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

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Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $y''(x)+ e^y(x) = 0$, $\lambda = 1$, $x\epsilon[0,1]$, $y(0) = y(1) = 0$ I have to solve this using the tridiagonal matrices. None of ...
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19 views

What's the general procedure to find roots of a funcion using numerical methods?

When I'm facing functions for which no formula exist to calculate the roots directly, what can I do with calculus to analyse it so that I can obtain information about the function's behavior? Suppose ...
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1answer
36 views

Trying to re-write Simpson's Rule: mistake?

Pre-Question (edited): Thanks Arthur Orignal Problem: The standard form of Simpson's Rule requires an even value of n so that you can make a series of parabolas Parabola 1 has area ...
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0answers
20 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
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0answers
15 views

Roots of characteristic polynomial have negative real parts implies positive coefficients of the polynomial

Can you help me prove that if all the eigenvalues $\lambda_i$ of a square n-dimensional matrix $A$, have a strictly negative real part then prove that all the coefficients $a_j$ of the characteristic ...
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0answers
23 views

solve complicated system of non-linear equations numerically

I have two algebraic equations I am trying to solve in MAPLE. They are: $14\,{a}^{26}{b}^{2}-91\,{a}^{24}{b}^{4}-364\,{a}^{22}{b}^{6}-1001\,{a} ...
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29 views

question in Numerical analysis

please guide me how to start and I will continue the another steps
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7 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
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2answers
34 views

When $a\ll b$, how to approximate $f = \int_0^a \sqrt{b^2+x^2}/\sqrt{a^2-x^2} \, \, dx$?

Suppose $a\ll b$. How do I then approximate $$\int_0^a \frac{\sqrt{b^2+x^2}}{\sqrt{a^2-x^2}}dx$$ ? I think that maybe Taylor approximation may help, but I am not sure how to proceed. My physics ...
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1answer
13 views

Gauss Curvature…Product of Minimum and maximum values

The function g(ϑ ) = cos2 (ϑ ) fxx (x0 , y0 ) + 2 cos(ϑ )sin(ϑ ) fxy (x0 , y0 ) + sin2 (ϑ ) fyy (x0 , y0 ) represents the Gauss curvature of the surface f (x, y) at the critical point (x0 , y0 ) in ...
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1answer
23 views

Find the lowest value of a parameter for which two functions intersect

I am attempting to an equation to determine the lowest value of $\lambda$ for which $f(x) = \lambda \sin ( \pi x)$ and $y = x$ intersect outside of 0 on the interval $[0,1]$ for some numerical ...
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1answer
17 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
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1answer
18 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
30 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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1answer
35 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
23 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
54 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
9 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
41 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
43 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 ...
2
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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
29 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 ...
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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
14 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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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
17 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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0answers
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
47 views

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

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
15 views

minimize the total cost of transportation [closed]

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
31 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
57 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
98 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
19 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
96 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
48 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
48 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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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 ...