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http___mathonline.wikidot.com_riemann-stieltjes-integrals-with-increasing-integrators-revi.xml
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Итерационни методи за решаване на нелинейни уравнения. Анотация Да се дефинира понятието неподвижна точка на изображението ϕ и да се докаже, че ако ϕ е непрекъснато изображение на интервала [ a,b] в себе си, то ϕ има поне една неподвижна точка в [a,b]. Да се покаже, че решаването на уравнението f(x) = 0 може да се сведе към намиране на неподвижна точка. Да се дефинира понятието свиващо изображение и да се докаже, че ако ϕ е непрекъснато изображение на интервала [a, b] в себе си и е свиващо с константа на Липшиц q<1, то: а) уравнението x = ϕ(x) има единствен корен E в [a,b]; б) редицата ${x_n}$ от последователни приближения (при произволно $x_0 \in [a,b]$ и $x_{n+1} = ϕ(x_n), n = 0, 1,… ,$)клони към E при n→∞, като $| x_n – E| \le (b - a)q^n$ за всяко n. Да се получи като следствие, че ако E е корен на уравнението x = ϕ(x) и ϕ има непрекъсната производна в околност U на E, за която | ϕ’(E)|<1, то при достатъчно добро начално приближение $x_0$ итерационният процес породен от ϕ е сходящ със скоростта на геометрична прогресия. Да се дефинира понятието ред на сходимост. Да се дадат геометрична илюстрация, формула за последователните приближения и ред на сходимост при: - метод на хордите - метод на секущите - метод на Нютон. Да се докаже, че при метода на хордите сходимостта е със скоростта на геометричната прогресия (при условие, че коренът е отделен в достатъчно малък интервал). Числен Анализ, въпрос 20. Метод на свиващите изображения за числено решаване на нелинейни уравнения. Ред на сходимост на итерационен процес. Числен Анализ, въпрос 21. Методи на хордите, секущите и допирателните (Нютон) за числено решаване на нелинейни уравнения.
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Riemann-Stieltjes Integrals with Increasing Integrators Review We will now review some of the recent content regarding Riemann-Stieltjes integrals with increasing integrators. Let $f$ and $g$ be functions defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. - On the Upper and Lower Riemann-Stieltjes Sums page we considered a special type of Riemann-Stieltjes sum. For every partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$, for each $k \in \{1, 2, ..., n \}$ we defined the numbers $M_k(f)$ and $m_k(f)$ as: - Then we said that the Upper Riemann-Stieltjes Sum corresponding to the partition $P$ is: - Similarly, we said that the Lower Riemann-Stieltjes Sum corresponding to the partition $P$ is: - We noted that for all partitions $P$, that since $\alpha$ is an increasing function that then $\Delta \alpha_k \geq 0$, and for all $t_k \in [x_{k-1}, x_k]$, since $m_k(f) \leq f(t_k) \leq M_k(f)$, we see that: - On the Properties of Upper and Lower Riemann-Stieltjes Sums page we looked at two very important properties. We noted that if $P'$ is finer than $P$ then: - We also noted that if $P_1, P_2 \in \mathscr{P}[a, b]$ are ANY partitions on $[a, b]$ then we always have that: - We concluded that the set of upper Riemann-Stieltjes sums is bounded below by any lower Riemann-Stieltjes sum, and similarly, the set of all lower Riemann-Stieltjes sums is bounded above by any upper Riemann-Stieltjes sum. - With this in mind, on the Upper and Lower Riemann-Stieltjes Integrals page we defined the Upper Riemann-Stieltjes Integral of $f$ with respect to $\alpha$ on $[a, b]$ as: - Similarly, we defined the Lower Riemann-Stieltjes Integral of $f$ with respect to $\alpha$ on $[a, b]$ as: - On the Splitting Upper and Lower Riemann-Stieltjes Integrals page, using some basic properties of the supremum and infimum of a bounded set, we proved that if $c \in (a, b)$ then: - On the Upper and Lower Riemann-Stieltjes Integrals of Sums of Integrands we also proved that: - On the Riemann's Condition Part 1 - The Existence of Riemann-Stieltjes Integrals with Increasing Integrators and Riemann's Condition Part 2 - The Existence of Riemann-Stieltjes Integrals with Increasing Integrators pages we proved a very important theorem that has an extremely useful condition known as Riemann's Condition which provides a necessary condition for a function $f$ with an increasing integrator $\alpha$ to be Riemann-Stieltjes integrable on $[a, b]$. We proved that if $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then Riemann's condition is satisfied, that is, for all $\epsilon > 0$ there exists a $P_{\epsilon} \in \mathscr{P}[a, b]$ such that for any $P$ finer than $P_{\epsilon}$ we have that: - We then proved that if Riemann's condition holds, then $\overline{\int_a^b} f(x) \: d \alpha(x) = \underline{\int_a^b} f(x) \: d \alpha (x)$, i.e., the upper and lower Riemann-Stieltjes integrals of $f$ with respect to $\alpha$ on $[a, b]$ are equal to each other. We proved that the equality of the upper and lower Riemann-Stieltjes integrals then implies the existence of the regular Riemann-Stieltjes integral of $f$ with respect to $\alpha$ on $[a, b]$ which completes the equivalence of these three statements. - On The Comparison Theorem for Riemann-Stieltjes Integrals with Increasing Integrators page we saw that if further $f(x) \leq g(x)$ for all $x \in [a, b]$, then we can compare the Riemann-Stieltjes integrals of $f$ and $g$, and: - Using Riemann's condition, on The Absolute Value of Riemann-Stieltjes Integrals with Increasing Integrators page we saw that further $\mid f \mid$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and that: - Also using Riemann's condition, on The Squares of Riemann-Stieltjes Integrable Functions with Increasing Integrators page we also saw that $f^2$ is then Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ - With the theorem directly above, on The Product of Riemann-Stieltjes Integrable Functions with Increasing Integrators page we were able to show that the product $fg$ is also Riemann-Stieltjes integrable by noting that $f(x)g(x) = \frac{1}{2}[f(x)]^2 + \frac{1}{2}[g(x)]^2 - \frac{1}{2}[f(x) + g(x)]^2$ and noting that Riemann-Stieltjes integrals of all terms on the righthand side of this equation exist by previous theorems we've looked at.