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

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28 views

How to establish a lower bound on this difference operator?

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
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2answers
71 views

Is there a general formula for estimating the step size h in numerical differentiation formulas?

Using three-point central-difference formula $$ f^{\prime}(x_0)\approx \frac{f(x_0+h)-f(x_0-h)}{2h} $$ and for $f(x)=\exp(x)$ at $x_0=0$ we have $$ \begin{array}{c, l, r} h & f^{\prime}(0) ...
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1answer
21 views

How to verify the gradient of a symbolic function using numerical gradient?

I have a function $f$, which takes as inputs a three arrays and returns an array. I have written a symbolic function $g$ to calculate the gradient of this function and I want to verify that it ...
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1answer
82 views

Numerical integration fails

I am doing something wrong. This is my algorithm to evaluate the integral $$\int_0^1 \frac{1}{1+x}dx= \log(2).$$ with the Newton Cotes algorithm (Simpson and 3/8). Both give me that for large n ...
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1answer
17 views

Computational complexity of numerical integration of gaussian function

$ \int^{b}_{a} \exp(-x^2)\,dx$. I have the following two questions regarding the above integral expression of the Gaussian function: Is there a numerical method we can use to solve the above ...
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0answers
64 views

Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$ f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi) $$ I tried to get this formula using Lagrange polynomial. ...
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1answer
23 views

What is a tensor-product Chebyshev grid?

What is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?
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1answer
37 views

Using Finite Difference to compute derivative in the Newton-Raphson root finding Algorithm

In the Newton-Raphson method we come across the following equation: $$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}$$ Can you please let me know if we can calculate the derivative term like this - $$f'(x_n) = ...
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1answer
28 views

Difference between derivative and its approximation

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
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1answer
109 views

Fast algorithm for approximating Eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16}$x$2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather ...
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0answers
17 views

Closest intersection between a ray and 2 variable function

Exact solution of this problem for an arbitrary ray and function does not exist. What i am interested in is a high quality numeric solution or something similar. Also, are there solutions that put ...
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0answers
45 views

Galerkin-Approximation of first Eigenfunction

I'm currently trying to understand a certain proof of an error estimate for the first eigenfuntions gained by a Galerkin-Approximation with Finite Elements of the Potential Equation $-\Delta u$ with ...
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1answer
17 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
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1answer
29 views

Differentiating the $QR$ decomposition?

Let $A(t)$ be a smooth family of invertible $n \times n$ matrices with $A(0) = I$, and let $A(t) = Q(t) R(t)$ be the $QR$ decomposition. Given $\dot{A}(0)$, what is an algorithm to compute ...
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1answer
32 views

Euler's method vs midpoint method

Are the following methods equally accurate and if not, why? Using Euler's method with a step size of $h$. Using the midpoint method with a step size of $2h$. Even though Euler's method has a ...
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3answers
156 views

optimal way to approximate second derivative

Suppose there is a function $f: \mathbb R\to \mathbb R$ and that we only know $f(0),f(h),f'(h),f(2h)$ for some $h>0$. and we can't know the value of $f$ with $100$% accuracy at any other point. ...
2
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1answer
59 views

Change of variable

I have to approximate the following integral, using Simpson's Composite $1/3$ Rule: $\displaystyle \int\limits_{0}^1 \mathrm{\frac{e^{2x}}{\sqrt[5]{x^2}}}\,\mathrm{d}x$. The only problem is that ...
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1answer
101 views

Why is the numerical solution of this equation unstable? Is this equation stiff?

I am trying to solve the following equation with an explicit fourth-order using the Runge-Kutta method: $$y' = t(y - t \sin t)$$ with initial conditions $y(0) = 1$ over the interval $[0, 10]$. The ...
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0answers
16 views

characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
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0answers
43 views

Continuation fixed points of parameter dependent Newton

Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in ...
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1answer
22 views

Numerical evaluation of the first (K) and second (E) complete elliptic integrals

To get a numerical evaluation of the first (K) and second (E) complete elliptic integrals: $$K(k)=\int_0^1\frac{dt}{(1-t^2)^{1/2}(1-k^2t^2)^{1/2}}, \ \ \ \ \ ...
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0answers
32 views

Matrix rank when solving using Gram Schmidt

Can I deduce the rank of a matrix (i.e. of size 3x2) when solving an over determined set of equasion using QR Gram Schmidt method? If not, is there a QR-related or other numerical method to find a ...
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1answer
24 views

Interpolation based on $n$ uniformly distributed points

We are given $n+1$ uniformly distributed points in the segment $[0,1]$: $x_i=\frac{i}{n}$, $i=0,1,...,n$ and a function $f(x)=e^{-x}$ $P(x)$ is the interpolation polynomial of $f(x)$ where ...
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0answers
30 views

Comparison of trapezoidal , Simpson's 1/3 ,Simpson's 3/8 and Boole's rules.

These rules are often used in numerical integration. How do we analyze the given support points or function and select the most suitable one for best approximation?
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2answers
29 views

Numerical precision of product of probabilities (normal CDF)

I'm trying to calculate $\prod_k{p_k}$ where $p_k$ are (potentially) very high probabilities of independent, zero-mean, standard normal random variables and $k>100$. However, I'm running into ...
2
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1answer
36 views

Numerical computation of power differences: $x^a - y^a$

I want to calculate a power difference, $x^a - y^a$, where $a$ can be large, and the numbers $x,y$ are of similar magnitude. What's a sound numerical way to approach this? Note: $x,y,a$ are all ...
1
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1answer
16 views

Computing $\mathrm{B}\left(x,y;\alpha+1,\beta\right) / \mathrm{B}\left(x,y;\alpha,\beta\right)$ numerically

I need to compute numerically ratios of the form: $$\frac{\mathrm{B}\left(x,y;\alpha+1,\beta\right)}{\mathrm{B}\left(x,y;\alpha,\beta\right)} \tag{1}$$ where ...
0
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0answers
28 views

Order of **convergence** of a multistep method

State the Dahlquist equivalence theorem regarding convergence of a multistep method. The multistep method, with a real parameter $a$, $$y_{n+3} + (2a-3)(y_{n+2}-y_{n+1}) - y_n = ...
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1answer
35 views

Finite Element method Implementation

I have written a program for the finite element method for an elliptic one dimensional problem.Initially I assumed a mesh that had only 10 points , but since the error was far above my tolerance ...
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1answer
107 views

Solving Differential Equations theoretically and using matlab

i am trying to solve the initial value and elliptic boundary value problems below. but now i need some help solving them using matlab. for the elliptic problem, any method is ok, but for the initial ...
1
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1answer
32 views

Newton's method linear convergence proof

How would you show that if f'(a)=0 then the Newton's Method is linear convergent when 1. $f''(a)\neq 0$ 2. $f''(a)=0, f'''(a) \neq 0$? I am having some trouble getting it to the point where you can ...
0
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1answer
92 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
0
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0answers
33 views

Interpolation using four nodes

Suppose there are four points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ my target is to interpolate any point $x_I$ between $x_2$ and $x_3$. Is there any Interpolation method which gives linear ...
1
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0answers
32 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
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1answer
38 views

Normalizing a dataset from the interval [0,1] with fixed properties.

So I have a rather large dataset where values are from the interval $[0,1] \in \mathbb{R}$. But the problem is that a big portion of the values are extremely close to $0$. So firstly I'm looking for ...
0
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1answer
42 views

Plot as you go in MATLAB

I'm self studying some numerical analysis and I'd like to get a feel for how an ODE solver speed varies as you move forward in time. Is it possible to use matlab to numerically solve an equation ...
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1answer
52 views

Where comes the +1 from in this formula?

I'm working on two papers ([1] equation 8, [2] equation 2.3) and I can't figure out why there is an identic formula I can't explain on both. $p(z) \in \mathbb{C}[z]$ is a monic polynomial with simple ...
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1answer
63 views

inequality of same distance points

I got this problem: Let $x_i=\frac{i}{n}$ for $i=0,1,...,n$. Prove that for all $$ x\in[0,1]:|\prod_{k=0}^n (x-x_k)| \le \frac{n!h^{n+1}}{4} $$ where $h:=\frac{1}{n}$. I tried to find maximum of ...
0
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1answer
20 views

How to evaluate the accuracy for sparse linear system solver

I'm currently trying to do some experiments on linear solver. However, it's a little hard to get the sense of the numbers. For example, I know large condition number is bad, but how large is bad? ...
0
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2answers
52 views

exact or numerical value of an improper integral

i am dealing with an improper integral which has been arised in my research. i will be greatful if you have any idea about the numeric value of this integral. $$ \int_{0}^{\frac{1}{4}} \frac{u^{4} ...
0
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1answer
34 views

Sum of Lagrange polynomial

I have to calculate $\sum_{i=0}^n x^k_i*l_i(x)$ for $k=0,1,2,...,n$ . I've proved that $\sum_{i=0}^n l_i(x) = 1$, but I cannot figure out how it may help me calculating $\sum_{i=0}^n ...
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1answer
239 views

Fast inverse square root trick

I found what appears to be an intriguing method for calculating $$\frac{1}{\sqrt x}$$ extremely fast on this website, with more explanation here. However, the computer-science lingo and ...
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2answers
29 views

MATLAB Newton non-linear equation

I have the following non-linear equation: where $w0=0.25,w0=0.5,w0=0.75$. I have to prove that if $k$ is a root, then also $−k$ is a root and that there exists only one $k∈(0,1)$ root, but my ...
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1answer
194 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
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0answers
24 views

Condition number composite function

I have a composite function $h(t)=g(f(t))$ and have to evaluate the condition number for $h(t)$ through the condition numbers of $g$ and $f$. I know that the condition number formula is ...
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3answers
132 views

How to calculate the square root of a number? [duplicate]

By searching I found few methods but all of them involve guessing which is not what I want. I need to know how to calculate the square root using a formula or something. In other words how does the ...
0
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0answers
30 views

Stiff differential equations without using Jacobian matrix

I want to solve a stiff system of differential equations. Its Jacobian matrix isn't constant and its determinant is close to zero so I cant inverse of it. Please tell me does exist a method that solve ...
0
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1answer
27 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
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1answer
58 views

Stiff differential equation

I'm trying to solve a system of differential equations with Runge-Kutta method. When I use the step size $h=1$ my problem has true answer but when I use the smaller $h$ (for example $h = 0.1$) my ...
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0answers
38 views

Numerical solution of non-linear differential equation with MATLAB

I need some information to know if I can solve a nonlinear integral equation with terms $ u_{x} $ , $ u_{x}.u_{y} $ , $ u_{xx} $ , $ u_{xy} $ $u_{yy} $ $ u_{x}^{2} $ $ u_{y} ^{2} $ By numerical ...