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

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41 views

Divided differences and Newton's method for Lagrange interpolant

Using the recurrence relation for finite differences, and the Lagrange interpolant , I need to show that . Then, from the following points, I need to compute the Lagrange interpolant using ...
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1answer
41 views

Showing that multiplication is stable in finite arithmetic

I need to show that multiplication is stable in finite arithmetic if $x$ and $y$ are real and $$\tilde{x}=x\left(1+\epsilon_x\right),\hspace{5mm}\tilde{y}=y\left(1+\epsilon_y\right)$$ are their real ...
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1answer
38 views

Calculating roots using Newton's Method for multiplicity $> 1$

I have programmed Newton's method in Matlab and it is working quite good. However if the multiplicity of the found root is greater than one, the method doesn't work very well. Therefore I did some ...
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0answers
22 views

Singular Value Decomposition with Jacobi Method

I asked a question regarding this method about an hour or two ago, but another quick question if someone understands this algorithm. It is Algorithm 6 on this page: ...
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36 views

numerical analysis of partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
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1answer
34 views

Singular Value Decomposition using Jacobi Method

First time user of the site, so I apologize if my question isn't worded properly. I'm trying to implement the SVD of a square matrix using Algorithm 6 found on this website: ...
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1answer
36 views

Linear Regression Application

I have a linear equation as follows: $B_0*x_0 + B_1*x_1 + ... + B_8*x_8 = result$ And i have about 200 different situations that are categorized into two different groups, depending on whether ...
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1answer
19 views

DFT by $n$ samples of a continuous periodic signal with more than $n$ frequencies

It is known that if we only have $n$ samples and take DFT, we only get at most $n$ distinct frequency data. But let's say that there is a continuous periodic signal with more than $n$ frequencies, ...
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25 views

What is the condition of $\Delta t$ or $\Delta x$ in FEM

Recently, when I solved the convection-diffusion problem numerically, I found that it often showed NaN in my screen. :( The problem is: \begin{align} u_t + u_x - u_{xx} &= 0\\ u(x,0) &= ...
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1answer
15 views

Consistent but not covergent

I have been asked to prove that the method: $$x_{n+3} + x_{n+2} - x_{n+1} - x_n = h\left(f_{n+3} + f_{n+2} + f_{n+1} + f_n\right)$$ is consistent, but not convergent. I have been able to show that ...
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1answer
32 views

Upper bound for the error magnitude

for the function $f(x) = e^x$ on the interval [0,1] by using polynomial interpolation with $x_0 = 0, x_1 = 1/2, x_2 = 1$ find the upper bound for the magnitude $\max_{0 \leq x \leq 1} |e^x ...
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1answer
26 views

Find $f(x)$ given $f(0), f(1)$ and $f[x1,x2,x3]$

I need to find f(x) given $f(0) = 0$, $f(1) = 2$, and the divided difference $f[x_1,x_2,x_3] = 1$ for any three points $x_1, x_2, x_3$ How do I go about solving this?
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2answers
27 views

If a signal is periodic, can the error of approximation by Discrete Fourier Transform be avoided when using finite number of samples?

As title says, if a signal $f(t)$ is periodic, can approximation errors of approximation by discrete Fourier transform (DFT) be avoided when only finite number of samples are used?
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1answer
24 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
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1answer
47 views

Is my matlab code solving the problem?

I'm solving an assignment in numerical analysis where I use this model function for soundwaves under the water after fitting the model function in a least-squares sens and finding the coefficients. ...
4
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1answer
66 views

Prove $\min_{i}|\lambda_i| \leq |r_{jj}| \leq \max_{i}|\lambda_i|$

Let A be a normal $n \times n$ matrix with the eigenvalues $\lambda_1,...,\lambda_n$ |A| = |QR|, $|Q^HQ| = I$, $|R| = [r_{ik}]$ upper triangular matrix. Prove: $$\min_{i}|\lambda_i| \leq |r_{jj}| \leq ...
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0answers
52 views

the algorithm and computation cost for truncated SVD in rank k

It seems that the time cost of truncated SVD in rank k for matrix $A\in R^{m\times m}$ is $O(m^2 k)$. Could anyone show me some algorithms to calculate truncated SVD with the above time complexity?
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0answers
22 views

Stationary distribution of multidimensional birth-death process

I am considering a 2D birth-death process with a rate matrix $A$, with (1) state space: each dimension can take an integer value from $1$ to $V$ and there are two dimensions. Therefore the size of ...
1
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1answer
26 views

Euler's method - Order of accuracy

Theorem Let $f \in C([a,b] \times \mathbb{R})$ a function that satisfies the Lipschitz condition and let $y \in C^2[a,b]$ the solution of the ODE $\left\{\begin{matrix} y'=f(t,y(t)) &, a \leq t ...
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0answers
27 views

Proove the convergence of the Gauss-Seidel iterative method when the matrix is diagonally dominant

I'm reading about this proof here: However, I don't understand this part: "...from which (3.3) immediately follows" (in the upper half of page 3). Does it mean that: $||y|| \leq \gamma ||x||$ then ...
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2answers
32 views

About annuity immediate calculation

Q1: Find $s_{12}$ if the nominal interest rate payable monthly is $5%$ per annum. What I have done is: $$i^{(12)}=0.05$$ $$1+i=(1+i^{(12)}/12)^{12}$$ which leads to $$i=0.0512$$ ...
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2answers
73 views

Prove that $q_{ki} = \lambda_1[1+ \mathcal{O}((\frac{\lambda_1}{\lambda_2})^k)] \; \text{for all } i \; \text{with} \; (x_1)_i \neq 0$

Let A be a real symmetric $n x n$ matrix having the eigenvalues $\lambda_i$ with $$|\lambda_1|>|\lambda_2| \geq ... \geq |\lambda_n|$$ and the corresponding eigenvectors $x_1...x_n$ with $x_1^Tx_k ...
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18 views

Computing the CFL number for non-linear advection

Consider $u_t -(1-2t)u_x=\phi(t,x) $ with $u(x,0)=u_0(x)$ for integrable $\phi$. The solution for this PDE is given by $ u(x,t)= u_0(x+t+t^2) + \int_0^t \phi(x(z),z) dx$. For a consistent scheme ...
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1answer
37 views

Rearranging equation to prevent denominator blowup in C

I know the title mentions C, because this is a programming related problem, but I think this specific issue is more pure mathematics so I figured here would make more sense! For a homework ...
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0answers
55 views

General analytical solution for first order time varying system of ODEs

I asked a question related to this previously, but not as explicitly as I should have, I'm restating it more concisely here. Assume we are given some matrix $\mathbf{A}(t)$, which is time dependent. ...
2
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2answers
67 views

Wolfram Alpha result for infinite series summation .

Consider the infinite sum $s=1+1/2^2-1/3^2-1/4^2+1/5^2+1/6^2-...$. We can see that the series is absolutely convergent and hence convergent. But WolframAlpha seems to give me a different answer. When ...
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0answers
21 views

Value of an intermediate while using Rk4 method

I am using rk4 method to solve the following equations. \begin{equation} \frac{dx}{dt} = v \end{equation} \begin{equation} m .\frac{dv}{dt}= F_{p + w}(x)+F_{g}(v,x) \end{equation} My problem ...
3
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1answer
63 views

Relationship between the solution to $Ax=b$ and $(A+I)x=b$

I have have a symmetric, tridiagonal, Toeplitz matrix $A$, where $A_{11} = -\frac{1}{2}$ and $A_{21} = 1$, and I need to solve the system $$ (A+I)x=b, $$ numerically where $b$ does not necessarily ...
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1answer
24 views

Calculating amperage

So this question might be OT, if it is please re-direct me to a better forum. I am currently interested in calculating the amperage used by an Android phone. As this is not currently supported on ...
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0answers
28 views

What does invariant exactly mean and how does it get the invariant?

I have read many journal about simulation of regularized long wave. In numerical test section,many researcher use invariant of mass,momentum and energy to check accuracy of their method but i found ...
1
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1answer
22 views

Trying to show convergence of a Forward Euler method based on step size restriction

I have shown that for the given ODE system, that when we apply the forward Euler method to something like \begin{align} \mathbf{y'} &= A\mathbf{y} \\ \mathbf{y}(t_{0}) &= y_{0} \\ t &\in ...
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1answer
24 views

Sign change in Newton's Method lecture example unclear

In an example in my lecture notes there's a sign change at one of the steps. I assume find our $f(x)$ to compute the zeroes of our $g(x)$ and I have no idea why it happened. Newton's method for ...
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0answers
81 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
3
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2answers
90 views

Example of a function, so that $g(x)\neq x$

I'm trying to find an example of a function $g:\mathbb{R}\to \mathbb{R}$ (or $g:[1,\infty) \to \mathbb{R}$), so that $$|g(x_1)-g(x_2)|<|x_1-x_2|$$ for all $x_1, x_2\in \mathbb{R}$ ( or $x_1,x_2\in ...
3
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1answer
94 views

Value of an integral involving the fractional part function

I have difficulties in evaluating the double integral defined in the following. Let $$\left\{ t \right\} = t - \lfloor t \rfloor, $$ $ t> 0$ be the fractional part function, where the ...
2
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0answers
38 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
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0answers
35 views

What is meaning of $FFT(\vec{E}(x,y ))$

What is the meaning and how one takes fourier transformation of vector that has spatial distrubution. Let say electric field (with transfer x, y distibution) with direction $$FFT(\vec{E}(x,y ))$$ ...
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2answers
44 views

Modified central difference formula

Prove or disprove the assertion: If $f$ is differentiable at $x$, then for $\alpha \neq 1$ \begin{equation} \lim_{h \to 0} \frac{f(x+h)-f(x+ \alpha h)}{h- \alpha h} = f'(x) \end{equation} I first ...
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0answers
11 views

Powers of matrices via the generalised Lanczos process

At each iterative step of the generalised Lanczos process for the pair of matrices (A,B), we obtain the following factorisation: $$ A Q_k = B Q_{k+1} \widehat{T}_k, $$ where $Q_k^T B Q_k = I_k$ and ...
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0answers
43 views

Calculate a state not by its derivative in ODE

I have a system with state space representation. This system intakes input $u_1$ and $u_2$. There are two nonlinear blocks. One generates state $x_3$ and two internal states $x_1$, $x_2$. And the ...
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1answer
27 views

why conventional approximation method is true?

why the text book method for finding the fitting curve is right ? we have n data we want to approximate with a polynomial of degree m $P_m(x)$ (m < n-1). and of course $E = \sum_{i=1}^m ...
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1answer
16 views

Milstein scheme for stochastic differential equation with constant drift

The Milstein scheme to approximate the solution of an SDE is $$ Y_{n+1} = Y_n + a\Delta_t + b\Delta W_t + \frac{1}{2} bb' ((\Delta W)^2 - \Delta) $$ where $\Delta_t$ is the time step size (usually ...
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0answers
12 views

Recovering a continuous density function from its discretized version.

The probability density function is defined on [0,10], and it's discretized by taking integration over short intervals [0,0.01],[0.01,0.02], etc. Is there any kind of interpolation method ( I'm not ...
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0answers
12 views

Lower bound for a difference

Let $x_1,\dots, x_n$ be $n$ positive real numbers. And $k$ be an integer with $k<n$ (in practice $n=k^2$ if it helps). I want to compute the smallest difference between the sum of two sets of $k$ ...
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2answers
25 views

Bisection method guessing interval

I know that generally the bisection method is used given a certain function and an interval where we know a root exists within it. What if we don't know the interval? Is there a way of "guessing" the ...
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2answers
21 views

Show that polynomial is at least of order 3 for iterative method

Show that the iterative method $ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} - \frac{f''(x_n)}{2f'(x_n)} (\frac{f(x_n)}{f'(x_n)})^2$ used to approximate $s$ such that $f(s)=0$, is of at least order 3 ...
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0answers
39 views

Optimizing a factorization algorithm

In the paper A ONE LINE FACTORING ALGORITHM the following algorithm is presented: ...
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2answers
17 views

Number of significant figures relative to true value of x

I've been stuck on this relatively simple matter for a while and I'd really appreciate some insight into what the actual answer should be. Say I'm given an approximate x value $x_A = 28.271$, and a ...
1
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0answers
41 views

Bernoulli monosplines

Please help me with Bernoulli monosplines. Let's consider $2\pi$-periodic cubic spline, which is consist from $N$ ranges $0<x_1<x_2<\cdots<x_N<2\pi$. We can introduce a periodic ...
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0answers
72 views

Space-Time FEM for parabolic problems

I am trying to solve a parabolic problem (an IBVP) in one spatial variable using the Galerkin method. After searching for inspiration, I find that the typical approach is to discretise the temporal ...