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
1answer
27 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...
0
votes
0answers
21 views

Approximating the Digamma fucntion near 1

Peace be upon you, I had the following system of equations to be solved \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ \psi(\beta)-\psi(\alpha+\beta)=c_2 \end{cases} \end{align*} ...
2
votes
0answers
22 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
1
vote
0answers
19 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
1answer
23 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*} ...
2
votes
1answer
22 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
16 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 ...
0
votes
1answer
25 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 ...
0
votes
0answers
23 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
0answers
12 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 ...
0
votes
1answer
64 views

Bisection Method [closed]

Please help me justify the accuracy of this method in approximating the solutions for a function. Thanks a lot!
0
votes
0answers
22 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'' ...
2
votes
0answers
46 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
1answer
28 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
31 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
23 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 ...
5
votes
2answers
133 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
30 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
15 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
24 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 ...
0
votes
2answers
21 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 ...
1
vote
1answer
37 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 ...
1
vote
0answers
29 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
1answer
34 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 $x_0$.
0
votes
0answers
26 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
25 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 ...
0
votes
0answers
21 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 ...
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
51 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 ...
1
vote
3answers
51 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
0answers
20 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: $\displaystyle\int_a^b \! f(t) \, \mathrm{d}t \simeq f(a + h/2)h$ ...
0
votes
0answers
28 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
33 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
30 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} ...
1
vote
0answers
27 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?
1
vote
0answers
19 views

System of PDE's with unknown functions

So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be ...
0
votes
0answers
23 views

Newton Divided Difference Table , if $f''(x)$ is given

Here is the problem: Suppose $f(0)=1$, $f(1)=f'(1)=f"(1)=0$, and $f(3)=16$. Compute the Hermite interpolation polynomial $P$. How would the divided difference table look for this, in order to ...
2
votes
2answers
124 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 ...
3
votes
2answers
32 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
2
votes
1answer
55 views

Derivation of power method

POWER METHOD Let $x_0$ be an initial approximation to the eigenvector. For $k=1,2,3,\ldots$ do Compute $x_k=Ax_{k-1}$, Normalize $x_k=x_k/\|x_k\|_\infty$. Then $\|x_k\|_\infty$ approaches the ...
0
votes
1answer
33 views

Prove $\frac{dy}{dx}$ is approximated by $\frac{y(x+h)-y(x-h)}{2h}$ to $O(h^2)$

I tried to solve it by truncating the Taylor series expansions for $y(x+h)$ and $y(x-h)$ but I couldn't find a way to relate it to the derivative. I wasn't sure where the appropriate place to truncate ...
0
votes
0answers
37 views

Finding the proper g(x) for fixed point iteration on $2\sin{\pi x} + x = 0$

After spending over an hour trying to get this problem I realize my trig is weak. I found: Fixed point iteration .Numerical method. The selected solution is informative, but lacking detail to really ...
1
vote
0answers
29 views

Higher-order difference quotients

The Mean Value Theorem for Divided Differences says that if $f$ is $n$ times differentiable, and $x_0< x_1 < \dotsb < x_n$, then there is a point $\xi\in (x_0, x_n)$ such that $f[x_0, x_1, ...
0
votes
0answers
9 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
0
votes
0answers
17 views

lagrange interpolation of a point

Let $f(x)=\sqrt[]{x}$ be our function. Let $P_n$ be the lagrange interpolation polynom of $f$ by $n$ points and $a$ be an element in the domain of $f$. Can rational points be chosen such that ...
1
vote
0answers
18 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
2
votes
2answers
47 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
1
vote
1answer
28 views

For what values of c will the iteration $x_{n+1} = g_c(x_n)$ converge to $\alpha_c$

Consider the equaction $x=g_c(x)\equiv cx(1-x)$, with c a nonzero constant. This equation has two solutions, and we let $\alpha _c $ denote the nonzero solution. What is $\alpha _c$?For what values of ...