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
0answers
11 views

Solve set of poorly conditioned linear equations in block matrix form

I would like to solve the following set of linear equations where A, B, C and D are each 4x4 matrices. K is then an 8x8 matrix The values in A and D have magnitudes of $\approx 10^{17}$, B has ...
0
votes
0answers
10 views

Understanding de Casteljau algorithm

I have a problem understanding the de Casteljau algorithm. For example, let these be the given Beziér nodes \begin{align*} d_0 = (0,2)^T && d_1 = (0.5,1)^T && d_2=(1,3)^T \end{align*} ...
0
votes
1answer
23 views

How to find convergence point for a given iterative scheme

The equation $x^2+ax+b=0$ has two real roots $\alpha$ and $\beta$. Show that the iterative method given by $\displaystyle x_{k+1}=-\frac{(ax_k+b)}{x_k}$ is convergent near $x=\alpha$, if ...
2
votes
1answer
16 views

Increasing Function or Polynomial with Prescribed Values

Consider $n$ points $(a_1,b_1), (a_2,b_2),\cdots, (a_n,b_n)$ in Euclidean plane with $a_1<a_2<\cdots < a_n$ and $b_1<b_2<\cdots < b_n$. It is easy to construct a polynomial of degree ...
1
vote
0answers
13 views

difference between runge kutta methods of same order

I recently read about runge kutta methods for solving differential equations. So far I understood the idea but up to know nobody could answer me following question: If we consider the explicit rk ...
3
votes
2answers
10k views

Implement a program in Matlab for LU decomposition with pivoting

I need to write a program to solve matrix equations Ax=b where A is an nxn matrix, and b is a vector with n entries using LU decomposition. Unfortunately I'm not allowed to use any prewritten codes in ...
12
votes
3answers
209 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
0
votes
1answer
240 views

Bisection method example

I'm writing a small program to resolve functions using bisection method. I want to test the case when the method finds 2 roots, but I can't find examples. Can anyone give me an example of a function ...
9
votes
1answer
271 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
0
votes
0answers
19 views

Use fixed point iteration to find root of equation $2x-\tan x=0$

I've tried $g(x) = \tan(x)/2$ and $g(x) = \operatorname{arctan}(2x)$, but neither of them satisfied the convergence condition. I guess I have a misunderstanding of the convergence condition. I ...
0
votes
1answer
45 views

Bisection Method [on hold]

Please help me justify the accuracy of this method in approximating the solutions for a function. Thanks a lot!
2
votes
1answer
43 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
2
votes
0answers
41 views

Numerically solve for maximum root

I am looking for an efficient algorithm that can numerically solve a piecewise function for its maximum zero root. The piecewise function will normally take the form of the plots below where by below ...
0
votes
0answers
16 views

Finite difference method for nonlinear partial differential equations

I have the following partial differential equation (PDE) $ \forall (x,t)\in(0,L)\times(0,\infty) $ \begin{equation} \begin{split} m_{z}\ddot{w}&+EIw'''-Tw''-f+c_{1}\dot{w}-EAv''w'-EAv'w'' ...
0
votes
0answers
9 views

Looking for a motivating example for backward error analysis

For many people, it seems self-evident that backward error is a powerful tool in numerical analysis. But for me, it is hard to imagine a situation in which backward error analysis provides any useful ...
2
votes
2answers
118 views

How to efficiently solve a series of similar matrix equations using the LU decomposition

This is the problem I'm dealing with: Let $\sigma_1,\dots,\sigma_n \in \mathbb{R}$ and $b_1,\dots,b_n$ be column vectors of length $n$. Consider the system $$ (A - \sigma_jI)x_j = b_j, \quad ...
0
votes
2answers
20 views

Starting guess for a Boundary Value Problem in a Non-linear Ordinary Differential Equation

I have read around the Internet and tried to find answers but inevitably, people just chalk the problem up to an application of Newton-Gauss method and then call it a day. My problem is finding a good ...
0
votes
1answer
22 views

Scilab : simulating model of general equilibrium equations

Hi i'm new to scilab !!! I have a static general equilibrium model with 8 endogenous variable and 8 independent equations.Can anyone guide me how to do simulation of such model in scilab ? I want to ...
0
votes
1answer
30 views

Determining Rate of Convergence

I have a question from the homework here: Show that the following sequence converges linearly to 0 $$P_n = \frac{1}{n^2}; n \ge 1$$ So we know $$\lim\limits_{x \to \inf} \frac{|p_{n+1} - ...
0
votes
0answers
15 views

Properties of Lagrange Interpolation [on hold]

Using Lagrange to show that (a) $\sum x^k l_j(x)=x^k$ and (b) $\sum (x_j-x^k)\cdot l_k(x)=0$
19
votes
1answer
302 views
+500

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
5
votes
2answers
123 views

Wikipedia wrong? Convergence of finite difference

Update: I have edited the Wikipedia page, so that the mistake no longer appears. On the Wikipedia article for "Finite difference" there is the claim Assuming that $f$ is continuously ...
0
votes
0answers
20 views

Newton Raphson Method Overestimating Parameters

I have implemented an almost plain vanilla algorithm to find the MLE estimates of 3 parameters in a log-likelihood function (in R.) When I test my algorithm with some simulated data it does pretty ...
0
votes
0answers
29 views

How fast can we approximate the sum of the tangent?

So Wikipedia gives the sum of the tangent on this page as: \begin{align} \sum_x{\tan{(x)}} &= ix - \psi_{e^{2i}}(x + \pi/2)+C \\ &= -\sum_{k=1}^\infty{ \psi(k \pi - \pi/2 + 1 - x)} \\ &- ...
1
vote
0answers
11 views

Convergence order of Runge-Kutta methods: proof requested

I have been told that: The convergence order of an explicit Runge-Kutta method with $s$ stages is at most $s$. Furthermore, for $s>5$ there is no explicit Runge-Kutta $s$-stage method of order ...
1
vote
0answers
21 views

Runge-Kutta methods and butcher tableau

What does the Butcher tableau of a Runge-Kutta method tell me about the method, besides the coefficients in its formulation? In particular, what requirements about it guarantee consistency and ...
1
vote
1answer
33 views

Sufficient conditions for the convergence of Newton's Method

Suppose we are employing Newton's method: $$ x_{k+1}=x_k - \frac{f(x_k)}{f'(x_k)}. $$ Suppose $f$ is twice differentiable, $f(c)=0$, $f'(x) \neq 0$ on $(c-h, c+h)$, and $x_1 \in (c-h, c+h)$. Let ...
0
votes
0answers
17 views

Second derivative approximation at the endpoint of a bounded function

I have a function defined on [a, b] and trying to approximate its second derivative using finite differences method. The centered finite difference formula works for interior points, but not for ...
0
votes
1answer
33 views

Finding the Newton map

Start with $p(x)=(x-x_0)^k g(x)$. I need to find the Newton map, which is $Np(x)=x−p(x)/p'(x)$. Is $p'(x)=k(x-x_0)^{k-1}g(x)+(x-x_0)^kg'(x)$? I'm having a tough time with $k$ and $x_0$.
1
vote
0answers
28 views

$f'>0$, $f''>0$ is sufficient for Newton's Method

I'm doing problem 22-14 in Spivak's Calculus, 4th edition. Here they outline Newton's method. They assume for convenience that $f'>0$ and $f''>0$, and that $f(x_1)>0$. They note that in this ...
0
votes
0answers
20 views

Order of gradient of basis function

In finite element methods (for example), with $\varphi_{i}$ the Lagrange basis function associated to node i, why do we have: $\mid\nabla\varphi_{i}\mid=\mathcal{O}(h^{-1})$
0
votes
0answers
23 views

how to prove this sparse coding equation

How can I prove the following? $\sum_i \frac{1}{2} \|\mathbf{x}_i - D\mathbf{\alpha_i}\|^2 = \frac{1}{2}Tr(D^TDA_t) - Tr(D^TB_t)$ where, $A_t = \sum_{i=1}^T \mathbf{\alpha}_i\mathbf{\alpha}_i^T\\ ...
0
votes
1answer
30 views

help with a math problem that applies the product rule

I am trying to apply the product rule to get a percentage rate. my problem is $0.32\times0.43\times0.05=0.00688$ or $0.68\%$ I rounded off to $0.69$ in fraction form I got $69/100$ people. Now my ...
1
vote
2answers
48 views

Exercise about Newton´s Method

I´m study numerical methods and some applications of calculus, and I need some help here: Consider $f$ and $g$ are real functions differentiable such that $g'(x)\neq 0$ for all $x$ $a)$ Show that ...
1
vote
0answers
11 views

Implementing periodic Gaussian

I am trying to implement periodic Gaussian in C. What is the correct way to evaluate the periodic Gaussian function as defined below : I am currently evaluating it as : Thanks in advance.
0
votes
1answer
47 views

Numerical way to deal with Dirac delta.

I have been wondering about this: I have a differential equation $y'(t) = y(t) + n \delta(t) y(t)$ with $y(-1) :=y_0$ Thus I want to apply a short delta pulse at some particular point $0$ to my ...
0
votes
0answers
38 views

Taylor Polynomial Accuracy [closed]

How does accuracy depend on the degree of the Taylor Polynomial and the distance from the point its being expanded about (say $x=0$)? I'm considering the function $1/(1-x)$ centered at $0$. I have ...
1
vote
3answers
50 views

How many numbers can a typical computer represent?

I couldn't find this elsewhere so I thought I'd give it a try to figure out exactly how many numbers a typical desktop computer can represent in memory. I'm thinking about this in the context of ...
0
votes
1answer
508 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
0
votes
0answers
22 views

Compute the experimental order of convergence

Compute the experimental order of convergence for a root finder with errors in 3 consecutive iterations of $10^4 , 10^7 ~\text{and}~ 10^{14} $ I'm having trouble understanding can someone give me a ...
0
votes
0answers
19 views

Calculate the order of error for the (summed) Midpoint rule?

I'm reading a comparison of the summed rectangle and and midpoint rules for estimating the value of an integral. The midpoint rule: $\displaystyle\int_a^b \! f(t) \, \mathrm{d}t \simeq f(a + h/2)h$ ...
-1
votes
1answer
30 views

Polynomial Interpolation - Prove that if g interpolates the function f and if h interpolates f then the function interpolated f [closed]

Prove that if g interpolates the function f at x0, x1, …., xn-1 and if h interpolates f at x1, x2, …., xn, then the function g(x) + (x0 - x)/(xn – x0) [g(x) – h(x)] interpolates f at x0, x1, …., ...
1
vote
3answers
139 views

Runge Kutta 8(5,3)

This is actually three small very related questions about Runge Kutta methods. 1) I have programmed a RK 7(8) method also RK 4(5). At the beginning I was assuming that the RK 7(8) uses two ...
0
votes
0answers
32 views

Applicantions of Newtons Method for $f(x) = \dfrac{e^x}{x^2+1}$

I´m study some applications of calculus and see that question: If $f$ and $g$ are real functions differentiable such that $g'(x)\neq 0$ for all $x$. $a)$ Show that $f(x)$ and $g(f(x))$ has the same ...
2
votes
0answers
26 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
2
votes
2answers
669 views

Deducing that the rate of convergence of the Secant method is 1.6.18 (Golden Ratio?

I have a function $$f(x) = e^x - (2-x)^3$$ And I am using the Secant method to find a root between $0$ and $5$. I know that the value of this root is $t = 0.7261444\ldots$ Here's my output from ...
2
votes
0answers
136 views

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
2
votes
1answer
120 views

One Sided Approximation for Mixed Derivatives

Consider the function u(x,y,z) I am trying to approximate the partial derivative at point (i,j,k) by one sided finite difference method. Now using one sided 2nd order finite difference approxmation ...
1
vote
1answer
460 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
1
vote
0answers
23 views

What is the difference between perturbation theory and numerical analysis?

What is the difference between perturbation theory and numerical analysis? Both subjects are trying to obtain the approximate answer. What are they study specifically?