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

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3answers
65 views

If you want to know $\sin (x)$ within $0.5$ of its true value, then how accurately do you need to know $x$?

If you want to know $\sin (x)$ within $0.5$ of its true value, then how accurately do you need to know $x$? I don't really understand how to think about this question.
1
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1answer
303 views

cube root of positive definite matrix

Suppose that $A$ is a real symmetric positive definite $20\times 20$ matrix with condition number $\kappa\le 1000$. I want to solve the system of linear equations $$A^{1/3}x=b$$ with $10$-digit ...
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1answer
1k views

Cubic Spline Interpolation practice

Going over practice problems for our final exam. I'm stuck on a problem involving cubic splines. In fact, I don't even know where to begin. I need to find the natural cubic spline $S(t)$ at $t_0=0, ...
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0answers
44 views

Is numerical stability preserved under basis transformation for pde

I'm using a Backward Time Central Space method to solve the heat equation in polar coordinates. In Cartesian coordinates, it is easy to show this is unconditionally stable (by assuming solution of the ...
3
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1answer
74 views

Proving conservation of mass for linear advection

Reading through some course notes about conservation of mass in linear advection approximation schemes, given $\phi(x, t) \in \mathbb{R}$ and is defined for $0 \leq x < 1$ with periodic boundary ...
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1answer
41 views

Showing that the composite trapezoid rule is invariant to certain symmetry

I am trying to show that if $f$ holds certain symmetric properties than the composite trapezoid rule $$ T(n) = (b-a)\frac{f(a)+f(b)}{2n} + \frac{b-a}{2} \sum_{k=1}^{n-1} f(x_k) $$ is invariant over ...
2
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3answers
47 views

Algebra question / conversion of ranges

Greets All Forgive me if I'm using the wrong terms but I'm trying to sync up two number ranges together. Example: I have two x axis (ranges) I would like to equate with each other ...
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3answers
102 views

Why does it help to use $640320^3 = 8\cdot 100100025\cdot 327843840$ when you calculate $\pi$?

A Ramanujan-type formula due to the Chudnovsky brothers used to break a world record for computing the most digits of $\pi$: $$ \frac{1}{\pi} = \frac{1}{53360 \sqrt{640320}} \sum_{n=0}^\infty (-1)^n ...
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1answer
244 views

Numerical Analysis: Bisection method proof

Prove that $C_{n+1} - C_n = 2^{-n-2}(B_0 - A_0)$ where $C_n$ is the $n^{th}$ computed value of $C$ in the bisection method. Not really sure where to start.
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0answers
19 views

Iteration technique ambiguity

I'm not sure what iteration technique I should use for this problem? We have learned Jacobi and Gauss-Seidel, but it is not specified here. Also, all of the values are variables, so I'm not sure how ...
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0answers
48 views

Conjugate Gradient Methods problem

I have one proble to solve. I have to determine $x$ that minimizes $d$. $C$ is the centroid of the figure. This is the problem given problem my numerical analysis class which is something that I did ...
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0answers
37 views

Finding a least squares approach

Let $c \in \mathbb{R}^n$ and $\mu \in \mathbb{R}$, $\mu > 0$. Find the matrix $A$ and vector $b$ to solve this problem using the least squares approximation: $$\min \left\{ x \in \mathbb{R}^n : ...
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2answers
40 views

Showing a limit for the mean value property

I need to understand a proof and i have a problem with the limit process at a specific point in this proof. I want to know, what is needed to get the following result: Let $f \in C^1(G)$, $\bar{x} ...
3
votes
3answers
334 views

Find an accurate value of $f(x)=\sqrt{4x^2+x}-2x$ for large values of x. Calculate $\lim_{x\to\infty}f(x)$

My works: $x^2$ can be very large if x is large, thus the function has lose-of-significance error and we need to reformulate it. $$ ...
1
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1answer
139 views

Use Taylor polynomials with remainder term to evaluate the following limits $\large\frac{e^x-x-1}{x^2}$

My work: Since $\large e^x=\sum\limits_{j=0}^\infty \frac{x^j}{j!}$, then $\large\frac{e^x-x-1}{x^2}=\sum\limits_{j=2}^\infty \frac{x^{j-2}}{j!}=\sum\limits_{d=0}^\infty \frac{x^{d}}{(d+2)!}$. (Let ...
2
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0answers
325 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
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0answers
75 views

Finding roots/zeros for collision detection in a video game

For the most simple of 2D games, I have implemented a posteriori collision detection (overlapping rectangles) on the $xy$ Cartesian plane, but am now interested in understanding the basics of a priori ...
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2answers
123 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$ \int_L K(x,y)U(y)dy = f(x) $$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
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1answer
27 views

Does existence of derivatives imply continuity? In regard to usage of Bernstein theorem for approximation using polynomials.

I have a function $f$ defined on interval $[a,b]$ and I know that all of it's derivatives exist and $|f^{(k)}(x)|>0$ for all $x \in [a,b]$ and $k \in \mathbb{N}$. Does this imply that $f$ is ...
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0answers
79 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
3
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1answer
64 views

Numerical evaluation of an integral “similar” to an exponential integral

What is an efficient and stable numerical algorithm to evaluate the integral: $\int_0^L e^{-\alpha x}\frac{e^{\frac{i\beta}{(x+x_0)}}}{(x+x_0)}\mbox{d}x$ with $i$ the imaginary unit, ...
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1answer
671 views

Jacobi method convergence for a symmetric positive definite matrix in $\mathbb{R^{2 \times 2}}$

I have to prove that a symmetric positive definite matrix $A \in \mathbb{R^{2 \times 2}}$ converges for the Jacobi method. Any ideas? The matrix $A$ is said to be positive definite if $x^t A x > 0 ...
3
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1answer
91 views

Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the Question for $$ b^{p-1} \equiv 1 \pmod{ p^m} \qquad \text{ with $p \in \mathbb P $, $1 \lt b \lt p$ and ...
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1answer
101 views

Numerical integration for integrals 7th order.

I need to calculate integral which looks like below, with some numerical method: ...
2
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1answer
515 views

computing the degree of exactness for a quadrature formula

I am reading Numerical Analysis and I have some problems on how to use in practice the definition of the '$\textbf{Degree of Exactness}$'. $\textbf{Definition:}$ Let $I_n(f)$ be a quadrature formula ...
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0answers
58 views

Quadrature errors of midpoint and trapezoid formulas

I have this question in my homework in Numerical Analysis but I can't figure it out. Can someone have a look and help me? Question: Let $E_0(f)$ and $E_1(f)$ be the quadrature errors of the midpoint ...
2
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1answer
158 views

Understand a weird method of calculus

I see this method of calculus on youtube and my question: is this method valid? How we can understand it? Thanks.
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1answer
60 views

Rate of convergence for series.

1)What is convergence rate of a series \begin{equation} K(k) = {\frac{\pi}{2}} \hspace{1mm} {\sum_{m=0}^\infty}\binom{-1/2}{m}^2 k^{2m} \end{equation} Note that the presence of squares of ...
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2answers
991 views

finite difference method for nonlinear ode

I am trying to use finite difference method to solve $y'''+y^2y''-y'=0, y(0)=y'(0)=0, y''(1)=1$. I let $u=dy/dx$ so the new problem is $u''+y^2u'-u=0, u(0)=0, u'(1)=1,y=\int u$. To try and solve this ...
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1answer
89 views

Simpson Rule special case of Gauss quadrature

Prove that the Simpson Rule for integrating the function $f$ on the interval $[a,b]$: $$I_2(f)=\frac{b-a}{6}(f(a)+4f(\frac{a+b}{2})+f(b))$$ is actually the Gauss quadrature for the weight $g=1$ and ...
2
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7answers
123 views

Solve for $n$ in $ \left(n+1 \right) 0.5^n=0.05$

$$ \left(n+1 \right) \times 0.5^n =0.05 $$ Is there a way to solve this directly for $n$? I know that by taking logs we can simplify it but we still do not get a value as far as I can see. A solution ...
3
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2answers
163 views

Approximating an Integral for Numerical Computation

I have a program that involves computing a definite integral many times, and have been struggling to find a way to do so quickly. The integrals I need to solve are of the following form: ...
2
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1answer
87 views

Quadratic Root Equation Error

Suppose a machine with the floating-point system $\beta = 10$, $p = 8$, $L = -50$, and $U = 50$ is used to calculate the roots of a quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ ...
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0answers
184 views

A 2D secant method?

I've recently had occasion (providing an engineering colleague with a little mathematical help) to solve a non linear system $\begin{align*}f(x,y)&=0,\\ g(x,y)&=0.\end{align*}$ If ...
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1answer
29 views

How to determine coefficients of $p(x) = x^6$ with the Chebyshev processing

I want to calculate the coefficients of $p(x) = x^6$ with the Chebyshev processing. How to do that? Following question would be, how to estimate the error in $[-1,1]$, if i only use terms until ...
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2answers
51 views

Numerical plot of function which cant be integrated.

How to plot numerically function $F(x)=\int e^{-x^2}dx$ ?
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1answer
29 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
2
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1answer
212 views

Series of natural numbers which has all same digits

For which x exists sum 1 + 2 + 3 + ... + n, where n > 3, which has notation xxx...x? So I am looking for a sum of natural numbers which gives a result which has all same digits, e.g. 5555555 or ...
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1answer
69 views

Proving boundedness for a FTBS numerical scheme

Given an FTBS scheme $\phi_j^{(n+1)} = \phi_j^{(n)} - c \left(\phi_j^{(n)} - \phi_{j-1}^{(n)} \right)$ where $c$ is the courant number, $n$ is the timestep, and $x$ is the spatial index, how can I ...
1
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1answer
280 views

Huffman Coding Integer Identifier Table

this might be trivial but I have a brainfog here! I am working through a JPEG compression problem using Huffman codes. According to our book, the process for coding a quantized matrix for a jpeg has ...
1
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1answer
173 views

Is implementation of method possible?

Can I implement the Jacobi and Gauss-Seidel method,at a matrix, even if its determinant equals to zero? I use Matlab and want to find the convergence of the method for the Hilbert matrix. I wanted to ...
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0answers
137 views

What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot ...
0
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2answers
94 views

Solving an initial value problem using Runge-Kuttas

I am trying to solve the equation Using Runge-Kutta methods, but the lack of y- values on the right hand side is confusing me. Any help would be much appreciated. My initial conditions are: dy/dx = ...
0
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1answer
78 views

Use Lagrange interpolation to prove $\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$

Suppose $f\in C^2([a,b])$ and $f(a)=f(b)=0$,use Lagrange interpolation to prove $$\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$$ I tried to use the theoretic error to prove ...
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0answers
60 views

Gaussian quadratures: Adaptive method

In my book I was asked to find the gaussian quadrature on the form $$ \int_0^1 f(x)\mathrm{d}x \approx W_0 f(x_0) + W_1 f(x_1) $$ With weight $w(x)=1$ and where $x_0$ and $x_1$ are the zeros of ...
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0answers
91 views

Discrete Harmonic Functions

Let $\Omega$ be a Lipschitz domain, $\Gamma\subset\partial\Omega$ with Lebesgue measure $>0$. Let $u_\Gamma\in V^h|_\Gamma$ be the trace of a finite element function. Then the following holds: $$ ...
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1answer
191 views

Solving Poisson Equation Finite-difference using maple

How do I solving Poisson Equation Finite-difference using maple consider Poisson equation $$\frac{\partial^2u}{\partial x^2} (x,y)+ \frac{\partial^2u}{\partial y^2} (x,y) = x*e^y$$ $0<x<2$ ...
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1answer
57 views

Question about Notation. What does this means? $f[0]=1, f[0,1]=-1$

Question about Notation. What does this means? $f[0]=1, f[0,1]=-1, f[0,1,2]=2$ (The values are exact, which is pretty confusing too, if they are refering to intervals) This question is from a ...
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1answer
835 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
2
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1answer
248 views

Iterative update of pseudo inverse solution

I have an overdetermined linear problem of the form $A x = b$, which is solved in least squares sense using the Moore–Penrose pseudo invers. The issue now is, that over time additional constraints and ...