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

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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 $(i,j)$.
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20 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 ...
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1answer
27 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 ...
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1answer
170 views

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 ...
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0answers
34 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 ...
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0answers
11 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, ...
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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 ...
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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 ...
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12 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 ...
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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 ...
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1answer
46 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 ...
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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; ...
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0answers
53 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 ...
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2answers
35 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 ...
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1answer
30 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 ...
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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, ...
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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 $$ds^2 = dx^2 + dy^2$$ ...
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0answers
29 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 $B$ ...
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2answers
39 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
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1answer
177 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}\,: ...
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4answers
105 views

Solve expression without loss of precision

Given the expression $$2675394361153184*(A+B+C+D)$$ Where... $$A=\frac{873892798365919}{334424295144148}\approx2.613125933$$ $$B=-(\sqrt{2}*\sqrt{(2+\sqrt{2})})\approx−2.613125930$$ ...
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1answer
93 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 $$ ...
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2answers
41 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 ...
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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 ...
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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) ...
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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 ...
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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 $x_1=2$.
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1answer
99 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
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0answers
64 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 ...
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40 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 ...
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30 views

Complexity of the power method

I'd like to find out what the complexity of the power method is depending on the size of the matrix $A \in \mathbb{R}^{n\times n}$ given that the algorithm runs until a certain stop criterion. I.e. ...
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0answers
23 views

Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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0answers
18 views

Correctly rounding an error to one significant figure

I'm running a process which produces several possible values for a given parameter and returns its mean and standard deviation, both given with an arbitrary precision. I find myself with the ...
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24 views

Blended surface

Partially blended surfaces are extensively used in the literature for shape preserving interpolation. Most of these shape preserving partially blended surface interpolation is based on the result that ...
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1answer
44 views

Solving 2x2 diagonally dominant matrix systems (non-symmetric)

I have a linear system of the form $Ax=b$ where $A\in \mathbb{R}^{2\times2}, b\in \mathbb{R}^{2\times1}$. A is diagonally dominant and non-symmetric. This is a "kernel" that I am using to solve a ...
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1answer
38 views

Gauss Seidel Method - How do I avoid calculating $L^{-1}$?

I'm trying to write a matlab code that gets a diagonal dominant matrix $A$, vector $b$, and finds an approximate solution $x$ to $Ax=b$ using Gauss-Seidel Method. I understand the theory. Suppose ...
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4answers
128 views

The equation $x-\sin x = 0$

If we have the equation $x-\sin x=0$, then we can trivially or numerically find the solution to be $x=0$. However, I rearrange the equation algebraically and get $$\frac{\sin x}x=1.$$ If I plug $x=0$ ...
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2answers
44 views

Firstly what is an $O(h^3)$ formula? Also I am not quite sure how to answer the question?

The forward-difference formula can be expressed as $$f'(x_0)=\frac{1}{h}(f(x_0 +h)- f(x_0))-\frac{h}{2}f''(x_0) - \frac{h^2}{6}f'''(x_0) + O(h^3).$$ Use Richardson's extrapolation to derive an ...
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0answers
53 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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0answers
22 views

efficient least squares A = BX+CXD (solving for matrix X)

I am interested in solving a least-squares solution of the form $$ \operatorname{argmin}_X \| A - BX - CXD \|_F^2 $$ for large (rank in hundreds to thousands) matrices $A,B,C,D,X$ I know this is ...
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1answer
34 views

Application of Conjugate Gradient Method to non-symmetric matrices

I am currently working on a problem in which I am using the Conjugate Gradient method to solve for the steady state solution of a continuous time Markov chain. I am applying the algorithm found in ...
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0answers
14 views

Initial value problem test cases

I am working on some materials about numerical solution of initial value problem for ODEs. Are there any state of art test cases used to test properties of methods? I have found one in Wikipedia and ...
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21answers
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Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
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7 views

Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
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0answers
11 views

Partial integro-differntial equation

Which method is best suited to solve an elliptic partial integro equation? Is finite difference for the derivatives and composite trapezoidal rule for the integral part stable?
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18 views

Find Lipschitz constant for the equation

I am still confused about the theorem relating to find the Lipschitz constant. I have the following equation: $$y' = \frac{1+t}{1+y},\quad 1\le t\le2,$$ where $ y(1) = 2$ and $h = 0.5$. If I were to ...
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0answers
32 views

Approximations for finite n in limit-based definition of the exponential function

The exponential function can be defined via: $$ e^x = \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} = \lim_{n \rightarrow \infty} g(x; n) $$ In my problem, I actually have the right ...
4
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2answers
68 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
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1answer
21 views

normal equations of $ y(t) = \gamma e^{\lambda t} $ for minimizing the error

Let $ y(t) = \gamma e^{\lambda t} $ and we have the points $(0,2)\ (1,0.7)\ (3, 0.3)$. The task is to get the parameter so that error is minimal. So we need to get the matrix for the normal ...