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

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0
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2answers
29 views

Determine Value of Constant for Iterative Convergence

The question I am trying to answer is stated as follows: The iteration $x_{n+1} = 2 - (1+c)x_n + cx_n^3$ will converge to $\alpha = 1$ for some values of $c$ (provided that $x_0$ is sufficiently ...
1
vote
0answers
392 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
2
votes
1answer
177 views

Numeric solution of third order ODE

I need to solve the following third order (non-linear) ODE by numerical methods: \begin{equation}\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1. \end{equation} By assumption, the solution should approach $ ...
0
votes
2answers
428 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
7
votes
9answers
264 views

How do I prove that $\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$

How do I prove that $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$ without using the calculator?
0
votes
1answer
443 views

How to evaluate a condition number for a function of several variables?

I'm trying to get the condition number of a multivariate function $f(a,b,c)$ to see if it is stable. I am reading the information here. I know how to do it for a $1$-dimensional function. But for a ...
0
votes
1answer
41 views

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
12
votes
1answer
447 views

Stochastic gradient descent for convex optimization

What happens if a convex objective is optimized by stochastic gradient descent? Is a global solution achieved?
2
votes
1answer
66 views

A Variant of Gradient Descent

Suppose I have some objective function $f(\beta)$ which I would like to minimize for $\beta$. A standard gradient descent would be $\beta^{(t+1)}=\beta^{(t)}-\alpha \nabla f(\beta^{(t)})$, where $\...
16
votes
4answers
9k views

Gradient descent with constraints

In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size ...
0
votes
1answer
392 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial c}{\...
0
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1answer
30 views

Integrating Over a Product of (Non-Separable) Piecewise Functions (Hyper-Solid Angle of a Convex Polyhedral Cone)

My problem is as follows: given a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ where $n$ is some integer of order 10 and $f$ is defined by a product of (non-separable) linear piecewise functions, ...
4
votes
1answer
42 views

What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
2
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0answers
13 views

General Reaction Diffusion Equation Cranck Nicolson

How can i write matlab code that general reaction-diffusion equation by using Crank Nicolson Method? ∂u(x,t)/∂t= D(u(x,t))+R(u(x,t)) where D and R spatial discretizations (matrices) of linear or ...
0
votes
2answers
37 views

Identify what value of $x$ may have issues with cancellation error

I am leaning numerical analysis and getting a hard time to understand cancellation error. For example, suppose we have $\ln(x+1)-\ln(x)$, this is the same as $\ln\left(\frac{x+1}{x}\right)$. Suppose ...
15
votes
5answers
27k views

What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best and why? Advantage and disadvantage of them?
0
votes
0answers
15 views

Projected gradient descent with momentum

Can we apply momentum to projected gradient descent? If so, how should we do that? In the domain I'm working on, momentum greatly speeds up gradient descent. However, I want to do projected ...
0
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0answers
17 views

Counting number of multiplications and divisions in the Cholesky decomposition

To compute the Cholesky decomposition for an $nxn$ symmetric and positive definite matrix A, use the following algorithm: $A=LL^T$, $L=[l_{ij}]$ $$l_{ii}=(a_{ii}-\sum_{k=1}^{i-1}l_{ik}^2)^{1/2}$$ $$...
0
votes
0answers
17 views

Relationship between Newton's method in root finding and optimization

In both root finding and optimization, there are Newton's method. Wikipedia has 2 links here and here. Root finding is using first order derivative and optimization is using Hessian. What's the ...
0
votes
2answers
31 views

What is the computational complexity of Newton Raphson method to find square root.

I am not a math student, so I don’t fully understand the complexity as mentioned on Wiki for Newton Raphson method for finding square root. But I wrote a computer program for Newton-Raphson’s method ...
0
votes
1answer
55 views

Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
0
votes
1answer
23 views

What's the relationship between automatic differentiation and gradient method?

I'm learning about shape optimization and in the numerical methods of shape optimization I've seen the terms automatic differentiation and gradient method. Doing a Google search gives an impression ...
0
votes
1answer
34 views

merging two power equations into one

Let, we have two power equations: $$y=k_1x^{a_1}$$ $$y=k_2x^{a_2}$$ Is there any way (analytical or numerical) to combine these two equations into one i.e. into the form: $$y=kx^a$$ Infact,what I am ...
0
votes
1answer
13 views

How to concretely interpret big O bounds on error for forward euler?

On the wiki page for forward euler (https://en.wikipedia.org/wiki/Euler_method#Local_truncation_error), it describes the local truncation error like so: $\mathrm{LTE} = y(t_0 + h) - y_1 = \frac{1}{2} ...
0
votes
1answer
1k views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
2
votes
1answer
31 views

PDE Existence and Uniqueness through discretization

This is a question I have been thinking about, but I'm not sure where to look to find an answer. Have a PDE in space and time $(x,t)$. Have a time discretization of the PDE, this results in a ...
2
votes
1answer
33 views

finite volume methods: what do I have to do with the cell averages after each step?

I'm having a hard time understanding finite volume methods. If I take for example the scalar advection equation $$\partial{u}_{t}+a\partial{u}_{x}=0, a>0$$ with suitable initial and bondary ...
2
votes
1answer
64 views

Upwind differencing scheme in Finite Volume Method (FVM)

I have some troble in understanding how I can assess the direction of the flow for the upwind differencing scheme. Lets say we have the following ODE: $$a(x)\phi '(x)+b(x)\phi ''(x)=f(x)$$ Now ...
0
votes
1answer
22 views

“Best” solution to incompatible system of linear equations

I'm comparing data to a theory I've developed and right now I have to do some parameter fitting. Say I have two unknown parameters $x$ and $y$ such that $a_{1}x+b_{1}y=c_{1}$, $a_{2}x+b_{2}y=c_{2}$, $...
0
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0answers
30 views

Des chiffres et des lettres: proof that a random “right count” has a 94% success, how to prove it? (rules inside)

I am not sure whether this is the correct forum. Anyway, here goes... The game "Des chiffres et des lettres" is the most long living television game in French history; in its current form, it dates ...
0
votes
0answers
26 views

runge-kutta 4th motion law [on hold]

I want to apply the runge kutta 4th to Newton's equation of motion: $$ \left(\frac{dv}{dt}\right) = a(t) $$ $$ \left(\frac{dx}{dt}\right) = v(t) $$ I am solving using c language. I am not sure ...
2
votes
0answers
37 views

Convex optimization with $\ell_0$ “norm”

I have an optimization problem of the form $$\begin{align*}\text{minimize }\;&f(x)\\ \text{subject to }\;&||x||_0 \le t,\end{align*}$$ where $t$ is a given constant and $f:\mathbb{R}^d \to \...
-1
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1answer
22 views

Rate of convergence vs number of iteration

Can anyone explain to me the difference between rate of convergence and number of iterations for a numerical algorithm? Is it correct to say rate of convergence measure how fast the sequence approach ...
4
votes
1answer
940 views

Find the error bound

Hey guys I am unsure how to find the error bound. Use the langrange interpolating polynomial of degree 3 or less and four digit chopping arithmetic to approximate $\cos(.750)$ using the following ...
1
vote
2answers
32 views

Finding a linear combination with constraints on coefficients

Let there be $n$ unit vectors $\{\boldsymbol{u}_i\}_{1\leq i\leq n }$ in an $m$ dimensional space. The vectors are not necessarily a basis of the space. Let $\boldsymbol{v}$ be a unit vector in the $m$...
1
vote
1answer
22 views

numerical-methods, Fixed point theorem.

I am just looking to apply a result, so can someone confirm the following for me. Let say I have a equation of the form below: $V_1(x) = a + bV_0(x)$, where in theory $V_1(x) = V_0(x)$. I have an ...
0
votes
0answers
24 views

Linear regression of matrix elements to get the minimal polynomial to perform a matrix inversion?

So each matrix $\bf A$ fulfils an equation for it's minimal polynomial $P_m({\bf A})$: $$P_m({\bf A}) = 0 \Leftrightarrow \sum_{k=0}^{k_n}c_k{\bf A}^k = 0$$ We can by multiplying with $A^{-1}$ and ...
1
vote
2answers
26 views

Numerically Solving a 3d PDE with Stochastic Terms

I'm getting a bit confused if the procedure I'm doing is correct so any feedback would be great! It's just a standard deterministic PDE for the price of a theoretic option, even if it's quite a ...
1
vote
4answers
165 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
0
votes
2answers
40 views

calculating log

how do I solve the following problem: given log(x) for x=310,320,330,340,350 and 360 as 2.4913,2.5051,5.5185,2.5440 and 2.5563 find the value of log 337.5
1
vote
0answers
20 views

2D integration using modified Gauss Hermite/Stratified Sampling

I'm a beginner, trying to numerically integrate $$ \int\int f(x_{1}, x_{2})g(x_{1}, x_{2}) \,dx_{1}dx_{2} $$ where $g$ is the standard 2D normal density function. Seemingly obvious choice would be ...
2
votes
3answers
76 views

Gauss-Laguerre quadrature

I am trying to compute this integral: $$ \int_{0}^{\infty}\prod_{k = 1}^{d}\left(1 - \,\mathrm{e}^{-a_{k}\,t}\right) \,\mathrm{e}^{-t}\,\mathrm{d}t,\quad \mbox{where}\quad a_{k} > 0, \forall\ k. $$ ...
2
votes
1answer
1k views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
2
votes
1answer
25 views

Properties of matrix stable (numerical) rank

I happened to notice that there is concept "stable rank" that people used a lot in matrix computation theories, such as the work of Rudelson & Vershynin (2005). It is defined to be the ratio ...
6
votes
1answer
86 views

Compute a diagonalizable matrix close in matrix exponential

It is known that for any matrix $A$, one can perturb $A$ slightly so that the resulting $A(\epsilon)$ is diagonalizable. I am wondering whether for any matrix $A$, $\epsilon>0$, there is an ...
0
votes
1answer
21 views

What should be the expected error of a 4th order Runge-Kutta integration in a multivariable (non-linear) context?

I am writing a program where the user input $n$ variables, their initial values and differential equations. They may be non-linear. To find the value of the variables at a time $(t_0 + T)$, I use 4th ...
2
votes
3answers
76 views

To find a real root

find real root of given equation $$5x - 2 \cos x -1=0 $$ I only know that I should use $$x_n = \frac{ x_{n-2} f_{n-1} - x_{n-1} f_{n-2} } { f_{n-1} - f_{n-2} }$$ And I applied it, but didn't get ...
10
votes
3answers
2k views

Big-O Interpretation

I have trouble understanding what the "Big O" notation, or asymptotic notation means. For instance, if you have $\sin(x)=x+O(x^3)$, what does this mean? Can anyone describe it in a simple way? I tried ...
-3
votes
0answers
36 views

What's the name of this method?

Here are the formulas use in the method, but I don't know the name of it. yn+1= yn + h * 'n + (h^2/2) * y''n y'n+1= y'n + h * y''n y''n= f(xn, yn, y'n) xn= xn+h and the form: n | xn | yn | y'n | ...
1
vote
2answers
53 views

Optimization with box constraints - via nonlinear function

I have the following convex optimization problem over $\mathbb{R}^n$ with box constraints: $$\begin{align}\text{minimize }&\;f(x)\\ \text{subject to }&\;x \in [-1,1]^n\end{align}$$ I can see ...