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

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96 views

How can I use rationalized haar functions to solve differential equations?

How can I use rationalized haar functions to solve differential equations? I do not have any idea what are haar functions and how to use them please help!
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1answer
85 views

Convergence in $L^{\infty}((0,T),L_{loc}^1(R))$

I studied a theorem, which says, that under certain conditions, using a numerical method, we can choose a subsequence, so that this subsequence converges to u(x,t) (the solution). The convergence is ...
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173 views

The existence of real SVD

I have known that if matrix A $\in C^{m*n}$,then there exists a SVD $A=U\Sigma V$.My question is if A is real,does there exist SVD which U and V real?
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0answers
207 views

Practical defn of “Order of Convergence”

Given an algorithm producing the sequence of iterates $\{x_n\}$ converging to some value $x$, it is customary to express the manner in which the sequence converges using the concept of the order of ...
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1answer
65 views

variant eigenvector problem

I have the following problems when solving a linear equation. Let $A=(a_{i,j})_{n \times n}$ be a non-negative matrix with $a_{i,j} \in (0,1)$, and let $0<r<1$ be a scalar. Now we define a ...
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1answer
69 views

Modification of the continuous time Lyapunov Equation

For my research I have been working with different continuous-time Lyapunov equations of the form \begin{equation} M R + R M^\text{T} = G \end{equation} where all matrices are real and $n\times n$ ...
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2answers
555 views

Fourier transform of heat equation

I need to solve following partial differential equation with Fourier transform numerically. $ \frac{\partial T}{\partial t} = \nabla(c\nabla T) $ where T is temperature, c heat conductivity and t is ...
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0answers
75 views

A basic question about randomly generated matrix

I have read in many research papers related with iteration methods to find the generalized inverses. Where to show efficiency of the methods randomly generated matrices of higher order have been ...
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0answers
216 views

Solving nonlinear ODE system with continuous Newton method

Is anyone knows how to apply continuous Newton method for solving nonlinear ODE systems? For example let the system is: $$\left|\begin{array}{cc} y'= \frac{z}{x} \\ z'= ...
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1answer
514 views

How to determine stopping criteria in iterative methods?

I am reading an iterative methods to compute generalized inverses of a matrix. Note that generalized inverse of a matrix $A$ is a matrix $X$ which satisfy $AXA = A$ I am using matlab program to code ...
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1answer
409 views

Efficiently estimate a 2D integral from irregularly sampled, limited data

I have measured data of the following form: $f(3.2, 2.5) = 10$ $f(3.7, 2.6) = 9$ $f(3.1, 2.8) = 9.1$ (etc)... That is, I know $f(x, y)$ for certain irregularly spaced values of $x$ and $y$. I ...
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1answer
250 views

Numerical derivative of sensor data in the presence of noise

I have a sensor producing bandlimited data at a predictable periodic rate, corrupted by IID white noise (at least over relatively short periods of time). There is also a slowly time-varying bias, ...
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2answers
1k views

Quadratic Equation Error

For the floating point system $(B, t, L, U) = (10,8,-50,50)$ and for the quadratic equation: $ax^2 + bx + c$, I need to show error that arises in various cases and how to fix those. $$a=10^{-30}$$ ...
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1answer
190 views

Numerical Analysis

I am trying to determine some numerical difficulties that arise from a couple problems, and a good way to re-write them to avoid those errors. For instance, I have: 1) $\sqrt{x+\dfrac{1}{x}} - ...
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2answers
475 views

Condition number matrix

For the identity matrix $I$, the condition number of the matrix always equals 1. My question is: are there any other matrices out there that have a condition number equal to 1, but are not the ...
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1answer
628 views

Separating equation into real and imaginary parts and eliminating integral

I have the following equation: $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} ...
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2answers
125 views

Different Representations of Numbers in Subsets of $\mathbb R$

I think I've mentioned sometime before about logarithmic number system. In this system, a real number $r$ is represented by $(\text{sgn}(r), \log|r|) \in \{-1, 0, 1\} \times \mathbb R$. If $\mathbb R$ ...
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4answers
261 views

What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?

I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
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1answer
297 views

Numerical Solution of Systems of PDE

Could someone give me some reference to the Numerical solution of a System of PDEs of following type.. (which also encompasses strongly elliptic system of PDEs) in 2D or 3D. $$\left\{ ...
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2answers
167 views

Triangular grid with 4 edges per vertex

I am trying to create a triangular grid/mesh for a rectangular domain in $\mathbb{R}^2$ with the property that each vertex is shared by (at most) four edges. Is this possible to accomplish?
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2answers
284 views

Need help understanding shooting method

I'm trying to solve a set of differential equations that all depend on a parameter, $\kappa$. I can use the system of ODEs to reduce the four equations into one second order differential equation, for ...
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1answer
63 views

Confidence in the first $k$ decimal places of a product after multiplying $N$ irrational numbers together

If I multiply $N$ irrational numbers together to generate a product $P$, where the irrational number are specified to a working precision of $m$ decimal digits, how many decimal digits, $k$, should I ...
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6answers
202 views

Is there any way to determine the first $3$ digits of $2^m-2^n$ ($n\leq m\leq 10^{100}$)

It's a problem in my ACM training. Since $n,m$ are really huge, I don't think the algo $2^n=10^{n\log2}$ will work. Also, it's not really wise to calculate the value of $2^n$, I think. So I stack. ...
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1answer
508 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 ...
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2answers
399 views

GMRES algorithm

Can you suggest me a reference (besides Wikipedia) where the GMRES (Generalized minimal residual method) algorithm is explained in full detail, in a nice and easy way to understand? A clearly written, ...
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2answers
2k views

Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
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1answer
83 views

Calculating the divisor, known to be small, of two Stirling approximations of the logarithmic Gamma function without overflows

Earlier, I asked a question on MathOverflow regarding how one might analytically approximate a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ for $a \ge 0$, $(ai) < 1$, and $n > 10^5$ ...
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0answers
65 views

A basic question related with the order of convergence of iterative method

I am working on an iteration problem for computing inverse of a non singular matrix $A$ I have got following relationship between error matrix defined by $E_k = X_k-A^{-1}$. $\|E_{k+1} \|\leq ...
5
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1answer
162 views

Trying to solve $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$ using Newton-Raphson method

This is what I've atempted so far in solving $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$. The following are the steps: step 1: $f(\lambda) = \lambda^3 - 3.250\lambda^2 + \lambda - 0.063 $ ...
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3answers
1k views

Understanding algebraic method of successive approximations to solve quadratic equations

I have straight away copied and pasted a worked example from john birds higher engineering mathematics {page 80 problem 4} which reads as follows. Use an algebraic method of successive approximations ...
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1answer
489 views

A problem for Matlab solver

I can numercally solve a system of two ODEs for x and p: dy1 = -y1 + y2 dy2 = -1 + y2 with the following Matlab code: ...
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1answer
67 views

Finite element method - Dual functional Error estimate

I have an equation -$u''=1,~~x\in(0,1)$. I solved it numerically by Finite element method. and find an approximate solution. $u_{h}$. As you know to do this I defined bilinear and linear functionals ...
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1answer
338 views

Identifying when exponential function argument has doubled or tripled

Suppose that I have a function $f(k) = U\exp(k)$. Suppose also that I know $f(k)$, but not $k$ or $U$. I modify this function as shown below, and take $k= k_0$ and $U$ as fixed constants: ...
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1answer
55 views

Estimates of Gaussian Logarithms

I've been implementing logarithmic number system and I came across these functions called Gaussian logarithms: $f(x) = \log(1 + e^x)$. $g(x) = \log(e^x - 1)$ for $x > 0$. $h(x) = \log(1 - e^x)$ ...
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2answers
215 views

Stability analysis of this differential equation

Below is a problem I recited from an exam I took. I wasn't able to solve it on time. Could someone show me how to simplify it? If $$f(t,y(t)) = y'(t)$$ $$y'(t) = -\lambda y(t)$$ where $\lambda$ is a ...
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0answers
190 views

Runge function error second factor

I'm currently learning about the Runge function. On Wikipedia, I read the following: Consider the function: $ \dfrac{1}{1+25x^2}$ Runge found that if this function is interpolated at ...
2
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2answers
86 views

Newton algorithm for a function in $\mathbf{R}^n\rightarrow \mathbf{R}$

I am curious on how the Newton algorithm would work to solve an equation of the type: $f(x_1,\dots,x_n)=0$. As far as I understand, in dimension $1$, one solves $f(x)=0$ by starting with some $x_0$, ...
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1answer
318 views

Finding an interpolating polynomial and natural cubic spline for a given accuracy

I'm trying to make an exercise but I don't know how to start. Is there somebody that can give me a hint so that I can start with the exercise. The exercise is: Consider the function $f(x) = \sin(x)$ ...
2
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3answers
716 views

Constrained optimization: equality constraint

I have this very general problem (for $n>2$): $$ \begin{align} & \max Z = f(x_1,\ldots ,x_n) \\[10pt] \text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt] & x_i \geq 0 \end{align} $$ Assume ...
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1answer
926 views

Numerical Integration of a Gaussian Distribution in Polar Coordinates

I want to numerically evaluate a 2D-integral of a specific probability distribution over some given area (I use MATLAB so all the code below is MATLAB code). I broke down the problem so that it ...
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1answer
105 views

Choosing initial approximation for computing Moore-Penrose inverse

I am trying to compute the Moore- Penrose inverse of a given $m\times n$ matrix $A$. I did convergence analysis then I came across to the following condition. For convergence of my method: $\max\mid ...
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1answer
1k views

Newton's method - error bounds

I just have a very brief question regarding the formula for error bounds in Newton's method. Depending on where you look, this will either be written as: $$e_{n+1} \approx \frac{f^{\prime ...
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1answer
1k views

Can I only apply the Gauss-Hermite routine with an infinite interval or can I transform the interval?

Short version of my question Can I only apply the Gauss-Hermite routine with an infinite interval or can I transform the interval? Long version (reason I'm asking) I am interested in solving ...
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2answers
361 views

Solving linear system of equations when one variable cancels

I have the following linear system of equations with two unknown variables $x$ and $y$. There are two equations and two unknowns. However, when the second equation is solved for $y$ and substituted ...
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1answer
430 views

Why does relative error give number of correct digits?

I learnt that if the relative error is 5*$10^{-s}$ then the number of correct digits the result has $s$. Why is this so? Can you illustrate with an example and/or a proof? Another way to put it ...
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1answer
831 views

Existence of non-trivial solution of Sylvester equation.

I'm trying to solve a special case of Sylvester equation in my case it looks like $$A*X=X*B$$ so it can be written in form $$A*X+X*(-B)=C$$ where C consist of all 0 items. I tried to solve it in ...
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2answers
474 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 ...
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1answer
454 views

how to prove iteration formula in newton raphson formula for multiple root

well, could any one tell me how to prove this in the relation with newton raphson method? if $y$ is a root of $f(x)=0$ with multiplicity $p$,then iterative formula becomes ...
3
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4answers
830 views

Newton's method - finding suitable starting point

I have some trouble solving a problem in my textbook: Given the following function: $$f(x) = x^{-1} - R$$ Assume $R > 0$. Write a short algorithm to find $1/R$ by Newton's method applied to ...
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0answers
228 views

Large number of Linear equation solving with diagonally dominant matrix

For a certain problem I am modelling, I have an MCMC sampler at the moment. It draws samples from the ($n-1$)-dimensional simplex (in this case, from a Dirichlet distribution) and evaluates the ...