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

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19 views

Solving modular inequalities efficiently

Is there an efficient algorithm (polynomial in $n$ and $N$? What about subexponential in $n$ and $N$?) to find the set of all solutions of the equations \begin{eqnarray} a_1 < &k_1x& < ...
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32 views

Cholesky decomposition and rotation matrix inverse

I implemented three methods for inversion of a matrix, all are classic. I wanted to test for the most generalized method, while taking efficiency into account. For Cholesky decomposition, which is ...
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0answers
37 views

Chebyshev spectral derivation with 16 nodes for $\,f(x)=e^{\,\text{sin}^{2}\,(x)+\cos(x)}\,$ defined in $\,[0,2\,\pi].\,$

I'm making the following exercise in Matlab, and I'm having trouble expresing my result in $x\in[0,2\pi]$ not in $x\in[-1,1]$. I first done this (as shown below) in Gauss-Lobatto points, but I don't ...
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1answer
22 views

A problem with Maple Polynomial Curve Fitting

I've been encountering a problem while trying to use the method of least squares to fit a quadratic polynomial. Below is the question 1) Use the method of least squares to fit a quadratic polynomial ...
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1answer
11 views

solving a stationary problem with matlab: descritization

The given equation is $$ \Delta(-\phi+\phi^3-\Delta\phi)=\frac{\partial \phi}{\partial t}. $$ I think it is equivalent to $\Delta\phi = \phi^3-\phi$ so I arrived to: $$ ...
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1answer
30 views

Proof of negative covariance (for the inverse transformation method)

Given a continuous strictly monotone distribution function $F$ and it's quantile function $F^{-1}$, with $U\sim\mathcal{U}(0, 1)$, the inverse transformation method takes $F^{-1}(U)$ and $F^{-1}(1-U)$ ...
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1answer
23 views

Finite difference method for a PDE in 2-dimensions?

Is it possible to use finite difference methods to solve a 2-dimensional wave equation (PDE) with boundary conditions? I know finite difference methods can be used to solve a 1-d wave equation, but ...
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14 views

Question regarding the approximant of a function

This is given as a sample exercise for our final exam in our Numerical Analysis class. However, nowhere in the course has the notion of approximant been defined or how to approach this kind of problem ...
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2answers
24 views

Finding vector which have a certain norm.

Let $$A=\begin{pmatrix} 9 & 5\\ 11 & 6 \end{pmatrix}$$ Find the conditionnumber $K(A)$ (for $\|\cdot \|_\infty$), and then try to find $b, r \in \mathbb{R}^2$ such that (for $Ax= b$ ...
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16 views

colvolution function

I am trying to understand the equaliti denoted in the attache picture. Any help? Thank you!
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1answer
55 views

Differential equation with Euler's method

Unfortunately from online classes i missed this lesson and now have an assignment question that has to be solved however im struggling to work this one out! Any help and answer would be appreciated ...
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1answer
38 views

Gauss rule Derivation

I am working through some exam prep questions, and need a little guidance on this one: The 2 point Gauss for weight $e^{-x}$ on the interval $[0, > \infty]$ has the form: ...
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1answer
61 views

Higher Order PDE using Finite Difference

How to approximate higher-order partial differential equation using finite difference method? $$\frac{\partial^{2} y}{\partial t^{2}}+\frac{\partial^{4} y}{\partial x^{4}}=0$$
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1answer
36 views

Finding the frequencies of vibration of a drum; PDE

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have ...
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1answer
27 views

Gram-Schmidt orthogonalization, determine zeros of the orthogonal functions.

Gram-Schmidt orthogonalization theorem The set of polynomials $\{ \phi_0(x), \phi_1(x), \ldots, \phi_n(x)\}$ in $[a,b]$ related to $w(x)$ as defined below is orthogonal: $$\begin{align} ...
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12 views

Enforcing additional constraints in linear equation

In a finite element context, I come up with a sparse "stiffness matrix" $A$ and a corresponding RHS $b$. The goal is now to solve $$Au = b$$ Where $u$ is a coefficient vector of the solution. Now I ...
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35 views

Can any one help me to check this question is right or wrong?

Taylor method in Numerical analysis to solve this question(4th order): Question is $y'=2t^2 +y^2 -1 $ $$ 0 \lt t \lt 2 \qquad y(0)=0 \qquad h=1$$ Solution : I solve this question is it right ...
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1answer
37 views

IVP using TS and Euler

I am just working through some exam practice problems, and I am a bit stuck with this one: Consider the IVP: $$ \frac{dy}{dt} = f(t,y), \space y(0)=y_{0} $$ (a) Expand solution $y = ...
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11 views

Methods for Extrapolating Data Close To Observations

I am aware of many different ways to go about interpolating between the values of known data points. However, whenever I come upon (I work in Quantitative Finance) the need to extrapolate data I find ...
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1answer
20 views

Third-order differential equation with initial values using Euler method

The problem I have is the initial value problem $$y''' = x + y$$ with $$ y(1) = 3, y'(1) = 2, y''(1) = 1$$ that should be solved with Eulers method using the step length, $h = \frac{1}{2}$. The ...
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2answers
10 views

Discretization of integral on infinite domain.

Let $[a, b]$ be a closed interval of the real line and let a sequence as $$a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!$$ This partitions the interval $[a, ...
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1answer
29 views

Show that $x^{m+n}-x^{m}=…$ in Gauss Seidel Iteration Scheme

I am just working through some exam practice questions and I would like to show that for non-negative integers, $m,n$, $$x^{(m+n)}-x^{(m)} = S^m(I+S+...+S^{n-1})(x^{(1)}-x^{(0)})$$ in the ...
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29 views

Relation between parallel transport and Jacobi field II

Before I asked a question here: Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic The current question is related, and actually arise from numerical ...
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1answer
26 views

What does $\|u\|_{\mathcal{C}^2(\bar{\Omega})}$ mean?

What might $$\|u\|_{\mathcal{C}^k(\bar{\Omega})}$$ mean? $u$ is a sufficiently often differentiable function $\Omega \rightarrow \mathbb{R}$ and $\Omega \subset \mathbb{R}^n$ a bounded domain. It ...
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1answer
21 views

Uniqueness of function approximation over three points?

Given a function $f(x)$, we want to approximate $f$ using $P(x)$, such that: $P(x_0) = f(x_0)$, $P(x_2) = f(x_2)$, $P'(x_1) = f'(x_1)$. Prove that such a $P$ is unique $\iff$ $x_1 \neq ...
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33 views

Newton backward and forward interpolation (for ODEs) intuition.

For Newton's backward and forward formulas, I understand everything algebraically, but can someone please explain me this formula intuitively, especially intuition how "powers of the forward ...
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44 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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1answer
47 views

Trying to use the method “Stiff” (Rosenbrock method implementation) from the book “Numerical Recipes in C”.

The program is compilable but I don't think it works correctly. According to the book, we need also method "odeint" for adaptive stepsize adjustment and fully implement Rosenbrock method. I used the ...
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0answers
12 views

Which method of estimating graph can be used here?

I am making an experiment and I need to estimate a graph about the results I get. The problem is, I don't know what results my experiment will give me. For example these are my experiment results (x = ...
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1answer
38 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
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1answer
40 views

Efficient algorithm to find the maximum of a sum of $m$ sines

Is there an efficient algorithm to find the maximum of a sum of $m$ sines? That is, find an $x \in \mathbb{R}$ such that $$f(x) = \sum_{k=1}^m \sin(\alpha_kx)$$ is maximized? By efficient, we mean an ...
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3answers
25 views

relationship between Big $O$ notation and limit

If I have a function $f(n)$ such that $f(n) \geq 0$ for all positive integers $n$ and that $\lim\limits_{n\to \infty} f(n) = 0$, then can I conclude that $f(n) = O\left( \dfrac{1}{n^k}\right)$ for ...
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1answer
26 views

Let $f:[-1,1]\to \mathbb{R}$ by $f(x)=x^4$. Determine the polynomial $p_2$ of degree less than or equal to 2 such that $||f-p_2||_2$ is minimal

also compute $||f-p_2||_2$. Write $p_2$ with respect to $\{P_0,P_1,P_2\}$ and $\{1,x,x^2\}$ I know its helpful to show what I have so far but I really don't know where to start. I'm looking at ...
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1answer
31 views

Approximation formula for third derivative, is my approach right?

Derive by using Taylor approximation up to 4th degree (in $h$) of $f$ in $x_0 \pm h$, $x_0\pm 2h$ at $x_0$, an formula for approximation of $f'''(x_0)$ with an error term of order $h^2$. Could ...
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1answer
97 views

Scale vector in scaled pivoting (numerical methods)

In the scaled pivoting version of Gaussian elimination, you exchange rows/columns not only based on the largest element to be found, but rather the largest relative to the entries in its row. You ...
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21 views

Derive by Double False Position

The equations for this problem are $3x = y$ and $2(x + 15) = y + 15$. I know the answers from doing algebra. $x = 15$ and $y = 45$, but I'm not sure how to calculate that using the method of double ...
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2answers
29 views

Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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23 views

Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
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1answer
30 views

Complexity of Newton iteration problem for a d-dimensional problem

If we assume that we have $f:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and we want to use the Newton iteration method to solve $f(x)=0_{\mathbb{R}^{d} }$. Is there any theorem regarding the ...
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3answers
91 views

Calculate an integral depending on n

Is there a way (simple or not) to calculate the following integral? $$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$ Thanks
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3answers
20 views

Efficient algorithm for maximum of a differentiable function

Is there an efficient algorithm which can be used to find the global maximum of a differentiable function (of one variable) on a given interval?
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1answer
57 views

Complex stationary point of $\frac{z}{1-e^{-z}}+z$?

I apply the method of steepest descents I need to know the stationary points $z_0$ of the function $$ p(z)=\frac{z}{1-e^{-z}}+z, $$ such that, $ 0 <\mathrm {Im} (z)<2 \pi$. That is, I want $z_0$ ...
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1answer
57 views

How to Find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
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14 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
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3answers
51 views

How to solve these simultaneous equations using numerical methods?

How to solve these simultaneous equations for $\alpha$ and $\lambda$ using numerical methods? $\lambda * [(\frac{3}{4})^\frac{-1}{\alpha} - 1] = 11$ $\lambda * [(\frac{1}{4})^\frac{-1}{\alpha} - 1] ...
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34 views

When to use iterative methods for solving systems of linear equation

Iterative methods such as Jacobi, Gauss-Seidel method and successive over relaxation have a very limited field of use - for diagonally dominant matrices. So how they could be used on practice? What is ...
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1answer
16 views

Numerical method with a time derivative boundary condition

I'm trying to reproduce a result from a paper I'm reading using a numerical scheme that I'm coding myself. The equation is a reaction diffusion PDE. $$\frac{\partial M}{\partial t}=\frac{\partial^2 ...
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11 views

Error estimation - spline interpolation

I got a question regarding error estimation and spline interpolation. I got a parabola shaped graph that I've used spline interpolation on to get more accurate data. I've used a much smaller step on ...
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13 views

Iterative methods monotonically decreasing of the residual

For a question on Iterative Methods I have to show that the 2-norm of the residual is monotonically decreasing. We are given the following formula: $r^{(k+1)} = r^{(k)} - \alpha^{(k)} A z^{(k)}$ where ...
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1answer
29 views

Does any numerical diff.eq. solver give correct results given small step-size?

I've seen that there are less stable numerical differential equation solving methods, like using plain Euler steps $y(x+h)=y(x)+hf(x)$. For a given $h$ there are better methods. But when solving ...