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

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0
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1answer
34 views

Sum of Lagrange polynomial

I have to calculate $\sum_{i=0}^n x^k_i*l_i(x)$ for $k=0,1,2,...,n$ . I've proved that $\sum_{i=0}^n l_i(x) = 1$, but I cannot figure out how it may help me calculating $\sum_{i=0}^n ...
10
votes
1answer
236 views

Fast inverse square root trick

I found what appears to be an intriguing method for calculating $$\frac{1}{\sqrt x}$$ extremely fast on this website, with more explanation here. However, the computer-science lingo and ...
1
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2answers
29 views

MATLAB Newton non-linear equation

I have the following non-linear equation: where $w0=0.25,w0=0.5,w0=0.75$. I have to prove that if $k$ is a root, then also $−k$ is a root and that there exists only one $k∈(0,1)$ root, but my ...
2
votes
1answer
188 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
1
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0answers
24 views

Condition number composite function

I have a composite function $h(t)=g(f(t))$ and have to evaluate the condition number for $h(t)$ through the condition numbers of $g$ and $f$. I know that the condition number formula is ...
2
votes
3answers
130 views

How to calculate the square root of a number? [duplicate]

By searching I found few methods but all of them involve guessing which is not what I want. I need to know how to calculate the square root using a formula or something. In other words how does the ...
0
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0answers
29 views

Stiff differential equations without using Jacobian matrix

I want to solve a stiff system of differential equations. Its Jacobian matrix isn't constant and its determinant is close to zero so I cant inverse of it. Please tell me does exist a method that solve ...
0
votes
1answer
26 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
0
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1answer
57 views

Stiff differential equation

I'm trying to solve a system of differential equations with Runge-Kutta method. When I use the step size $h=1$ my problem has true answer but when I use the smaller $h$ (for example $h = 0.1$) my ...
1
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0answers
32 views

Numerical solution of non-linear differential equation with MATLAB

I need some information to know if I can solve a nonlinear integral equation with terms $ u_{x} $ , $ u_{x}.u_{y} $ , $ u_{xx} $ , $ u_{xy} $ $u_{yy} $ $ u_{x}^{2} $ $ u_{y} ^{2} $ By numerical ...
0
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0answers
27 views

Integral with Inverse error function

I have a challenging integral to solve involving the inverse error function, $\rm Erf^{-1}$, $\mathcal{I}(x,\beta)=\int\,_{x_c(\beta)}^x\,{\rm d}x\,\exp\left[\sqrt{2}\sigma\,{\rm ...
1
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2answers
78 views

Questions about the field scientific computing

I have heard about the field of Applied and Computational Mathematics, Scientific Computing and want to get some information. Is this a combination of computer science and mathematics? What subjects ...
2
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0answers
76 views

Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
1
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2answers
47 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: ...
2
votes
2answers
98 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 ...
0
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0answers
24 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 ...
1
vote
1answer
33 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 ...
0
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0answers
29 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 ...
0
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0answers
18 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 ...
2
votes
0answers
61 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 ...
-1
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0answers
40 views
+50

Hermite Interpolation with a missing value

How can I do the Hermite interpolation with the following function, with given ; http://imgim.com/img_20140514_124209-1.jpg Using this schema ...
1
vote
1answer
55 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 $e^x$ ?
0
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1answer
76 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 ...
1
vote
1answer
36 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 ...
1
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3answers
44 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 ...
1
vote
1answer
113 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') = ...
0
votes
0answers
24 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 ...
0
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0answers
13 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 $x=x_j$. ...
0
votes
0answers
35 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 ...
0
votes
1answer
51 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 ?
0
votes
1answer
24 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 ...
-1
votes
2answers
46 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
2
votes
1answer
108 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 & ...
1
vote
1answer
19 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, ...
1
vote
1answer
34 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 ...
1
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0answers
23 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, ...
1
vote
1answer
40 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 ...
1
vote
1answer
114 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 ...
0
votes
2answers
45 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 ...
4
votes
1answer
65 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 ...
0
votes
1answer
25 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 ...
5
votes
0answers
90 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 ...
0
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0answers
18 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 ...
3
votes
1answer
48 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 ...
0
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0answers
27 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 ...
0
votes
0answers
22 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 ...
1
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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 ...
0
votes
0answers
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 $E\varepsilon_1 = 0,$ $E\varepsilon_1^2<\infty$ ...
0
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0answers
23 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 $f(u,v,\phi_i)=0$, $i=1,2,3,...$. ...
2
votes
1answer
28 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 ...