative of y means the amount by which y changes if x changes in a very short extent. So mathematically it is given by .

according to that

f(x)=

so here

y = f(x) .. means

that ?

the amount of y changes if x changes in a very short extent .. ?

Where did the square root go?

 

You start out with the big square root, take a derivative of it, and then multiply by the derivative of whatever is inside it (chain rule).

But as you do that, you have another square root. Don't let that scare you, all it means is that you need to multiply the derivative again (chain rule). But as you do THAT derivative, you encounter yet another square root. Again don't fear, just continue the chain rule.

 

It is a chain rule problem. It's just a longer one that requires three chains.

 

Start it out, and I can guide you better once I see how you do the first few steps.

the big square root and derivative .. ok .... step (1)

X (into)

derivative of two more square root of sinx ?

how many terms are we dealing with here ..

you know if it was a ploynomial .. there would be two or three terms .. right ?

and we usually find the derivative of each term .. yea ?

This is a chain rule problem. Go from "outwards" and then "inwards" in sequence.

 

[math]y=\sqrt{sinx+\sqrt{sinx+\sqrt{sinx}}}[/math]

 

Start with the differentiation of the big square root, then multiply by the derivative inside it, which includes a square root, etc.

 

If you post how you started it, we can help you continue. We don't give final answers here, just help you get there.

 

~mooey

lets see ..

a single y is sinx+sinx+sinx

The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change)

so ..

am i trying to find the second y ?

first let me clear that .. and will continue ..