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
33 views

Finding a function using first derivative

I have some data about just first derivative of a function. Also, I know a point of this function(e.g. (x1,y1)). How can I obtain the function? All my date are numerical. dev f(x)=[ 580.00 , 479.7308 ...
0
votes
2answers
46 views

Is there a proving of $a_{n+1}=\frac{a_n}{1.5-\frac{(a_n^2-6)^2}{32}}$?

Is there a proving of $$a_{n+1}=\frac{a_n}{1.5-\displaystyle \frac{(a_n^2-6)^2}{32}}?$$ the above formula gives $\sqrt{2}$ We know that the main formula of finding any square root of number depends ...
1
vote
0answers
24 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
0
votes
0answers
14 views

Diagonal matrix of $D_L$ and $D_R$ [on hold]

Find Diagonal matrix of $D_L$ and $D_R$ such that $C^{-1} = D_LC^TD_R$ I was thinking $C$ can be Cauchy matrix with general form $C = \frac{1}{x_i + y_i}$, but even so, I still don't know how to do ...
1
vote
2answers
18 views

Maximum timestep for RK4

I'm trying to find the maximum timestep that can be used when applying a RK4 numerical method to solve this system. I know how I would do this for single equation but have no idea how to solve it for ...
6
votes
3answers
85 views

Parallel lines divide a circle's area into thirds

When I was young I came up with a geometry problem and drew it in a notebook: Suppose we have a circle with radius $r$ and area $A$. Let two parallel lines be equidistant from the center of the ...
0
votes
1answer
35 views

Numerical method to calculate sum of infinite series?

for example: I have a series is there numerical computation method to find it ? thanks
0
votes
1answer
36 views

How to use newton's method on a function of multiple variables?

I have a function $f \colon R^3 \to R$. I want to find $x$, $y$, $z$ such that $f(x,y,z)=0$. I'm using the method from here: http://en.wikipedia.org/wiki/Quasi-Newton_method#Search_for_zeroes. ...
0
votes
0answers
28 views

What exactly are divided Differences?

I'm reading a numerical analysis textbook, and we have a definition of something called "divided differences". I have read the notation and know what the definition is, but what exactly is it ...
1
vote
0answers
22 views

Proof subtraction is not forward stable

I've been taught that the "subtraction operation" is not accurate/forward stable as the relative error can be arbitrary large. I tried to prove it formally but I end up with a contradiction. What ...
0
votes
1answer
10 views

Iterative method to compute $c^{1/p}$

Project the locally convergent iterative method to compute: $$c^{1/p} , p = ..., -2,-1, 1,2,... $$ I thougt about Newton's method, but I got stuck. Please hint me.
-1
votes
0answers
15 views

Numerical Analysis - Newton method, solve linear arrangement

Using Netwon method for linear arrangements solve $$ xy-z^2=1 ,\\ yz-x^2+y^2=2 ,\\ e^x-e^y+z=3 ,$$ where initial point $(x,y,z) = (0,0,1)$, one iteration. Could anyone show me the solution of this ...
0
votes
1answer
36 views

Very High degree Polynomial Roots: How to Cope with Large Values?

I hope I'm not duplicating! I'm wondering how it is possible to find all roots of a polynomial of very high degree (100,1000,1000000, ...) numerically. In all numerical methods, the polynomial is ...
2
votes
0answers
39 views

Express Lagrange polynomial in term of Cauchy matrix

Given 2n distinct real numers $s_1,s_2, \dots, s_n$ and $t_1, t_2, \dots,t_n$ define the $n \times n$ Cauchy matrix $C = C(t,s)$ by $C_{ij} = \frac{1}{t_i - s_j}$. Express the Lagrange interpolation ...
2
votes
1answer
19 views

Question about formulation of initial value problem for ordinary differential equations

Consider the following initial value problem $y'(t) = f(y(t)), $ $0 < t$ $y(0) = y_0$, where $y_0$ is a fixed constant. Here, $y'(t)$ is given only for $t > 0$, not including $t = 0$. That ...
2
votes
1answer
14 views

Least squares problem, using derivative to find the normal equation?

for a matrix $A \in \mathbb{R^{m\times n}}$ and for $x,\epsilon \in \mathbb{R^n}$ and $\epsilon$ small we have that ...
0
votes
3answers
37 views

Solving first order initial value problem numerically

I want to find the solution $y_0(t)$ of the linear first-order ordinary differential equation $$y'-(1/t)y= t \text{sin}(t)$$ satisfying the initial condition $y(\pi /2)=0$. I know how to do this ...
0
votes
2answers
20 views

Interpolation using rate of change

I have a set of data with missing points, which I estimated using spline interpolation. I've now been given the rates of change at each data point. How will this change/improve my current ...
1
vote
1answer
26 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
2
votes
1answer
38 views

Solving $Ax_2 = \lambda x_1$ and $A^Tx_1 = \lambda x_2$ using SVD

Please using only SVD, I have solved the problem using other methods Solving $Ax_2 = \lambda x_1$ and $A^Tx_1 = \lambda x_2$ using SVD: I am solving this to find $\lambda$ and $x_1,x_2$ To find ...
0
votes
0answers
15 views

How to solve constrained ode problem

Currently I'm facing question in which let's say I have 3 coupled eqn. \begin{align} x = f(x', z', y', t) \\ y = f(x', y', z', t) \\ z = f(x', y', z', t) \\ \end{align} There is initial ...
0
votes
1answer
41 views

Solving $y' = 2xy^2$ using backward Euler

$$ y' = 2xy^2, \quad y(0) = 1$$ The formula for backward Euler is $$ y_{n + 1} = y_n + hf(x_{n+1}, y_{n+1}) $$ Where $h$ is the step size. Plugging in for $f$ $$ y_{n + 1} = y_n + h \times 2 ...
0
votes
0answers
44 views

How would you integrate numerically this function?

How do you understand this integral ignoring the rect function? dx and dz are the pixel size of the numerical grid.
1
vote
0answers
20 views

Good convergence criterion for stochastic optimization?

This is a question that has bothered me quite long, as I have faced it many different optimization and equation solving problems. The basic idea is that one wishes to minimize $F(x)$ and has one ...
0
votes
1answer
16 views

Finding parameters for a quadrature formula

To compute the integral $\int_0^1f(x) dx$ numerical I want to use the following quadrature formula: $$Q(f)=\omega_0f(x_0)+\omega_1f(1)$$ The question is how one should choose $\omega_0,\omega_1 ...
1
vote
0answers
18 views

How to study Numerical PDE's using finite element from Mathematical Point view [closed]

I want to ask you to recommend me some video lectures to study finite element method in Numerical PDE's from mathematical point view based on the knowledge of functional analysis. Thank you very much. ...
2
votes
2answers
38 views

Differentiate a Differential equation

Given the Differential equation $y'=-2xy^{2}$. Find the derivative $\frac{d(y')}{dx}$! My approach, which is not correct according to Wolfram Alpha: Plugging in: ...
0
votes
1answer
15 views

Infinity Norm calculation $\| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|$

I have the following infinity norm: $$ \| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|_\infty. $$ Computing from [1,2]. I know that I can compute this in matlab and I get .072. However, how would one go ...
3
votes
2answers
31 views

Lotka-Volterra model with two predators

In this, Lotka-Volterra model, we have two predators: $$\frac{dp}{dt} = ap\left(1-\frac{p}{K}\right) - (b_1q_1+b_2q_2)p$$ $$\frac{dq_1}{dt}=e_1b_1pq_1-m_1q_1$$ ...
1
vote
1answer
35 views

Müller's Method

I have these question and I cannot solve it. Can somebody help me? Use Müller’s method to determine the roots of $$ f(x)=2x^5−2x^4+6x^3−6x^2+8x−8 $$ Choose $x_2=0.8 $, $x_0=0.808$ ...
0
votes
1answer
17 views

numerical solution of partial differential equations by the finite element method claes johnson p

Let us now consider the following abstract minimization problem (M): Find $u \in V$ $$F(u)=min_{v \in V} F(v)$$ where $$F(v)=\frac{1}{2} a(v,v)- L(v),$$ and consider also the following ...
0
votes
0answers
32 views

Finding a Bezier curve, given the control points [closed]

Find the Bezier curve where the control points are P(2,3),P1(2,0),P2(3,1),P3(4,4)
-1
votes
0answers
21 views

Boson system Shooting newton

$$ \frac{d^2y}{dx^2}-\frac{(D-1)}{x}\frac{dy}{dx}+(y-1)(2-3y^2)y=0 $$ $D$=dimension of problem. $D=1,2,3$ boundary condition: $$ dy/dx(0)=0, $$ $$ y(\infty)=1 $$ how do we solve the above ...
0
votes
0answers
21 views

How to transform $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ into a system of first order PDE's and finite difference matrix

So I have this equation: $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ and I need to transform it into a system of first order PDE's. I was thinking like this: ...
0
votes
1answer
21 views

In the finite difference formulas, how can we pick h to give a certain tolerance?

There's lots of questions on here about finite differences. In particular, picking the 'best' h value. But what if I want to find the biggest 'h' which bounds to a given tolerance? On first glance, I ...
1
vote
2answers
35 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} ...
1
vote
1answer
29 views

Prove graphically that the Lambert equation has exactly zero, one or two roots

I need some help on the below problem. Consider the Lambert equation: $xe^x = a$ for real values of x and a (a) Show graphically that the equation has exactly one root $ \xi(a) \ge 0 $ if $ ...
1
vote
2answers
53 views

How to solve a matrix equation for a scalar?

Given matrices $Q, P \succeq 0$, a vector $q$, a real number $\gamma$. How can one solve the equation $ q^T (Q+\lambda P)^{-T}P(Q+\lambda P)^{-1} q = \gamma$ for the scalar $\lambda$ in an efficient ...
0
votes
1answer
8 views

Calculation of two dimensional Fourier transform on a disk using FFT?

How to calculate the FFT of the function $f(x,y)$ defined on a disk ($\sqrt{x^2+y^2}<r$)? It seems all published FFT code deal with 2-dimensional Fourier transform for functions defined on ...
2
votes
0answers
23 views

A trajectory for shortened k-space data acquisition MRI

Given a real function $f:\mathbb{R}^n \to \mathbb{R}$, denote by $\hat{f}$ its Fourier Transform. I have shown that $\hat{f}(\vec \omega)=(\hat{f}(-\vec \omega))^*$ where $^*$ denotes complex ...
0
votes
1answer
28 views

Is the lagrange interpolation polynomials a linear functional?

Im taking numerical analysis and abstract algebra and I think that the Lagrange interpolation polynomials is a linear functional, I did notice that such polynomials are in the dual basis, since it ...
1
vote
1answer
26 views

system of First-Order ODES

I am looking at the following exercise: Consider the initial value problem $\left\{\begin{matrix} x''(t)=x(t)\\ x(0)=a\\ x'(0)=b \end{matrix}\right.$ Write it as a system of First-Order ODES with ...
1
vote
1answer
20 views

Thomas Algorithm for Tridiagonal System

A professor gave us an assignment to solve a Tridiagonal system using Thomas Algorithm. Here is the exercise: I am lost as to what to do with that $(0.2\pi)^2$ and do I just calculate the ...
3
votes
1answer
43 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
0
votes
0answers
9 views

Finding an analytic form of a function that satisfies asymptotic conditions

I have a family of functions that I obtain numerically. They depend on $x$ and parametrically also upon a certain parameter $L$. I would like to find an analytical form for this family of functions so ...
6
votes
0answers
48 views

fixed point iteration

I am trying to find the root of $f(x)=\arctan(x)$ by using successive iteration. There are some conditions to apply this in successive iteration . 1) The function has to be continuous. 2) ...
0
votes
0answers
16 views

Approximating an integral with a change of integral

(I have previously found out $x_1 = -\frac{1}{\sqrt{3}}$ and $x_2 = \frac{1}{\sqrt{3}}$ ) Approximate an integral using the 2-point rule, with an appropriate change of integral, to approximate ...
0
votes
1answer
12 views

How do I discretize a parabolic partial differential equation?

I have the following homework question: To keep my long sob story as short as possible, my awesome applied numerical methods teacher had a personal emergency and is replaced for the rest of the ...
1
vote
0answers
38 views

Name of this PDE: $\frac{\partial^2u}{\partial t^2}=\frac{\partial^3u}{\partial x^3}$

So I got an exercise to try some numerical methods on the following PDE: $$\frac{\partial^2u}{\partial t^2}=\frac{\partial^3u}{\partial x^3}$$ I tried to find some information about it, but I do not ...
2
votes
3answers
51 views

Why do I get a big relative error for my function? (Numerical Analisys - floating point)

When evaluating on the computer the following function: $$f(x)=\frac{x^2}{(\cos(\sin(x)))^2-1}$$ there is a big relative error for values $x\approx0$ (values very close to zero). I used the Taylor ...