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

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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
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0answers
21 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 ...
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1answer
38 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
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0answers
63 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 ...
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1answer
34 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
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1answer
16 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 ...
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2answers
48 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 ...
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2answers
23 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 ...
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1answer
29 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
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1answer
39 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 ...
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0answers
16 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 ...
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1answer
53 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 ...
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0answers
48 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.
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0answers
26 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
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1answer
19 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 ...
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0answers
40 views

What is he easiest way to approximate γ as a decimal number?

What is the easiest way to give the numerical value of the Euler-Mascheroni constant? The mathmetical way to give that value? Thanks lot!
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2answers
40 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
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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
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2answers
40 views

Lotka-Volterra model with two predators

In this, Lotka-Volterra model, we have two predators: $$\frac{dp}{dt} = ap\left(1-\frac{p}{K}\right) - (b_1q_1+b_2q_2)p$$ $$\frac{dq_1}{dt}=e_1b_1pq_1-m_1q_1$$ ...
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1answer
40 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 $, $x_0=0.808$ ...
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1answer
25 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 ...
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0answers
24 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: ...
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1answer
22 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 ...
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2answers
41 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}} ...
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1answer
36 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 $ ...
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2answers
58 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
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1answer
9 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
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0answers
29 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 ...
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1answer
32 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
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1answer
27 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 ...
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1answer
31 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
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1answer
46 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} ...
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0answers
10 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
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0answers
53 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) ...
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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 ...
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1answer
16 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 ...
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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 ...
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3answers
54 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 ...
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1answer
26 views

Advice to solve a system of 8th order univariate polynomials

I am struggling to solve a least square problem in which the tedious part is the initialization. Grid search methods are out of question. The initial problem I've stated my problem in a previous ...
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2answers
32 views

If root is very near to the max/min, what happenns with Newton Raphson method? Does it diverge?

If root is very near to the max/min, what happens with Newton Raphson method? Does it diverge? Or converges slowly? I know if some iteration involves a stationery point then we can not go further. But ...
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2answers
33 views

Newton-Cotes formula problem

Please help me to solve this problem... By the method of undetermined coefficients I found $a=c=1/6$ and $b=2/3$ and $\alpha=\gamma=2/3$ and $\beta=-1/3$. Also that both are exact for polynomials of ...
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0answers
18 views

(xλ)→(Ax−λxxxT−1) write down Newton's method for this equation

this question is taken from Rainer Kress (numrical Analysis). i coud not translate this question into newton's method form. Because there is matrix and vector. it is hard to take derivation of this ...
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1answer
27 views

The order of accuracy of the implicit Euler method is equal to $1$

I want to show that the order of accuracy of the implicit Euler method is equal to $1$. That's what I have tried: We have the initial value problem $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a ...
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1answer
23 views

Backward Euler method- How do we get the approximation?

Approximating $y'(t^n)$ at the relation $y'(t^n)=f(t^n,y(t^n))$ with the difference quotient $\left[\frac{y(t^{n+1})-y(t^n)}{h} \right]$ we get to the Euler method. Approximating the same derivative ...
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1answer
19 views

How does one bound computational error for a finite difference approximation of the second derivative?

I'm trying to wrap my head around ways to minimize total computational error (defined as a sum of the bounds on the truncation and rounding errors) by taking a differentiable function $f : \mathbb{R} ...
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0answers
26 views

Solving system of delay differential equations

Are there any numerical methods for solving systems of delay differential equations with time-dependent delays? For example, I have a system: $$\frac{dP_1}{dt} = f_1(t) P_2(t-\tau(t)) P_3(t)$$ ...
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1answer
24 views

Help with method(s) show an iterative method converges to a known fixed point

Are there any general techniques that can be used to show that an iterative method converges to a (known) fixed point?. In my current situation, I know the exact fixed point, but I am unaware of a ...
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53 views

How to solve this using computer.?

Given $B = \begin{pmatrix} 0.3 & 0 \\ 0 & 0.4 \\ \end{pmatrix}$, and $\pi = \begin{pmatrix}0.4\\0.6\end{pmatrix} $, I need to find the elements of the stochastic matrix (the rows sum to ...
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0answers
18 views

$A(\theta)-$ stable method, region of absolute stability

We have to look for numerical methods for the numerical solution of $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ that have 'great' regions of ...
1
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1answer
31 views

Proof of an alternate Matrix Condition Number Representation

I'm currently looking over a section in my textbook on Matrix Condition Numbers and it's given the definition $cond(A) = ||A|| \cdot ||A^{-1}||$ but it's also equated this definition of a condition ...