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)

0
votes
2answers
14 views

What is the intuition behind matrix splitting methods (Jacobi, Gauss-Seidel)?

Descent Methods, like Gradient and Conjugate Gradient ones, have a nice geometric interpretation and I really love them. What about Jacobi, Gauss-Seidel or other matrix splitting methods? I can't see ...
2
votes
0answers
88 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
-1
votes
0answers
16 views

Possible to proof/disproof this statement?

Given 2 different function $E=f(p,v)$ $\frac{dE}{dv}=g(p,v)$. $E=f(p,v)$ $\frac{dE}{dv}=f_p(\frac{dp}{dv})+f_v=g$ $\frac{dp}{dv}=\frac{g-f_p}{f_v}=x(p,v)$ From this formula got the ...
0
votes
1answer
6 views

Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map. Then, is it true that the spectral radius of the Jacobian $J_F(x)$ at $x = x^\star$ is less than 1 if and only if $x^\star$ is a fixed ...
-1
votes
0answers
15 views

Adomian method (how was the solution in this problem obtained?) [on hold]

can someone please help explain to me how the y terms in problem 1 of this paper were obtained in detail paper: http://www.ccsenet.org/journal/index.php/jmr/article/view/45923/24853 thanks here is ...
1
vote
1answer
61 views

Numerical series

Consider the series below that consist of 2 different formula $P_aV_a^{1.4}=P_bV_b^{1.4} $ and $P_aV_a=P_bV_b$ that keep repeating itself in the whole sequence. By assuming $P_1$ and $V_1$ both=1, ...
-2
votes
0answers
19 views

Gauss Chebyshev formula [on hold]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral. $$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1. Also compare the result with true value, where the zeros and the ...
0
votes
1answer
18 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
1
vote
0answers
21 views

Explicit numerical method for solving second order PDE

I'm interested in solving PDEs of the following form $\frac{\partial^2}{\partial t^2} G(t,t^{\prime}) = f\left(t,t^{\prime},\frac{\partial}{\partial t} G(t,t^{\prime}), G(t,t^{\prime})\right), \qquad ...
2
votes
0answers
21 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
2
votes
2answers
52 views

Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for ...
1
vote
1answer
30 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
0
votes
0answers
11 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
-1
votes
2answers
24 views

calculate the magnitude of the normal reaction [on hold]

A solid sphere of 1.45 kg and radius 0.1 m, is rolling down a rough plane that is inclined at an angle $\frac{\pi}{6}$ to the horizontal. The figure shows the three forces acting on the sphere (its ...
0
votes
0answers
8 views

Generating volume constrained splines

Suppose I have a set of points in $\mathbb{R}^3$, $\{\vec{r}_1,\vec{r}_2, ...,\vec{r}_n\}$, suppose between points $\vec{r}_i$ and $\vec{r}_{i+1}$ there is an associated volume $V_i$. I want to ...
0
votes
1answer
10 views

Central Difference Method

Solve the following using the central difference method: $y(x)= y'+ y + 2x$ where $0 < x < 4$ with $n=4$ subintervals (thus $h=1$). Given that $y(0)=0$ and $y(3)=1$, find $y(1)$. Really ...
-1
votes
0answers
23 views

fast computation of complete elliptic integral matlab

I'm using matlab to compute complete elliptic integrals of first ($K$) and second kind ($E$). I'm having an issue with computational speed evaluating these integrals using matlab function ...
3
votes
2answers
57 views

Find the smallest $N$ such that $\sum_{k=1}^N\frac{1}{p_k}>\pi$. (The $p_k$'s are the prime numbers.)

How to solve the following problem? Let $\{p_k\}_{k=1}^\infty$ be the set of primes (in increasing order). What is the smallest integer $N$ such that $$\sum_{k=1}^N \frac{1}{p_k}>\pi?$$ We ...
0
votes
0answers
26 views

Numerical Triple integral with three other parameters in R

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
3
votes
0answers
21 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
3
votes
1answer
54 views

Find desired root (based on certain constraints) while using Newton-Raphson method

My function is a vector of dimension 6, and on using Newton Raphson method, the solution usually converges to the nearest root. However, I know that my function has multiple roots (it's an Inverse ...
1
vote
1answer
19 views

O(h) operator over uniform grid

For a uniform grid $$x_n = -1 + nh$$ where $h = \frac{2}{N}$ and the integration rule $$I_N(f) = h\sum_{n=0}^{N-1}f(x_n)$$ which corresponds to a left hand Riemann sum or to integrating an ...
0
votes
1answer
39 views

Newton-Raphson Method used in a real engineering/physical/mathematical situation

I've been using the Newton-Raphson Method in my Numerical Methods course for a while now, blindly solving non-linear equations and systems of equations . This makes me somehow lose motivation, as I ...
0
votes
0answers
8 views

quadrature schemes for integral equations fredholm

I am trying to solve this fredholm integral equation using numerical schemes. Can anyone please suggest a quadrature rule for this. The singularity makes it difficult here. $f(x) = \int_0^1 ...
4
votes
1answer
43 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
1
vote
1answer
23 views

Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
5
votes
3answers
30 views

Numerical integration of a data set with uncertainties

I have a 1D data set {xi, yi} with no uncertainties in xi and with uncertainties dyi in yi. The resulting discrete function is monotonic and relatively smooth and I would like to integrate the ...
1
vote
0answers
19 views

convergence of a numerical method

given a function $f:\mathbb R\to\mathbb R$ in of class $C^3$. We suppose that there exists $s\in \mathbb R$ such that $f(s)=0$ and $f'(s)\neq 0$. Let $\beta$ be a real number s.t. $\beta \neq 0$. We ...
2
votes
0answers
21 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
1
vote
0answers
7 views

How to define boundary conditions for a sphere to run reaction-diffusion equations on its surface?

I'm in a Biology lab, and we managed to simulate reaction-diffusion equations on a torus using periodic boundary conditions for a 2D matrix. We want to try doing the same on a sphere, but I'm a ...
0
votes
0answers
19 views

prove this using lagrange and newton divided difference error!

suppose f(x) is polynomial with degree of three.prove $f[{x}_{0},{x}_{1},{x}_{2}] = \frac{1}{2}{f}^{(2)}(\frac{{x}_{0}+{x}_{1}+{x}_{2}}{3})$ and ${x}_{0},{x}_{1},{x}_{2}$ are distinct point. I ...
0
votes
0answers
17 views

A question about the condition of quadrature formula

I am reading through my numerical mathematics script and I am currently in the chapter 4 (see listing) computer arithmetic direct solution of linear systems of equations polynomial interpolation ...
1
vote
0answers
54 views

Finding $ \max_{x \in [2,4]} \left| 2 x \cos(2 x) - (x - 2)^{2} \right| $.

This is a problem taken from Burden’s and Faires’ Numerical Analysis. Define $ f: \Bbb{R} \to \Bbb{R} $ by $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
0
votes
1answer
909 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
2
votes
1answer
619 views

Derive error term by using Taylor series expansions.

Using Taylor series expansions, derive the error term for the formula \begin{equation} f''(x)\approx \frac{1}{h^{2}}\left [ f(x)-2f(x+h)+f(x+2h) \right ]. \end{equation} I've tried it on my own ...
3
votes
1answer
21 views

Is there a meaningful distinction between “direct” and “iterative” methods for solving equations?

I'll motivate this question with an example. The Abel-Ruffini theorem states that there is no general "formula" for the roots of polynomials of degree greater than 4. (Specifically it states that ...
0
votes
0answers
9 views

Distribution of SDE numerically from Fokker-Planck.

I'm aware of some numerical methods related to SDEs such as Euler-Maruyama, Milstein etc. However, couldn't one also simulate the equivalent Fokker-Planck equation via finite element methods? This ...
2
votes
1answer
72 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
0
votes
1answer
42 views

Interpolating $n$ points by piecewise quadratic polynomial

Given $n$ data points. Is it possible to interpolate them by piecewise quadratic polynomials with knots at the given data such that the quadratic interpolant is: (a) Once continuously differentiable? ...
1
vote
1answer
16 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, ...
0
votes
1answer
34 views

Runge-Kutta force at each time-step

Consider that I am solving a second order ODE using RK2/RK4. The ODE represents simple equations of motion: Equations of motion I am trying to solve: \begin{align} \frac{dx}{dt} &= v \\[.3em] ...
0
votes
0answers
18 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
1
vote
1answer
44 views

Quick question that I can't find anywhere online about Runge-Kutta

I'm writing a presentation on modelling fluid flow. We used Runge-Kutta second order to describe the flow as a numerical method. I just want verify that Runge-Kutta fourth order would be of a higher ...
2
votes
1answer
99 views

Orthogonality of Lagrange Polynomials in Hermite Inner Product

My question is as shown above. I have churned through the first part, but I am stuck on showing the orthogonality of the Lagrange polynomials. My first hope was to use the fact that the Lagrange ...
3
votes
1answer
39 views

What is the justification and intuition behind Muller's method's quadratic equation?

Usually we write the quadratic formula like this: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ But Muller's is written like this: Muller's method. Why is that?
-3
votes
1answer
33 views

Deriving differentiation rule [closed]

Assuming I know the values of a $C^{\infty}$ function $f(x)$ at $x_0 = -h, x_1 = -\frac{1}{2}h, x_2=\frac{3}{4}h, x_{3}=2h$ where $h$ is a small parameter. I need to derive an $O(h^3)$ ...
1
vote
0answers
35 views

Really confused about LU decomposition and Doolittle algorithm

I'm really confused about the Doolittle algorithm, so I need some help. At the end of the description at wikipedia, it says It is clear that in order for this algorithm to work, one needs to ...
0
votes
0answers
16 views

Shooting method with non-robin (or derivative) boundary conditions

I am trying to solve a problem in which I have to find all the values of $\lambda$ for which the boundary value problem has just one solution for each $a,b\in\mathbb{R}$. The problem is the following: ...
2
votes
1answer
22 views

Second order to first order equations

I need to write $$\frac{d^2\theta}{dt^2} + 4\sqrt{k}\,\frac{d\theta}{dt}+g\sin(\theta)=0$$ as a first order equation. What I have done so far is: Let $z = \frac{d\theta}{dt}$ Then $z' = ...
0
votes
0answers
27 views

How do I make a Maclaurin series expansion faster?

Suppose I want to approximate to e using the Maclaurin series. In this case, increased accuracy comes with at trade off of computation time. How do I make the Maclaurin series expand faster/ using a ...