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

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13 views

Local truncation error of a one-step method

I have the following one-step method: $y_{n+1} = y_{n} + \frac{1}{2}h(f(y_{n}) + f(y_{n} + h f(y_{n})))$ I need to calculate the local truncation error: $\sigma_{n+1} = y(t+1) - z_{n+1}$ where ...
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1answer
44 views

Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem: Real numbers, all are known ...
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0answers
14 views

Second order differential equation with unknown coefficient numerically

I have the following differential equation $$x''(t)+p(t) x'(t)=0,\qquad t\in[0,1]$$ I need to solve it numerically (find $x(t_i)$, where $\displaystyle t_i=\frac{i}{N}$ for $i=1,...,N-1$) using next ...
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1answer
11 views

Numerical methods question about Euler method solution notation?

My nonlinear dynamics prof posted a solution to a problem about the Euler method. I understand everything in his solution except for one statement he makes. Say we are looking at the ODE $\dot x = ...
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7 views

How to find a conservation scheme for a second order ODE.

Give a second order ODE: $y''(t)+y(t)+g(y)=0, t>0$, with initial data $y(0)=0,y'(0)=1.$ Define $$E(t)=\frac{[y'(t)]^2}{2}+\frac{y^2(t)}{2}+\int_0^{y(t)}g(s)ds.$$ How to find a second order finite ...
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1answer
12 views

Orders of data in Divided Differences and Lagrangian Interpolation

As we know that the order of data points i.e. x values do not matter in Divided Differences and The Lagrangian Interpolation. Why is that? What happens if we arrange them in order? better ...
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13 views

Half-precision floating-point conversion

if anyone can tell me how to calculate the decimal value if you know hex. I'm writing a program for a device where I get the value in hex ​​format and need to transfer them in decimal form. I know ...
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21 views

Specialized numerical method for transcendental equation

Is there any specialized, very fast, numerical method for solving equations of a type $$ e^{-px-q} = \frac{ax^2 + bx + c}{kx + l} $$ wher all $ a, p, q $ are strictly positive? To be more precise, ...
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2answers
39 views

Weierstrass Approximation Theorem

Does it matter what the interval is in the Weierstrass Approximation Theorem? Is it possible that the interval be any possible numbers within the function f(x)? How much the interval matter?
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1answer
30 views

Conditioning of the calculation of roots for cubic polynomial

Let $P(x)=x^3+qx+r$. I have to show that the calculation of the three roots $\lambda_i(q,r),i=1,2,3$ can be extremely ill conditioned. For this I looked at the implicit derivative of ...
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26 views

Solving ODE by finite differences and Newton's method.

Given this boundary value problem $y'' = (x^2(y')^2 - 9y^2 + 4x^6)/x^5, \quad 1 \leq x \leq 2, \qquad (1)\\ y(1) = 0, \; y(2) = \ln 256$ I have to solve the problem using finite differences, for 21 ...
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1answer
28 views

3D Finite Difference Matrix

I have been working with a finite difference code for the case in which my problem is axysimmetric. This means that all the code I have so far is for 2D In this case the coefficient matrix isn't ...
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1answer
26 views

Numeric calculation of partial derivative: proper sequence of operations?

I am calculating a second order mixed derivative by the following formula $$\frac{\partial^2 f(x, y)}{\partial x \partial y} \approx \frac{f(x + h, y + h) - f(x - h, y + h) - f(x + h, y - h) + f(x - ...
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31 views

Show that $\displaystyle\sum_{i=0}^r\frac{\gamma_i^{\star}}{r+1-i}=0$

Let $\displaystyle\gamma_i=\int_0^1\frac{s(s+1)...(s+i-1)}{i!}ds$ and $\displaystyle\gamma_i^{\star}=\int_{-1}^0\frac{s(s+1)...(s+i-1)}{i!}ds$ If ...
2
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2answers
43 views

Why Runge-Kutta methods cannot find the solution of Lorenz system?

The solution of the following Lorenz system s=10; r=28; b=8/3; f = @(t,y) [-s*y(1)+s*y(2); -y(1)*y(3)+r*y(1)-y(2); y(1)*y(2)-b*y(3)]; in the interval $[0,8]$ ...
2
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1answer
32 views

Solving a non-linear system of equations

Studying for finals I have come across a result that I understand how the system is derived but I cannot solve the system. I feel like this should be trivial, but I do not know where to go. Through ...
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1answer
31 views

How to find the zeros of an integral?

I am having trouble finding the roots of an integral. For example $F(c)=\int_a^b{(x^2-c^2)}dx$ for some finite interval $[a,b]$. The problem is that I am trying to do this using numerical analysis. ...
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1answer
29 views

Problem with computing numerical Gradient with Matlab

I am playing with numerically computing gradient of some function in matlab, and I have this weird result which I could not figure why. Here is my simple matlab code ...
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2answers
19 views

Absolute error in machine-precision terms.

I am trying to wrap my head around errors in floating point calculations. Let me denote absolute error as follows: $e = |x - \hat{x}|$, where $x$ is the exact number and $\hat{x}$ is its floating ...
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0answers
7 views

Can you give some information for rothe method

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
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1answer
10 views

How can I calculate the argument of amplification factor?

For example, I have an amplification factor of upwind scheme for hyperbolic conservation law, $$\lambda(k)=1-\nu(1-e^{-ik{\triangle}x})$$magnitude of which is ...
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1answer
53 views

Trying to show $\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$

I am using Laplace's Method and I am trying to show $$I =\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$$ $h(t) = -t$ has a maximum at $0$ and as it is a simple function there is no need to expand it. ...
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1answer
27 views

Fixed Point Iteration - Numerical Analysis

please help me solve the following question. Qsn: Solve the following system by Fixed Point Iteration. $$ x^2-2x+y^2-2y=3$$ $$x+y=-1$$ Progress: So I know that we have to assume one of the ...
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0answers
17 views

Introductory text on numerical analysis [duplicate]

I was wondering if anyone has a good suggestion for a textbook on numerical analysis. I am an undergraduate with little prior knowledge about topics in numerical analysis since I have never taken a ...
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0answers
20 views

Fifth order BVP using Daftardar Jafari method-Research paper example query?

Please check the research paper regarding numerical solution of non linear 5th and 6th order BVP using Daftardar Jafari Method (section 6: numerical results, example 1) ...
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0answers
22 views

Solution for a function using Taylor series

How shall I evaluate $0.7^{0.7}$ using the first five terms of the Taylor series for $\ln(1+x)$ and $e^x$?
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2answers
24 views

Finding the function when the Newton-Rapson formula is give.

The question is, "Show that the Newton-Raphson method of the form $$x_{n+1}= \frac{12x_n-5x_n^3}{8}$$ can be used to estimate $\sqrt{0.8}$. Show that this method will converge if the initial estimate ...
2
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1answer
30 views

Can we take negative step size in Euler's method?

Thus far we've taken the step size $h$ to be positive, and therefore we've developed solutions to the right of the initial point. Is Euler's method valid if we use a negative step size $h<0$ and ...
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6 views

Get stuck with some statements of convergence rate of the iteration method from “Iterative methods for sparse linear systems (2nd edition) ”

Here are the statements I get from the book and the two highlight parts are what I can not understand well. The questions are: Why we can conclude that $\rho=\rho(G)$ from"the above analysis"? ...
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20 views

Interpolation question about polinomials?

Let $f(x)=x^n$. show that for each $n$ distinct point $x_0,... x_n$ we have $f[x_0,...x_n]=0$ and $f[x_0,...,x_{n-1}]=\sum_{i=0}^{n-1}x_i$. also show that if $x_{i+1}-x_i$ is fixed for each $i\ge 0$ ...
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0answers
7 views

What is the difference between zero and absolute stabilty?

In numerical ODEs, what is the difference between these two types of stability in multistep methods? From what I can gather, zero stability places a bound on how much future approximations can ...
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1answer
49 views

Recursive formula for integration by parts of given functions

I need to find, if it actually exists, a recursive formula for the following evaluations of indefinite integrals: \begin{align} I_{1,n}(x,R) &= \underset{n \,\text{terms}}{\underbrace{\int dx ...
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23 views

Least Square Apply Non-Linear Function

I am study numerical methods and I see that question. Considering that $f(1)=0.6065, f(1.5)=0.8825, f(2)=1, f(2.5)=0.8825, f(3)=0.6065.$ Utilizing the method for least squares, approximate the ...
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1answer
20 views

A Theorem about Interpolation Method?

I have a question about interpolation. I think that question is a theorem, but I don´t find nothing about that. Anyone can help me? Show that, if $g$ is the polynomial of degree $m<n$ that ...
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0answers
14 views

Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
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0answers
13 views

How to work with difference-of-elements penalty in optimization

I am trying to solve the optimization problem $$\min_{H,S>0} \|W(H+S)-X\|^2_F+Q(H)+\eta\|S G\|_F^2$$ where $X\in\mathbf{R}_+^{m\times T}$, $W\in\mathbf{R}_+^{m\times k}$, ...
1
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1answer
27 views

Not able to use fzero function in Matlab

I am new to Matlab. I am trying to solve a non-linear equation using this inbuilt Matlab function called fzero() but it's not giving me the results. The main file ...
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0answers
52 views

SOLVED Implementing Euler's Method step

Given: $$ \frac{dy}{dx} = -2y + 4e^{-x} $$ $$ \frac{dz}{dx} = -\frac{yz^{2}}{3} $$ $$ y(0) = 2, z(0) = 4, x\in[0,1],h = 0.2 $$ Do I have to implement the step of h = 0.2 any differently for z'? That ...
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0answers
10 views

Proving if the graph of F is over it's tangent line at all points then Newton Raphson converges.

Here's the problem: Let $F : \mathbb{R} \rightarrow \mathbb{R} \,, F \in \mathbb{C}^1$ so that $F'(x) < 0 \, \forall \, x$ and there's a unique $r$ so that $F(r)=0$ and let $L_{x_0} (x)$ be the ...
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0answers
23 views

A lower bound for the condition number of Hilbert Matrix

I am trying to prove that $\operatorname{cond}_{\infty}(H_n)\ge n^2$ for all natural $n$, where $H_n$ is the Hilbert matrix in $\mathbb R^{n\times n}$, i.e. $$(H_n)_{i,j} = 1/(i+j-1)$$ I am trying ...
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0answers
32 views

Approximate solutions for quintic equation

The other day I asked a question in here about deriving the equations $$u^2\left(\left(1-s_1\right)+3u+3u^2+u^3\right) =\alpha\left(s_0+2s_0u+\left(1+s_0-s_1\right)u^2+2u^3+u^4\right),$$ where ...
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0answers
17 views

Integrate for 3 step method

I need to integrate $$\int_0^{3h}\frac{(t-t_{i-1})(t-t_{i-2})}{2h^2}f(t_i,y(t_i))- \frac{(t-t_i)(t-t_{i-2})}{h^2}f(t_{i-1},y(t_{i-1}))- \frac{(t-t_{i-1})(t-t_i)}{2h^2}f(t_{i-2},y(t_{i-2})) dt$$ where ...
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1answer
88 views

Finding minimum point of banana function using Newton's Method

I am using the banana function $F(x_1,x_2)=(1-x_1)^2+100(x_2-x_1^2)^2$ over $x_1,x_1 \in \Re$. I am using $f_1(x_1,x_2)=0, f_2(x_1,x_2)=0$ to find the minimum point of $F$ which it is (1,1). Now I ...
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0answers
32 views

Matlab project - Jacobi method for tridiagonal matrices…

I have to do a project in Matlab to my University and I don't quite understand what I should do. I was given script that solves systems of equations with Jacobi's method with given tolerance and ...
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3answers
73 views

Prove that $x^3 -3x^2 +6 = 0$ has only one real root

I know that if I take the derivative of $$x^3 -3x^2 +6 = 0$$ and prove it is always greater than zero, I'll find that this functions is always increasing, and therefore if I find an interval where ...
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1answer
20 views

Integral over the unit ball in $\mathbb{R}^n$

Let $f(x)=|x|^r$ on $B_1(0)$ real valued function.Where $B_1(0)$ is the unit ball in $\mathbb{R}^n$. I am trying to show that if $r>1-n$ f has a weak derivative. ATTEMPT: I know from the ...
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1answer
61 views

Newton's method convergence implementation

How can I solwe this problem: Experimentaly examine convergence Newton's method for conformation: \begin{align} 2x^3-y^2-1=0 \\ xy^3-y-4=0 \end{align} for various loaded inputs with start points ...
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1answer
17 views

Large error after factoring h from fourth-order Runge–Kutta method

Consider $$\frac{dy}{dx} = 2x-y, \qquad y(0)=1$$ That has an exact solution $y(t) = 2t+3 e^{-t}-2$. But I want a numerical solution. So I decided to use forth-order Runge–Kutta method ...
0
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1answer
35 views

Solve a viscous Burgers' equation with a Newton-GMRes method

I implemented a preconditioner for a GMRes method. To test this preconditioner I want to solve this one dimensional viscous Burgers' equation $$\partial_t u(x,t) + u(x,t) \partial_x ...
1
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0answers
25 views

3 step ODE method using interpolating polynomial

I am trying to find a 3-step method of the form $y_{i+1}=y_{i-2}+b_0f_{i-2}+b_1f_{i-1}+b_2f_i$ to solve the ODE $y'=f(t,y)$ by using an interpolating polynomial and then finding $\int_0^{3h} ...