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

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Backwards Stability of systems

Let $A$ be a nonsingular matrix, let $x_{k+1}$ be an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$. Show that $x^{k+1}$ is $\epsilon$-backward stable approximate of ...
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21 views

Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= ...
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3answers
48 views

Linearizing an equation containing both $x$ and $\ln x$

The equation of interest is of the form: $$ k_1 \ln(y/x) = k_2 x $$ And I am wondering how can one linearize this equation for $x.$ Splitting the $\ln$ function would give something along: $$ k_1 \ln ...
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44 views

Convergence Newton method convex function

Is the Newton method convergent for convex differentiable functions and system of convex differentiable functions? Assuming you don't start with a stationary point and there exists a root. If not are ...
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1answer
31 views

numerical methods for ODEs

I am working on this equation: $$\frac{dx}{dt}=Ax+b$$ $$c'x=d$$ Where $x$ is a vector ,A is a constant matrix, b c are constant vectors. d is a constant number. i.e. $c_1x_1(t)+\cdots+c_nx_n(t)=d$ ...
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49 views

Proof that equation is non-convex function

I have a objective function as following $$E(\phi)=\int_{\Omega}(I(x)-m_1)^2H(\phi(x))dx+\int_{\Omega}(I(x)-m_2)^2(1-H(\phi(x)))dx+\int_{\Omega}|\nabla H(\phi(x)|dx$$ where $I$ is an image; $I: \Omega ...
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1answer
15 views

Numerically stable calculation of multinomial probabilities

I'm looking for a numerically stable method to compute expressions of the form $$\frac{(a+b+c+d)!}{a!b!c!d!}\left(\frac{1}{4}\right)^{a+b+c+d}$$ So far I've been using a compensated sum algorithm to ...
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1answer
21 views

System of equations from weighted Gaussian Quadrature

I've been working on a weighted Gaussian Quadrature problem for a Numerical Analysis class and have been having the hardest time. The problem boils down to solving a system of four equations: $$ ...
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1answer
137 views

Evaluate $\int_0^2 \sqrt[3]{x^2 + 2x - 1} \, dx$

Calculate the value of the integral $$ \int_0^2 \sqrt[3]{x^2 + 2x - 1} \,dx $$ with measurement uncertainty not larger than $10^{-3}$. I know we can evaluate integration using the ...
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22 views

Looking for polynomial to represent approximate 2D matrix.

I am looking for a polynomial that similars Legender polynomial(a set of orthogonal polynomial basis function. Could you suggest to me some polynomial? Because my goal is that I want to approximate 2D ...
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1answer
39 views

Reaction diffusion numerical scheme: discrepancy between devised scheme and scheme that works

I've been writing a PDE solver for a reaction diffusion equation in one dimension. To test it I'm using the "method of manufactured solutions" (that's the term my supervisor gave it - I'm not sure if ...
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1answer
76 views

Show that $\limsup$ of $\sin n$ is 1

I want to prove that the $\limsup\limits_{n\rightarrow \infty} \sin n=1$. I know that $1$ is an upper bound for $\sin n$ but I cannot find a subsequence of $\sin n$ that converges to $1$. Can somebody ...
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1answer
19 views

Obtaining a numerical method of the form $y_{n+2}-y_n=h\left(\beta_0 y_n'+\beta_1 y_{n+1}'+\beta_2 y_{n+2}'\right)$ given an identity.

I am stuck in part B of this exercise (I am giving the whole exercise though). I would really appreciate it if someone could explain to me how to solve this type of integral. Thanks a lot! Question: ...
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1answer
30 views

Obtain a higher order of accuracy for a particular Runge Kutta method than the maximum order for general problems

I am going through a past exam paper, but there is a solution which I don't understand. Thanks a lot! The problem has 2 parts, and the part I don't understand is in part B, but it is related to part ...
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25 views

Even least squares approximation

Can anyone help me with this problem or give me a tip on where to start. Let's consider $\theta_n$ a class of approximations with the following properties: all functions $\varphi \in ...
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1answer
19 views

Why are quadrature points given by the zeros of orthogonal polynomials?

We know there exists unique Gaussian quadrature formula. Its quadrature points are given by the zeros of the orthogonal polynomial. Why do we use only the zeros of the orthogonal polynomials in ...
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2answers
33 views

Using Lagrange interpolation to determine canonical form of polynomial

I have some difficulty of determining the canonical form of a polynomial. Here is the problem: Suppose $P$ is a polynomial with degree $5$, and value of $P$ at $-2, 0, 1, 4, 5, -3$ are $2, 4, ...
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3answers
106 views

What are two easiest numerical method to calculate $\sqrt[23]{123456789}$ by hand?

What are two easiest numerical methods to calculate $\sqrt[23]{123456789}$ by hand? I want to compare that two methods myself Just name the methods
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1answer
11 views

Triangle Inequality for SPD Matrix Norm

We define a symmetric, positive-definite matrix $A$ to be one such that $A = A^T$ and for $x \neq 0$, $x^TAx > 0$. If we have a norm $\|x\|_A = \sqrt{x^TAx}$, how can we show the triangle ...
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1answer
23 views

How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to ...
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1answer
18 views

Proof using smaller step size and increasing step, Euler method tend to exact solution(solution verification)

Please help to verify is the proof below contain any error. I start by considering a differential equation $\frac{dy}{dt}=f(t)$ and using a step size of $\frac{h}{n}$ where n is consider to be a very ...
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1answer
39 views

How to take derivative of matrix inside integrate $\frac {\partial \int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx}{\partial A}$

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+\lambda_1 A^2+\lambda_2 B^2$$ where $A^T$ is transpose of vector $A$. $A$ is a column vector such as $A= \begin{bmatrix} ...
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1answer
14 views

Using y'' in numerical iteration

Lets say I know $y(x_1), y'(x_1), y''(x_1), y(x_2), y'(x_2), y''(x_2)$ values exactly I know only 2 points. What is the best way to find $y(x_3)$? I don't know how I would include the second ...
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2answers
75 views

Prove that $f^{-1} (F)$ is closed

A set $F \subset \mathbb R$ is closed if for any convergent sequence $\{x_n\}$ in F converges, we have $\lim_{n \to \infty} x_n=x \in F $. How to Prove that if $f :\mathbb R \to \mathbb R$ is ...
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2answers
114 views

Accurate Formula and One Old-Exam Questions?! [closed]

We get stuck in a problem on old-exam. \begin{equation*} A=\sqrt{x+ \frac {2}{x}} -\sqrt{x- \frac {2}{x}}~\text{and}~x>>1. \end{equation*} For calculating $A$ which of the following option ...
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0answers
18 views

A question of Adams-Bashforth formula

This problem is from Numerical Analysis 3th (David Kincaid & Ward Cheney) problem 8.4.8 a. Using the method of undetermined coefficients, determine A and B in the following Adams-Bashforth ...
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3answers
63 views

$17!=3556xy428096000,$then $(x+y)$ equals?(without using a calculator)

$17!=3556xy428096000,$then $(x+y)$ equals? a)$15$ b)$6$ c)$12$ d)$13$ With help of calculator $(x+y)$ can be easily calculated as $15$.But without a calculator,I can only conclude that the sum of ...
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1answer
61 views

Numeric integration of Greens Function over singularity

I'm currently using python to numerically evaluate the follow expression at various values of $r$ and $\theta$. \begin{equation*} f(r,\theta) = \int_{-\pi}^{\pi}\int_{0}^{R}\frac{\exp(ikS)}{2 \pi ...
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155 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...
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0answers
27 views

Can this equation be solved given integer $n$ and constraint on $k$ and $e$ to be rational? $2k (e^2-n) + k^2 = 2n e^2 - 2n^2$

I am looking for a general way to solve for rational $e,k$ given integer $n$ $$2k (e^2-n) + k^2 = 2n e^2 - 2n^2$$ Repeating fraction methods are fine I just need a rational number for $k$ and ...
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1answer
18 views

relationship between forward and backward error?

Let $\tilde{f} $ be some algorithm: we have: $$ \| f(x) - \tilde{f}(x) \| = \| f(x) - f(\tilde{x}) \| \leq \|f'(x) \| \|x - \tilde{x} \|$$ I'm curious on the last step, how did they get the ...
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2answers
42 views

Convergence multi variable newton method with polynominals

I have n polynominals in n variable. Are there any sentences that gurantee convergence if the jacobian is not singular at the start point? I have one linear polynom in n variables. And n-2 ...
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1answer
24 views

interpolating and difference table, an old mid exam?!

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, how many fraction was used? ...
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38 views

How to solve this ODE numerically?

I have a question about how to solve this ODE numerically: $$\frac{C}{4}y'^2+\frac{C}{4}y''y+(0.098)^2y''y'''=0$$ where $C$ is a constant and the initial conditions are $y(0)=y''(0)=0$ and $y'(0)=1$. ...
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6answers
120 views

how can i find roots of $4{x^3}+24{x^2}+74{x}+50 $? [closed]

There is one root between 0 and -1. than how can i find that root by hit and trial method. or any other method that can be used.
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1answer
20 views

Determining convergence of multistep method

I'm a bit confused on how to show that whether multi-step scheme such as $$u_{n+2}-3u_{n+1}+2u_n=-hf(t_n,u_n)$$ (modified from these lecture notes) is convergent. Neither my lecture notes nor my ...
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0answers
13 views

sparse, complex, unymetric test-matrix

Can anybody recommend me a sparse, complex, unsymmetric test-matrix (maybe from MartixMarket) which is solvable with a transpose-free QMR without preconditioning in under 1000 iterations?
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1answer
12 views

forms of the Romberg Method equation

My teacher wrote the this equation for the Romberg method $ I_{j,k}=\frac{4^j I_{j-1/k+1}-I_{j-1/k}}{4^j-1} $ Is this the right equation? Most the equations I looked at online for the Romberg ...
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3answers
15 views

Fixed Point Iterations - Root Finding

Given any function $f(x)$, how can you come up with the corresponding $g(x)$ such that $g(p)=p$ (where p is the root)? Say, $$f(x)= sinx -\frac{x}{1.4}$$ my professor told me to simply isolate for ...
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33 views

Distance from a point to the involute of a circle

I know that the involute of circle of radius $r$ centered at $(0,0)$ is given by the following parametric form: $$\begin{cases} x(\theta) = r \big(\cos(\theta) + \theta\ \sin(\theta) \big),\\ ...
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2answers
67 views

Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
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1answer
26 views

Finding the solution to an equation using trial and improvement.

Using trial and improvement to find this solution to 2 decimal places. The equation $x^3=10-3x$ has a solution such that $1 \le x\le 2$.
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6answers
116 views

Showing that $\sin(x) + x = 1$ has one, and only one, solution

Problem: Prove that the equation $$\sin(x) + x = 1$$ has one, and only one solution. Additionally, show that this solution exists on the interval $[0, \frac\pi2$]. Then solve the equation for x with ...
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1answer
24 views

Origin of divergence in a divergent field (2D)

I have a field of measured vectors, see example of four vectors in image below. If there was no noise they would all point outward exactly from one "central point". i.e. there would be a circle whose ...
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0answers
26 views

Order of convergence of nonlinear iterative solver

I'm given a sequence $x_n \rightarrow \alpha$ of a nonlinear solver such that $$\lim_{n\rightarrow\infty}\frac{x_{n+1}-\alpha}{x_n-\alpha}=c$$ converges linearly (i.e. $c\in(0,1)$). Now, I need to ...
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1answer
14 views

How to find out transient response of z-transform (discrete)

Given z-transform transfer function $H(z) = \frac{Y(z)}{X(z)}$, with the corresponding linear ODE, how does one find out transient response of such a transfer function given a certain input?
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16 views

How to calculate u1-modified parametric iteration method

Please refer to paper "A modified parametric iteration method for solving nonlinear second order BVPs, link: http://www.scielo.br/pdf/cam/v30n3/a02v30n3. In example $5.1$, I am unable to understand ...
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1answer
22 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
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0answers
25 views

Region of absolute stability

We have the problem $$\left\{\begin{matrix} y'=\lambda y &, t \in [0,+\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\ y(0)=1 & \end{matrix}\right.$$ Applying the Backward Euler method ...
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1answer
29 views

Is the system stiff or not?

Let the problem of initial values $\left\{\begin{matrix} y_1'(t)=-0.5y_1(t)+0.501 y_2(t), t \in [0,10^3]\\ y_2'(t)=0.501y_1(t)-0.5y_2(t) \\ y_1(0)=1.1 \\ y_2(0)-0.9 \end{matrix}\right. \\ $ The ...