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

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Numerical methods of optimal design

Consider an optimisation problem $$ \frac{\partial^2}{\partial x^2}\left( \sigma(x) \frac{\partial^2 u(x)}{\partial x^2} \right) = f(x),\; +\; \text{boundary conditions,} \;\;\; 0 \leqslant x ...
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164 views

Finding a force function from bodies in equilibrium

(This is an edited version of the original question, since I'm starting a bounty) I'm trying to find a function $y$ from given data. Reverse optimization, so to speak. Say we have two ...
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105 views

How to calculate the numerical integral of the type $\int_a^b e^{x^2}dx$ efficiently?

How to calculate the numerical integral of the type $ \int_a^b e^{x^2} dx $ efficiently? My problem is: we need to compute repeatly the integral: $$ \frac{\int_{|m|=1}mm ...
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116 views

Extrema in two variables of a sum of logs, or equation with sum of rational functions

I am trying to find numerically $\arg\min_{x\in(1, +\infty),y\in(0, 1)}\sum_i\log(xy+\alpha_ix+\beta_iy+\gamma_i)$, where the sum has a large number of terms, and the coefficients are such that the ...
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114 views

Solving $A''(x)=f(x) A(x)$ via Numerical Methods

I was looking speedy and simple algorithm to solve second order differential equation in numarical way. I thought an idea to solve it via geometrical equalities. See my approach to the problem ...
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148 views

Regular Perturbation Series Problem

I want to find the a 3 term perturbation soln of (i) $(1+x)^3 = ex$ where $e\ll1$ Direct substitution of the regular perturbation series $x = x_0 + ex_1 + e^2x_2$ into (i) does not work I ...
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147 views

integral of the sum

I am trying to integrate the following sum. I need to get at least first 5 terms (using math or computer). I've tryed wolfram alpha online-did not work. I should find $$ \int_0^\infty ...
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64 views

reference request

Can anybody help me to find the books on numerical solutions of partial differential equations including examples on irregular geometry (specially books or links on matlab code examples in this case)? ...
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1k views

How to use Richardson extrapolation on Euler's method with step size h and h/2 in order to derive the modified Euler method?

Suppose that we want to numerically solve the initial value problem $x'(t)=f(x,t), x(0)=x_0$. The modified Euler's method $$x(t+h)=x(t)+hf(t+\frac{1}{2} h,x(t)+\frac{1}{2} hf(t,x(t)))$$ My ...
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337 views

Second order central difference of the Nth order

I'm trying to find some tabulated data in some big-and-smart-book with regards to second order central difference of a function of just one variable: f''(x). I did find formula for 7th order [1], but ...
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2k views

Understanding Power Method/Inverse Iteration in Linear Algebra

For a linear algebra class, we are currently learning about finding the largest/smallest eigenvalues of a matrix using the power method and inverse iteration methods. I just want to make sure that I ...
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810 views

How to derive Hermite polynomial from a given data set?

The problem is asking to find a Hermite polynomial to predict the position of the car and its speed when t = 10s. The Hermite polynomial formula is defined as: $$H_{2n+1}(x) = f[z_0] + ...
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202 views

Condition number of DA and D'A: row equilibrated matrix

I am with an exercise that first asks me to show that for any regular matrix $A$, there exists a diagonal matrix $D$ such that $A$ is transformed into a row equilibrated matrix by a left ...
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55 views

Solving an equation in several variable of the form $a_1x_1x_2+a_2x_2=c$

Consider the following equation in two variables $$a_1x_1x_2+a_2x_2=c$$ where $x_i$ are variables and the other constants can be any real numbers. In three variables, this is ...
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82 views

Stability of numerical schemes for difference equations

When estimating the solution to an ordinary differential equation, how can you tell if your difference scheme is stable? For example, consider the following scheme: $$ u_{i+1} = 2u_{i} + hf(x_i,u_i), ...
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54 views

Initial Condition for Numerical calculation of differential equation with 3 steps

Say we discretize some differential equation with the following iteration equation: $\phi ^{n+1}(x,y)= \phi ^{n-1}(x,y) +f(x,y)\phi ^n$ (if you'd like some more specific example, let me know) The ...
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85 views

What exactly is a “representation singularity”?

I've heard the term "representation singularity" in a few contexts about numerical instability of algorithms to find Gröbner bases, but I can't seem to find a precise definition for what it actually ...
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728 views

Shifted Power Method

Using the shifted power method I find the eigenvalue (of the matrix A) farthest from a number $\mu$ and the corresponding eigenvector . In the method I follow the below steps: I first compute the ...
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162 views

solution for related function

If we have $f(x)=(2A(A+\delta)-1)x^{2}-2Ax+1$, then the values of $x$ are equal to $\frac{1}{A+\sqrt{1-A(2\delta+A)}}$ and $\frac{1}{A-\sqrt{1-A(2\delta+A)}}$. The question is how to find the ...
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164 views

Composite midpoint and trapezoid quadrature of twice differentiable function

This is a homework problem that I am having trouble with. Given $f: \mathbb{R} \to \mathbb{R}$ is twice differentiable ans $f''(x) \geq 0$ on the interval $x \in [a,b]$. Prove that: $$ ...
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150 views

Spline function: Parametric representation of a curve

I have this problem: I have a curve/figure on a sheet of graphpaper. Then I have to select points, read the x,y-values and label them $t_{0} = 1.0$ and $ t_{1}=2.0$ ect. I now have to obtain a table ...
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273 views

Comparing numerical methods given a system of nonlinear first-order ODEs

I have a system of nonlinear first-order autonomous IVP ordinary differential equations for which I'll solve numerically since I can't obtain a closed-form solution. What are the notions that ...
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100 views

calculating multivariable integrals

I having a look at how to calculate using PC a multivariable integrals. I am reading about the Quasi Montecarlo methods using the following (t, m, s)-Nets and (t, s)-Sequences Faure sequences My ...
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308 views

Integrating pde backwards in time

I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation $$ \begin{align} p_\text{new} & = p_j + ...
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441 views

Stability condition for explicit scheme in finite differences

I've the following explicit scheme in finite differences (for a one dimensional non uniform diffusion problem), being $k$ the time step, $h$ the space step, $A$ the thermal conductivity at position ...
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1k views

Teach me a simple, efficient division algorithm

I want to implement arbitrary-precision arithmetic in JavaScript for non-negative integer numbers. Long division isn't efficient if instead of the usual 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) there ...
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467 views

Find the first odd multiplicity root of a function

I'm trying to find the "first" (greater than some initial $t_0$) odd root (that is, a root after which the sign of the function changes) of a function $f(t)$, if there is one, that is also less than ...
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311 views

How effective is this alternative to integration?

I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following: $f(x) = ...
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14 views

How to derive the error when approximating divergence using the Gauss divergence theorem?

I am trying to derive the error for approximatively computing the divergence of a vector field $\mathbf{a}$. The Gauss divergence theorem states $\int_V \nabla \cdot \mathbf{a} dx = \oint_{\partial ...
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17 views

Finite difference for variable coefficient with Neumann Boundary

The equations is the same as this post, but with respect to the Neumann boundary. The physically correct boundary conditions for this equation are \begin{equation} A(x)\frac{\partial u(x)}{\partial ...
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34 views

Is it possible for the Simpson's method to converge faster than Rombergs method?

I have the following integral: $\int_{0}^{100} \frac{x^{3/2}}{\cosh{(x)}}dx$ I am running code for the Simpson's method and Romberg method to evaluate the integral numerically and the results show ...
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Clarification of matrix equation needed in recursive least squares example.

I was looking at the answer to the post entitled "simple example of recursive least squares" and I would like to post a question concerning the matrix equation that is presented in the answer. First ...
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17 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
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57 views

Crank-Nickolson

I have these two equation $$ \frac{\partial q(t,x)}{\partial t} = \frac{\partial^2 q(t,x)}{\partial x^2} - \frac{L_1 a(t, x) q(t, x)}{1 + \frac{L_2}{L} (1 - q(t,x))}\\ \frac{\partial ...
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20 views

Quantify difference between regularity and irregularity

I am solving an equation numerically on a 1D-domain using the finite-element method. I am solving it using two different domains, one regular and one irregular. Naturally, the solution varies slightly ...
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13 views

numerical solution for nash equillibrium

I have the following setup. $\pi_1=f_1(q,r)$ and $\pi_2=f_2(q,r)$ are the real valued payoff/profit functions of the two players. Player 1 gets to pick $q$ and player 2 gets to pick $r$. I also know ...
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12 views

How to Solve Using Recursive Least Squares Approach

We start with the initial point $\hat{P}_0\!=\left(x_0,y_0\right)$ and the function $f\!\left(x,y\right)=K$ where $K$ is a constant real number and where $f\!\left(x_0,y_0\right)\!{\ne}K$. We are ...
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21 views

Trying to model a substance settling in water using an advection equation?

I am trying to model a substance dispersed in a container of water gradually settling at the bottom. I am considering only one dimension. The top is at $z = 1$, and the bottom is at $z = 0$. So at $t ...
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16 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
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23 views

How can I modify this simple code to include the pressure term? (1-D Navier Stokes)

I have a mathematical model that involves a cylindrical container that is being modeled with a one dimensional simplification as the system is isotropic with respect to the z-axis. As part of the ...
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8 views

von Neumann stability of the implicit downwind scheme

I want to investigate von Neumann stability of the implicit downwind scheme for this PDE: $u_t-u_x=0$. I got $\frac{\Delta t}{h} \geq 1$. It seems odd. I also checked the CFL condition of implicit ...
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59 views

How to I approximate $I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$ s.t. the error is bounded?

Edit: Because the original question was pretty trivial, I want to ask the same question but with:$I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$. How to I approximate $I = \int_{-1}^{1} ...
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13 views

How to obtain the stability function of $y_{n+1}=y_n+h[\theta f(y_n)+(1-\theta)f(y_{n+1})]$?

I am going through a past exam paper but I don't know how to obtain the stability function in this case, I know how to do it when I have the matrix or when I have an explicit method, but not in this ...
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10 views

how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
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24 views

Intuitive explanation for error in Newton's Divided Differences?

When interpolating a smooth function $f$ using $n+1$ points, the error in the interpolation is bounded by $e(x) \leq$ $f[x_0,\ldots,x_n,x] \cdot \prod_{i=0}^n(x-x_i)$. This seems kind of interesting ...
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21 views

Finding Local Linear Basis Functions

Seeking the linear basis functions for a finite element solution, I was given the paper shown by my professor and was asked to find the remaining local basis functions and then compute all the global ...
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32 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
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15 views

When do I use a specific interpolation method?

I am having a course on Numerical Analysis and I was wondering if I can use any interpolation method to interpolate any data, or one method has some specific advantages over another. Here are some of ...
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18 views

Little Doubt in Secant Method

Given Question is : A root of equation $xe^{x}-1=0$ lies in interval $(0.5,1.0)$. Determine this root correct to three decimal places using secant method DOUBT I know method, but my problem is how ...
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35 views

To determine the interval of unit length which contains the smallest positive root of $x^{3}-5x-1=0$

I am doing Bisection method of numerical analysis. The question I encountered is as follows To determine the interval of unit length which contains the smallest positive root of $x^{3}-5x-1=0$. Hence ...