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

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1answer
296 views

Clarification when using the Bisection method

I understand how the Bisection method works: you take an interval and test the end-points and the mid-point. Somewhere in those intervals, there will be a root. You keep swapping the mid-points and ...
1
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1answer
152 views

convergence / fixed point method

Any help with the following: Problem: Consider the fixed point problem: $x=f(x)$ and given: $x_{n+1}=\frac{n}{n+1}f\left ( x_{n} \right )$. If $x_{0}$ is a fixed point where $\left | f^{'}\left ( ...
4
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2answers
419 views

Numerical analysis textbooks and floating point numbers

What are some recommended numerical analysis books on floating point numbers? I'd like the book to have the following In depth coverage on the representation of floating point numbers on modern ...
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1answer
340 views

Strange behaviour with ode45 in matlab, probably some numerical error.

I'm having a problem with a parameter estimation in a non-linear model. I think the culprit is that ode45 (an ode solver in matlab) is not properly solving my ode. It's the in red highlighted part, ...
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0answers
334 views

Problem with Newton's Method in solving a System of Equations

I'm trying to use Newton's method to solve the following system of equations, where f and g are functions of x and y. (h,a,f,c,d,b and k are just constants). $f(y,x)=\left[\begin{array}{c} y^{1}\\ ...
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0answers
145 views

integral of the sum

I am trying to integrate the following sum. I need to get at least first 5 terms (using math or computer). I've tryed wolfram alpha online-did not work. I should find $$ \int_0^\infty ...
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1answer
162 views

Cubic spline: Help understanding Wikipedia markup

On the cubic hermite spline Wikipedia page, the formula for interpolating between $x_k$ and $x_{k+1}$ is given by ...
4
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1answer
70 views

How to safely solve a pair of elementary equations in a floating point computing system?

I wrote a simple short computer program to solve a pair of equations of the format , $y = a1 * x + b1$ $y = a2 * x + b2$ . But , it outputs clearly wrong answers sometimes when $abs(a1)$ or ...
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1answer
115 views

Non-analytical equation

Suppose I have integrated a differential equation numerically using some time-interval $\delta t$. How might I determine the accuracy of the result? Does solving simultaneous equations reduce the ...
1
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1answer
121 views

on the number of roots of harmonic polynomials

Maybe it is not a standard term, a harmonic polynomial is $h_{n,m}(z,\bar{z})=f_n(z)+\overline{g_m(z)}$ where $f$ and $g$ are polynomials in $z\in\mathbb{C}$ of degrees $n,m$ respectively.We may ...
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0answers
60 views

reference request

Can anybody help me to find the books on numerical solutions of partial differential equations including examples on irregular geometry (specially books or links on matlab code examples in this case)? ...
2
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1answer
233 views

Is the Hessian of the 'generalized least square function' positive semi-definite?

Let $F:\mathbb{R}^n\rightarrow \mathbb{R}$ a scalar field which has a quadratic from, $$F(\mathbf{x}) = \frac{1}{2}\mathbf{d}(\mathbf{x})^\top\Lambda\mathbf{d}(\mathbf{x})$$ with ...
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5answers
1k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
3
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2answers
480 views

Computing a volume (area) of intersections

The task should be very common, what are the best and easiest to implement algorithms to compute the volume of union/intersection of given bodies? Or union/intersection area for 2D figures. I don't ...
0
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1answer
481 views

Could explain me how eigenvector helps with compute gradient and how make differentiate operation on decrete space like digital image?

Could you explain me how eigenvector helps with compute gradient and how make differentiate operation on descrete space like digital image? I know that this question is so connected with computer ...
2
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3answers
961 views

Newton's method and trig functions on a computer

I'm trying to use Newton's method to find roots for the function $A \cos(\Theta_2 - \Theta_1) + B \sin(\Theta_1)$. (That is, iterate $x_{i+1} = x_i - f(x_i) / f'(x_i)$). I've got a working ...
2
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1answer
367 views

Failing to check if a number is a perfect square

To check if a natural number is a perfect square, programming in Python, I check if int(math.sqrt(n))==math.sqrt(n), id est if the decimal part of the square root is zero. Here my question: are there ...
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1answer
125 views

Expanded form of Divided differences

I am studing numerical analysis and I'm reading In wikipidia about it. I want to understand the Expanded form - I tried to prove this($f[x_0,\ldots,x_n]=\ldots$) by induction and failed. Any Ideas on ...
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1answer
1k views

Backward Euler for a system of equations

I have tried to solve a system of equations in the form: $$\eqalign{ {dy_1\over dt }&= y_1 +dt*f(y_1,y_2)\cr {dy_2\over dt} &= y_2 +dt*g(y_1,y_2) } $$ using different schemes such as ...
2
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1answer
117 views

Divided differences of $x^n$

Denote $f(x)=x^n$ ($n$ is natural), I am trying to prove that $f[x_0,x_1,\ldots,x_n]=1$, $\{x_i\}$ are distinct $n+1$ real numbers. I tried doing this by finding an interpolation of the function and ...
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2answers
2k views

Clarifying the definition of “unstable”

I would appreciate a definition clarification. if a numerical method is "unstable", does it mean that if we introduce a small random error in one of the steps, the error would be magnified greatly ...
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1answer
120 views

Algorithm to approximate roots using blackbox root counting function

I am looking for an algorithm to do the following. Given: An interval $[a,b]$. A black box function $n(x)$ which returns the number of roots (zeros) of a function that lie to the left of $x$. That ...
3
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1answer
51 views

How can we show that $x_{n+1}=f(x_{n})$ will converge if $|f'(x)|\leq\lambda<1$ on the interval $[x_{0}-\rho, x_{0}+\rho]$?

It is an exercise from Kincaid and Cheneys's book. How can we show that $x_{n+1}=f(x_{n})$ will converge if $|f'(x)|\leq\lambda<1$ on the interval $I=[x_{0}-\rho, x_{0}+\rho]$ where $\rho = ...
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0answers
1k views

How to use Richardson extrapolation on Euler's method with step size h and h/2 in order to derive the modified Euler method?

Suppose that we want to numerically solve the initial value problem $x'(t)=f(x,t), x(0)=x_0$. The modified Euler's method $$x(t+h)=x(t)+hf(t+\frac{1}{2} h,x(t)+\frac{1}{2} hf(t,x(t)))$$ My ...
8
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1answer
1k views

Using the Arnoldi Iteration to find the k largest eigenvalues of a matrix

I'm trying to obtain a general understanding of this algorithm which determines the k-largest eigenvalues of a matrix $A\in \mathbb{R}^{n\times n}$. How I see it: power iteration: take random ...
0
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1answer
235 views

functional iteration

This is a question from D Kincaid & W Cheney, Numerical Analysis (3ed), Brooks-Cole 2002; Find the condition on $\alpha$ to ensure that the iteration will converge linearly to a zero of $f$ if ...
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1answer
1k views

What is the fastest numeric method for determinant calculation?

I have a C++ matrix class which can do the following operations on a square matrix related to determinant calculation: LU Decomposition Calculation of eigenvalues Calculation of determinant by ...
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0answers
270 views

Second order central difference of the Nth order

I'm trying to find some tabulated data in some big-and-smart-book with regards to second order central difference of a function of just one variable: f''(x). I did find formula for 7th order [1], but ...
5
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2answers
2k views

Newton's method for roots of multiplicity $> 1$

I'm studying numerical analysis and I read in Wikipedia that "If the root being sought has multiplicity greater than one, the convergence rate is merely linear". The article in Wikipedia also ...
2
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1answer
579 views

Trouble Understanding Newton-Raphson Iterative Method

Currently i am reading this page which discusses the newton-raphson method of approximating the roots of an equation. It says given a function $f$ over the reals $x$, and its derivative $f$,we begin ...
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1answer
135 views

Improper integrals over infinite ranges.

I'm trying to understand how to numerically integrate over infinite ranges using the composite trapezoidal rule, like in this book. Namely, I am having the problem with the series starting with ...
1
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1answer
157 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 ...
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0answers
39 views

Functional Iterations and order of convergence [duplicate]

Possible Duplicate: Functional Iterations (numerical analysis) Let F be a function F:R->R and Let x0 be (any) real number. we define : x_n+1=F(x_n). (n>=0) The book I'm reading on ...
3
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2answers
148 views

Fast inversion of a triangular matrix

I need to inverse a matrix $A$ given its $QR$ decomposition. It's a numerical task. It is told that the inversion should be "possibly cheap". But it does not look like I can do something more ...
1
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1answer
943 views

Trapezoidal Rule (Quadrature) Error Approximation

I'm trying to prove the error bound from the classical trapezoidal rule integral approximation, which states the error is $-(b-a)^3f''(c)/12n^2$ for some $c$ within the limits of integration. ...
2
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2answers
674 views

Multivariate function interpolation

I have a (nonlinear) function which takes as input 4 parameters and produces a real number as output. It is quite complex to compute the function value given a set of parameters (as it requires a very ...
4
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1answer
412 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
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1answer
91 views

Solution of a system of second order algebraic equations in complex numbers

What is the simplest solution for this set of equations: $ \sum_{i=1,3,5,..}^{N-1} \left | x_i \right |^2=c_1,\ \sum_{i=2,4,6,..}^{N} \left | x_i \right |^2=c_2,\ \sum_{i=1,3,5,..}^{N-1} ...
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0answers
177 views

Double integration involving polynomial functions and sinc function

I encountered a problem which I can't seem to simplify/solve. I was wondering if any mathematicians or specialists knows how to approach this problem? $$\int^{0.5}_{-0.5} \int^{0.5}_{-0.5} \; ...
2
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1answer
528 views

Monte Carlo - Control Variates & Antithetic method

Supposing $g(x)=\sqrt[3]{x}$, I want to calculate the expected value of g, $E(\sqrt[3]{x})$, using Monte Carlo method, by generating $x_i$ from a Weibull distribution with parameters $(1,5)$. After ...
3
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2answers
137 views

Accuracy of solution?

I have a question that asks me to get the solution to the equation $x+\arcsin(x)={\pi\over 2}$ by using a calculator. (Repeatedly pressing cos?) Then it asks to justify the accuracy of the answer. ...
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0answers
1k views

Understanding Power Method/Inverse Iteration in Linear Algebra

For a linear algebra class, we are currently learning about finding the largest/smallest eigenvalues of a matrix using the power method and inverse iteration methods. I just want to make sure that I ...
8
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2answers
566 views

A function for which the Newton-Raphson method slowly converges?

I'm doing a MATLAB assignment in which you work out and implement a better version of Newton-Raphson using a second degree Taylor polynomial instead of a first degree one. I have the algorithm worked ...
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2answers
838 views

Library for Jacobi eigenvalue algorithm [closed]

I am looking for a C or C++ or fortran library that implements the Jacobi eigenvalue algorithm: http://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm do you know if it is available?
2
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2answers
882 views

Modified Cholesky factorization and retrieving the usual LT matrix

I have been looking at the modified Cholesky decomposition suggested by the following paper: Schnabel and Eskow, A Revised Modified Cholesky Factorization Algorithm, SIAM J. Optim. 9, pp. 1135-1148 ...
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1answer
66 views

Does the solution to $u_t=-uu_x+0.1u_{xx}$ decay in time?

Consider the following PDE: $$ \begin{align} &u_t+uu_x=0.1u_{xx},\qquad 0<x<1,t>0\\ &u(x+1,t)=u(x,t),\qquad t\geq 0 \\ &u(x,0) = \sin 2\pi x,\qquad 0\leq x\leq 1 \end{align} ...
3
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1answer
336 views

Why Monte Carlo integration is not affected by curse of dimensionality?

What is the common sense explanation behind that fact that MC integration is free of "curse of dimensionality" in contrast to deterministic integration rules (e.g. trapezoidal rule)?
3
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0answers
900 views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...
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2answers
106 views

High order methods for solving ODEs

I would like to know about really high order methods for solving ODEs. Say of order 30 and higher. What are they? Any surveys/reviews?
5
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1answer
362 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. ...