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)

1
vote
2answers
10k views

Solving coupled 2nd order ODEs with Runge-Kutta 4

I'm having a hard time figuring out how coupled 2nd order ODEs should be solved with the RK4 method. This is the system I'm given: $x'' = f(t, x, y, x', y')$ $y'' = g(t, x, y, x', y')$ I'll use the ...
1
vote
1answer
2k views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
1
vote
1answer
440 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
0answers
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 that ...
1
vote
2answers
127 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: $d_{k+1}=u(t_{k+1})-\underset{u_{k+1}}...
1
vote
0answers
218 views

Reducing or avoiding the Gibbs phenomenon.

What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms. This could mean pre-processing or post-processing or altering the transform. With ...
1
vote
1answer
212 views

Functional Iterations (numerical analysis)

(I'm opening a new post because the claim has been found in another book and now I have an exact description) I wish to prove this claim I found in a numerical analysis book : Assume that $\varphi(x)...
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 ...
0
votes
1answer
3k views

estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
0
votes
2answers
34 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
0answers
44 views

Adams-Bashforth-Moulton two-step predictor-corrector

Can someone help me this question please, this is from past years exam.
0
votes
2answers
142 views

First-degree spline interpolation problem

I have been spending hours working on a problem from Cheney & Kincaid on first-degree spline interpolation problem but am heading nowhere. Any explanation, pointers or hints from you would be bery ...
0
votes
2answers
777 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
1answer
57 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 ...
-2
votes
1answer
68 views

How was this formula for differentiation derived?

[![enter image description here][1]][1]Please tell me how this formula for numerical differentiation derived. I think it has something to do with Vandermonde Matrices but I am not quite sure how to go ...
-2
votes
2answers
37 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.
79
votes
24answers
14k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never ...
22
votes
2answers
607 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 ...
30
votes
1answer
831 views

What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
9
votes
3answers
3k 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?
10
votes
4answers
295 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 ...
9
votes
2answers
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 ...
25
votes
1answer
532 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
12
votes
7answers
4k 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 ...
11
votes
2answers
785 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, ...
8
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 ...
5
votes
1answer
357 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 ...
10
votes
1answer
640 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
8
votes
1answer
471 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 $$...
3
votes
3answers
1k 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
11k 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 $\...
2
votes
2answers
23k 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 ...
11
votes
1answer
813 views

Bounding the basins of attraction of Newton's method

In general, Newton's method for root finding has a "bubbly" boundary between basins of convergence for different roots. This is where fractals are usually created from. But outside these "bubbly" ...
8
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 ...
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 ...
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 $2500\times 2500$. ...
6
votes
1answer
921 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 ...
5
votes
1answer
206 views

How to compute the pade approximation?

Like $\log(1+x)$? Is there any algorithms? I have read many materials but doesn't have an idea
5
votes
1answer
922 views

Kullback-Leibler divergence based kernel

I'm looking to paper "A Kullback-Leibler Divergence Based Kernel for SVM Classification in Multimedia Applications". Author suggest to use kernel function for two distributions $p$ and $q$: $k(p,q)= \...
3
votes
1answer
191 views

Singularities of an integral

We have the integral : $$I(t)=-i\int_0^\infty \frac{\log\left[\frac{\sin(t\log\sqrt{1+ix})}{\log(1+ix)} \right ]-\log\left[\frac{\sin(t\log\sqrt{1-ix})}{\log(1-ix)} \right ]}{e^{2\pi x}-1} \, dx$$ I ...
1
vote
2answers
79 views

When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $...
10
votes
3answers
6k views

fast algorithm for solving system of linear equations

I have a system of linear equations, $Ax=b$, with $N$ equations and $N$ unknowns ($N$ large) If I am interested in the solution for only one of the unknowns, what are the best approaches? for ...
9
votes
1answer
462 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 ...
8
votes
3answers
258 views

Exact result of a series using Euler-Maclaurin expansion.

This is a variant of Exercise 64 in Chapter 9 of concrete mathematics. Prove the following identity \begin{equation} \sum_{n = -\infty}^{\infty}' \frac{1 - \cos( 2\pi n k )}{n^2 } = 2 \pi^2 ( k - k^...
7
votes
1answer
166 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(W_t)_{t\...
7
votes
2answers
212 views

What is the general definition of time-scale for a differential equation?

For a differential equation like $\dot{x}=ax$ where $x$ is a function on an independent variable $t$, and $\dot{x}=\frac{dx}{dt}$, and $a$ is a constant, we define the time-scale $\frac{1}{a}$, which ...
7
votes
3answers
720 views

Parallel lines divide a circle's area into thirds

When I was young I came up with a geometry problem and drew it in a notebook: Suppose we have a circle with radius $r$ and area $A$. Let two parallel lines be equidistant from the center of the ...
6
votes
3answers
116 views

Digits of $\pi$ using Integer Arithmetic

How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as ...
5
votes
1answer
764 views

Efficiently calculating the logarithmic integral with complex argument

My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
5
votes
1answer
544 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...