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

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0
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1answer
41 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, ...
0
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0answers
39 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) &= u_0(x)...
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1answer
21 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 ...
0
votes
1answer
186 views

Upper bound for the error magnitude

For the function $f(x) = \mathrm{e}^x$ on the interval $[0,1]$, by using polynomial interpolation with $x_0 = 0$, $x_1 = 1/2$, and $x_2 = 1$, find the upper bound for the magnitude $$ \max_{0 \leq ...
0
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1answer
41 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?
0
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2answers
56 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?
1
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1answer
55 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 ...
1
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1answer
130 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
81 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 ...
3
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1answer
352 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?
1
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0answers
66 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
222 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 \...
0
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0answers
270 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
67 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 $$i=0.0512$$ $$s_{12}=((1+i)^{12}-...
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2answers
87 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 =...
1
vote
1answer
58 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 assignment,...
1
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0answers
133 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
163 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 ...
3
votes
1answer
71 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 ...
-1
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1answer
41 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 ...
1
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0answers
62 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
32 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 (...
0
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1answer
27 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 ...
3
votes
1answer
187 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
votes
2answers
131 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
votes
1answer
146 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
214 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) &...
0
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0answers
48 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 $$FFT(\vec{E}(x,y ))$$ ...
3
votes
2answers
60 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 ...
0
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0answers
13 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 $\...
0
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0answers
59 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 ...
2
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1answer
40 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 (y_i-P_m(...
0
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1answer
53 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 ...
0
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2answers
231 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 ...
0
votes
3answers
49 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
46 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 ...
1
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0answers
103 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 ...
0
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2answers
162 views

Simpson's 3/8 rule formula

I am trying to work with Simpson's 3/8 rule, but I wanted to double check my formula: $$I(f) = \int_a^bf(x) dx \ \approxeq \ \frac{3h}{8}\left(f(a) \ + \ 3f\left(\frac{a+b}{3}\right) \ + \ 3f\left(2\...
0
votes
1answer
51 views

Exact power of prime that divides an (unrelated) number

I am trying to understand a paper where a numerical algorithm is described. I do not understand the point where the expression "exact power of a prime that divides a number" is used. Here is the ...
2
votes
3answers
160 views

Exponential curve fit

I want to fit a curve of the form $y = ab^x +c$ where a, b and c are constant whereby i have a data of points $(x_i, y_i)$ I can reduced my primary equation into a form $log(y - c) = log(a) + xlog(b)...
0
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1answer
32 views

Re-use of Linear Finite Elements

I am putting together a little FEM code at the moment. It is just meant to let me play around with some toy problems (e.g. Laplace, etc.) My rather daft question is this: do the formulations for an ...
1
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1answer
59 views

Quadratic Formula expressed with Taylor's Theorem

I am having trouble solving the problem below. I think I understand the first part by just doing a taylor expansion of $f(\delta - a)$ where $a=0$ and the function equals $\sqrt{1-\delta}$. But I do ...
1
vote
2answers
51 views

Show Newton's method can go wrong with two roots

If $f:\mathbb{R} \to \mathbb{R}$ is differentiable with at least two roots, I wish to show that Newton's method will not converge for some $x_0$. I know that $f'(x)$ has a zero, say at $z$. It ...
1
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0answers
48 views

Are numerical approximation always possible in ODE if an solution exists? [closed]

Are numerical approximation always possible in ODE if an solution exists? Numerical approximation defined as the method to find solution in any degree of accuracy I don't know which theorem talk ...
0
votes
1answer
22 views

Claim: The complex conjugate of $ \omega_N $

My textbook says that $ \omega^{N-k}_N = \bar \omega^k _N $, where the bar denotes the complex conjugate. Why is this true? Sidenote: I believe $\omega^{N-k}_N=(e^{-2\pi i}/N)^{N-k} $.
1
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1answer
143 views

Numerical differentiation: 2-point vs 5-point method

I want to compare the following two numerical differentiation schemes: 2-point numerical differentiation: \begin{equation} \dot{\omega}_t = \frac{1}{dt} \left [ \omega_{t} - \omega_{t-dt} \right ] + \...
0
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1answer
39 views

Numerically solve ODE with boundary conditions

If I want to solve the eigenvalue problem $-y''=\lambda y$ with either periodic or antiperiodic boundary conditions on $[0,2\pi]$, how can I enter the boundary conditions? I mean, in general I would ...
2
votes
1answer
71 views

Numerical error analysis

I stand before the following task and I do not know how to solve it. The input parameter $$a=10^6, b=10^6 + 10^{-2}$$ will be round internally to $$a^*, b^*$$ with $$a=a^*(1 + \epsilon_1)$$ $$b=b^*(...
6
votes
1answer
142 views

Floating point arithmetic operations when row reducing matrices

A numerical note in my linear algebra text states the following: "In general, the forward phase of row reduction takes much longer than the backward phase. An algorithm for solving a system is usually ...
1
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2answers
167 views

Explaining roundoff error when row reducing matrices

In my linear algebra textbook (in the context of row reducing and obtaining a matrix in echelon or reduced echelon form), there is a numerical note that reads as follows: "A computer program usually ...