Hallo, All!
Please, could anyone explain me what are linear and quadratic convergence?
I just dont understand following: if f(p) is from interval (0,1) then there is a linear convergence.
Can someoe help?
Best regards, M
Hallo, All!
Please, could anyone explain me what are linear and quadratic convergence?
I just dont understand following: if f(p) is from interval (0,1) then there is a linear convergence.
Can someoe help?
Best regards, M
Are you sure the place where you read this fact does not have the relevant definitions? It is probably talking about the rate of convergence.
If it does not , then ok. I just foudn it on net and that confused me.
I wil cut and paste:
Limit (mathematics) - Wikipedia, the free encyclopedia
"Convergence and fixed pointA formal definition of convergence can be stated as follows. Suppose as goes from to is a sequence that converges to a fixed point , with for all . If positive constants and exist with
then as goes from to converges to of order , with asymptotic error constant
Given a function with a fixed point , there is a nice checklist for checking the convergence of p.
1) First check that p is indeed a fixed point:
2) Check for linear convergence. Start by finding . If...."
"As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 1/5, 2/5, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √(1⁄5), √(2⁄5), and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely..."
PLEASE, HOW DO WE GET SQUARE FRM 1/5 ETC? I just dont get it. It shoudl be the surface under the curve.
$\displaystyle \sqrt{\frac{1}{5}}$, $\displaystyle \sqrt{\frac{2}{5}}$, ... are the values of $\displaystyle \sqrt{x}$, which is the function that is being integrated, at the approximation points $\displaystyle \frac{1}{5},\frac{2}{5},\dots,1$. These values are used to find the area of yellow rectangles in the picture. This area, in turn, is an approximation of the area under the graph of $\displaystyle \sqrt{x}$.
It would help if you explain the transition from the rate of convergence to integrals. They are not directly related. E.g., you don't need the definition of the rate of convergence to define integrals.
No, the area of green rectangles is an approximation to the area under the curve. Note that I did not mention green rectangles, only yellow ones. You like to change the subject without explanation: first from the rate of convergence to integrals, then from yellow to green, don't you?
By "quadrat square" do you mean square root? I already explained this:
In other words, $\displaystyle \sqrt{1/5}$ is the height of the leftmost yellow rectangle. This height is multiplied by the width 1/5 to get the area of the rectangle.
To approximate the area under the graph, the height of the rectangle is chosen to be the y-coordinate of the point on the graph above 1/5. Do you know what the graph of a function is? The graph of f(x) is a set of points with coordinates (x, f(x)) for various x. Here the function is $\displaystyle f(x)=\sqrt{x}$. Which point is located on the graph above 1/5? By definition of the graph, it is $\displaystyle f(1/5)=\sqrt{1/5}$.
$\displaystyle \sqrt{1/5}\approx0.45$.
Please (untill U still have nerves!!!) how did we get that height is square root from 1/5? That is the main problem!
(P.S. Please, excuse me, I am in a terreble hurry, and still this, cause I am not gifted for mth at all!)