Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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1answer
13 views

find estimation of interpolation error for non differential function

Given $f(x)=|x|^{1/2}$ , $-1\le x\le 1$ , I have found the interpolating polynomial $ p(x)=x^2$ for $x_{0}=-1,x_{1}=0,x_{2}=1$. How to estimate $$\max_{-1\le x\le 1}|f(x)-p(x)|$$ now that $f$ is not $...
0
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1answer
22 views

interpolation error using higher derivatives

Given $x_{0},x_{1},x_{2}\in[a,b] $ each one different from the others,$f \in C^{4}[a,b]$ and $p\in\mathbb{P}_{3}$ so that $$p(x_{i})=f(x_{i}), i=0,1,2 $$ and $$p'(x_{1})=f'(x_{1})$$ prove that: $$\...
-1
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1answer
61 views

How to find the numerical error when we don't know the exact solution? [closed]

When some quantity $x$ (e.g., the values of a solution of a PDE, using a finite difference method) is calculated numerically, we get its approximate value $x^*$. The error is $|x-x^*|$. But since we ...
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1answer
23 views

Is $\frac{\rm d}{\rm d\omega(t)}\int_{t_0}^t\omega(t')\rm dt'=\int_{t_0}^t\frac{\rm d \omega(t')}{\rm d\omega(t')}dt'=t_0-t?$

I'm trying to calculate an error propagation, but the expression in the most LHS of the equation in the title crops up. Are you allowed to simply exchange the order so that the operations cancel? $\...
-1
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1answer
40 views

How do you derive the backward differentiation formula of 3rd order using interpolating polynomials?

It was my exam question, and I could not answer it. How do you drive the backward differentiation formula of 3rd order (BDF3) using interpolating polynomials? I only knew how to derive it using the ...
2
votes
0answers
31 views

Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-...
0
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1answer
30 views

What is the appropriate way to use the Runge-Kutta method to calculate neural network node activities?

I am a cognitive scientist, and am modelling a particular kind of neural network called a "masking field", where the change in activation of a particular node in the masking field is: $${dy_i(t)\over ...
0
votes
2answers
144 views

Quadrature integration: calculating the weights

We've started today the Integration part on out Numerical Method course. Our professor wrote this exercise on the blackboard and we've been told to start thinking about it. It's the next one: ...
0
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1answer
12 views

Normalizing vector which causes overflow

Let's say I have a vector of the form $$[a_1e^\frac{r_1}{a_1},\hspace{2mm} a_2e^\frac{r_2}{a_2}],$$ which I can't compute because $\frac{r_i}{a_i}$ is large enough that when I raise $e$ to it, it ...
0
votes
0answers
34 views

Calculate number of trials reaching $p_k$ probability for $k$ successes given the $p_t$ probability of each trial success

Basically, I'd like to be able to answer questions in the form of "What is the number of trials needed to have at least $p_k$ probability of at least $k$ successes, given that on each trial the ...
0
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1answer
33 views

verification: Newton method convergence

P(x)=x$^2$-x-1 has a root of $\frac{1+\sqrt5}{2}$=1.61803398 My iterations are: x$_0$=2 x$_1$ =$\frac{5}{3}$=1.6666 x$_2$=$\frac{34}{21}$=1.619 x$_3$=$\frac{1597}{987}$=1.618 So it took me 3 ...
0
votes
1answer
28 views

Verification: fixed point formula

I have $2+\sin(x)-x=0$ and I need to formulate so that it converges in $[2,3]$. I have verified that the formula $x=2+\sin(x)$ gets a convergence in this region. Now to verify the assumptions my ...
1
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1answer
65 views

Root of function

Can you find root of the equation $$f(k)=1+(1-k^2)\ln(1+\frac{1}{k})?$$ I tried to use Matlab command but it does not give me any result. Can you suggest a method to find root of equation.
0
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1answer
44 views

Newton method iteration

I am trying to solve non-linear systems and since I can't download matlab on this device, I was wondering if there is a way I can set it up in excel. I know the formula for x$^{k+1}$=x$^k$-[Df(x)]$^-...
0
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1answer
36 views

Matrix-free conjugate gradient

In the conjugate gradient method for solving $Ax = b$, to update the search direction $p$ you would need to evaluate the matrix-vector product $Ap$, i.e. making sure that each search direction are A-...
0
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2answers
51 views

Solving $u'' - 5u = 6$ with finite difference methods.

I have a task: For an equation: $$u'' - 5u = 6, x \in (0, 1)$$ $$u(0) = 0, u'(1) - 3u(1) = 1$$ construct a recurrence relation("scheme" in the original) with second order approximation on a two-point ...
5
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3answers
115 views

Approximating the normal CDF for $0 \leq x \leq 7$

In answer http://stackoverflow.com/a/23119456/2421256, an approximation of the complementary normal CDF (ie $\frac{1}{\sqrt{2 \pi}} \int_x^{+\infty} e^{-\frac{t^2}{2}} dt$) was given for $0 \leq x \...
3
votes
3answers
409 views

What is the sum of this alternating series?

I need to find the sum of an alternating series correct to 4 decimal places. The series I am working with is: $$\sum_{n=1}^\infty \frac{(-1)^n}{4^nn!}$$ So far I have started by setting up the ...
2
votes
1answer
37 views

How to show the fixpoint iteration of a function converges? EDITED

Let $g(x)=2x-e^{-x}$. For a function $G(x)$ such that $G(x)=x-\alpha g(x)$, how to show the fixpoint iteration of $G(x)$ converges for all $\alpha$ with $0<\alpha<2/3$? I couldn't find any ...
0
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1answer
15 views

How to find the number of iterations needed within a certain degree of accuracy in the bisection method

I know how to find a zero of a function by the bisection method. But I am not sure how to find the number of iterations needed within a certain degree of accuracy. Let's say, when we use the ...
-2
votes
4answers
27 views

How to show a zero of a function is a fixpoint of another function?

$g(x)=x \log(x+1)+x-1$ where the log has the base $e$. Set $G_1(x)=1/(\log(x+1)+1)$ and $G_2(x)=1-x\log(x+1)$. Show that the zero $x^*$ of $g(x)$ is a fixpoint of $G_1(x)$ as well as $G_2(x)$. My ...
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3answers
39 views

How do you graph an inequality on Real/Imaginary plane?

Suppose we have $z$ as a complex number, $z \in C$, how do you graph an inequality which has $z$ in it? This kinds of inequalities arise when we need to graph the shape of stability region of a given ...
2
votes
1answer
60 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
0
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1answer
32 views

Clarification on linear vs. quadratic convergence

I'm having some trouble in understanding the different definitions for the rate of convergence of a sequence. I was playing around with following two functions, but even after experimenting for a ...
1
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0answers
33 views

Least square polynomial interpolation

Given an arbitrary continuous function f(x), let Pn(x) be the polynomial of degree at most n that approximates f(x) in the least squares sense. Is it true that Pn(x) interpolates f(x) at n + 1 points? ...
1
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0answers
35 views

Using Finite Differences and Integration to prove result

If $f(x)$ is a polynomial in $x$ of third degree and: $$u_{-1}=\int_{-3}^{-1}f(x)dx\ ;\ u_{0}=\int_{-1}^{1}f(x)dx\ ; u_{1}=\int_{1}^{3}f(x)dx$$ then show that $$f(0) = \frac{1}{2}\Bigg(u_0-\frac{\...
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0answers
14 views

Weighted Difference Schemes

I hope most of you are familar with weighted finite difference approximations of derivative. e.g; $$u_x=\frac{s(u(x+h)-u(x))}{h} +\frac{(1-s)(u(x)-u(x-h))}{h}$$, where $0\le s\le1$. I want to know ...
0
votes
0answers
22 views

Runge kutta method

In Runge Kutta method, we have seen things like $u_{n+1}^1, u_{n+1}^2, u_{n+2}^1$, I am confused about $u_{n+1}^2, u_{n+2}^1$, I kinda of mix them together. Can someone please explain the meaning of ...
4
votes
3answers
99 views

Prove $(x_n)$ defined by $x_n= \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}$ converges when $x_0>1$

$x_n= \dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}$ I know it converges to $\sqrt2$ and I do not want the answer. I just want a prod in the right direction. I have tried the following and none have ...
2
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1answer
53 views

Newton-Côtes closed formula for $\int_{0}^{2n} x^{2n+1}\cos(2{\pi}x) dx$

Given $n \in \mathbb{N}$, I want to determine the value obtained by approximating the integral $$ \int_{0}^{2n} x^{2n+1}\cos(2{\pi}x) dx $$ using the closed Newton-Côtes formula of $2n + 1$ points. ...
2
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1answer
37 views

Runge Kutta method example

Hi, can someone plz explain where the formulas for $w_{i+1}$ come from? Thanks!
0
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1answer
29 views

Comparing truncation errors from RK4

Consider the ODE $y'=f(x,y)$ with $x_0 = 0$, $y_0 = y(x_0) = 0$. We wish to approximate $y$ on the interval $[0,1]$. Let $h_1$ be some reasonable step size and $h_2 = \frac{1}{2} h_1$. These two ...
2
votes
2answers
48 views

If $A\in \mathbb{R}^n$ is symmetric and satisfies [the following] then $A$ is positive definite.

The following being: $$A(i,i) >\sum_{j\ne i} |A(i,j)| \quad \text{for} \quad i=1,2,...,n $$ How can I prove this?
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0answers
20 views

Order of Lobatto IIIa Method w.r.t MATLAB's bvp4c/bvp5c

I'm writing an overview about the use of MATLAB's bvp5c boundary value problem solver. In the literature (for example [1], pg 36), it is stated that the Lobatto IIIa methods are order 2s-2 for an s ...
2
votes
1answer
43 views

Difference between Backward and Forward differences

In numerical methods we are all familiar with finite difference table where one can identify backward and forward difference within same table e.g. given any entry in finite difference table, one can ...
0
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0answers
35 views

Why $\lambda$ is a complex number in the topic of “Stablity in numerical methods”?

I am studying the stability of numerical methods. In this topic we take a numerical methods such as Forward Euler and we try to find the condition that make it stable or unstable. for Forward Euler ...
0
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1answer
49 views

Determining the order of convergence of $ X_{n+1} = \frac{(X^3_n + 3aX_n)}{(3X^2_n + \alpha)} $

I need to find the order of convergence for: $$ X_{n+1} = \frac{(X^3_n + 3aX_n)}{(3X^2_n + \alpha)} $$ In a previous part we are told $\alpha$ = 2 and $x_0$=1. I know the first step is to take the ...
1
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0answers
44 views

Differentitaion of a non-linear equation using FDM method

I'm a Ph.D student of Hydraulic structures. I'm reading a paper in that the equation $(II)$ below is obtained by differentiating the equation $(I)$ using FDE (Finite Difference Equation) method and '...
1
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1answer
20 views

Show Newton's iteration to compute the root $x^*$ of $|x|^{\frac{3}{7}}$ does not converge

I need to show that Newton's iteration to compute the root $x^*$ of $$f(x) = |x|^{\frac{3}{7}}$$ does not converge for any starting guess $x_0 \neq 0$. The first thing I did was to create the Newton'...
1
vote
1answer
35 views

What can we say about the convergence of these fixed-point iterations for $\phi:\mathbb{R}\to \mathbb{R}$

Let $\phi: \mathbb{R}\to \mathbb{R} \in C^2(\mathbb{R})$ and let $x^{*}$ be a fixed-point of this function. Further assume that $|\phi'(x^{*})| \neq 1$. We define two sequences $\begin{align} &...
2
votes
1answer
47 views

Understanding steps to obtain derivative of $|x_n|^{\frac{3}{7}}$

I was trying to solve the following derivative $$|x_n|^{\frac{3}{7}}$$ as follows $$(|x_n|^{\frac{3}{7}})' \\= \frac{3}{7}(|x_n|^{\frac{3}{7} - 1}) \cdot (|x_n|)' \\= \frac{3}{7}|x_n|^{\frac{-4}{...
1
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0answers
23 views

Show that if all row-sums of a square matrix $A$ are equal to $0$, then $A$ is singular [duplicate]

I need to show that if all row-sums of a square matrix $A$ are equal to $0$, then the matrix is singular. My idea was that to represent the situation, I can do as follows: $$A\vec{x} = \vec{0}$$ ...
1
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0answers
37 views

Error of the Numerov Method

The Numerov method is an iterative algorithm for solving second order differential equations. A full derivation is here on the Wikipedia page: https://en.wikipedia.org/wiki/Numerov's_method. I am ...
0
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1answer
63 views

Please explain the solution

Use a Taylor series expansion to compute an error estimate in approximating the derivative of the function $f:\mathbb R\to\mathbb R$ using the formula $$ f'(x_0) \approx \frac{f(x_0-2h)-4f(x_0-h)+3f(...
1
vote
1answer
18 views

Runge Kutta error estimation

I am trying to solve a numerical analysis dealing with Runge Kutta methods. The problem is in solving the differential equation: $$\frac{d \vec{y}(x)}{dx} = \vec{F}(x,\vec{y}).$$ Defining the error ...
0
votes
1answer
26 views

Newton's method for unconstrained minimization

Let $f(x) = \frac{1}{2} x^T Q x + b^T x + c.$ Prove that Newton's method finds a critical point after a single iteration. Here $Q$ is positive definite. For this: I need to find first of $\nabla ...
1
vote
1answer
17 views

Solving a polynomial equation along a set of lines numerically.

Assume that I for some reason want to solve multidimensional polynomial equations $$p(x_1,x_2,\cdots,x_k) = 0$$ or possibly (if there is no solution) $$\min_{\forall x_{.}} \{p(x_1,x_2,\cdots,x_k)\}$$ ...
0
votes
1answer
40 views

Proving that $\log x$ is Big Oh of $x^k$ for every positive k

Can I know a way to prove the above condition purely by the definition (and may be Taylor Series) and without using L'Hospital's rule? It is obvious for k greater than or equal to 1 but how can you ...
2
votes
1answer
71 views

Riemann Sum Approximations: When are trapezoids more accurate than the middle sum?

We can approximate a definite integral, $\int_a^b f(x)dx$, using a variety of Riemann sums. If $T_n$ and $M_n$ are the nth sums using the trapezoid and midpoint (middle) sum methods and if the second ...
1
vote
1answer
8 views

Does LU factorization needs pivoting?

Today I have a numerical methods exam,and of course i tried some exercices, but today I heard something that messed my mind, I always do LU fact. Like this : I take Lower triangular matrix, and then ...