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

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39 views

Determine which one more accurate approx $f''(x)$

Derivatives can be written 1.) $$f'(x) = lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ 2.)$$f'(x) = lim_{h\rightarrow 0} \frac{f(x)-f(x-h)}{h}$$ Also $$f''(x) = lim_{h \rightarrow 0} \frac{f'(x+h) - ...
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198 views

How do you numerically solve a multivariable ODE system with different time steps per state variable?

If you have a large multivariable ODE system, and certain processes occur at a much smaller time scale, how can you implement a solver that uses smaller time steps for state variables involved in fast ...
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134 views

Analytical solution(root) for a tenth order polynomial?

is it possible to develop an analytical solution (root) for such a polynomial: $f(x)=\left(x^{10}-c_1^2\right)*\left(c_2-x\right)^2-0.2*\left(x^2-1\right)*c_1^2$ with $c_1$ and $c_2 >0$. Numerical ...
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68 views

Algorithm for finding power

I has been searching for a high precision library in PHP to do calculations like $$232323232323^{121212.2232323232}$$ etc (ie, with very large numbers, including decimals), but failed to get any. ...
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49 views

numerical implementation of the resolvent kernel of an integral equation

I started exploring implentation of Volterra equations only recently. The iterative kernel for my problem looks like this: $$L_i(x,y) = \int\limits_x^y L_1(y,t)L_{i-1}(t, x)dt. $$ I have been trying ...
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66 views

Solving systems of differential algebraic equations: Is it legitimate to hold some variables constant?

I have a system of linear differential and algebraic equations, along with some non-linear equations. That is, I have a system of equations that can be written in the form $\mathbf{A}y'(t) + ...
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233 views

L-stability and Stiff decay

In my Numerical Methods for PDEs textbook by Ari Uscher L-stability and stiff decay are introduced by considering a generalized test equation: $y' = \lambda (y - g(t)), 0 < t < b$ where g(t) ...
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59 views

A method called “incorrect method”

Good night. Is there a method called "incorrect method" to calculate second order differential equations? If so, please, is there a web page about it, as I have to investigate this method? Thank ...
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58 views

What are the weights of the quadrate formula with weight function $x\mapsto (1-x^2)^{-1/2}$

I'm trying to solve this numerical analysis exercise: I was able to prove everything until the part marked in red. I think I need to use this: So we get an exact result for $T_0$: ...
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152 views

Numerical Analysis Using Nonlinear Least-Square Fit function of the form f(t) = Asin(Bt) + Ccos(Dt) to fit 24 data points

can anybody help me solve this problem. As I am stuck at the $E(A,B,C,D)=\sum_{k=1}^{24} (f(t_k)-y_k)^2 $ also $E_2 (f)=? $ is the root mean square error I am having difficulty finding the correct ...
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196 views

Matlab.Compute $f(x) = \sin(x) + \cos(x)-1$

Write a procedure to compute $f(x) = \sin(x) + \cos(x) - 1$ The routine should produce nearly full machine precision for all $x$ in the interval $[0, \frac{\pi}{4}]$ Hint: $\sin^2 \theta = ...
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187 views

Non-linear ODE for which backward Euler becomes unstable?

One way to solve initial value problems of the type $\dot{x} = f(x), \; x(0) = 0$ numerically is to use the backwards Euler method $x_{n+1} = x_{n} + \Delta t f(x_{n+1}), \; n = 1,\ldots \\ x_{1} = ...
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44 views

Gauss-Seidel Smoother for Problems with jumping coefficients

for a project I am working on a "robust" multigrid implementation, i.e. a implementation which achieves fast convergence even for problems with discontinious coefficients. My first problem is: $ - ...
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39 views

Show that $g'(p) \approx ({p_n-p_{n-1}})/({p_{n-1}-p_{n-2}})$

Suppose the sequence {${p_n}$} is generated by the fixed point iteration scheme $p_n = g(p_n-1).$ Further, suppose that the sequence converges linearly to the fixed point $p.$ Show that $g'(p) ...
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75 views

Order of a linear multistep method

Given a linear multistep method $\sum_{j=0}^{k}\alpha_jy_{n+j}=h\sum_{j=0}^k\beta_jf_{n+j}$ how to show using Taylor series expansion that the method is of order $p$ if and only if ...
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216 views

Formula for intersection of “power” curve and parabola.

EDIT I have edited this question to make it more clear. I have spent quite some time trying to find this on Google, but haven't succeeded. I need the formula(s) to determine the intersection ...
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78 views

Solving the difference equation in the stability analysis of a multistep method.

So I am confused in going from a difference equation to a linear ODE. To make this concrete let's look at the second order Adams-Bashforth method we have: $$ Y_{n+1} = Y_n + h(\tfrac34 f_n - \tfrac12 ...
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122 views

Formula for monthly payment of mortgage

What is the formula for monthly payment of mortgage including Term, Interest Rate, Cost of Home Down, Payment Insurance, Property Tax, HOA Fee. I'm a programmer and want to add this functionality to ...
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99 views

Algorithm of projection

Suppose $S$ is a compact surface in $\mathbb{R}^{3}$ defined by a sufficiently smooth level set function $f$, that is, $S=\{s: f(s)=0\}.$ I am studying an algorithm that projects a point $x_{0}$on ...
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82 views

When examining global error bounds for Euler method, can I rescale the domain limits?

I'm looking at provable global error bounds of the Euler method for the first time and I was surprised to find that the bound grows exponentially in the amount of time (the domain size) propagated ...
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302 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
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353 views

$N$ equally spaced points on an ellipsoid

I would like to find a algorithm for determining the $(x,y,z)$ co-ordinates for evenly distributed $N$ points on the surface of an ellipsoid. These points must be spaced from its nearest neighbour ...
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432 views

Algebraic ellipsoidal least squares fit

I'm looking to perform a least squares fit in 3D to a quadratic surface of the form: \begin{equation} Ax^2 + Bxy + Cxz + Dy^2 + Eyz + Fz^2 + ax + by + cz + d = 0 \end{equation} by minimizing ...
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160 views

Convergence of the Jacobi iteration method

I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm $||I-Q^{-1}A||_\infty < 1$ and ...
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27 views

Variational formulation and change of coordinate

I was wondering if a change of coordinate (e.g cylindrical change) could affect the variational formulation with respect to the metric. I mean, does the metric $\mathbf{dx}$ which appears in the ...
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131 views

Differentiating a multivariable function

In numerical mathematics, we looked at the "Taylor series method" to construct one-step methods. Let $\mathbf{\dot{y}} = \mathbf{f}(\mathbf{y})$ be a system of differential equations. We define a ...
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132 views

Stable solution of nonlinear equation

I'm trying to numerically solve equation for $t\in \mathbb{R}$ $$ n \cdot e^{At}x_0= c $$ $A \in \mathbb{R^{3\times 3}},n,x_0 \in \mathbb{R^3}, c \in \mathbb{R}$. I'm looking for smallest positive ...
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217 views

Matrix spectral decomposition

Let $A$ be a square matrix $(N \times N)$ and $a_{ij} \in \mathbb{R}$. Suppose A has N eigenvalues $\lambda_{1} < \lambda_{2} < ...\lambda_{n} \in \mathbb{R}$. $A$ = $R \Omega R^{-1}$ its ...
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52 views

Projection of fine mesh with low degree polynomial to projection of coarse mesh with higher degree polynomial of a funtion $u$.

One way of improving some finite element approximations is to take the approximate solution and project it (mesh element-wise) using $L^2$-inner product onto a space of piecewise polynomials of higher ...
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79 views

How do you interpolate the local maxima of a set of points in more than 3 dimensions?

I have a set of about 400 points each with 6 coordinates and one scalar value. How can I find out where the local maxima are?
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112 views

Mysterious term in semi-implicit Euler scheme

In the paper i'm currently working with I don't understand the role of the term $C^m$ in the following semi-implicit numerical Euler scheme they use, which consists of following two recurrence ...
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52 views

How to conserve probability using a numerical integration scheme?

I have an iterative operator that conserves probability given by $P_{n+1}(z_j) = \int_a^b P_B(x+z_j)P_n(x) dx$, where $P_n$ is the PDF at time step $n$ and $P_B$ is a PDF that is fixed with compact ...
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44 views

$H^1$ estimate by $L^2$-norm

Let $(\tau_h)$ be a shape regular triangulation. Prove that there exists a constant $c>0$ such that $$\|v\|_{H^1(\Omega)}\leq \frac{c}{\min_{T\in\tau_h}} \|v\|_{L^2(\Omega)}$$ for all $v\in V^h$ ...
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314 views

Gradient Descent in a 3D parameter space

I'm trying to computationally implement a gradient descent algorithm in 3D to find the maxima of a function. I want to use a recursion scheme like $$\textbf{x}_{k+1} = \textbf{x}_k + \alpha \nabla ...
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179 views

closed-form solution for 1/tanh(x) - 1/x that can be evaluated at/near x=0?

I'm looking to evaluate $\frac{1}{\tanh x}-\frac{1}{x}$ over a range that includes x=0. Is there an alternate form that is both exact, and numerically stable at/near x=0? For now I'm using the Taylor ...
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223 views

Best method of interpolation?

I am learning different interpolation methods, and their pros and cons. Which interpolation method do you think is the best for practical use? If you can give me links to research papers about various ...
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214 views

How to solve the projectile problem with numerical method in matlab

i wanna ask how to solve the projectile problem using matlab? could you give me the source code in matlab? the equations is x"=-(1+0.1*x)^2 , with x(0)=0, x'(0)=1. thanks before.
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59 views

Help setting up non-linear parabolic BVP for Newton's method for Non-linear Systems

I am trying to apply Newton's method for non-linear systems to this equation: $$\frac{\partial u}{\partial t}=\frac{\partial ^{2} u}{\partial x^{2}}+(1-u^{2})u+f(x,t) , x \in [-1,1], t>0$$ ...
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120 views

Bilinearly Blended Coons Patch

I have four quintic Bézier curves and I want to create the Coons Surface that interpolates them. How can I obtain the 25 control points of the surface? Thanks.
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34 views

A journal for the article related to methods for solving Cauchy equations

I'm a postgraduate student in physics, but I have achieved interesting results in Cauchy equation. I found a reason why Adams method for solving differential equations gives rise to divergent ...
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217 views

How to Diagonalize an Extremely Large Sparse Matrix in SLEPc/PETSc

Dear Friends, Recently I have started with learning SLEPc/PETSc, but I didn't find a way to solve my problem. I have to solve a big sparse matrix which is a two dimensional quantum ...
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148 views

Approximate the solutions to the following parabolic partial-differential equations.

Use the: a)forward difference method b)backward difference method c)Cranck-Nicolson algorithm to approximate the solution of the parabolic partial differential equation: $$\frac{\partial u}{\partial ...
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1k views

Matlab code for fixed point iteration

I want to write in Matlab a function that appreciates the fixed point iteration for a system of equations. The idea is: $\begin{bmatrix} x{_{1}}^{m+1}\\ x{_{2}}^{m+1} \end{bmatrix}= ...
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20 views

Bivariate Quasi-interpolants

Consider the bivariate spline quasi-interpolant $S2$ defined on a bounded rectangle $R$ with simple interior knots. It is known that, if $f \in C^k(R)$, then $||f-S_2f||_{\infty}$ behaves like ...
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599 views

Runge-Kutta and Butcher table?

In the Wikipedia article on Runge-Kutta methods, there is a notation explained using a Butcher table with a $c_{i}$ vector (nodes), a $b_{i}$ vector (weights) and a runge-kutta matrix $a_{ij}$. My ...
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34 views

von Neumann Stability help

Using the forward time centered space scheme, I transformed the equation: $u_t-2u_{xx}-u_{yy}=0$ to ...
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57 views

Optimize fill factor by move objects between areas

I have a optimization problem which is about several small rectangles inside one outer rectangle. We have, let say, three outer rectangles which are in following order (similar to weeks). Each ...
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55 views

Howuse this $R_{l}=\frac{1}{n}\left(\frac{(-1)^l}{2n}+\sum\limits_{m=1}^{n-1}\frac{1}{m}\cos{\frac{ml\pi}{n}}\right)$ and MATLAB get this four fig?

we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization and ...
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45 views

What are the available libraries or programs for finding extremes of a function with no symbolic definition?

In my current mathematical inquiry, I would like to gain insight on behaviour of a $d$-dimensional continuous function by locating its maximum over the hyperplane $\sum_{i=1}^d x_i = 1$ for $x_i$ ...
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510 views

Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...