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

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6
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0answers
159 views

A new type of Arithmetic-Harmonic mean for $n$ numbers

Let's introduce the following iterative procedure. Take two numbers $x_0$ and $y_0$. $$a_0=\frac{x_0+y_0}{2}~~~~~~~~~~~b_0=\frac{2x_0y_0}{x_0+y_0}$$ $$x_1=\frac{x_0+a_0+b_0}{3}~~~~~~~~~~~y_1=\frac{...
1
vote
1answer
32 views

The convergence of an infinite radical involving $\cos(\alpha/3)$

By using the triple angle formula for the cosine, $\cos 3\alpha$, we get the cubic equation $ 4x^3-3x = \cos \alpha $. Now, by expressing $ x $ as $ x = \frac{1}{2}\sqrt{3+\frac{\cos \alpha}{x}}$ ...
0
votes
1answer
54 views

Check numerically the definite-positiveness on linear subspaces

I have a given matrix $W\in \mathbb{R}^n$ with known fixed entries. I would like to check the definite-positiveness of $W$ on appropriate linear subspaces. Typically I would like to show (...
1
vote
1answer
37 views

interpolation polynomial error

We have points $x_0=a \lt x_1 \lt x_2 ....x_n=b $ and $\;w_{n+1}(x)=\prod_{k=0}^{n}{(x-x_k)}$. Let $h=max_{j=0...n}|x_j-x_{j-1}|$ Let $f \in C^{n+1}[a;b]$ and $p_n\in \mathbb P_n$ be the ...
0
votes
0answers
45 views

Adams-Bashforth-Moulton two-step predictor-corrector

Can someone help me this question please, this is from past years exam.
0
votes
1answer
22 views

Preconditioning : ILU($\emptyset$) factorization and SSOR relation?

Let's suppose we have a matrix A = D - L - U , where D,L,U are diagonal , strictly lower and strictly upper triangular,respectively. Generally the preconditioning matrix, according to SSOR(Symmetric ...
0
votes
0answers
24 views

Interplanetary Optimisation using a simulator with PyGMO or SciPy

I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ...
1
vote
1answer
74 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
0
votes
1answer
35 views

Intersection of a helicoid and a line

I have a Helicoid described by the following parametric equations: $$x = u\cos(v)$$ $$y = cv$$ $$z = u\sin(v)$$ The helicoid revolves around the y-axis: Eliminating $u$ and $v$, we obtain the ...
1
vote
2answers
68 views

Error in trapezoidal rule via integral mean value theorem

During a class, I saw the following analysis of the error term in the trapezoidal rule For $f \in C^2([a,b])$, $\int_a^b f(x) \,dx - \frac{b-a}{2}[f(a)+f(b)] = -\frac{(b-a)^3}{12}f''(\eta)$ for ...
0
votes
1answer
36 views

Estimate counts with different sample sizes

Given an arbitrary time period, lets say one week, but it could be five days, one month etc.., I have a sample from a population. My sample consists of shoppers at a store. For week one my sample is ...
0
votes
0answers
33 views

Matrices over integer fields to solve complex polynomials.

Inspired by the fruitful answer to this question regarding numerically solving polynomial equations in terms of simpler fields (in that case representing real numbers as fractions of integers), I ...
0
votes
0answers
44 views

Solving differential equation with numerical method

How do I solve this differential equation numerically for $1<\beta<\inf$ knowing $\sigma$ ? $\frac{\gamma(u)}{du}=\beta\cdot\gamma(u)\cdot\gamma(\beta\cdot u)$ $\gamma(0)=\sigma$ Thank you ...
0
votes
0answers
15 views

Maxwell-type system: how to solve?

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: $$ ...
1
vote
0answers
29 views

Newton conjugate gradient algorithm

In this video, the professor describes an algorithm that can be used to find the minimum value of the cost function for linear regression. Here, the cost function is $f$, the gradient is $g_k$ where $...
-3
votes
1answer
46 views

Proof Bisection method [closed]

Denote the successive intervals that arise in the bisection method by $[a_1\,, b_1], [a_2\,,\, b_2], [a_3\,,\, b_3],$ and so on. Let $c_n$ be the midpoint of $[a_n, b_n]$. Show that $|c_n − c_{n+1}| = ...
0
votes
0answers
30 views

Second-order differential equations methods

I'm looking for a method called 'Inexact Method' Idk if it goes by another name, here's what I do know: It's one of the two 2nd order differential equations methods. The other method is called '...
1
vote
1answer
35 views

To solve large systems of multivariate polynomial equations

Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (...
1
vote
1answer
49 views

Iterations with matrices over simple fields approximating solutions for more complicated fields.

Inspired by this question I started wondering if there exist some systematic way to construct approximation to any number one can find using matrices over a preferrably simpler field. In the question ...
0
votes
2answers
53 views

Numerical solutions of partial differential equations

I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g., $\frac{\...
1
vote
2answers
25 views

2nd Order Runge-Kutta Method

Could someone please help me with the next step of this 2nd order Runge-Kutta method. I am solving the ODE \begin{align*} x'=-\frac{x(t)}{2}, \ \ x(0)=2. \end{align*} I wish to use the second order ...
1
vote
1answer
28 views

Reconstructing a matrix

Before reading on, let me acknowledge that this problem is solveable generally, however I am interested in knowing if a certain form of solution exists. If I have a square complex unitary $n\times n$...
0
votes
0answers
38 views

Newton's method

Let $a > 0$. Starting from a convenient equation and using Newton's method, deduct a method to approximate $1/\sqrt a$ without division. How do you choose the starting value? Which is the stop ...
1
vote
3answers
51 views

For how many weeks can I group my students?

I thought of an interesting question that I don't know how to solve. I imagine there are numeric results out there somewhere, but I don't know if this question has a formal name; if anyone could link ...
4
votes
0answers
19 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
0
votes
2answers
70 views

Square root of x : $\sqrt{x}$ (Numerical Method)

$$f(x) = \sqrt{x}$$ has to be approximated by polynomial interpolation $p(x_n) = f(x_n)$ with the positions $\{x_n\} = \{1,4\}$. For such problem which method is the fastest? And find $p(2)$. My ...
0
votes
1answer
21 views

Condition number interpretation

I have a (nonlinear) problem with two variables, for which I computed a relative condition number as $$K_{rel}(x_1,x_2) = \max\{1, c\},$$ where I had $$\Bigg| \frac{f(x_1, x_2) - f(\tilde{x_1},\tilde{...
0
votes
0answers
33 views

Interpolation and Interpolationerror - how to compute ?

I want to compute the greatest $a>0$ for given $\epsilon>0$ such that $$max_{x\in [-a,a]}|f(x)-p_2(x)| < \epsilon$$ where $a$ is the distance between two grid points and the maximum is the ...
0
votes
0answers
30 views

Question about Moment generating functions, precision, rounding errors

Hello I have a question in regard to moment generating functions and something I noticed but wasnt to clear on. Say that a random variable $X$ has $$MGF=M_{X}(t)=\frac{4}{4-t^2}$$ for $-2 \lt t \lt 2$...
1
vote
0answers
34 views

Sensitivity of Eigenvalues with invertibte matrix

Let $A$ a matrix having a set of eigenvectors $\{v_1,\ldots,v_n\}$ linearly independent with $\{\lambda_1,\ldots,\lambda_n\}$ eigenvalues associated. Let $\lambda$ eigenvalue of the perturbed matrix $...
1
vote
1answer
20 views

Matrix similar and unitarily diagonalizable

Let $A,B \in R^{n \ x \ n} $ similar and unitarily diagonalizable. Prove that there $Q$ unitarily such that $Q^{H}AQ=B$
0
votes
0answers
27 views

Heat Equation 1D in Cylindrical Region

I have heat equation 1D in cylindrical coordinates: $$u_{\rho\rho}+\frac{1}{\rho}u_{\rho}=u_{t},\;0<\rho<1,t>0 $$ with boundary conditions $u(1,t)=0,\;t>0$ and initial condition $u(\rho,0)=...
0
votes
1answer
20 views

proving an inequality involving a linear spline / piecewise polynomial

I have $n+1$ sample points $x_i = \left(\frac{i}{n}\right)^4$ and want to approximate the function $f(x)=\sqrt{x}$ by a linear spline $f_n \in S^{1,0}(\mathcal{T_n})$ on the interval $[0,1]$. I know ...
2
votes
1answer
29 views

tangents at unit circle - parametrization leads to strange result

I'm thinking about the tangents at the unit circle in the upper right corner and I do so in terms of the parametrization $$ \begin{eqnarray} v:&[0,1]& \rightarrow \mathbb{R}^2\\ &t&...
0
votes
1answer
38 views

Bisection Method and midpoint

If $f(a) < 0$ and $f(b) > 0$, then prove that the point $c$ computed in the bisection method is the point where the line through $(a, \operatorname{sign}(f(a)))$ and $(b, \operatorname{sign(f(b))...
1
vote
1answer
47 views

Understanding definition of explicit Euler method

I'm quite new to ODE and the IVP (initial value problem), so I've some doubts related to these topics. I'm reading a draft provided by my professor which talks about the "polygon" or explicit Euler ...
2
votes
0answers
29 views

How to prove that one formula is numerically better than another

If $\mathbf{u}$ and $\mathbf{v}$ are vectors in real 3-dimensional space, here are two formulas for computing the angle between them: $$\theta = \operatorname{atan2}\left( \|\mathbf{u}\times\mathbf{v}...
2
votes
1answer
37 views

Falsi regula using maple

Hi I'm using falsi regula algorithm in maple. For first function it worked fine : restart; epsilon := 1e-3: f := x->x^3+x^2-3: a:= 1: b:=2: step:=infinity: while abs(step) >= ...
1
vote
1answer
25 views

Can someone help me understand using the Jacobian matrix with Newton's Method for finding zeros?

I'm struggling to understand approximating solutions to non linear equations using a Jacobian matrix. I understand intermediate steps, but I'm unsure how everything comes together. I want to use ...
0
votes
0answers
30 views

Differential equation method

I'm looking for an specific method called "Inexact Method", our numerical analysis professor told us that it is relationed with the differential equations, but I haven't found anything yet. Is there ...
0
votes
1answer
25 views

Rate of Convergence vs Tolerance

I have this confusion that if I know that a numerical method has a rate of convergence equal to $O(c^k)$ (see this link, page no .8), then how to find the number of iterations to reach a tolerance of $...
3
votes
1answer
84 views

Evaluate the improper integral $\int _1^\infty x^{-10/9} \coth(x) \,\mathrm dx$

Could someone help me to evaluate this integral please: $$\int _1^\infty \dfrac{1}{x^{10/9} \tanh(x)} \,\mathrm dx$$ I tried using change variable method in order to change the integral bound.
0
votes
1answer
286 views

Evaluation of a set of Integrals involving fractional part

I need help in evaluating the following integrals involving the fractional part: \begin{equation} I_{k} = \int_{t = 0}^{1}\int_{y = y_{k}}^{y_{k+1}}\int_{x = x_{k}}^{x_{k+1}} \left\{\dfrac{y}{t}\right\...
0
votes
1answer
33 views

calculating $\sin x$ in floating point arithmetic

I would like some help in the following exercise: In floating point arithmetic we want to calculate $\sin 30$ using the type $$\sin x=\sum_{k=0}^{N}t_{k}$$ where $t_{0}=x,t_{k}=-t_{k-1}\frac{x^2}{(2k+...
0
votes
1answer
22 views

Relation between iterative methods and preconditioning , your thoughts

Let's suppose we have an invertible matrix P ( P from preconditioning ) : $Ax=b \Leftrightarrow Px = Px -Ax + b$ or $ Px = (P-A)x + b$ The iterative method which produced by the above is : $Px^{k+1}= ...
0
votes
1answer
36 views

pade approximation

How do I use pade approximation to transform a given function? Pade approximation is the best approximation in mathematics, compare to Taylors approximation & others. I have learnt this ...
3
votes
2answers
69 views

Least Squares Alternates- approximating functions

I was given this least squares problem to solve: Find a linear function $\ell(x)$ such that $\displaystyle\int_0^1(e^x-\ell(x))^2{\rm d}x$ is minimized. As an answer, I got $\ell(x)=0.5876+0....
0
votes
0answers
22 views

Refinement of the trapzoid rule

Suppose that $f:[y,x]\to\mathbf{C}$ is a continuously three times differentiable function, where $y<x$ are integers. Show that $$\sum_{y\leq n\leq x} f(n)=\int_y^x f(t)\,dt+\frac{1}{2}f(x)+\frac{...
1
vote
1answer
44 views

Data structure for a symmetric $n\times n$ matrix

Suppose you are given a symmetric matrix $A\in\mathbb{R}^{n\times n}$ and consider the computation of the matrix vector product $A u \rightarrow v$ where $u\in\mathbb{R}^n$ is given and $v\in\mathbb{R}...
0
votes
0answers
26 views

Computing unknown matrix norm

Suppose that we have a unknown vector norm but there is a machine that get a vector and correctly tell us the norm of the vector. Is there any algorithm to compute the matrix norm corresponding to ...