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

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43 views

Numerically stable Lanczos process? I need to compute Elements of inverse in sparsity pattern of A

I have a large sparse symmetric positive definite matrix NxN matrix $A$. Let $s$ be the average number of non-zeros per row (i.e. $sN$ total non-zero elements). I would like to compute the elements ...
2
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0answers
34 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
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0answers
24 views

Does eigenvalue theory remain in numerical applications?

If I have a symmetric matrix $A \in \mathbb{Q}^{n \times n}$ in matlab, then in theory it is guaranteed that $A$ has a orthogonal basis of eigenvectors and real eigenvalues. Does this remain in ...
2
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51 views

We are learning about LU Decomposition .. because?

I know what LU Decomposition is but I don't know why we have have to learn about it. What are we using it for? (What's the point to know about it?) Thanks.
2
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35 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
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102 views

lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
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33 views

Is numerical stability preserved under basis transformation for pde

I'm using a Backward Time Central Space method to solve the heat equation in polar coordinates. In Cartesian coordinates, it is easy to show this is unconditionally stable (by assuming solution of the ...
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235 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
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29 views

Metastable solution for system of nonlinear equations

System of nonlinear equations: $$E_i=\epsilon_i+\sum_{j\neq i}^N \left(\frac{1}{1+\exp(E_j/T)}-\frac{1}{2}\right)\frac{e^2}{r_ij} \tag 1$$ where $T=0.05$, $r_{i,j}$ is given symmetric matrix with ...
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201 views

Runge-Kutta method accuracy

I got Runge-Kutta method here and I solve this system using it. So here's Runge-Kutta stuff $k_1 = f(t_n, y_n)$ $k_2 = f(t_n + h/2, y_n + hk_1/2) $ $k_3 = f(t_n+h, y_n - hk_1 + 2hk_2)$ $y_{n+1} ...
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76 views

Numerical Methods for estimating divergence over an improper integral

Problem given a function $f(x)$, defined on $[ \epsilon, \infty )$. Is there a way to "numerically estimate" whether the integral of the function diverges over the domain $[ \epsilon, \infty )$? ...
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22 views

Nontrivial Matrix-estimate

I try to proof the following estimate: \begin{align} h' W^{-1} H W^{-1} h \geq c h' H h \qquad c>0, \qquad\qquad (1) \end{align} where $h\in\mathbb{R}^{K-1}$ and ...
2
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107 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 ...
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110 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
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37 views

Is scalar product a well-conditioned operation?

I'm reading a course and one of the exercises is about establishing whether scalar product is a well-conditioned operation. Here's their solution. They disturb each element of the vector by ...
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46 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
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75 views

how to choose point spacing to approximate a parametric curve using line segments?

Suppose I have a parametric equation for a curve $\vec{r} = f(t)$, which I wish to draw using line segments between some set of points at times $t_0, t_1, t_2,$ etc. If I want to achieve a given ...
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43 views

When solving PDEs is there an alternative to interpolation for out-of-grid point?

I'm numerically solving a PDE where the space domain is huge. So, I often need to interpolate to get out-of-grid points needed by the finite difference algorithm. As a result, I've a lot of numerical ...
2
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116 views

Solution of an implicit Fourier transform equation

How does one solve the following equation ($\hat{a}(k)$ denotes the Fourier transform of $a(x)$ and $q$ is real positive): $$\hat{a}(k)=f(k)\widehat{a^q}(k).$$ This equation appeared in some paper. ...
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753 views

Show that the averaged vector field one step method is well-defined

Let $\dot{\mathbf{y}} = \mathbf{f}(\mathbf{y}), \;\mathbf{f}: D \subset \mathbb{R}^d \to \mathbb{R}^d$ be an autonomous differential equation with $\mathbf{f}$ smooth. We define the averaged vector ...
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56 views

To calculate a derivative of a set of points, is it more correct to interpolate finite differences or to derivate the interpolation?

I have a series of points extracted from numerical simulations. I also recently discovered the amazing power of finite differences. Nevertheless, I was used to estimate my derivatives from the ...
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47 views

Finding $k$ unknowns given the sum of their first $k$ powers

Motivation: The motivation for this question came from a Computer Science problem of finding duplicates in a list in constant time and constant space. If the list of numbers was $i_1, i_2, \ldots, ...
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65 views

automatization of numerical derivation

I would like to know if there is an automized or fast way to numerically derivate a large number of tab-delimited files (derived from the program kaleidagraph) and to automatically extract some key ...
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130 views

monotonic smoothing fit to be implemented (in python or other language)

In a post that already exists, implementation-of-monotone-cubic-interpolation, there is a good method for fitting data which necessarily includes all of the given points. But, what if I need to ...
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68 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
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82 views

van der pol and liapunov

i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated. a) show that the second-order differential equation for ...
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265 views

Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
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73 views

How to can I transform the 2D cuasi Laplace equation with variable coefficients to finite difference scheme?

I want to solve $$\frac{\partial}{\partial x}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial ...
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49 views

What kind of numerical methods are best applicable to this?

I'm wondering: what would be the best numerical method for solving a nonlinear integral equation of the form $$f(x) = a(x) + \int_{-A}^{A} K(x, t, f(t)) dt$$ where $f$ is the unknown function, a ...
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130 views

Induction proof of polynomial interpolation theorem

show if $\phi (x) = f(x)g(x)$, this is valid: $\phi [x_0,x_1,...,x_n]=\sum\limits_{r=0}^n f[x_0,x_1,..,x_r]g[x_r,x_{r+1},...,x_n]$ by induction. I have tried to prove it by the divided differences ...
2
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0answers
41 views

maximal m-elements of the matrix inversion

Suppose the $n\times n$ matrix $A$ is invertible, and all its elements are between 0 and 1. The existing matrix inversion operation of $A^{-1}$ will take $O(n^3)$ time. Now I just want to find the ...
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157 views

Optimization problem about large matrices

I'd like to solve the following optimization problem: Find non-negative scalar $a$, $b$, $c$ to minimize $\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$ where ...
2
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349 views

Consistency order of backward Euler method

How can I proof that backward Euler method has consistency order 1? Implicit function theorem states that for a sufficiently small $h$, $$ \vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1) $$ has a unique ...
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173 views

all eigenvalues of a large sparse symmetric matrix

my question is similar to how to diagonalize a large sparse symmetric matrix, to get the eigenvalues and eigenvectors however i wish to be more concrete and ask if one can, on a standard PC (e.g. a ...
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0answers
91 views

Suggestions for projects in mathematics of finance

I am looking for computational projects related to mathematics of finance suitable for a senior level independent study for a student who has seen the green light, (or the light of the green!). I had ...
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0answers
53 views

Solution to pertubed linear system

Suppose one has the following system of linear equations $$(A + \Delta A) x = b$$ where $A$ and $\Delta A$ are large sparse matrices and $\Delta A$ is "small" compared to $A$, furthermore vector $x$ ...
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29 views

Approximate a constant function with sequence of spline functions

Suppose that for a constant $c \in \mathbb{R}$ $$\sup_{t \in [a,b]}\Big|c- \sum_{l=1}^{m}a_{l}\ B_{l}(t;q)\Big|< \epsilon.$$ The $B_l$ form a B-spline basis of degree $q$ on the interval $[a,b]$ ...
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144 views

Check for Ill Conditioned matrix

How can I efficiently check if a tridiagonal system with 1's in diagonal is ill-conditioned or not ? The common way is to get the ratio of largest and smallest singular values and see if its greater ...
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0answers
65 views

Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
2
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0answers
415 views

Understanding Fourier Transform and FFT

First off, I'm sorry if this is a repost. I am currently writing my thesis, and I've been thrown into some Fourier analysis, which I know nothing of. So, even if this question has been posted before, ...
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0answers
508 views

Interpolating polynomial with Chebyshev nodes

I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points. ...
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89 views

Numerical Analysis Gauss-Lobatto

I am trying to find the expression of the weights and nodes for the Gauss-Lobatto quadratures with 4 nodes. I am guessing this is a sum of weights? Does anyone here have experience working with this ...
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0answers
234 views

Helmholtz and Biharmonic equation examples with exact solution

I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare results of my numerical solutions with it. I was able to find quite ...
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0answers
364 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
2
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0answers
94 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
2
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0answers
75 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
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0answers
322 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
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264 views

Fourier transform for Neumann boundary condition

I need to solve system of two coupled partial differential equations numerically. $\frac{\partial x_1}{\partial t} = c_1\nabla ^2 x_1 + f_1(x_1,x_2) \\$ $\frac{\partial x_2}{\partial t} = ...
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90 views

Apply Watson's lemma to Gamma function

I was wondering is there anyone know how to show that the gamma function $\Gamma(z)$ satisfies the conditions of Watson's Lemma, where z is on the right half plane. After I changed variable t=xv ...
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0answers
219 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'= ...