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

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3
votes
1answer
515 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow ...
3
votes
3answers
3k views

what is “kink”?

Pleas tell me that what a "Kink" is and what this sentence means: Distance functions have a kink at the interface where $d = 0$ is a local minimum.
2
votes
1answer
43 views

Partial Derivative of Newton's Divided Difference

Let $x_0<x_1<...<x_n$, and let $f$ be continuously differentiable. Show that $$ \frac{\partial}{\partial_{x_i}} f[x_0,x_1,...,x_n]=f[x_0,x_1,...,x_i,x_i ,x_{i+1},...,x_n] $$. I have the ...
2
votes
3answers
145 views

Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods

I have to calculate approximations of the solution with the method $$ y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0 $$ for various ...
2
votes
1answer
65 views

Is the following scheme for generating $p_n=(1/3)^n$ stable or not. $p_n=(5/6)p_{n-1}-(1/6)p_{n-2}$.

Is the scheme for generating $p_n=(1/3)^n$ stable or not? $$p_n= \frac{5}{6} p_{n-1} - \frac{1}{6}p_{n-2}.$$
2
votes
1answer
497 views

Quick way of finding the eigenvalues and eigenvectors of the matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$

Matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$ has the eigenvalue: $\lambda_i=\alpha-2\cos(i\theta),$ $i=1,\dots,n$ and the corresponding eigenvectors are: ...
2
votes
1answer
647 views

butcher tableau runge kutta methods

Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me Write the Butcher Tableau for the 1-stage $\theta$ method: $$U^n ...
1
vote
1answer
36 views

Runge-Kutta methods: step size greater than 1

The local truncation error of Runge-Kutta 4 is $O(h^5)$, while those of RK1 is $O(h^2)$. I wonder then what happened when the step-size of RK methods is greater than 1: will the accuracy be improved ...
1
vote
1answer
202 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
1
vote
2answers
98 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
1
vote
2answers
42 views

Taylor Series approximation

Let $f(x) = (1-x)^{-1}$ and $x_0=0$. Find the $n$-th Taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$. Find a value of $n$ necessary to approximate $f(x)$ within $10^{-6}$ on $[0,0.5]$. I am ...
1
vote
2answers
396 views

Method of False Position (Regular Falsi) - Pros/Cons

Could anyone provide and explain some drawbacks and benefits of the method of false position against say newtons method. I know one of benefits is that it doesn't require the derivative and one of ...
1
vote
1answer
166 views

Non-negative matrix factorization with a regularization term

Given a matrix $X_{M*N}$ with non negative values, I would like to factorize it into $U_{M*K} , V_{N*K}$, where both of them have also non-negative values. Additionally I would like to have a minimum ...
1
vote
1answer
214 views

How to make recursive computation of $I_n=\int_0^1 \frac{x^n}{x+\alpha}\,dx$ stable?

This is homework, so please only hints. We see that the recurrence $$I_n=\frac{1}{n}-\alpha I_{n-1}$$ can be used to compute the value of the integral $I_n=\int_0^1 \frac{x^n}{x+\alpha}\,dx$ ...
1
vote
0answers
167 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
1
vote
2answers
859 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
1k views

How to fit non-linear matlab data?

I'm working on a problem in scientific computing namely fitting data to this equation $c(z) = 4800 + p_1 + p_2 \cdot z/1000 + p_3 \cdot e^{ -p4 \cdot z/1000}$ The data is in a background question ...
1
vote
1answer
2k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
1
vote
3answers
1k views

Computing the square root function with Newton's method

Show that Newton's method can be used to compute the square root function $\sqrt a$ using the formula $$x_{n+1} = \frac{1}{2}\left(x_{n} + \frac{a}{x_{n}}\right)$$ show that the error is $$\sqrt a ...
1
vote
1answer
387 views

Chebyshev polynomial question

Consider the Chebyshev polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined by $$T_0(x) = 1; \quad T_1(x) = x; \quad T_n(x) = 2x\cdot T_{n−1}(x) − T_{n−2}(x)$$ for $n = 2, 3, \ldots$. ...
1
vote
1answer
768 views

existence and uniqueness of Hermite interpolation polynomial

What are the proofs of existence and uniqueness of Hermite interpolation polynomial? suppose $x_{0},...,x_{n}$ are distinct nodes and $i=1 , ... ,n$ and $m_{i}$ are in Natural numbers. prove exist ...
1
vote
0answers
334 views

Integrating pde backwards in time

I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation $$ \begin{align} p_\text{new} & = p_j + ...
0
votes
1answer
47 views

Convergence of the fixed point iteration for sin(x)

Sin(x) converges at x=0 as can be seen with a graphical illustration. When attempting the fixed-point iteration for sin(x) on a computer, the number gets closer and closer to zero but it does not ...
0
votes
2answers
32 views

Find the first order system of linear equations

Regard the diff equation: $mϕ′′+aϕ′+(mg/L)ϕ=0$ $ϕ(0)=0.1$ $ϕ′(0)=0$ where $m=0.1,L=1,a=2,$ 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ...
0
votes
1answer
57 views

Dilating sinusoidal period at certain abscissa $x$?

Lets take a look at this function : $$f(x) = \sin\left(\frac{\pi x }{2}\right)$$ when $x$ tends to $1$ this functions get closer to $1$ by bigger values now look at this one : ...
0
votes
2answers
44 views

We have x=0.2234 and Δx =0.2*10^-3.Find how many accurate digits does the number have?

So we know that $Δx≤(1/2)*10^{-n}$ where n is the number of the accurate digits. Now I just have to replace $Δx =0.2*10^-3$=$(1/2)*10^{-n}$ and find n. But why have I been given x=0,2234 since I dont ...
0
votes
2answers
129 views

finite elements-exercice

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
0
votes
2answers
694 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
2answers
839 views

Newton's method - determine accuracy in calculation

I have almost managed to solve a problem (I think), but I am a bit unsure if my procedure is correct, and my answer is not quite the correct one. Would appreciate any input! The problem is as ...
-2
votes
2answers
33 views

Use taylor series to arrive at the expression f'(x)=1/h[-3*f(x)/2+2f(x+h)-f(x+2h)/2]

I'm not really sure how to go about this.. any help is appreciated.
22
votes
2answers
580 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
8
votes
3answers
2k views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
12
votes
7answers
3k views

How to derive a function to approximate $\sqrt{3}$?

The problem is to used fixed-point iteration method to find an approximation to $\sqrt{3}$. The equation from the book is $f(x) = \dfrac{1}{2}\left(x + \dfrac{3}{x}\right)$. It make senses to me ...
10
votes
4answers
288 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x^{(n)} dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1)\: dx$$ to five significant digits. I've used ...
7
votes
2answers
2k views

Numerical method for finding the square-root.

I found a picture of Evan O'Dorney's winning project that gained him first place in the Intel Science talent search. He proposed a numerical method to find the square root, that gained him $100,000 ...
11
votes
2answers
760 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
9
votes
2answers
1k 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
3answers
858 views

Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
7
votes
1answer
392 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $D$ be the diagonal matrix w/alternating in sign diagonal entries: $$D_{kk}=(-1)^{k+1}\tan(\frac{k\pi}{2n+1}),$$ where $k=1,2,\dots n\in N$, and let $B$ be the $n$ by $n$ square $(0,1)$-matrix ...
7
votes
1answer
2k views

Numerical methods book

I'm looking for an introductory book on numerical methods. I'm beginning to learn to program (in Haskell, a functional language, if that would affect the recommendations). The reason I want such a ...
2
votes
2answers
19k views

How to solve simultaneous equations using Newton-Raphson's method?

I understand how to find roots of a polynomial equation programmatically using Newton-Raphson method as explained here. How to find the values of $x$ and $y$ from the simultaneous equation given ...
9
votes
1answer
454 views

Is there a binary spigot algorithm for log(23) or log(89)?

The Bailey-Borwein-Plouffe formula yields a binary spigot algorithm for π, and related formulas give the bits of log(2) and those of the logarithms of some other integers. I got stuck (over a year ...
7
votes
1answer
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
6
votes
1answer
158 views

Please, help to identify this numerical constant

I'm trying to find an answer to this question. Let $K(k)$ be the elliptic integral of the first kind and $K'=K(\sqrt{1-k^2})$. According to Abel's theorem (see this link) we know that if ...
6
votes
1answer
9k views

Convergence rate of Newton's method

Let $f(x)$ be a polynomial in one variable $x$ and let $\alpha$ be its $\delta$-multiple root ($\delta\ge2$). Show that in the Newton's $x_{k+1}=x_k-f(x_k)/f'(x_k)$, the rate of convergence to ...
6
votes
1answer
874 views

Finding all roots of polynomial system (numerically)

I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton ...
6
votes
1answer
3k views

how to diagonalize a large sparse symmetric matrix, to get the eigenvalues and eigenvectors

How does one diagonalize a large sparse symmetric matrix to get the eigenvalues and the eigenvectors? The problem is the matrix could be very large (though it is sparse), at most 2500*2500. Is there ...
5
votes
1answer
321 views

Conjugate Gradient Method and Sparse Systems

What is it about conjugate gradient that makes it useful for attacking sparse linear systems. Why would steepest descent be significantly worse? Please keep in mind that I am still trying to fully ...
5
votes
1answer
631 views

How should a numerical solver treat conserved quantities?

Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers? On the one hand, one can observe ...
5
votes
3answers
3k views

rewriting to avoid catastrophic cancellation

why is rewriting $x^2 -y^2$ as $(x+y)(x-y)$ a way to avoid catastrophic cancellation? We are still doing $(x-y)$. Is it because the last operation in the second form is a multiplication?