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

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31 views

Runge kutta 4th order computation of force solving 2nd order ode

\begin{equation} \frac{dx}{dt} = v \end{equation} \begin{equation} m .\frac{dv}{dt}= F_{p }(x)+F_{g}(v,x) \end{equation} Conside I am solving the above two equations using runge kutta 4th order ...
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1answer
31 views

intuition about cubic splines vs quadratic splines (degree 3 vs degree 2).

my intuition about quadratic(degree 2) splines is that by the help of its three variables (in each sub-interval) you can make a piecewise differentiable function on the whole interval. in the process ...
4
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1answer
38 views

Laplacian solvers for inversion of large matrices?

I have a large matrix L of size 400,000 $\times $ 400,000 . I'm using this L matrix in the following way. Lin = L$^{-1}$ C = D - B * Lin * B'; B,D are of appropriate sizes. L matrix is ...
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2answers
265 views

Integral for which numeric methods will always give an incorrect result?

I was thinking about the following function: $$f(x) = \begin{cases}0 & x\;\mathrm{computable}\\1&\mathrm{otherwise}\end{cases}$$ And the following definite integral: $$I = \int_0^1 ...
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2answers
53 views

Integral approximation - [closed]

Whole day I can not figure out how can be proved the equality: $$\int_0^1 x^2 dx = \frac{1}{n} \sum_{i=1}^n \left(\frac{2i-1}{2n}\right)^2 + \frac{1}{12n^2}$$ Can someone help me, what should I use ...
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1answer
36 views

The bubble function

In the finite element method and more precisely the MINI element method in two dimensions, they use a function called the "bubble function" which is related to a triangle K of the space meshing and is ...
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0answers
31 views

Fast algorithm to invert a large sparse matrix

I am interesting in sparse matrix that defined at here. I am looking for a fast algorithm to invert the matrix (better than Gaussian Elimimation). Could you suggest to me some methods that reduce ...
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1answer
28 views

How to estimate parameters in trigram?

A popular method of computing trigram in NLP is linear interpolation: The question is how to estimate the three linear interpolation parameters to maximzie the following expression? Any form of ...
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1answer
39 views

Expressing a function's value using finite differences

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and let $x = (x_0, x_1, x_2, \dots)$ be a sequence of pairwise distinct real numbers. For every $n \in \{1, 2, \dots\}$ and every ordered $(n+1)$-tuple ...
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3answers
62 views

Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
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0answers
17 views

How to make a 9-point two-dimensional stencil for a elliptic operator?

I want use a finite difference schem to discretizate the elliptic operator: $$ \nabla \cdot \left( k(x, y) \nabla p\right), $$ where $k(x, y)$ a positive scalar function and $p$ is the unknow. We can ...
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1answer
23 views

A proof on forward difference operator

Show that $u_0-u_1+u_2-u_3+...=\frac{1}{2}u_0-\frac{1}{4}\Delta u_0+\frac{1}{8}\Delta^2 u_0-\frac{1}{16}\Delta^3 u_0+...$, where $\Delta$ is the forward difference operator. My attempt: ...
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0answers
8 views

How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
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1answer
18 views

Lipschitz inequality required for Broyden convergence proof

I'm trying to understand the proof of the convergence of the Broyden method through the book Numerical Methods for Unconstrained Optimization and Nonlinear Equations, and at some point the proof ...
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2answers
31 views

Find the first order system of linear equations

Regard the diff equation: $mϕ′′+aϕ′+(mg/L)ϕ=0$ $ϕ(0)=0.1$ $ϕ′(0)=0$ where $m=0.1,L=1,a=2,$ 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ...
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1answer
27 views

Absolute stability theoretical and empirical comparison

Regard the diff equation: $m \phi'' +a\phi'+(mg/L)\phi=0$ $\phi(0)=0.1$ $\phi'(0)=0$ where $m=0.1, L=1, a=2,$ 1) Rewrite the second order diff equation as a system of first order linear ...
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1answer
33 views

How can I compute $\sum_{i=1}^n x_i \log(x_i)$ in a stable manner?

Given a vector in $\mathbb{R}^n$ I have an algorithm to compute $$\sum_{i=1}^n x_i \log(x_i)$$ However for my application the norm of $x$ must be 1, hence for big $n$ the components tend to be too ...
2
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1answer
68 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
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2answers
15 views

Absolute stability for ODEs

I know the formula for asbolute stability is $$|1+h\lambda|<1$$ but how does it work when $\lambda$ is non-negative? I get a negative step which is wrong. Thanks!
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0answers
16 views

Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
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1answer
16 views

How to stabilize cyclic tridiagonal matrix algorithm?

I've received a task which is: Solve equation by cyclic tridiagonal matrix algorithm: $$ \frac{\partial{f}}{\partial{t}} = \lambda*\frac{\partial{f}}{\partial{x}}, \\ x\in[0,1]\ t\in[0,1] \\ ...
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0answers
22 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
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0answers
15 views

Setting up a linear equation to solve using LU decomposition..

I have these to linked equations which i have to solve a variable for. The equations are a(x) = c1 + d1 $\cdot \int_{\alpha}^{\beta} c(x,y) \cdot b(y) dy$ b(y) = c2 + d2 $\cdot ...
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1answer
38 views

Cubic splines better than quadratic splines?

I have read in a number of places that cubic splines are of more practical use than quadratic splines in general (there are exceptions of course). Anyone know specifically why they are more ...
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0answers
12 views

the matrix equation with signum functions

May I ask do you know any way to solve the matrix equation with signum functions: Given the known matrices $D_n \in R^{n\times n}$, $D_m \in R^{m\times m}$ and $f \in R^{m\times n}$ and parameter ...
2
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2answers
174 views

Solving equation with rational exponent

I have this equation: $$\mathrm{r} (x-1) = x^{8/9} - x^{1/9}$$ where $\mathrm{r}$ is a constant. Is there a general technique to solve such equations? Raising it to the 9nth power: $$\mathrm{r}^9 ...
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2answers
55 views

$\sum a_n$ converges conditionally

If we assume that $\sum a_n$ converges conditionally then How can we comment that $\sum a_{2n} $ does not converges, While it does when $\sum a_n$ converges absolutely ?
3
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1answer
56 views

Root finding and Bolzano Theorem

Suppose that $f:\mathbb{R} \to \mathbb{R}$ is continuous on the closed interval $[a,b]$, and that $f(a)<0$ and $f(b)>0$. Then it must be $0$ at some point. How to Mathematically proof this ...
2
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1answer
44 views

How to write ODEFUN's with y prime's? [closed]

I'm attempting to solve: $$ y'' = 3y-2y' $$ My work is basically... $$ y' = u $$ $$ u' = 3y - 2*y' $$ Now, I'm trying to write a solver for ODE45... but I can't figure out how I should write the ...
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1answer
32 views

What is the norm of a complex vector?

I have two arrays $a$ and $b$ containing complex values. Now I one of my target operations is the following: $$||a-b||$$ The result should be a single real number. Now I am a bit confused how to apply ...
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1answer
39 views

Numerically find the minimum distance between given point and the curve of given function.

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ and $\mathbf{x}_0\in\Bbb{R}^n$. How could I (numerically) find the minimum Euclidean distance between the curve $f(\mathbf{x})=0$ and $\mathbf{x}_0$, granted that $f$ ...
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1answer
26 views

Which numerical method should I use to solve these equations?

While analyzing the chemical equilibrium of a combustion reaction products and finding their dissociation ratios, I got the following set of equations (5 unknowns & 5 equations): $b = 2.6 - a$ ...
3
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1answer
53 views

How small is $x$ when $x\ll{1}$

When $x\ll{1}$, then how close to zero is $x$. I mean what value of $x$ makes $x\ll{1}$. Are the values $x = 0.5,\; 0.001, \;0.00001$ OK? I am solving a complicated equation that contains the ...
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2answers
57 views

Dividing a sphere into thirds with two planes

Earlier this week, I asked this question about dividing a circle into thirds, and we were able to use numerical methods to come up with the desired result. Now, I want to extend that question to a ...
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0answers
34 views

Rate of Convergence of $A_{s,k}=\prod_{p}\left(1-p^{-1}\right)^{s-k+1}\sum_{m=0}^{k-1}{s\choose m}\left(1-p^{-1}\right)^{k-1-m}p^{-m}$

I'd like to know how fast the infinite product $$A_{s,k}=\prod_{p}\left(1-p^{-1}\right)^{s-k+1}\sum_{m=0}^{k-1}{s\choose m}\left(1-p^{-1}\right)^{k-1-m}p^{-m}$$ converges, where the product is taken ...
0
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2answers
39 views

It is possible to orthogonalize a set of linearly independent vectors via SVD?

Let's say I have a set of linearly independent vectors, collected in a square matrix $\mathbf{M}$. I know that I could orthogonalize these vectors with the QR decomposition, $\mathbf{M} = ...
1
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1answer
26 views

Find local truncation error

We have the method $$ y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0 $$ I have to calculate the local truncation error for the method ...
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2answers
102 views

$\sum a_{2n} $ converges [closed]

Prove that if $\sum a_n$ converges absolutely, then $\sum a_{2n} $ converges. I know this part, How it can be done but i am having problem in proving the later part of the question T o show that this ...
0
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0answers
11 views

What is the difference between newton interpolation polynomial and interpolation polynomial with Neville scheme?

I am trying to find the interpolation polynomial by using Neville scheme. It looks like divided difference . What is the difference between newton interpolation polynomial and interpolation polynomial ...
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1answer
25 views

Newton-Raphson iterative method when derivative is $0$?

Let's say I have a function with with a root a of order m. I know I can write my function, $f(x)$, as $f(x) = (x-a)^mh(x)$, where h(x) doesn't have a zero in a neighborhood of a. I want to prove that ...
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1answer
37 views

Is there numerical evidence supporting the predicted density of the primes of the form $x^2+1$?

A famous conjecture (due I think to Hardy and Littlewood) states that if $P(x)$ denotes the number of primes of the form $n^2+1$ less than or equal to $x$, then $$P(x)\sim \frac{C\sqrt x}{\log x}$$ ...
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0answers
24 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
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2answers
41 views

Conditions needed for a unique root to also be a “clear-cut” root

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a continuous function that has a unique root $r_{0} \in (0,1)$. I want $r_{0}$ to be a ``clear-cut root" (not sure what to call it) in the following ...
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0answers
19 views

How to solve a semilinear PDE numerically using COMSOL MULTIPHYSICS

I want to solve a semilinear parabolic PDE given by, $u(T,x)=0$, while on $(t,x)\in[0,T)\times(0,1)$, \begin{eqnarray} \frac{\partial u}{\partial t} +a(x)\frac{\partial^2u}{\partial ...
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1answer
33 views

Computing differentiation rule with error bound

I have values for $x$, $f(x)$ and fixed error bounds for $f^{(n)}(x)=c_n$ for $n=\{1,2,3,4,5\}$. I want to compute $f^\prime(x)$ using $f(x-h),f(x)$, and $f(x+h)$. Since the function is continuous in ...
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0answers
30 views

Selecting nodes for spline interpolation

Is there a general method to determine the best sample points for spline interpolation (whether for piecewise linear or piecewise cubic Hermite) given $x$, $f(x)$, and estimating $f^\prime(x)$? Does ...
2
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3answers
124 views

Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods

I have to calculate approximations of the solution with the method $$ y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0 $$ for various ...
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0answers
20 views

Approximating the integral of the exponential of a quadratic function

An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and ...
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1answer
15 views

Need help for construct DLSQ-spline in $B$-form

In Schumaker, Larry L. "Computing bivariate splines in scattered data fitting and the finite-element method." Numerical Algorithms 48.1-3 (2008): 237-260 (Link to journal, link to author page for ...
0
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1answer
43 views

Lebedev grid integral with $\cos(2\theta)$.

This is just a part of the problem I have noticed while dealing with numerical integration in spherical coordinates. I want to evaluate integral ...