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

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6
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2answers
94 views

How do people calculate values for trig functions?

This may sound like a stupid question, but I'm wondering how people originally calculated specific values for trig functions before calculators existed. Did they just draw circles and manually measure ...
0
votes
2answers
34 views

The error formula for the Romberg integration [closed]

I am just wondering if there exists a error formula for the Romberg integration. Since it just applies Richardson extrapolation to Trapezoidal rule, is its error formula the same as that of the ...
1
vote
1answer
17 views

An example of hermite interpolation [closed]

I found this example on wikipedia. What I don't understand is that on the right hand side, the column starts with $-10$. Why isn't the column $-10, -4, 4, 10$? Why do the numbers in the middle place ...
0
votes
1answer
19 views

Using Maple to find forward error and error magnification factor.

I am trying to use Maple to easily find a forward error and error magnification factor. Is there a specific command for this?? I know to find forward error the equation is ...
0
votes
0answers
18 views

Solve an integral of a Multinomial with Gaussian prior

I have a Multinomial whose parameters evolve over time following a random walk with diagonal covariance. I have a sample of the multinomial for each time. $$p(X_t) = Multinomial(\theta_t)$$ ...
2
votes
3answers
61 views

Intersection point of two functions - one linear, the other with logarithmic and sqrt terms

I would like first to appreciate everything that is being done on this forum and to greet you all! I have namely two functions and the goal is to find the intersection point of them. $y_1 = a + ...
1
vote
2answers
83 views

Disadvantage of Newton method in optimization compared with gradient descent

In optimization with Newton method in wikipedia, there is a diagram showing Newton's method in optimization is much faster than gradient descent. What is the disadvantage of Newton's method compared ...
1
vote
1answer
18 views

Neumann boundary conditions at corner of rectangular domain

If we have a rectangular domain $\Omega$ and we are approximating the derivative of $u(x,y)$ by a finite difference $$u_{xy} \approx \frac{(u_{i+1,j+1} + u_{i-1,j-1} - u_{i+1,j-1} - ...
0
votes
1answer
15 views

Solving a low rank symmetric system

I'm working on a problem where I need to solve a large set of systems of equations, where each has a structure that looks like: $\left( M^\top_{n\times p}\Sigma_{p\times p} M_{p\times n} + ...
3
votes
1answer
52 views

If $\{x_n\}$ converges to zero and we have $|x_{n+1}/x_n|$ is bounded then $|x_{n+1}/x_n|$ has a limit.

This is a situation that appears frequently in numerical analysis say $\{x_n\}$ converges to zero and we have $|x_{n+1}/x_n|$ is bounded. Then I need to prove (not convinced it is true though) that ...
0
votes
1answer
26 views

solution to integral of polynomial $\int_{-1}^{1}(\frac{1-\xi}{2}H_1^{e}(t)+\frac{1+\xi}{2}H_2^{e}(t))^{1/3}d\xi$

$\int_{-1}^{1}(\frac{1-\xi}{2}H_1^{e}(t)+\frac{1+\xi}{2}H_2^{e}(t))^{1/3}d\xi$ Do you have any ideas how to solve this integral (numerically or analytically)? any good approximations?
2
votes
0answers
44 views

'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
0
votes
0answers
53 views

Why is numerical integration not working well on logarithm function with bounds $[-1,1]$

When I try to integrate function $x(\log(x)-1)$ from $-1$ to $1$, analytically I get $0.0000 - 1.5708i$ When I try to integrate it numerically, using $10$ points gaussian quadrature I get $0.0000 - ...
0
votes
3answers
82 views

Nonlinear Equation involving a matrix

I have a matrix $A$ whose entries are each a function of a variable $\epsilon$, with $\epsilon>0$. This matrix arises from Radial Basis Function (RBF) interpolation, and is symmetric ...
1
vote
2answers
43 views

Optimization over linear combinition of min functions

Assume we are given these six variables: $x_{12},x_{21},x_{13},x_{31},x_{23},x_{32}$. Then if, $A_{ij} = min\{x_{ij},x_{ji} \}, B_{ik} = min\{x_{ik} - A_{ij}, x_{ki} \}, C_{jk} = min\{x_{jk} - ...
1
vote
0answers
32 views

Is this a correct way to approximate a derivative

I have some function $S(\hat y)$ which I want to approximate its derivative with respect to a vector. It's a tad complicated, I'll try and explain. $S$ is a function of $\hat y$ only, but $\hat y$ is ...
1
vote
1answer
24 views

Implicit Euler local error issue

I'm not sure I get what's going on here, and online resources are not helpful, at least I didn't find any helpful ones. For the problem: $$ \frac{dy}{dt} = f(t, y(t))$$ a numerical solution for ...
0
votes
0answers
92 views

Numerical Computation - Taylor series

I'm taking numerical computation course, and I have a problem with this question: Apply Taylor's formula to obtain a power series approximation about $a=0$ to $\sin(\pi x/2)$. Find the remainder, ...
0
votes
1answer
41 views

Why $\int_{0}^{1} \frac{x^n}{x + 10}$ can be approximated with this unstable algorithm?

We want to evaluate the integral $$\int_{0}^{1} \frac{x^n}{x + 10}dx$$ and we develop an approximation recursive algorithm, which is based on the fact (1) $$y_n + 10y_{n-1} = \int_{0}^{1} \frac{x^n ...
2
votes
0answers
23 views

Which error does one usually need to consider in numerical analysis?

When analyzing performance of a numerical method, I have considered and plotted $$\| x_n - x^* \|$$ where $x_n$ is the $n$-th iteration and $x^*$ is the true value (in our toy examples, it wasn't hard ...
1
vote
1answer
30 views

Asymptotic notation: What does $o(\epsilon_\text{mach})$ mean?

I'm having serious problems to understand what people mean when they write $o(\epsilon_\text{mach})$, where $\epsilon_\text{mach}$ stands for the machine epsilon. I'm seeing this in backward analysis ...
1
vote
0answers
22 views

characteristic method for wave equation in a non-uniform string

wave equation in a non-uniform is $u_{tt}=c(x)u_{xx}$, $c(x)=1$, when $0<x<1/2$, $c(x)=2$, when $1/2<x<1$, Does $u(x,t)$ has the form like $u(x,t)=f(x-t)+g(x+t)$ when $0<x<1/2$, ...
0
votes
1answer
45 views

Simulating a rocket in Matlab

I want to simulate a rocket. I found this code in a book. For the past two days I have been trying to understand it. For instance there is a line: ...
1
vote
2answers
31 views

Showing Lagrange polynomials form a basis for $\Pi_3$.

Here's the question: Let $x_0=-2, x_1=0, x_2=1, x_3=4$ and let $L_j$ be the Lagrange polynomial for $j=0,1,2,3$. Show that $L_0, L_1, L_2, L_3$ form a basis for $\Pi_3$. So I've calculated ...
0
votes
0answers
14 views

Limit Definition for Norms

Let $X_n\in \mathcal{M}_{n\times n}(\mathbb{R})$ be defined as an iterative sequence, $B\in \mathcal{M}_{n\times n}(\mathbb{R})$ and $\|\cdot\|$ be an operator norm. If we are given that $\|X_n-B\|\to ...
2
votes
2answers
64 views

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: $$f(y) = \int_{0}^{y}\frac{4}{1+x^{2}}dx$$ I want to estimate $f(\infty)$. I ...
0
votes
1answer
46 views

Does the norm of the product give information about the norm of the matrices?

Let $\|\cdot\|$ be an operator norm subordinate to $\|\cdot\|_{\infty}$ and $A,B\in \mathcal{M}_{n\times n}(\mathbb{R})$. Also, let us assume that $\|AB\|\to 0$. Now, by the multiplicative inequality ...
-2
votes
1answer
75 views

Help with Lebesgue integration [closed]

I want to solve integration of $\sin(x)$ from $0$ to $\pi$ ,with the help of measure ..
2
votes
2answers
59 views

Fixed point method

I am trying to find why when using fixed point method to find a root, I cannot find the convergent value that I need. I have the function: $\dfrac{\cos(x\pi)}{9}+\sin(x\pi)$ I solved for x, and I ...
0
votes
0answers
27 views

Is $ \| \sum_{i \in [k]} \otimes^3 v_i - T \|_F^2 + \theta \| \sum_{i \in [k]} \otimes^3 v_i \|_F^2$ convex?

I am trying to find the minima of the following equation with respect to $v_i$, $i \in [k]$, to solve an optimization problem but I can't manage to make (stochastic or not stochastic, neither of them) ...
0
votes
2answers
24 views

approximating functions via a piecewise combination of linear and constant functions

I am curious if anyone has encountered any literature on approximating functions via a piecewise combination of linear and constant functions. I have seen a couple of papers which use piecewise ...
3
votes
1answer
54 views

Solving $x^{1/x}=y^{1/y}$ iteratively by power towers, convergence

Suppose we have no idea that $x^{1/x}=\exp \left( \frac{1}{x} \ln x \right)$ or don't know anything about exponents and logarithms. But we can certainly compute roots, for example by this method. ...
4
votes
3answers
38 views

Fixed Point Iteration doesn't converge, how to find its convergence?

$g(x)=(2/3)(\cos x-\sin x)$ $x_n=g(x_{n-1})$ with initial guess $x_0=1$ I need to compute for n= 0,1,...,8 When I try it, my sequence diverges: $x_1=-0.20078$, $x_2=0.78623$, $x_3=-0.00079$, ...
0
votes
3answers
38 views

Most accurate finite difference result for first derivative [closed]

I got a problem in my assignment: obtain the most accurate finite difference results possible for the first derivative of f (x) = exp(cos(x)) at x=1, h = 0.5, 0.25, 0.125,...2^{16}. I have to do this ...
1
vote
0answers
27 views

adjoint method for computing derivatives

I am curious if anyone has heard of this problem before: Suppose that $u(x,p)$ is a function of $x$ and $p$. These arguments need not be scalars. Let $u(x,p)$ satisfy some differential equation, ...
0
votes
0answers
22 views

What are 2*0.3-0.6 and 3*0.3-0.9 in decimal floating point numbers?

Investigate the values $a=2*0.3-0.6$ and $b=3*0.3-0.9$. What do we expect to get on a computer using decimal floating point numbers? My reason and doubt: It doesn't really mention how many degree ...
0
votes
0answers
48 views

Prove Norm Theorems

I have the following as given: Let $A \in C^{m\times m}$. Then: 1) $$\lVert A\rVert_1 =\sup_{v\in C^m \setminus\{0\} }{\lVert A_v\rVert_1 \over \lVert v\rVert_1} = \max_{j} \sum_i |a_{ij}|$$ How can ...
1
vote
0answers
21 views

Coupled linear PDE equations (2nd and 1st order) - Numerical Method

I am trying to solve a coupled reaction-diffusion equations, using Crank Nicolson (implicit Finite Differences Method). I know how to solve them separately, but not simultaneously. Coupled PDE's ...
0
votes
2answers
30 views

Is there a way to measure how close a matrix is to being rank-deficient?

I'm working on a least-squares problem with an overdetermined matrix, and I've noticed that changing the data very slightly leads to a huge change in my solution (in this case, plotting an elliptical ...
1
vote
0answers
36 views

Finding the Surface Area Obtained by Rotating $\sin(\frac{\pi}{3})x$ from $x=0$ to $x=3$

I tried using this formula $$\int_a^bf(x)\sqrt{1+(f'(x))^2}dx$$ and managed to reduce the problem (after integrating by substitution twice) to: $$A=54(\pi)^2 \int_{46.32}^{-46.32} (\sec g)^3 \,dg$$ ...
0
votes
0answers
15 views

How to validate an error model for function approximation?

I have a model that i would like to experimentally validate. It is something like: $$\epsilon = \epsilon(q_1,...,q_n) = \epsilon(\vec{q})$$ This model describes the trend of the error bound of a ...
-2
votes
2answers
43 views

Improve the order of accuracy for second derivative using central difference approximation

The formula $$D^{m}(h)=(4^mD^{m-1}(h/2)-D^{m-1}(h))/3$$ for $m=2,3,\ldots$improves the order of accuracy for first derivative using central diff approx. where ...
0
votes
1answer
18 views

Octave false position

i'm try to write some code in octave based in false position method. So, here it is: And I get the follow error: "parse error near line 40 of file C:\Users\HP...falsa.m syntax error else" So, ...
0
votes
0answers
16 views

Measuring rank deficiency, and its implications for linear least-squares.

I'm working on a problem involving linear least-squares fitting on an overdetermined system. I've noticed that small changes in the data have a really big impact on the parameters that I'm solving ...
1
vote
1answer
40 views

Numerically Solving a Poisson Equation with Neumann Boundary Conditions

The Problem Suppose I have an equation of the form $\nabla^2 \phi(x) = f(x)$ on the interval $A \le x \le B$, where $f(x)$ is known and $\phi(x)$ is unknown. I have Neumann-type boundary conditions: ...
0
votes
0answers
16 views

conjugate gradient method - error estimate

According to book (Numerical optimization, Nodal, Wright), the error estimate of CG method is defined as following $$ \|x_k - x^*\|_A^2 \le \left ( \frac{\lambda_{n - k} - \lambda_1}{\lambda_{n - k} + ...
2
votes
1answer
73 views

Eigenvalue-related statements

How can I prove that the following statements are equivalent? $\lambda$ is an eigenvalue of $A+\delta A$, where $\|\delta A\|_{2}\leq \epsilon$ $\exists u\in \mathbb{C}^{m}$ such that $\|(A-\lambda ...
2
votes
0answers
17 views

Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
0
votes
0answers
14 views

Numeric Eikonal Solution

I am trying to solve an Eikonal problem explicitly, this is my first time encountering one. I have to post an external link to the equation because I cannot format it properly: ...
1
vote
0answers
11 views

Implicit scheme for a nonlinear PDE system

I have the next PDE system: $u_t=D_u u_{xx}-u+av+u^2v\\ v_t=D_v v_{xx}+b-av-u^2v$ Where $D_u,D_v,a,b$ are constants. I want to build or find a implicit SECOND ORDER scheme for this PDE system. I ...