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

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3
votes
3answers
408 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
36 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
votes
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 ...
0
votes
3answers
37 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
56 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
votes
1answer
30 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
vote
0answers
30 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
vote
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) = ...
0
votes
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
21 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
97 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
votes
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
votes
1answer
35 views

Runge Kutta method example

Hi, can someone plz explain where the formulas for $w_{i+1}$ come from? Thanks!
0
votes
1answer
28 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?
0
votes
0answers
12 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
34 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 ...
-1
votes
0answers
21 views

Interpolation of random data

I have points $(t_1,x_1(t)),(t_2,x_2(t)), \cdots , (t_n,x_n(t))$ and I would like to estimate values of $x_k(t)$ where $1 < 2 < \cdots < k <\cdots <n$. How can I do this. I have read ...
0
votes
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
votes
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
vote
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
vote
1answer
19 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 ...
1
vote
1answer
34 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|)' \\= ...
1
vote
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
vote
0answers
34 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
votes
1answer
62 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 ...
1
vote
1answer
16 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
16 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
65 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
7 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 ...
0
votes
0answers
16 views

Good derivation of Romberg's method?

I'm confused about the Romberg's method even after viewing numerous explanations of it. I understand the "high-level" view of what's happening, but not how it's derived/proven. Can anyone recommend a ...
-1
votes
2answers
17 views

Substituting $t=\tan(\theta)$ to $\int_0^{\infty} \frac{\cosh[(1+t^2)]^{-1/2}]}{1+t^2} dt $ [closed]

How does one substitute $t=\tan(\theta)$ to $$\int_0^{\infty} \frac{\cosh[(1+t^2)]^{-1/2}]}{1+t^2} \,dt \, ? $$
0
votes
1answer
17 views

Why does Euler-Maruyama method use a square root of the time step

Euler-Maruyama method is supposed to be an extension of the Euler method for ODE, but applied to SDE. This means that if we have an equation: $$ dY_t = Y_t dW_t $$ where $W_t$ is the Wiener process, ...
10
votes
0answers
785 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
1
vote
1answer
42 views

What degree should my hermite polynomial be

I constructed with a degree 6 polynomial but apparently even a degree 5 would suffice. $$\array{f(x)&=&ax^6&+&bx^5&+&cx^4&+&dx^3+ex^2+fx+g \\ f'(x)&=& ...
0
votes
2answers
39 views

Is my solution correct ?

Determine a fixed-point function $g$ in the interval $[0,1]$ that produces an approximation to a positive solution of $$3x^2-e^x=0$$ So I would rearrange and make $x=ln(3x$^2$)$ and then go on to do ...
0
votes
1answer
17 views

Where am I going wrong in my cubic spline, work out a,b and c

So I know $f^{'}_1(x)=6x+3ax^2$ and $f'_2$(x)=6x+3bx^2$ and $f^{"}_1(x)=6+6ax$ and $f^{"}_2(x)=6+6bx$ Now, $f_1(0)=0$ and $f_2(0)=c$, therefore $c=0$ And $f^{"}_1(0)=0$ and $f^{"}_2(0)=0$ But ...
1
vote
1answer
52 views

Please explain the last step of this newton method for system of equations

The step of working out x$^1$. I know the above is the formula but do they actually work out the inverse of the derivative matrix, is there a quicker way to do this?
1
vote
0answers
32 views

Computing a Green's function - where did I go wrong?

This is from a homework problem that was recently returned to me in a numerical analysis course. The grader even noted that he didn't know where I went wrong but the solution was marked as incorrect. ...
0
votes
1answer
48 views

Multistep Method: Gear's Formula Interpolation

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.
0
votes
0answers
14 views

Can I use Euler approximation for backwards computation?

I want to solve the following Ricatti Equation: in Matlab. I'm given the final value of P and as far as I understand I have to go backwards solving this. My question is, can I use the Euler ...
1
vote
3answers
96 views

Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
0
votes
1answer
45 views

Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...
0
votes
2answers
39 views

Numerically stable version of calculation with cancellation [closed]

What's a numerically stable way to compute $$ \frac{2^{1/n}}{2^{1/n}-1} $$ for large (integer) $n$?
2
votes
1answer
23 views

Prove multidimensional Newton's method converge at least quadratically

Newton's method for root finding is simply $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. The following is a theorem from my textbook. where 6.1.22 is shown below Now I want to prove a similar ...
0
votes
0answers
11 views

Intermediate value theorem for fixed point convergence error

Hi I am trying to understand the convergence analysis for fixed point, and this is what I am not getting. So Let r be a root i.e r=g(r) Iteration Xk+1=g(Xk) Error=Ek=|xk-r| Ek+1=|xk+1-r|=|g(Xk)-r| ...