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

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Weierstrass substitution for solving trigonometric eqations

I'm trying to solve a set of equations of a parallel robot . The equations can be writen as $x(\cos(\theta),\sin(\theta))$ $y(\cos(\theta),\sin(\theta))$ so to solve the equation I used Weierstrass ...
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21 views

finding eigenvector with its eigenvalue known

suppose I have a square matrix $A$ of large dimension(>100) with eigenvalue $\lambda$, what are the numerical methods to find its corresponding eigenvector without using the inverse of A?
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21 views

Finite difference method for elliptic pde stability condition

I am trying to implement a finite difference scheme for the elliptic pde $\nabla(a(x_1,x_2)\nabla\phi(x_1,x_2))=0$ where $(x_1,x_2) \in (0,l_1)\times(0,l_2)$ with $l_1 \neq l_2$ necessarily. I know ...
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26 views

4th order method

I am asked to solve a ODE using the 4th order Runge-Kutta method, and then given the analytical answer, 'show the method is 4th order numerically' . What does the question 'show the method is 4th ...
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17 views

preconditioning techniques for Householder transformation

I am using Householder transformation to get similar matrices for the original symmetric positive definite matrices of large dimensions. However the eigenvalues of the similar matrices are quite ...
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45 views

Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...
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50 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
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40 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
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38 views

Finding root by mean of Bisection Method

I have a problem with this function f(x) = e^-x - x^2 i try to find The roots by Bisection method , I'am using Maple 16 this the command i used for calling Bisection method : Bisection(f(x), x = ...
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24 views

Runge-Kutta for newton's law with dependency

I'm trying to determine the changes (position and velocity) on a mechanical system during a step of time. I have a mobile mass whose position (everything is 1D-only) is denoted $x(t)$, velocity $v(t)$ ...
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19 views
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61 views

Lagrange interpolation polynomial and error estimation

Given is a function $f(x)$ with $f(0)=1$, $f(\frac{1}{2})=2$ and $f(1)=-1$. Additionally is given that $max_{x\in \left [ 0,1 \right ]}f''(x)=1$. Find its Lagrange interpolation polynomial $P$ and ...
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34 views

How to design Boundary condition for Euler equations (CFD)?

I'm developing on the calculation of the euler equations using the finte volume method. As you may know each cell is calculated by the incoming and outgoing flux. That means I need in a 1D System the ...
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22 views

Solving a nonlinear volterra integral equation with two integrals each with a non separable kernel

I am trying to solve the nonlinear volterra integral equation ...
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32 views

Calculating log and trigonometric functions using only +,-,/,*

How to calculate logarithm and trigonometric functions (sin, cos etc.) on base n with using +,-,/,* ? Is there any way to do it?
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22 views

How to determine the upper bound of the global error when I calculate a ODE using Euler method

I have an ODE say, $u^\prime(t) = tan^{-1}u(t)$ $(0 \le t \le T)$, $u(0) = a$. Now I would like to decide the upper bound of the global error when using Euler method to solve this ODE. I know that the ...
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40 views

System of linrar equations and condition number

The relative error of the solution of a system of linear equation $Ax=b$, for any natural norm $\|\cdot\|$ is bounded by $$ \frac{1}{\| A\| \|A^{-1} \|} \frac{\|r\|}{\|b\|} \le \frac{\|e\|}{\|x\|} \le ...
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38 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
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45 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
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37 views

Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we ...
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17 views

Kernel density estimation of a divergent probability density function

I'm working with a 2D probability distribution function (pdf) that will be something like $$P\left(r,\theta\right)\approx\frac{3}{\pi^3}\frac{1}{e^{r}-1},$$ when written in polar coordinates (i.e. ...
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40 views

solving partial differentiation using finite difference method

I have been trying to solve right hand side (RHS) of the following one-dimensional partial derivative equation: $\frac {\partial p} {\partial t}=\frac {\partial} {\partial x} ({D(x)}e^{-\beta V(x)} ...
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55 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
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40 views

Solving solely continuous system of ode's with matlab

I'm working with the numerical integration of the system of differential equations, $\dot{x}=f(x)$ with the vectorfield, $f(x)$ being solely continuous. Examples of the systems which I'm working on ...
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18 views

Calculate the weights and the node in the integration formula

The problem is the following. Calculate the weights $w_1$ and $w_2$ and the node $x_1$ in the weighted integration formula $\int_0^1x^{\frac{3}{4}}f(x)dx\approx w_1f(x_1)+w_2f(\frac{3}{4})$ The ...
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43 views

Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
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14 views

Best uniforme approximation of nule function in the meaning of Tchebychev

I would be interest to know , why exactly approximate a nule function and it is in the same time nule ? I would be like someone give me enough (papers, link ...) about "The best uniforme ...
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79 views

Finding minimum of a distance function using matlab

I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $ D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m ...
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16 views

Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
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22 views

Numerical solution to a coupled differentio-algebraic system of equations

$$\frac{\mathrm{d}X_1}{\mathrm{d}t} = P \times ( \frac{I_a^n}{K_i\times exp(I_a*m) + (I_a)^n} ) \times ( 1-( \frac{A.X_2 + B}{ K_o})^z)$$ $$X_1 = X_2 -[ P' \times \frac{I_a^n}{(Ki*exp(I_a * m) + ...
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60 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
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35 views

Quadrature formula of type Gauss.

I am preparing for a test and I have to solve a type of exercise for witch I don't have any good examples. What I have to do is to find a quadrature formula for: $$\int_{-a}^{a} w(t)f(t)dt ...
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26 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
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31 views

Condition under which newton raphson converges

I see in a book that under the following condition newton-raphson method (for finding the zero of a function) converges: 1) The function is continuously differentiable 2) The function is positive in ...
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18 views

numerical representation of functions by integrals

this is a dum question but i would need to know , let be a function $$ f(z)= \int_{0}^{a}G(z,x)dx $$ (1) then if i replace the integral by a quadrature formula $$ f(z)\approx ...
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18 views

Comparison of LMM and RK methods

Is it true that the stability and precision of the Runge-Kutta methods is better then the Linear multistep methods?
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36 views

Predictor-Corrector

I'm constructing a Predictor-Corrector method in P(EC)^3E mode with a first order AB method as predictor and a fourth order AM method as corrector on the ODE $y'=-y$. By experiments on the value of ...
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123 views

Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$ f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi) $$ I tried to get this formula using Lagrange polynomial. ...
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22 views

Closest intersection between a ray and 2 variable function

Exact solution of this problem for an arbitrary ray and function does not exist. What i am interested in is a high quality numeric solution or something similar. Also, are there solutions that put ...
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54 views

Galerkin-Approximation of first Eigenfunction

I'm currently trying to understand a certain proof of an error estimate for the first eigenfuntions gained by a Galerkin-Approximation with Finite Elements of the Potential Equation $-\Delta u$ with ...
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48 views

Matrix rank when solving using Gram Schmidt

Can I deduce the rank of a matrix (i.e. of size 3x2) when solving an over determined set of equasion using QR Gram Schmidt method? If not, is there a QR-related or other numerical method to find a ...
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87 views

Comparison of trapezoidal , Simpson's 1/3 ,Simpson's 3/8 and Boole's rules.

These rules are often used in numerical integration. How do we analyze the given support points or function and select the most suitable one for best approximation?
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38 views

Interpolation using four nodes

Suppose there are four points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ my target is to interpolate any point $x_I$ between $x_2$ and $x_3$. Is there any Interpolation method which gives linear ...
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52 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 ...
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42 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 ...
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58 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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57 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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25 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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34 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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28 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 $x=x_j$. ...