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

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242 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
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2answers
181 views

What is the convergence rate of Brent's method (root-finding algorithm)?

As far as I know, Brent's method for root finding is said to have superlinear convergence, but I haven't been able to find any more concrete information. Is its convergence rate known to be at least ...
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52 views

Autocorrelation Function and Power spectrum from ACF

In my assignment I am required to write or use a C code to find the autocorrelation function of a given function and then find the power spectrum from it. The function is as follows: $$f(t) = \cos(10 ...
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1answer
41 views

Numeric Analysis Interpolation of $f(x) , f'(x) $

There is a problem i'm finding quite difficult to solve, i'd be grateful if anyone could point me to the solution : We want to interpolate the function $f(x)$ and it's derivative $f'(x)$ s.t ...
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48 views

Errors in numericaly solving hyperbolic PDE in matlab

I am a beginner for PDE and I want to solve a hyperbolic PDE using matlab's builtin function hyperbolic(). However I am facing some erros and I could not resolve them. Can someone suggest or comment ...
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22 views

Numerical evaluation of an infinite 3D sum of cosine?

Consider the following function: $$f\left(x, y, z\right) = \sum_{\left(n, m, l\right)\in \mathbb{N}_*^3}e^{-\alpha\left(n^2+m^2+l^2\right)}\frac{\cos\left(\omega nx\right)\cos\left(\omega ...
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87 views

Numerical integration of function with singularities

I am currently trying to solve a semi-infinite integral containing a set of singularities lying on the real axis numerically. The process I am using is breaking the integral into small steps $\Delta ...
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1answer
56 views

Is it possible to calculate $e^x$ given $2^x$?

Given a value $x$, if I have a microprocessor instruction that will give me the value of $2^x$, is it possible to calculate (or approximate) the value of $e^x$ ?
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78 views

Maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$

What is the maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$ on the interval $[-1,1]$, with $\bullet$ equidistant nodes $x_i$, $(x_0=-1,x_8=1).$ $\bullet$Chebyshev nodes, $\displaystyle ...
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1answer
70 views

Can Successive Over-Relaxation be used for Nonlinear Equations?

My question was whether or not successive over-relaxation (http://en.wikipedia.org/wiki/Successive_over-relaxation) could be used to find solutions to a nonlinear equation. In particular I am ...
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150 views

ill-conditioned matrix 'Matrix is close to singular or badly scaled'

in the equation of A*q=b, A is a NxN matrix in which the numbers can be up to 10^56 and the minimum is 1. the condition number of the matrix can be as large as 3.16e+064. The SVD, QR and LUP have ...
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1answer
137 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
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30 views

Numerical one-step method: initial value and non consistent method

We had to code a program solving a starting point problem. (Runge-Kutta 6th Order) The ordinary differential equation (first order) is: $y'*y= \cos x$ with $f(0)= 2$; $[0,10]$ I have 2 questions for ...
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20 views

Numerical integration and probability density functions

How to calculate the integrals of this type? Which method I can use? $$ I_1(t)=\int_{0}^{\infty} dy f(x,y,t)p(x_j,y,t)$$ where $p(x_j,y,t)$ is $p(x,y,t)$ for some $x=x_j$. ...
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45 views

Convergence theorem (interpolation)

I am trying to complete a proof of the theorem which we have considered in my numerical analysis course. The tutor made a short sketch, but for me it was not very clear how we prove the statement of ...
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1answer
53 views

How can Picard proved that his method was right?

In order to solve initial value problems .. We know that Picard's method is right , but i need to know how can Picard proved this ?
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1answer
32 views

How to fit parameters to given data of given function?

I am given a function $g(t) = a + b \cdot \exp(-c \cdot t)$ and a set of $(t_i, g(t_i))$ pairs (temperature measurements), and the task is to find values of parameters $a,b,c$ s.t. they fit given ...
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2answers
47 views

How to prove that there is no interval that maps to itself under a function

I have the function $ g(x) = x^3 + 3x^2 - 3 $ and I need to show that there is no interval $ [a,b] $ such that $ g:[a,b] \mapsto [a,b] $. How do I go about this? Thanks a lot
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1answer
121 views

Finding vector $x$ so that $Ax=b$ using Householder reflections.

Assume $n\times m$ matrix $A$ and vector $b$ are given. I am looking for $x$ that satisfies $Ax=b$ in terms of linear least squares problem. Let $A=\begin{bmatrix} 1 & 1 & 1 \\0 & 1 & ...
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1answer
39 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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1answer
51 views

Proof using the Contraction Mapping Theorem

How do you identify an interval [f,g] so that the Contraction Mapping Theorem guarantees convergence to the positive fixed point for the following: a) $\frac{14-x^3}{13}\ $ b) $e^{-x}$ I tried ...
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47 views

Initial approximation to inverse of beta distribution function / quantile of beta distribution

I'm interested in implementing an algorithm to find the quantile of the beta distribution, and I'm looking at this paper: Journal of the Royal Statistical Society Series C (Applied Statistics). 1973, ...
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1answer
52 views

Numerical Solution of an Equation with Multiple Roots

Let me consider an equation $f(x)=0$ which I know to have a solution $x=x_0$, but I need to find its another solution. So I might consider finding root for the equation $$\frac{f(x)}{x-x_0}=0$$ but I ...
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1answer
373 views

What initial guess is used for finding n-th root using Newton-Raphson method?

I would like to know what is an optimal initial guess for use with Newton-Raphson method when finding n-th root. I develop some program which uses GMP C++ library. GMP manual says: The initial ...
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2answers
74 views

Method of False Position (Regular Falsi) - Pros/Cons

Could anyone provide and explain some drawbacks and benefits of the method of false position against say newtons method. I know one of benefits is that it doesn't require the derivative and one of ...
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1answer
110 views

Due to numerical inaccuracy, the solution of a boundary value problems becomes negative

I treat a toy example to get my point across. In reality I have to deal with a much more complex model. Let us consider a one dimensional boundary value problem using the bvp5c solver in Matlab. Two ...
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1answer
40 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
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103 views

What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
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21 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
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1answer
87 views

Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
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41 views

like Gauss-Chebyshev integration formula using Lagrange polynomials

Suppose that $L_k(x)$ is Lagrange Interpolation Polynomial for points $x=1,0,-1$. How to show that: $$\int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx=\sum_{k=-1}^1C_kf(k)+E$$ where ...
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0answers
25 views

finite element space: $\{v\in V^h:\mathrm{div}(v)=0\}\Rightarrow v^h\equiv 0$

Let $\bar\Omega=[0,1]^2$ be the unit-square. If we triangulate the unit-square uniformly with triangles of length $h$, we can define $V^h_0$ as the finite element space with homogenous ...
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0answers
25 views

computing length of a curve given as set of points

Given a set of points $(x_i,y_i)$ from a simple curve. How can the length of the curve be computed approximately?? I understand these points can be connected by line segments and the sum of the ...
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20 views

Statistical or numerical solution of the equation

Consider autoregression AR(1): $$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$ $\{\varepsilon_t\}$ - i.i.d. random variables with $E\varepsilon_1 = 0,$ $E\varepsilon_1^2<\infty$ ...
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0answers
25 views

Plot an implicit function using codes in Mathematica

Suppose that $f(u,v,\phi)=0$ can only be solved implicitly and numerically to give $u=u(\phi)$ and $v=v(\phi)$. Each $\phi_i$ gives $(u_i,v_i)$ by solving numerically $f(u,v,\phi_i)=0$, $i=1,2,3,...$. ...
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1answer
41 views

Orthogonality of Lagrange Polynomials in Hermite Inner Product

My question is as shown above. I have churned through the first part, but I am stuck on showing the orthogonality of the Lagrange polynomials. My first hope was to use the fact that the Lagrange ...
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1answer
59 views

Writing in 1993, a researcher noted that it is hard to prove things about a cellular automata model - has this changed?

Leah Edelstein-Keshet in her 1993 article Cellular automata approaches to biological modelling writes: We do not believe that CA should be viewed as a replacement for rigorous mathematical models. ...
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3answers
58 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
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1answer
44 views

How to compute the pdf of a sum of iid random variable using discrete Fourier transform?

Suppose there are $n$ i.i.d random variables $X_1, X_2, ..., X_n$ sampled from the distribution $p(X)$. We can compute the characteristic function of $X$ by \begin{equation} f(t) = \mathbb{E}[e^{i t ...
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51 views

Francis Algorithm (Implicit QR Algorithm)

In Numerical Analysis, we are touching upon QR and Francis Algorithm. I understand that for Francis's Algorithm, we reduce the matrix to its upper Hessenberg form using Householder transform. What I ...
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1answer
64 views

how to find an upper bound of the spectral radius

I've given the real valued matrix $K=K_1+K_2$, with $K_1$ and $K_2$ symmentric and positiv defined. Further there are given this 3 matrices: with $\omega > 0$ and Now I tried the whole day ...
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1answer
31 views

Numerical Hodge decomposition with boundaries

For a fluid simulation, I'm using the algorithm proposed by Jos Stam (http://www.intpowertechcorp.com/GDC03.pdf). One step of the algorithm is a routine that projects the velocity vector field so ...
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13 views

efficiently solve for values of a coefficient in a function, so for those values, the function intersects another function a specific number of times.

This is my summer assignment for my freshman "Intro to Numerical Methods with Matlab: Unit 2" course. The task: "Write an efficient Matlab code, which will take any closed $f(x)$ and $g(x)$ and ...
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106 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
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3answers
78 views

About the calculation of decimal digits of series up to the nth digit

Considering that we don't know any of the digits of some number defined as the limit up to infinity of a sum, I want to know how many terms do I have to sum to get the correct decimal representation, ...
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2answers
103 views

Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

I have to integrate $$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$ I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I ...
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1answer
38 views

Gauss-Newton method — is this matrix product invertible?

In the Gauss-Newton method for solving overdetermined systems of equations, the iteration matrix is of the form $(J^t J)^{-1} J^t$, for a $m \times n$ Jacobian matrix $J$ with $m > n$. I was ...
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1answer
30 views

QR decomposition

Assuming that we have a QR decomposition of a matrix $A \in \mathbb{R}^{m \times n}$, $Q \in O(m)$ and $R \in \mathbb{R}^{m \times n}$ such that $A=QR$, where R is an upper triangular matrix. Now ...
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3answers
82 views

Approximating the length of an ellipse given equation

Can anyone help me with this problem in numerical analysis? Determine to within $10^{−6}$ the length of the graph of the ellipse with equation $$4x^2+9y^2=36$$ Thanks a lot.
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1answer
27 views

Accuracy of the result question short?

We have 1/(4,5) . When we do the divison,what accuracy does the result have?