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

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-1
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0answers
25 views

interpolation a function in too small numbers [on hold]

I have a problem that must interpolate function in very small points. I wrote this program by Lagrange method in matlab, but my result is NaN. please hep me? I take a photo from my a part of my ...
0
votes
0answers
24 views

interpolation the points that are so near [on hold]

I must interpolate a function in [0,1] interval. I just have function in x=0:.01:1. so I use Lagrange interpolation method. But for h=.001, my results are false. I think this happen because of the ...
2
votes
0answers
25 views

What is the (currently) optimal root finding algorithm for multivariate functions? [on hold]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
1
vote
1answer
25 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
0
votes
0answers
20 views

numerical (script) fit of function with 2 arguments

I would like to find the least-square fit for a 1D-function that takes two arguments. m(x,y) = d * (x-x0)^2 / (y-y0)^2 I would like to write a c++ routine to ...
2
votes
1answer
26 views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
0
votes
1answer
14 views

discretization of mixed boundary conditions in the advection diffusion reaction equation, the Crank Nicolson method

Sorry, I have some doubts regarding the discretization of mixed boundary conditions in a PDE. I have discretized my equation, but I doubt about the boundary conditions. I dont know if you have to ...
0
votes
2answers
33 views

Solving a differential equation numerically to plot particle path

I'm trying to plot the evolution of a particle in an accretion disk by solving the equation $$2X\frac{\partial X}{\partial\tau}=V_R(X,\tau)$$ where I have found $V_R$ numerically to be ...
2
votes
1answer
23 views

Online resources to learn numerical methods for PDEs?

I would like to get into a career that uses alot of applied math. I took a numerical analysis course in undergrad and liked it, so I plan to self-learn numerical methods for PDEs. Other than the MIT ...
1
vote
1answer
27 views

Approximation of values in [0,1] by sums of unit fractions

Let $U = \{(-1)^k\cdot\frac{1}{n}: n\in\mathbb{N}, k\in\{-1,1\}\}$ be the set of positive and negative unit fractions. For a positive integer $m\in\mathbb{N}$ and a real $x\in [0,1]$ we set $d(m,x) = ...
1
vote
1answer
27 views

Finding spectrum with Schur decomposition

let $\alpha$ be a real number and $A_n(\alpha)$ the set of $n\times n$ matrices such that $$\alpha A+A^HA+A^H=I$$ I'd like to find the set of all eigenvalues for the matrices in $A_n(\alpha)$ using ...
5
votes
0answers
31 views

Careers in applied math with an MS other than in finance and data/machine learning?

Since I like math, I would like a career that uses alot of applied math. I'm about to complete my Master's and could do my thesis in numerical solutions of PDEs I'm already aware of careers such as ...
1
vote
1answer
67 views

Non linear ordinary differential equation

How to solve the ordinary differential equation $\frac{d^2y}{dx^2}+\sin(x+y)=\sin x,y(0)=0,y'(0)=1$ Then its possible to solve it by numerical methods?
2
votes
2answers
23 views

Iterative integration algorthm

By iterative algorithm, I mean a numerical algorithm that works by improving on a previous approximation to obtain a more accurate approximation. An example is Newton's Method. The numerical ...
0
votes
0answers
52 views

Best approximating linear polynomial for $|x|$ on $[-1,5]$ [on hold]

How to find the best approximation polynomial $p_1(x)\in P_1$ for $|x|$ on $[-1,5] $ w.r.t. $\|\cdot\|_{\infty}$? I want to use the Equioscillation theorem but have no clue.
2
votes
1answer
25 views

best approximation polynomial $p_1(x)\in P_1$ for $x^3$

I want to find a best approximation polynomial $p_1(x)\in P_1$ for $f(x)=x^3$ in $[-1,1]$ w.r.t. $||\cdot||_{\infty}$. I want to use Chebyshev polynomial to do that, but I don't know how to hang on.
1
vote
2answers
27 views

Constructing the Normal CDF $Z$-tables?

One topic that is always bypassed (in my experience, at least) in an undergraduate statistics course is the construction of $Z$-tables, such as the one provided at this link ...
0
votes
1answer
30 views

How to discretize mixed partial derivatives?

How to discretize $\frac{\partial^3 f}{\partial x\partial y^2}$ at mesh point $(i,j)$? We should use mesh points which are nearest to $(i,j)$.
1
vote
0answers
19 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
1
vote
1answer
26 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
0
votes
1answer
118 views
+50

Bounding error when iterating a function

If I am iterating some function $f$ that goes to infinity as x goes to infinity with error $o(g(x))$, for example, is there anyway to bound the error? To be more specific, if I have some sequence ...
1
vote
0answers
30 views

Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$ \displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s $$ I tried solving it by collocation ...
-3
votes
0answers
16 views

numrical kinetic energy [closed]

can anyone knows how to evaluate the kinetic energy operator matrix numerically??? i need to evaluate each element in that matrix!!!! i have found something but its not exactly right (checked it on ...
2
votes
0answers
10 views

Finding Volume of Monte Carlo Integration

Suppose $\mathbf{X}\in\mathrm{R}^n$ is an $n-$ dimensional random vector having joint Gaussian distribution i.e. $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol\mu,\boldsymbol\Sigma\right)$, where, ...
0
votes
0answers
14 views

How to use Legendre Quadrature for Multiple Unbounded Integrals?

Legendre Quadrature as most other methods is designed for $[-1,1]$ interval and some variable change methods are used for extending them to $[a,b]$ interval. I found some method in a textbook for ...
0
votes
1answer
15 views

Is there a formula that can be used to determine the number of iterations needed when using the Secant Method like there is for the bisection method?

The formula used to find the number of iterations needed to find a root of a function using the bisection method is this; $$|c_n-c|\le\frac{|b-a|}{2^n}.$$ Is there a formula that can be used to ...
0
votes
0answers
10 views

Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute ...
0
votes
0answers
16 views

fifth and sixth order Runge-Kutta method

I want to solve one system of ODE problem with numerical methods. I used $RK_2$, $RK_3$, $RK_4$ and Adams_Bashforth method.Now I want to solve it with high order Runge Kutta methods. Do you have these ...
0
votes
1answer
44 views

derivative B-spline with own knot set

Define the spline function of degree $q$ on the interval $[\xi_0,\xi_K]$ $$f(t)=\sum_{j=1}^{K+q}b_j B_j(t)$$ where $B_j$ are degree $q$ B-spline basis functions determined by the knots ...
0
votes
0answers
38 views

Intersection of lines in 3D space [closed]

Given two or more pairs of points in 3D space, I should calculate the intersection of the lines passing through each of these pairs of points: for each pair of points, I get the linear system of two ...
2
votes
1answer
39 views

How do you derive the secant method formula from the equation below?

The Secant Method forumula is; $$ x_{i+1}=x_i - \frac{f(x_i)(x_i-x_{i-1})}{f(x_i)-f(x_{i-1})}.$$ Derive the formula from the equation below; ...
1
vote
0answers
47 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
1
vote
2answers
28 views

Why is Romberg integration usually based on trapezoidal rule?

The wikipedia article on Romberg Integration says that it's simply Richardson Extrapolation applied to either the Trapezoidal Rule or the Midpoint Rule. I'm reading out of a couple of textbooks on ...
3
votes
1answer
27 views

How to compute Fourier coefficients using a cubic spline-corrected FFT?

I'm not particularly experienced in numerical analysis, and so I recently had quite a massive shock when I discovered that sampling a smooth function and computing the FFT of the result does not ...
1
vote
1answer
24 views

How to modify Gauss-Hermite quadrature rule when the weight function is slightly generalized

hope this is the right forum. Consider a slightly modified version of the Gauss-Hermite quadrature rule, where the weight function is not $\exp(-\frac{x^2}{2})$ as in the standard Gauss-Hermite rule, ...
0
votes
0answers
19 views

compare Adams-Bashforth method and Runge-kutta method [closed]

I want to know which is this two methods is better? Adams-Bashforth or Runge-kutta? I say better when my solution is more accurate.can you sort this method for me by rate of accuracy? RK2, RK3, RK4 ...
1
vote
1answer
22 views

infinitisimal part and the directional integral

In the paper A circle detection approach based on Radon Transform by Erman Okman and Gozde B. Akar. I have a few questions on some basics. first of all what does $$ds^2 = dx^2 + dy^2$$ ...
2
votes
0answers
27 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
1
vote
2answers
33 views

Numerical integration using Birkhoff theorem

There is a method for numerical integration that uses Brikhoff ergodic theorem? For example if we have a irrational number $\alpha$ we know that for every continuous function $f \colon [0,1] \to ...
4
votes
1answer
153 views

What is the connection between $\rho$ and $\sigma$ if $\rho\rho^T=\sigma\sigma^T$?

I want to prove that there exists a Borel function $R(\rho,\sigma)$ with values in $M^{d\times d}$ defined on $D=\lbrace(\rho,\sigma)\in M^{d\times d}\times M^{d\times d}\,: ...
3
votes
4answers
84 views

Solve expression without loss of precision

Given the expression $$2675394361153184*(A+B+C+D)$$ Where... $$A=\frac{873892798365919}{334424295144148}\approx2.613125933$$ $$B=-(\sqrt{2}*\sqrt{(2+\sqrt{2})})\approx−2.613125930$$ ...
0
votes
1answer
92 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
0
votes
2answers
40 views

Determinant Formula for Tri-Diagonal Matrix

for an assignment in numerical analysis, I need to find the eigenvalues of a matrix with values only in the diagonal, upper diagonal and lower diagonal. I guess there is an easy formula for this sort ...
1
vote
1answer
33 views

Numerical approximation using Halley's method.

I'm working on an R exercise, but I'm having difficulty grasping the math behind the exercise in order to implement it properly. The exercise requires me to approximate a function using the first ...
0
votes
0answers
39 views

numerical calculation of an integral

I am having trouble finding the solution of this numerically and wondered if I could get some tips so that I can: $$ \int\limits^1_{0}\left[\min(ax, b) - \min(a x, c)\right] dF(x; p, \rho)$$ (1) ...
1
vote
2answers
38 views

algorithm to find the root of a real-valued function $f$

I see in a book the following algorithm to find the root of a real-valued function $f$ $$ \theta_{n+1} = \theta_{n} + \epsilon f(\theta_n); \epsilon >0 $$ with the condition that the initial ...
1
vote
1answer
33 views

I am not sure how to use the secant method formula without a function being given?

Calculate an approximation value for $4^{\frac34}$ using four steps of the secant method with the starting values of $x_0=3$ and $x_1=2$.
0
votes
1answer
82 views

Gradient descent (with line search) for convex functions viewed as alternation

I have fundamental confusion about gradient descent (with line search) and the reason it works. I try to explain my view here, and please tell me where it goes wrong. Let $f: \mathbb{R}^n \to ...
2
votes
0answers
63 views

Numerically approximate the maximum of an element of a vector after a series of matrix multiplications.

Where S is a sigmoidal function, A_i is a matrix, and x is an input vector, and ...
0
votes
0answers
39 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...