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

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$L_2$ norm of trigonometric interpolation

Let $I_N f(x)$ be the trigonometric interpolation of discrete function $f(x)$ with Fourier coefficient $g_l$. How can I prove this relation: $$\|f||_2=\|I_N f(x)||_2$$
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2answers
12 views

Implementing Adams-Bashforth of order 2 (AB2) algoirthm

Assuming we are given the initial condition for an ODE such that: $$ \begin{cases} x' = f(x,t) \\ x(t_0) = x_0 \end{cases} $$ We are going to solve it numerically using AB2. We know that the ...
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0answers
6 views

CG as an orthogonal projection

I have heard that the Conjugate Gradient method can be viewed as an orthogonal projection onto the Krylov subspace $K(A,r_0)$, but I can't find a reference that deal with it in this way. Could you ...
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2answers
63 views

Fixed point iteration question

This image was taken from this youtube video https://www.youtube.com/watch?v=OLqdJMjzib8 Since only one of them converges, how do we know in advance which formula to work with?
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0answers
10 views

Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
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0answers
20 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
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0answers
16 views

Help understanding a homework problem (Preconditioning matrices, numerical methods)

Below is a link to the problem (because I didn't want to have to go through the pain of TeXing it all out myself), the basic idea is we are supposed to be first showing that a specific matrix has a ...
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0answers
24 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
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3answers
48 views

Numerical solution of an ODE system of equations using RK4

I have given an assignment to find the solution to the ODE system of equations as follow: $$\begin{cases} x_1' = x_1 + x_2 \\ x_2' = -3x_1 -10x_2 + x_2 ^2\end{cases}$$ With initial conditions: ...
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0answers
55 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
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2answers
72 views

Solving equations like $xe^x = c$ via functional iteration

Yesterday I randomly thought of solving $xe^x = c$ via functional iteration (FI) after manipulating the equation into a form "$x = \cdots$" that gives the 'fastest' convergence rate regardless of the ...
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0answers
11 views

Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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0answers
22 views

How many Gauss points are required to provide exact value for the Gauss quadrature rule

How many Gauss points are required if the Gauss quadrature rule should provide the exact value of the integral $I=\int_{-1}^1f(x)dx$ for $f(x)=(x^2-1)^2$? I am really not sure what theorem to use to ...
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0answers
6 views

How to deal with the composite function in a numerical approximation problem?

Consider a quasilinear two-point boundary value problem: $$-(a(u)u'(x))' = f(x) , x\in (0,1)$$ with $a(u)>0$ and $u(0) = 0, u(1) = 0$. I am supposed to derive an algebraic system so that I can ...
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2answers
43 views

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix.

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix. So I need to show that $x^T(AA^T+\alpha I)x>0$ for all vectors $x$. I'm ...
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1answer
35 views

Simpson's rule is not producing better results than Riemann sums

I have to calculate RMS value $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ and I know from the maths that the Simpson's rule should provide better approximation of the definite integral than the Riemann sums. ...
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1answer
23 views

Some trivial but confusing terms about numerical integration

Some terminological questions about numerical integration: When a question states trapezoidal rule with 2 points, does that mean 2 subintervals or 3 subintervals? Since 3 subintervals have 2 points ...
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1answer
23 views

Proving error bound on Simpson's Rule, Numerical Integration

The approximation from "Simpson's Rule" for $\int_a^b f(x)\, dx$ is, \begin{equation} S_{[a,b]}f = \bigg[\frac{2}{3}f\Big(\frac{a+b}{2}\Big) + \frac{1}{3}\Big(\frac{f(a) + f(b)}{2}\Big)\bigg](b-a). ...
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0answers
22 views

Fourier series for absolute value of sin functiom

If we take the absolute value for sin function, then it becomes even. However, isn't period of this function pi? To find fourier series, 1.Even 2. period 2 pi. Can we just treat this function as ...
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1answer
21 views

Transforming an integral to a different domain

For a given $v(x)$ with $x\in[0,1]$, use the variable transformation $x=g(\eta)=\frac{1}{2}\eta+\frac{1}{2}$ to transform the integral $I=\int_0^1v(x)dx$ to an integral over $[-1,1]$. My doubts: ...
2
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1answer
49 views

Numerically Solve a Second Order ODE with singular coefficients

I need to solve the following numerically: $$xy''+y'+xy=x$$ with initial conditions $y(0)=0$ and $y'(0)=1$. I need the solution for $x:[0, 10]$. I've written the ode as a system of first order odes ...
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0answers
18 views

Estimate the number of candidates who obtained fewer than 70 scores.

In an examination, the number of candidates who obtained scores between certain limits are as follows: Scores $0—19$, $20—39$, $40—59$, $60—79$, $80—99$, Number of candidates $41$, $62$, $65$, ...
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0answers
22 views

Numerical stability of computational results

Let z be a function of a finite number of variables i.e. z=f(a,b,c,...). If we have the mathematical formula connecting z and the variables, we can determine how the value of z varies with a change ...
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0answers
12 views

How to build a matrix in MATLAB with the next characteristics?

Let $\lambda_1=\frac{k D_u}{2h^2}$ a constant value. How to generate a matrix in MATLAB with the next entries: $A= \begin{pmatrix} 1+\lambda_1 & -\lambda_1 & 0 & 0 & \cdots & 0 ...
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0answers
22 views

Numerical Analysis Stability or Condition

Im not sure if its the right place to ask this question since it may be more of an informatics problem and I apologize if its the case: I want to evaluate a certain function, $\frac{log(cos(x))}{x}$ ...
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0answers
20 views

Problem with code for numerical integration in matlab.

Hi I have problem with calculating this expected utility with means of numerical integration, using matlab. The matrix stock data is a 1000x8 matrix with columns representing 7 stocks (columns 2-8, ...
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2answers
35 views

Prove limit of a sequence in Newton's method

Given the $ f(x)=x^3+x-1 $, I have shown so far that $ f$ has a unique root $r\in(0,1)$ and that for the sequence $(x_{n}), n>=0$ produced by Newton's method we have $$\lim_{n\to\infty} x_{n}=r$$ ...
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1answer
19 views

Normal system of the least square method

I'm trying to show the following. $Pa$ is the approximation system of $y$. I want to show that finding the minimmum for the function $$f(a,y)=||Pa-y||_2^2$$ is equivalent to solve the normal system of ...
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0answers
12 views

Equation involving Bessel and Struve functions

I need to solve the equation $Z(\gamma) = r$ of the function $$Z(\gamma) = 1 - \frac{2}{\gamma} \left(J_1(\gamma) - i H_1(\gamma)\right),$$ where $J_1$ is the Bessel function and $H_1$ the Struve ...
2
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1answer
30 views

$LDL^t$ Factorization Algorithm to find a factorization of the form A

For $$ \begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 5 \\ \end{pmatrix} $$ I found that $$ L=\begin{pmatrix} 1 & ...
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0answers
24 views

LU factorization Algorithm

How to show that the LU Factorization Algorithm requires $n^3/3-n/3$ multiplications/divisions and $n^3/3-n^2/2+n/6$ additions/subtractions?
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1answer
21 views

Numerical solution of ordinary differential equations, multistep method

I try to solve the following question, but I have no clue why we have $x'$ in the RHS: The formula $ x_{n+1} = (1-A)x_n + A{x_{n-1}} + \frac{h}{12}[(5-A)x'_{n+1}+8(1+A)x'_n + (5A-1)x'_{n-1}] $ is ...
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0answers
19 views

Minimizing the average

Let's say I have a nice-behaving function $f: \Bbb R^n\to \Bbb R$, and I would like to find its maximum. Then I can apply gradient search algorithms to look for that, and to cope with possible ...
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0answers
16 views

Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
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0answers
16 views

Finding the error terms of the Legendre polynomial (Numerical Analysis)

(a)Let $x_0$,..., $x_k$ be $k+1$ distinct points. Let $P_k(x)$ be the Lagrange interpolating polynomial of a smooth function $f(x)$ using these points. Show that ...
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0answers
11 views

Prove Ladyzhenskaya- Babuska-Brezzi condition for Poisson problem with homogenoeus Dirichlet boundary condition

I'm considering the problem: \begin{equation} \label{eq:PM} \begin{cases} \mathbf{u} -\nabla p=0\quad \text{ in } \Omega\\ \mathrm{div} \mathbf{u}=-f \quad \text{ in } \Omega\\ p=0\quad \text{ in } ...
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0answers
18 views

Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
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1answer
24 views

For what maximum positive $k$ is $2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$ true?

I am trying to find the maximum value of $k$ such that the inequality $$2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$$ is satisfied. I impose restrictions that $n \in \mathbb{Z}$ with $n \geq ...
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0answers
26 views

Proof that $f[x_0,x_1,…,x_n,\epsilon,\epsilon]=\frac{f^{n+2)}(\eta)}{(n+2)!}$

Up to now i have the following rule for divided differences: Assuming $x_0 \le x_1 \le...\le x_n$ then If $x_0 \lt x_n$ then ...
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0answers
8 views

Forward difference problem

How to compute $\Delta^{2}(cosx)$ ? I try using relation $\Delta =E-1$ where $E$ is shifting operator. Please need help.
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1answer
15 views

Number of iterations for Gauss-Seidel

I am having some difficulty understanding the following solved problem: Question: Shouldn't we have $||T||^k_{\infty} ||e^{0}||_{\infty} \leq 10^{-6}$ instead? Where does the $5$ come from? And ...
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2answers
49 views

Evaluating integral with a singularity.

I want to evaluate an integral numerically that contains one singularity. The software I use for this is Python. The actual integral I want to evaluate is quite long with a lot of other constants so I ...
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1answer
24 views

Avoiding loss of significance without series.

How could the function $$f(x)=\frac{\sin x}{(x^2+1)^{1/2}-1}$$ be computed to avoid loss of significance? I know that $$f(x)=\frac{\sin x((x^2+1)^{1/2}+1)}{x^2}$$ But $x^2$ has a problem.... How to ...
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0answers
13 views

Why is it that if a numeric method has quadratic rate of convergence then it can reach d digits of precision in logd iterations?

I was trying to understand why a method with quadratic convergence can get close to a good solution in $\log d$ iterations. Assume we have a method that has the property that the number of digits of ...
3
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1answer
21 views

Numerical method with convergence greater than 2

It is well known fact that, for solving algebraic equations the bisection method have linear rate of convergence, secant method have rate of convergence equal to 1.62(approx.) and Newton-Raphson ...
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0answers
18 views

Weight Function in gaussian quadrature

My question is pretty simple, although I know of the properties that the weight function must follow , such as being well defined,positive,continuos and integrable on the interval . I do not know how ...
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0answers
36 views

Method check: numerically calculate 1D integral of a 3D function

I have a function $f(r)$ where $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $\forall x,y,z \geq 0$. I know the values of the function at many points, essentially I have a table of values with $r$ and the ...
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1answer
25 views

Intersecting three rays and a sphere of known radius

So I actually solved this problem using an iterative solver, but it annoys me because as far as I can tell it should be possible to do it directly. I have three known 3D "rays" that all start at the ...
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2answers
24 views

Solving a Linear IVP [closed]

I need help solving this linear Initial value problem: $$y'=-L(y(t)-\phi(t))+\phi'(t) \\ y(0)=y_0$$ where $\phi(t)=\cos(30t)$.
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1answer
14 views

Equivalents definition of linear convergence

Suppose that the sequence $\{x_n\}$ converges to $0$. I want to prove that these definitions are equivalent: a) We say that $\{x_n\}$ converges linearly to $0$, if there exists a number $q \in (0, ...