Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? S=+∞∑n=−∞1σ√2πe−12(nσ)2 I am looking for something better than 1<S<2. Answer If fact, there is an explicit solution to the expression S=+∞∑n=−∞1σ√2πe−12(nσ)2=1σ√2πϑ3(0,e−12σ2) where appears the elliptic theta function. The function decreases asymptotically to 1 but it goes to infinity for small values of σ. … Read more

Find a>1 s.t. ax=xa^x = x has a unique solution

What a makes {x∣ax=x} a singleton? (1.4444)x−x≤0 has real solutions. (1.4447)x−x≤0 has no real solutions. I guess 1.4444<a<1.4447 I tried running simulations using goal seek in Excel, but I think I’m doing it wrong because I keep getting a lot of values below 1.4. How can I approach this problem? Answer Note that ax=x⟺a=x1/x. So, … Read more

Proof that lagrange and newton’s interpolation are the same

Its known that newton’s interpolation and lagrange interpolation gives the same value All i need is to prove it Answer There is a unique polynomial of degree ≤ (number of points – 1) going through the points. Lagrange and Newton both produce such a thing, therefore the same thing. AttributionSource : Link , Question Author … Read more

Montecarlo estimate of a integrand from 0 to ∞\infty

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: f(y)=∫y041+x2dx I want to estimate f(∞). I know that with some calculation, the exact values are given by π and 2π. However, there is some confusion with respect to defining the bounds and area … Read more

Understanding convergence of fixed point iteration

I was reading some slides explaining the convergence of the fixed point iteration, but honestly I’m not seeing or having an intuitive idea of how fixed-point iteration methods converge. Assuming p<1 as for the fixed point theorem, yields |xk+1−x∗|=|g(xk)−g(x∗)|≤p|xk−x∗| This is a contraction by a factor p. So |xk+1−x∗|≤p|xk−x∗|≤p2|xk−1−x∗|≤⋯≤pk+1|x0−x∗|→0 The smaller the p, the faster … Read more

Finding an approximation of a function’s root

I have the polynomial function f(x)=x5+2×2+1. I am trying to find an approximation to its root in [−2,−1], with the precision of 0.1, and with a minimal number of steps. The answer I was given was −17/16. I find it incorrect, and I wish to ask for your assistance. I have calculated f(−2) and f(−1), … Read more

Lorenz attractor depending on the numerical solution method

I have a problem. I am using two different numerical methods to try to “solve” the Lorentz attractor. Those are the Euler method (RK(s=1)) and the trapezoidal method with fixed-point iterations. The problem is that providing both methods the same initial parameters and the same iteration parameters (step-size, time-interval) I get two different “solutions”: Euler … Read more

Is it possible that a Runge-Kutta method has a global error of 0?

In a quiz of a course I am taking, there was the following multiple choice question: Decide which of the following statements is false, and justify your election showing a counterexample: (a) There are numerical methods for solving an Initial Value Problem of fifth order (b) There are integrals that cannot be computed exactly using … Read more

Best derivative-free numerical optimization methods when lenghty function evaluation

I need to find the global minimum of an optimization problem with seven parameters (all linearly constrained). My problem is that one evaluation of my function takes about 15 minutes (my code is pretty optimized and I cannot reduce the computation time any further). As a consequence I need an algorithm which can detect a … Read more

Numerical integration of $\int_a^bf(x) \: \text{d}x$ for $f(x) \to \infty$ when $x \to b$

This is the function I am trying to approximate using Simpson’s rule: $$\int_0^1 f(x) \: \text{d}x =\int_0^1 \frac{e^x}{\sqrt{1-x^2}} \: \text{d}x.$$ Of course, Simpson’s rule is of the form $$\int_a^b g(x) \: \text{d}x \approx \frac{b-a}{6}\left( g(a) +4g\!\left( \frac{a+b}{2} \right) +g(b) \right),$$ but in this case $f(x) \to \infty$ as $x \to 1$. I’m not sure if … Read more