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

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7 views

Compute the following norm derivatives

I was wondering if anyone can explain me how to compute the derivatives of the following norms: $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$ $\bigtriangledown ...
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0answers
23 views

Trajectory With Air Resistance

For a video game, I am trying to calculate the angle needed for a projectile to hit coordinates x,y (both non-zero) with air resistance, i used equations from this site, and derived a function of y ...
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0answers
16 views

How to solve parabolic equation via implicit Euler in 2 dimensions?

I have the following parabolic equation: $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} $$ over domain $(x,y)\in [0,10] \times [0,10]$ ...
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0answers
9 views

Solving Duffing equation by Matlab ode23

How can I use Matlab to solve numerically this duffing equation with known $\kappa, \Gamma, \omega$..thanks.. $$x'' +\kappa x' +x -x^3 =\Gamma \cos\omega t$$ I have only few knowledge of Matlab..
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2answers
36 views

For what values of $x$ is the assignment $y=1-\cos x$ problematic, and why?

So I'm kind of stuck on this question and I don't exactly know how to describe this on the title header and I apologize... For some values of $x$, the assignment statement $y := 1-\cos(x)$ ...
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0answers
7 views

Polynomial Interpolation Existence and Uniqueness

The question I am attempting to solve is as follows: Let $f$ be a polynomial of degree $\le n$ and let $p_n$ be a polynomial interpolant to $f$, at the $n+1$ distinct nodes $x_0,x_1,...,x_n$. PROVE ...
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2answers
37 views

name of function $x_{n+1} = f(x_n)$ and the sequence it generates

this is a very trivial question, but I couldn't find what the proper name a function of the form $x_{n+1} = f(x_n)$, where $f: X \rightarrow Y$ for some initial $x_0$ is, and what the the sequence ...
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1answer
33 views

Explicit formula for the implicit Euler method

Given the problem; $\displaystyle\cases{ y'(t)=y^2(t) & \cr y(0)=1 }$ for $t\in[0,1]$ Using the implicit euler method, find an explicit formula to get $y_{n+1}$ HINT: The ...
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0answers
21 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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0answers
8 views

Multiplication of polynomials in Chebyshev basis

For polynomials in the monomial basis like $p_n(x) = \sum_{k=0}^N a_k x^k $, the product of 2 polynomials is can be either found though the convolution of the 2 corresponding polynomial vectors or ...
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2answers
29 views

estimate for highly oscillatory superexponential integral

I would like to estimate $\int_{-\pi}^{\pi} e^{i n y} e^{-b e^{c y^{2}}} dy$ to within a RELATIVE error of better than 1%, if possible. Here, $n$ is an integer ...
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0answers
27 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
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0answers
19 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
2
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0answers
32 views

How does this equation hold (Secant method)?

Consider we are approxinating a root by the secant method. Then, the interation is given by $x_{n+1}=x_n - f(x_n)\frac{x_n - x_{n-1}}{f(x_n)-f(x_{n-1})}$. In my text (Atkinson), it's written that: ...
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0answers
13 views

best fit straight line MAXIMIZING k-y values

I understand the minimization of the sum of the least squares approach to obtain a best fit straight line. This approach, however, unduly weights the "outliers" more than those points close to the ...
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0answers
16 views

Polynomial division/deflation with FFT

There is a need to divide a polynomial $p(x)$ by polynomial $q(x)$, whereas it is known that the remainder will be zero (i.e. the question is about polynomial deflation). A known method is to use the ...
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0answers
5 views

Bounding the Lebesgue constant.

This is a homework question, so I would prefer hints/suggestions as opposed to full-out solutions. Given the Lagrange polynomials $\ell_i(x)=\displaystyle\prod_{j=0;j\neq i}^n\frac{x-x_j}{x_i-x_j}$ ...
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0answers
37 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
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2answers
16 views

How to solve this using gauss jordan method?

I am trying to solve the following equation using gauss jordan method but unable to solve due to the type of equations.At the end i am getting unwanted zeros in 2nd and 3rd row.Here is my work... ...
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1answer
26 views

Is it possible to restore the missing entry by Newton forward divided difference method?

I've only seen the similiar problem but there are some entries on higher degree given.
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10 views

Integration by parts applied to weak form of boundary value proble

In my finite element textbook the proof for strong and weak form equivalence is determined as such: $$\int_0^1w_{,x}u_{,x}dx = \int_0^1wfdx + w(0)h$$ Integrating by parts and making use of the fact ...
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0answers
46 views

What's the Fibonacci number sequence? In other words, which pattern do Fibonacci numbers have? In other words again, what are their properties? [on hold]

I want to know how to use the Fibonacci numbers to make a sequence, but first, please explain to me what the Fibonacci numbers are. I'm very curious about hearing your answers and I'm sorry if this ...
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0answers
14 views

minimize the total cost of transportation [on hold]

can anyone help me to solve this question to minimize the total cost of transportation,how to use Vogel’s approximation method
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0answers
21 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
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1answer
30 views

Determine error in Neville's Algorithm calculation

I've been mulling over this problem for a while and I don't even know how to start it. The book is hopelessly vague. The problem states Neville's Algorithm is used to approximate $f(0)$ using ...
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0answers
10 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
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1answer
23 views

Is it possible to solve pde with 2 Neumann boundary conditions (Gaussian Elimination)?

I have the following equation: $$ \nabla^2u = f $$ over $\Omega: [0,10] \times [0,10]$ where boundary conditions: $$ \left\{ \begin{array}{ll} \frac{\partial u (0,y)}{\partial x} = 0 \\ ...
2
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1answer
32 views

Estimation of superexponential integral

I was wondering if anyone could give as precise an estimate as possible for the integral $$ \int_0^b e^{-a e^{-x^2}}\, dx, $$ where $a$ is positive. It is not related to any special functions as far ...
1
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1answer
50 views

All fixed points of a function are globally stable or unstable.

I am analyzing the iterated sequence of the function $\lambda \sin( \pi x)$ for $x, \lambda \in [0,1]$, where $x_n=f(x_{n-1})$ for a paper I am writing. I know that all fixed points of this function ...
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3answers
54 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
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0answers
94 views

Taking the Fourier transform of a Hankel function

Considering the following inverse Fourier transform $$ f(t) = -\alpha \int_{-\infty}^{\infty} F(\omega)H_0^{(2)}(k(\omega) \beta) \exp(+j\omega t) d\omega$$ where $F$ is an arbitrary function and ...
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0answers
11 views

Adams-Moulton and BDF methods

whats are the differences between Adams-Moulton and BDF methods. which one is better and which one computes the solution faster? i think adams moulton is a better method as it can get to the solution ...
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0answers
18 views

Numerical method of lines for solving PDEs

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
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0answers
23 views

Uniqueness of a differential equation

Let $I_o=[t_0,t_0+T]\subset\mathbb R$, where $T>0$, $f\in C^0(I_0\times\mathbb R;\mathbb R)$ and satisfying Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
5
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1answer
93 views

Difference table for interpolation

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 $n$, $n(n-1)/2$ fraction was used. I ...
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1answer
47 views

What's the formal difference between analytical and numerical?

While trying to wrap my head around differential equations in a practical way, I found a quite enlightening phrase about it Solving a differential equation can be done in three major ways: ...
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2answers
47 views

What is rule of this function?

I have these values.these are inputs and outputs of a function.I want to find rule of function.input is N. ...
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0answers
12 views

Minmax approximation

Let $f(x)=a_nx^n+....+a_1x+a_0, a_n\neq0.$Find the minmax approximation to $f(x)$ on $[-1,1] $by a polynomial of degree$\leq n-1 ,$and also find the error $\rho_{n-1}(f).$ This problem is from one of ...
5
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2answers
79 views

Exact result of a series using Euler-Maclaurin expansion.

This is a variant of Exercise 64 in Chapter 9 of concrete mathematics. Prove the following identity \begin{equation} \sum_{n = -\infty}^{\infty}' \frac{1 - \cos( 2\pi n k )}{n^2 } = 2 \pi^2 ( k - ...
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1answer
48 views

Solve quadric equation system

How to solve this? For given real and symetric matrices $A_1,A_2,A_3,A_4\in\mathbb{R}^{4\times4}$ find $x\in\mathbb{R}^4$ $$x^TA_1x=0$$ $$x^TA_2x=0$$ $$x^TA_3x=0$$ $$x^TA_4x=0$$
2
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0answers
41 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
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0answers
14 views

What is the theory behind numerical integration such as adaptive quadrature and laplace approximation? [closed]

I am trying to understand the theory behind the numerical integration. How it is done and what it results? Thanks !!!
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0answers
42 views

How many iterations of the Newton's method are needed to achieve a given precision

There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge ...
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2answers
28 views

Given $f(x)= e^x - e^ax$ with roots $P$ and $Q$,$0<P<1<a<Q$ , show that $g_1(x) = e^x/e^a$ and $g_2(x)= a + \ln x$ have exactly two fixed points each.

I have a midterm tomorrow and while I was looking through old exams from my professor I stumbled on a problem for which I'm not able to see the solution. We want to find the rots of $f(x) = e^x - ...
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1answer
35 views

Rewriting partial differential equation

I have some trouble rewriting a partial differential equation, more specifically the heat equation in one dimension: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t)\\ $ ...
2
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1answer
19 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
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4answers
110 views
+50

Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision. What are the techniques, tricks, ...
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1answer
11 views

Can anyone explain why reducing the stepsize h used in Euler's Method reduces the approximation of a function at a point?

Let $y'=t^{3}y^{2}$ where $y(0)=1$. Approximate $y(1)$ using Euler's method with h=0.25. I learnt online that reducing the step size h reduces the error of the approximation. Can anyone explain why ...
1
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1answer
33 views

Euler-Forward product rule

For a numerical approximation we use the Euler-Forward method, we have as definition $$ f'(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ Now we have that $f$ is the product of two other functions namely ...
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0answers
20 views

Runge-Kutta method for solving (2x2) system of odes

I need to implement a 4th order RK-method (Runge-Kutta method) to the very well know non-linear system of odes: predator-prey system or Lotka–Volterra equations. The explicit system is: $$dx/dt = ...