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

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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 ...
2
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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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0answers
10 views

Solving a matrix of ODEs with an invariant of the matrix as a variable coefficient

I have the following system of ODEs: $$ \dot{\mathbf A} (t) + c \thinspace I(\mathbf A(t)) \thinspace \mathbf A(t) = \mathbf 0,$$ where $I$ is, say, the second invariant of the symmetric matrix ...
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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
47 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 $$\lim_{n\to\infty} x_{n}=r$$ ...
1
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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
votes
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
17 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 ...
-2
votes
0answers
28 views

How can i get the Consistency Order of the follwoing multistep methods [closed]

For Example if i look at the euler method: \begin{align} y_{n+1} &= y_{n} + h·f_n \end{align} It is well known that euler method has order 1. \begin{align} y(x_{n+1};y_{n}) = y(x_{n}) + ...
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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 ...
0
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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 ...
0
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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 ...
0
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0answers
35 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 ...
0
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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 ...
0
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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 $\phi(t)=\cos(30t)$.
0
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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, ...
2
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1answer
54 views

How to Numerically Solve an integral equation.

First I really doon't have any background with integral equations! That said, I would like to solve the following: $$\int_a^b \frac{K(t)}{t-x} \phi(t) dt=f(x) , a<x<b$$ where$$\int_a^b \phi(t) ...
2
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5answers
65 views

Pade approximant for the function $\sqrt{1+x}$

I'm doing the followiwng exercise: The objective is to obtain an approximation for the square root of any given number using the expression $$\sqrt{1+x}=f(x)\cdot\sqrt{1+g(x)}$$ where ...
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2answers
19 views

What does it mean by the approximation $\int_a^bf(x)dx\approx\sum_{i=0}^nA_if(x_i)$ is exact for all polynomials of degree up to $2n+1$?

There is these notes about Gaussian Quadrature and I am trying to understand what does the sentence "is exact for all polynomials of degree up to $2n+1$" actually mean. Gaussian Quadrature - General ...
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1answer
17 views

Some doubts on Simpsons Rule by the Method of Undetermined Coefficients

There is this note about Quadratic Interpolation by Simpsons Rule that I don't quite understand how to get the LHS. Simpsons Rule by the Method of Undetermined Coefficients We seek an approximation ...
0
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0answers
11 views

Can someone help me solve this SDE?

I am having a problem about solving a SDE. The SDE is dx_t = dB_t + (v - x_t/(T-t)^2theta)dt, where T and v are fixed number, theta can vary. I have tried numerically solve this, but the ...
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1answer
34 views

Integrate $\int_{a}^{b}[\int_{a}^{x} (t-a)(t-b)(t-\frac{a+b}{2}) dt]dx $

The following integral arrives me while reading Atkinson's book on Numerical Analysis. It's pretty simple but I'm not sure what theorems should I be using: I have to integrate the following: ...
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3answers
34 views

Numerical partial derivative of an inverse function

We have a function whose inverse cannot be written in analytical form, such as: $$f(x)=kx^3+x$$ How to find $\frac{\partial}{\partial k}f^{-1}$ ? $f^{-1}(y)$ for a given $y$ can be easily found ...
0
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1answer
22 views

Number of continuous derivatives of a piecewise quadratic polynomial

I've been trying to reason through the following problem: Suppose that we interpolate $n+1$ data points with a piecewise quadratic polynomial. How many continuous derivatives can this interpolating ...
3
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1answer
24 views

Local Truncation Error of Implicit Euler

The LTE of an implicit Euler method is $O(h^2)$ because the method has order $O(h)$, but I'm not sure where to get started in proving this arithmetically. Any help would be appreciated. Thank you!
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1answer
59 views

Implicit Euler method and explicit Euler method

I wanna know what is the difference between explicit Euler's method and implicit Euler's method. And is the local truncation error for both of them is $O(h)$ and the coefficient of the $O(h)$ term is ...
1
vote
2answers
18 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...