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

learn more… | top users | synonyms (2)

2
votes
2answers
425 views

What is the meaning of “integral point”?

While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about ...
3
votes
0answers
91 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
1
vote
2answers
566 views

Minimum number of iterations in Newton's method to find a square root

I am writing an algorithm that evaluates the square root of a positive real number $y$. To do this I am using the Newton-Raphton method to approximate the roots to $f(x)=x^2-y$. The $n^{th}$ iteration ...
1
vote
1answer
198 views

Interpolation- Barycentric coefficients for nodes that are Chebyshev points of the second kind.

So I came across the following theorem: If the interpolation node are Chebyshev points of the second kind given by : $$ x_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq 0) $$ Then the ...
0
votes
1answer
43 views

How to approximate a function to Inegrate

How can I solve the following integral? or how can I approximate the function to solve the integral? $$\int\limits_0^{2\pi} \sqrt{1 + (N\cdot A\cdot\sin(N\theta))^2}\mathrm{d}\theta$$
1
vote
0answers
43 views

How do i discretize following Elliptic PDE-problem?

Given that we are looking at heat conducted trough a rectangular metal block with the region $\Omega= \{ 0\leq x\leq 4, \ 0\leq y \leq 2 \}$ Following elliptic problem is formulated $\nabla^2T=0, \ ...
1
vote
1answer
56 views

$p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,…,x_k]$, prove that q has the same leading coefficient.

So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. ...
1
vote
0answers
48 views

Literature investigating root finding of convex Functions

I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to ...
0
votes
0answers
70 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
1
vote
0answers
59 views

Difference Equations and displacement operator

For a Prep exam Exercise from the book: Numerical analysis of scientific computing. Section 1.3-3 Let $p$ be a polynomial of degree $m$, with $p(0) \neq 0$. If a sequence $x$ contains $m$ ...
1
vote
1answer
50 views

Newton's method for a given polynomial

Let $f(x)=\frac{1}{5}x^5+\frac{1}{3}x^3+x-1$ Show that $f$ has only one zero $r$ in interval $(0,1)$ To find approximation of $r$ we apply Newton's method ...
1
vote
0answers
106 views

Applications of Numerical methods

I'm in a course of Numerical Methods and part of an assignment is find an article about an application of numerical methods, explain this article and present a program (in matlab/octave) that ...
1
vote
1answer
142 views

Error estimates for the trapezoid rule

Say we require $$ \left|\int_{a}^{b}f(x)dx - T_{n}\right| < 10^{-4} $$ where $T_{n}$ is the composite trapezoid rule with $n$ subintervals. To guarantee that $T_{n}$ satisfies this error bound, we ...
1
vote
1answer
139 views

Bisection method absolute error

I know that $\varepsilon \le 2^{-n-1}(b_0 - a_0)$, how to conclude from this that I need $n = \lfloor log_2{\frac{b_0 - a_0}{2\varepsilon}}\rfloor+ 1$? Using logs I get $ n \le ...
1
vote
2answers
224 views

Difference between Quadrature Rules and Riemann Sum

I am reading and learning about Quadrature rules. Would it be correct to say that a Riemann Sum method enters in the category of quadrature rules? I am asking because the Riemann sum evaluates the ...
2
votes
1answer
818 views

Derive the error term of Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $I_f = \int_{a}^{b}f(x)dx$ is based on linear interpolation of $f$ at $x_0=a$ and $x_1 = b$. The Simpson rule is likewise based on quadratic polynomial ...
0
votes
2answers
58 views

Newton method to find $\frac{1}{\sqrt{a}}$ [duplicate]

What function should I use to find $\frac{1}{\sqrt{a}}$ without using division?
1
vote
2answers
31 views

Reverse of number (numerical)

You can find reverse number $R$ using $x_{n+1} = x_n(2 - x_n \cdot R) \ \ $ where $ n = 0,1,..$ Prove it using Newton method for finding $0's$ of some function $f$ Anyone have idea what that ...
0
votes
1answer
36 views

Compute $p \in P_2$ that minimizes $||x^{3}-p||$ in the '2'-norm

So far I've got an orthogonal base with: $\psi_0 = 1$, $\psi_1 = x$ $\psi_2 = x^{2}-\frac{2}{6}$ Am I supposed to calculate $p$ as: $\alpha_0\psi_0+\alpha_1\psi_1+\alpha_2\psi_2$ with: $\alpha_i ...
1
vote
1answer
50 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
0
votes
2answers
51 views

Help with numerical analysis proof

Let $u$ be a nonzero vector in $\mathbb{R}^n$, and define $\gamma=\frac{2}{||u||_2^2}$ and $Q=I-\gamma uu^T$. Prove Q is a reflector satisfying A) $ Qu=-u$ B) $Qv=v$ if $<u,v>=0$ My ...
2
votes
0answers
155 views

Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$ \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$ Applying the Woodbury matrix identity ...
0
votes
1answer
38 views

numeric differentiate: show that the relative mistake can be at 100%

i have $f(x) = x+1$, a physical size, and the values $\tilde{f}(x_i)$ are measured at equally spaced points $$x_i=ih, \qquad 0 \leq i\leq 10^3, \qquad h=10^{-3},$$ with a maximum relative mistake of ...
1
vote
0answers
66 views

Issues with quasi Newton method convergence

I have this issue with the convergence of the quasi newton method. I have a convex objective function which I need to minimize wrt some parameters. I generated some synthetic data using a defined ...
2
votes
1answer
26 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
1
vote
0answers
351 views

4th order Runge-Kutta for Coupled ODEs

here is my code: ...
0
votes
1answer
35 views

experimental sequence of number

I'm doing a small numerical experiment. I got, from the first simulations, the following sequence of numbers. I'm trying to imagine a mathematical law behind this sequence. It could be a geometric ...
0
votes
1answer
346 views

Application of the Backward Euler method to the DE …

I'm having trouble solving this question below and would like to have some help: Apply the Backward Euler method to the differential equation: $y' = -20y + 20\cos (t) - \sin (t)$, $0\leq t\leq 2$, ...
3
votes
1answer
77 views

Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$.

I want to Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$. Where $A$ is a matrix. I have a proof that involves Jordan Blocks. The proof is long and involved but it not ...
0
votes
1answer
169 views

Runge-Kutta-Fehlberg Method Problem

Suppose all infected individuals remained in the population to spread the disease. A more realistic proposal is to introduce a third variable $z(t)$ to represent the number of individuals who are ...
1
vote
1answer
53 views

an estimate for condition number: $\kappa(C^{-1}A)\leq \kappa(C^{-1}B)\kappa(B^{-1}A)$

I'm currently reading through "Domain Decomposition Methods" by Tosseli and Widlund and in the appendix I found the following Theorem: Let A, B, C be symmetric positive definite matrices. Let ...
4
votes
2answers
231 views

Using Newton's method calculate $ \frac{1}{\sqrt{a}} $ without division

Suggest algorithm for the numerical calculation $ \frac{1}{\sqrt{a}} \ a > 0 $ without division, use Newton's method. My idea is: $$ \frac{1}{\sqrt{a}} = (\sqrt{a})^{-1} = a^{-\frac{1}{2}}$$ $$ x ...
0
votes
1answer
25 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
1
vote
1answer
1k views

Water flows from an inverted conical tank with a circular orifice

I'm having trouble with a question for a Numerical Analysis class. It's a textbook question, but I can't seem to wrap my head around it. I've viewed over the Runge-Kutta method in my book, but I have ...
1
vote
1answer
631 views

Natural Cubic Spline 3 points

I am trying to do a natural cubic spline but I'm having trouble. f(-.0247500)=-.5, f(.3349375)=-.25, f(1.101000)=0 I tried doing the matrix, Ax=b where, h0=h1=.25 an a0=-.0247500, a1=.3349375, ...
1
vote
1answer
137 views

Can we effectively solve a system of polynomial equations (numerically).

It is known, and simple, that roots of a one polynomial in one variable $f(x)$ can be found numerically. If we restrict to the reals, one can readily give an explicit bound for any zero of $f$ in ...
0
votes
1answer
57 views

Dyadic rationals representation in the computer memory.

So I know what a dyadic rational is and that is represented with a finite number of binary digits in memory. The idea is that I want an explanation why is it represented in this way ? Thank you.
3
votes
1answer
92 views

Why below sequence is diverge?

This problem maybe simple for you,but i dont know that why below sequence is diverge?please help me about this: why $‎\lbrace\mid x_{k}\mid‎\rbrace$ with below definition is diverge? $x_{k+1}:= ...
1
vote
1answer
30 views

Estimating error in calculation

I'd like somebody to verify my solution of the following problems: Let's assume, that float arithmetics $fl()$ has precision $\nu$ for standard operations $(+\ -\ \cdot \ \div)$. a.) Estimate ...
0
votes
1answer
84 views

Gaussian Quadrature - derivation problem

Derive a formula of the form $$\int_{a}^{b}f(x)dx=w_0f(a)+w_1f(b)+w_2f'(a)+w_3f'(b)$$ that is exact for polynomials of the highest degree possible. Not sure how to derive this formula. Possibly ...
1
vote
1answer
150 views

Numerical integration of functions sampled with constant and non-constant rates

I have arbitrary functions of time $y(t)$ (sin, linear, sawtooth, triangular, etc) with two types of data sampling, namely constant sampling rate and variable sampling rate (in some parts the function ...
0
votes
0answers
780 views

Exact inversion of matrix complexity (by Gaussian elimination)

I would like to check if what I have done is correct. Please, any input is appreciated. Problem statement: Consider a non-singular matrix $A_{nxn}$. Construct an algorithm using Gaussian elimination ...
1
vote
1answer
38 views

Show, that $ \displaystyle \sum_{k=0}^n \lambda_k(0)x_k^j = \begin{cases} 1 \ (j=0)\\0 \ (j=1,2,\ldots n)\end{cases} $

Show, that $ \displaystyle \sum_{k=0}^n \lambda_k(0)x_k^j = \begin{cases} 1 \ (j=0)\\0 \ (j=1,2,\ldots n)\end{cases}$, where $\lambda_{k}$ is the 'helper' polynomial from Langrange Interpolation ...
0
votes
1answer
35 views

Determine the spacing $h$ of the x-coordinates so that the error of linear interpolation for this data is $10^{-3}$.

We want to construct a table of equispaced values to $f(x)=sin(x)cos(x)$. Determine the spacing $h$ of the x-coordinates so that the error of linear interpolation for this data is $10^{-3}$. I know ...
1
vote
1answer
57 views

Evaluation of Standard Normal Integral

I have always wondered how we calculate the percentiles of the Standard Normal Distribution given that the CDF cannot be obtained in closed form: $$F(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} ...
0
votes
2answers
38 views

Show that $(\phi_{n}^{(n)})^{-1}= -(\sum_{i=0}^{n-1}(\phi_{i}^{(n)})^{-1})$

So I have $n+1$ points $x_{0},x_{1},...,x_{n} \in \mathbb{R}$ and a following quasi-function: $\phi_{j}^{(n)}=\prod_{i=0,i \neq j}^{n}(x_{j}-x_{i})$ Show that $(\phi_{n}^{(n)})^{-1}= ...
3
votes
2answers
149 views

Approximation of pi

Given that $\frac{\pi^2}{6}=\sum_{n=1}^{\infty}\left(\frac{1}{n^2}\right)$, I have to write a program in C that finds an approximation of $\pi$ using the formula ...
0
votes
0answers
48 views

Numerically solving first order system of ODES

I am currently trying to numerically solve a first order system of ODES, of the form: $$\frac{d\rho}{ds}(s)=f_1(\rho,\theta,k_1,k_2)$$ $$\frac{d\theta}{ds}(s)=f_2(\rho,\theta,k_1,k_2)$$ ...
0
votes
0answers
31 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
0
votes
2answers
115 views

Calculating Integral via Hermite

Using the "First Hermite Approach" I have to calculate: $$\int \dfrac{x^3+x^2+x}{x^4+2x^2+1} \mathrm dx $$ I really have no idea how to use this so-called First-Hermite-Approach. I wanted to solve ...