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

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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 ...
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1answer
11 views

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

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
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2answers
22 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 ...
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0answers
4 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 ...
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2answers
51 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 ...
3
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
18 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 ...
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1answer
36 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 ...
2
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1answer
48 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 ...
4
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1answer
42 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 ...
2
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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 ...
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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 ...
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17 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 ...
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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 ...
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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 $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
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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 ...
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1answer
20 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 ...
2
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1answer
68 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 ...
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1answer
15 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, ...
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1answer
33 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] ...
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1answer
37 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?
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1answer
35 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? ...
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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: ...
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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: ...
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0answers
34 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 ...
2
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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' = ...
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1answer
19 views

The truncation error associated with linear interpolation of a function $f(x)$ using ordinates $x_0$ and $x_1$ is not larger in magnitude [on hold]

To show that the truncation error associated with linear interpolation of a function $f(x)$ using ordinates $x_0$ and $x_1$ is not larger in magnitude than $$\frac {(x_1 - x_0)^2}{8} \times \max_{x ...
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2answers
43 views

Which topic is this question from? (Root-finding)

Prove that if $x=a$ is an approximation to one root of the equation $f(x)=0$, then $x=a-\frac{f(a)}{f'(a)}$ is a closer approximation. How to solve this question? Is this asking us to prove ...
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0answers
23 views

Question on the uniqueness of LU decomposition

Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$. We define ...
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0answers
11 views

Optimization with KL-divergence in CVX [on hold]

I am trying to solve an optimization problem whose objective is KL-divergence (there are only linear equality constraints and nonnegativity constraints). In particular, my goal is to obtain a solution ...
3
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1answer
38 views

Name of the LU decomposition algorithm

On the wikipedia page of LU decomposition there is an algorithm that produce the decomposition. It is called Doolittle algorithm. I'm really interested who is Doolittle? Or from where the name comes ...
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22 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 ...
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0answers
21 views

Newton-Raphson for Discontinuous Spring Moment Balance

I am trying to solve the following problem where there should exist an equilibrium between spring forces and moment applied to a cylinder: To do so I am solving the equality: $ M_a - \sum_{i=0}^n ...
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1answer
26 views

Numerical phase plane?

In my Differential Dynamical Systems text book, I came across the following question: Sketch the local behavior you obtained in the phase plane and compare with a numerical phase plane plotter that ...
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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 ...
0
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1answer
8 views

tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
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0answers
36 views

Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $$ I_f=∫^b_af(x)dx $$ is based on linear interpolation of f at x0=a and x1=b=a+h. Consider now a a cubic Hermite polynomial, interpolating both f and its ...
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0answers
9 views

Accurate numerical integration for “data times an analytical function”

The Question is as follows: I have an algorithm/data that provides me the value of a function $f(x,y,z)$ on the points of a grid. On the other hand I have an analytical function ...
2
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1answer
30 views

Numerical approximation of differentiation

I have the following task to solve: Let $b>x$ be defined, determine $w_0,w_1$ and $w_2$ in dependency of $b$ such that the approximation $f''(x) \approx w_0 f(x-h) + w_1 f(x) ...
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3answers
28 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 ...
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30 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
3
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0answers
26 views

Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
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14 views

Simpson's rule and Hermite interpolant

For a uniform grid, $$x_n = -1 + nh$$ where $h = \frac{2}{N}$, I need to show that Simpson's rule is an $\mathcal{O}(h^5)$ integration rule. So far, I know to let $p(x)$ be the Hermite polynomial from ...
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1answer
31 views

Need help with a Crank Nicholson Method example problem.

I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have ...
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1answer
13 views

efficient computation of Cholesky decomposition during tridiagonal matrix inverse

I have a symmetric, block tridiagonal matrix $A$. I am interested in computing the Cholesky decomposition of $A^{-1}$ (that is, I want to compute $R$, where $A^{-1}=RR^T$). I know how to compute the ...
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29 views

How should I rewrite this code to not use feval? [migrated]

I recently finished a homework set in my Applied Numerical Methods class and did alright on it. However, my professor made a note to say I shouldn't use the feval() function because it's outdated. ...
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2answers
33 views

Analyzing derivative of function.

I have some function $g: [a,b] \to [a,b]$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $\forall x \in [a,b]: |g'(x)| \lt 1$. How can I find out if this is true or not? P.S. I ...