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

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38 views

What is the typical $\epsilon$?

In doing some self-learning in numerical methods I having come across the following a number of times. $\epsilon$ is the smallest computational unit such that, $x + \epsilon = x$. What value does ...
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1answer
15 views

Accuracy of $e^x$ approximation using remainder of Taylor Series Approximation.

The Problem: If we consider approximating $e^x$ on $[-1,1]$, the Taylor Theorem for $x_0 =0$ says we can represent $e^x$ using a polynomial with a (known) remainder: $e^x =\{1 + x + \frac{1}{2!}x^2 ...
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1answer
18 views

Find condition number for vector of roots

Consider $f(z)=z^2+az+b$ we put it's root in vector $[z_1, z_2]^T \in \mathbb{C}^2$. Find condition number in maximum norm of finding the root vector when changing variable a where $a=-2, b=3$ I know ...
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28 views

How one can approximate irrational raised to irrational power?

How one can evaluate irrational number raised to irrational power? Like is there an easy way to prove that $-0.685<\pi^e-e^\pi<-0.675$?
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1answer
21 views

Prove $f - p_*$ is orthogonal to all $p \in P_n[a,b]$

Let $p_* \in P_n[a,b]$ be the best $L_2$ approximation to $f \in C[a,b]$. Then $f - p_*$ is orthogonal to all $p \in P_n[a,b]$ I set: $$p_* = \sum_0^n c_i\Phi_i \text{ and } c_i = \langle ...
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1answer
48 views

Newton Method Simplification

Hi Everyone I have the following homework assignment problem that I am struggling with: Consider Newton's method $$ x_{k+1} = \frac{1}{2} \left( x_k + \frac{a}{x_k} \right), \qquad a > 0, $$ for ...
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13 views

Prove $\Phi_{n+1}$ is orthogonal to all p $\in P_n[a,b]$

Let $\Phi_j(j=0,1,2,...,n+1)$ be a system of orthogonal polynomials on [a,b]. Prove: $\Phi_{n+1}$ is orthogonal to all p $\in P_n[a,b]$ I'm going to solve like following: I set $p_n(x)=\sum_{k=0}^n ...
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14 views

Fitting with initial conditions

i try to do some fits but i have ten initials conditions and i think it will be difficult to evaluate the sensitivity of my conditions. Do you know some methods which allow to know the sensitivity of ...
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18 views

Numerically Integrating Singular Integrals

When using the changing of variables technique in Numerical Integration, is there a general rule/template for which substitution to use just by looking at the function. I am confused. For example if ...
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1answer
28 views

Proving an equivalent to a summation formula

Let us recall the summation formula $$\sum_{k=1}^nk=\frac{n(n+1)}{2}$$ How do we show that $$\sum_{k=1}^nk=\frac{1}{2}n^2+\mathcal{O}(n) ?$$ I started by stating the definition of "big-o" notation ...
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3answers
42 views

Use of Taylor series expansion to find second derivative for sixth order method

Use Taylor's expansion to derive sixth order method (i.e $\mathcal{O}(h^6)$) for approximating the second derivative ($f '' (x_0)$ ) for given sufficiently smooth function $f(x)$. I have this things ...
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11 views

Problem with the numerical PDE solving (possibly lattice-pinning)

I am solving quite complicated PDE's. The behavior was unexpected, and I started to simplify it. Finally I found out, that the problem is in the modified heat equation. The equation is: $$ ...
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15 views

Solving non-linear functional equations numerically by sequence of linear least-squares?

So I am experimenting with a linear systems solver to find new exciting applications for it. While it is possible to play around to solve some of the more basic functional equations, I am trying to be ...
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1answer
41 views

Integrating Lagrange polynomials

Could you suggest some efficient way to numerically compute $\int\limits_0^{t_i}l^N_j(t)dt$, where $l^N_j(t)$ is the $j$th N-point Lagrange polynomial and $t_i$ is the $i$th interpolation point?
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1answer
20 views

Numerically solving 1D 2nd order PDE Goursat problem

I am looking for help with solving the following Goursat problem (this is what the paper I am reading from calls it). I have been attempting a numerical solution in matlab but I do not fully ...
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1answer
17 views

Polynomial interpolation with evenly spaced data.

Given the following data table, x 1.20 1.25 1.3 1.35 1.40 1.45 1.50 f(x) 0.1823 0.2231 0.2624 0.3001 0.3365 0.3716 0.4055 ...
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14 views

IVP Using Numerical Methods

Suppose that $y(t)$ is the exact solution of the ivp $$y'(t)=f(t,y(t)), y(0)=y_0$$ and $u(t)$ is any approximation to $y(t)$ with $u(0)=y(0)$. Define the error $e(t)=y(t)-u(t)$. How can I show ...
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2answers
85 views

Numerical integration in Matlab (Simpson's rule)

A cubic polynomial is given by $$y=\frac{x^3}{6RL}$$ with $R$ and $L$ being constants. Use Matlab and numerical methods to find $x_l$ so that $$L=\int^{x_l}_0 \sqrt{1+(y')^2} dx$$ when $R=200$ and ...
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2answers
52 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
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37 views

Find the best approximation of $y = e^x$ in $P_2 [-1,1]$

$L_2$ norm weighted by $\omega$(x) = $\frac{1}{\sqrt(1-x^2)}$ The least-squares approximating polynomial Pn(x) of f(x) using Chebyshev polynomials is given by: $P_n(x) = a_0T_0(x) + a_1T_1(x) + · · ...
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27 views

The generalized formula for Runge-Kutta method

For p=1, Runge-kutta is just the euler method. I am just wondering if there exists a generalized formula for Runge-Kutta method when p is arbitrary. Thanks!
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2answers
41 views

Use least squares to find best fit value of angle phi

Given initial object $\begin{bmatrix}1 & 0 & 0\\0 & 0 & 2\end{bmatrix}$ and final object $\begin{bmatrix}0.9557 & 0.0965 & 0.9648\\-0.3381 & 0.0158 & ...
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1answer
44 views

Newton's method for a system of 7 equations

I want to use Newton's method to find two solutions to the system of equations using $L_\infty$ norm and an f-test with $tol_f = 10^{-6}.$ $$\frac{1}{2}x_1+x_2+\frac{1}{2}x_3 - \frac{x_6}{x_7}=0$$ ...
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0answers
29 views

The butcher array of explicit Runge- Kutta method

Just a quick question, for a family of explicit Runge-Kutta methods parametrized by order q, by applying up to $p-1$ passes of deferred correction to p steps of Euler's method. When $p=2$, should its ...
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0answers
39 views

Why is it difficult (and not precise) to compute the rank of large matrix numerically?

I have a general question. I have a large square matrix ($n> 1000$) and it is needed to compute the rank of this matrix. I am reading that the computation of the rank for large matrices, can make ...
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1answer
51 views

Finite Difference Method MATLAB Program

I am a beginner to MATLAB. I am trying to create a MATLAB program for the finite difference which is to calculate potential in a grid. I have to include a condition such that the iterations stop once ...
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0answers
33 views

Lagrange interpolation for ellipse

Consider the ellipse $$\frac{x^2}{4} + \frac{y^2}{2} =1$$ The line integral $I$ of the ellipse in the first quadrant is $$I=\int^2_0 \Big[ 1+(y'(x))^2 \Big]^{1/2} dx$$ Find the cubic polynomial ...
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40 views

Convergence of iteration scheme of solving matrix equations

Consider the equation $A\mathbf{x}=\mathbf{b}$. It is equivalent to $$S\mathbf{x}=(S-A)\mathbf{x}+\mathbf{b}$$ where $S$ is a splitting matrix. We now consider the iteration scheme ...
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16 views

The Heuristic Gauss-Kronrod Based Error Estimator in Quadpack

I'm trying to understand the local error estimate that Quadpack (and subsequently other libraries such as GSL, quadpack++, cubature, etc.) uses for it's general adaptive quadrature subroutine QAG. The ...
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1answer
29 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
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0answers
15 views

How to numerically solve a Green's function using mathematica? [migrated]

Suppose we have a Green's function $$LG(x)=\delta(x),$$ how to numerically solve it by mathematica? Can the mathematica read the delta function directly? For analytical calculation, we usually ...
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1answer
17 views

Numerical Analysis - Upper bound for $|R|$

I am asked to find an upper bound for $|R|$ valid for all $x\in[-1,1]$ that is independent of $x$ and $\xi$. Given that, $$R(x)=\frac{|x|^6}{6!}e^\xi$$ for $x\in[-1,1]$ where $\xi$ is between $x$ ...
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46 views

A difficult question about Runge-Kutta method and Euler's method

I found this question extremely difficult... Especially part b. For part b, I've tried several times. Is it correct that we need to use Taylor expansion to solve it? Since $e'(t) = f(...) + ...
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1answer
17 views

Problem applying Simpson's rule

I am having a problem applying composite Simpson's rule for the integral $$I=\int_0^2\dfrac{1}{x+4}dx$$ with $n=4$. The exact value of the integral is about $0.405$, however, Simpson's is giving ...
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34 views

Minimize/Maximze a function against its approximation.

Let $f \in C^{\infty}[1,2]$ be a function we would like to approximate, let be $g$ such approximation, you can assume $g$ is a spline function (at least quadratic). In literature I have seen that a ...
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0answers
18 views

Characteristic of Euler- Tricomi equation

Consider the Euler-Tricomi equation $$u_{xx}-xu_{yy}=0$$ Determine its characteristics. I know the characteristics is $x dx^2=dy^2$ from wikipedia, can anyone show me how to get it?
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25 views

In practice what is (modified) Gram Schmidt used for?

Modified Gram-Schmidt is known to be numerically less stable than methods like Householder orthogonalization and also not quite as fast at approximately $2mn^2$ flops. So in practice do we ever use ...
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26 views

Numerically solve a set of “coupled” ODE

I am looking for some numerical methods to solve a set of coupled ODE, coupled in the sense that $$ \frac{d\alpha}{dt} = F\left( \alpha(t), \beta(t), \frac{d\beta}{dt}, t \right) $$ $$ ...
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0answers
11 views

Iterative method to find a solution

Here is the equation i tried to solve and only got one answer : $101.33=x.exp(1.045(1-x)^2).74.218 + (1-x).exp(1.045x^2).101.05$ And $exp(1.045(1-x)^2)=g1$ , $exp(1.045x^2)=g2$ Which they show ...
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0answers
19 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
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1answer
9 views

Adding two functions represented by a table of values with a different step size?

Let $f(t)$ be some numerically obtained $T$-periodic function represented by a table of values over one period or a set of points $(t, y)$ with a time step $\Delta t.$ Now let's change the ...
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1answer
50 views

Rounding unit vs Machine precision

I'm not sure if this question should be asked here... For a general floating point system defined using the tuple $(\beta, t, L, U)$, where $\beta$ is the base, $t$ is the number of bits in the ...
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1answer
24 views

on a characterization of convergent matrices

Let $A\in \mathbb R^{n\times n}$ a matrix. It's known that the following statements are equivalent: 1) $A$ is convergent, namely $\lim_{k\to\infty}(A^k)_{ij}=0$ 2) $\lim_{k\to\infty}||A^k||=0$ for ...
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2answers
28 views

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 ...
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0answers
17 views

Order of convergence Secant method

As we know, the order of convergence of the secant method is 1.618... But, when I try to find the convergence const of the function: f(x)=(1+x^2)^(1/3)-1, it's not convergence... it's going to ...
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1answer
24 views

eigenvalues lesser than 1 implies affine maps are eventually contractive

Consider $(\mathbb R^n,d)$ where $d$ is the Euclidean metric. A map $w:\mathbb R^n\to \mathbb R^n$ is said $\textbf{contractive}$ if there exists $0<s<1$ such that for every $x,y\in \mathbb R^n$ ...
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1answer
26 views

Determining the most appropriate set of eigenmodes for a modal decomposition of an experimental data set

I have a complex vector of the transverse amplitude and phase distribution of a laser beam, derived from experimental data. When modelling these field distributions, ordinarily the eigenmodes of the ...
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1answer
30 views

Understanding this trigonometric identity $\frac{n}{2} (2 \cos \frac{\alpha_n}{2} \sin \frac{\alpha_n}{2}) = \frac{n}{2} \sin \alpha_n$

I was looking an example that motivates rounding errors using the quadrature of a circle (?). And at a certain point there's this identity: $$\frac{n}{2} \left( 2 \cos \frac{\alpha_n}{2} \sin ...
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0answers
31 views

How to find the ground energy state solution in a quantum harmonic oscillator?

Recently, I came across a question which asks to solve the Schrödinger equation for a harmonic oscillator on $ [a, b] $ : $-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = ...
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0answers
24 views

Eigenvalue equation and the diffusion equation

I am running some finite-element software on Matlab that generates solutions to the diffusion equation over a punctured, rectangular domain. This is the same as the $k \times k$ matrix system ...