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

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1answer
78 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
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3answers
78 views

How to turn a decimal into a number to divide something by into it.

So here is what things will convert to: 0.5 = 2; 0.25 = 4; + MILLIONS MORE 1 = The whole of a number ( / 1 ) 0.5 = Half of the number ( / 2) But what is the math to convert decimals into only a ...
3
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1answer
48 views

Finite differences and conservation law

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_{n+1,j} ...
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0answers
79 views

Matlab Newton's Method Non-linear system

There is something wrong with this program and I cannot seem to find it. I am trying to calculate the solution of a non-linear system using Newton's method. Matlab keeps saying there is a problem with ...
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1answer
33 views

Solving numerically a non-linear equation.

How is the more appropriate numerical method to solve the equation $$\cos(2\pi x)+\cos \left(\frac{2\pi N}{x}\right)=2,$$ for a given $N$? Notice that if $N \in \mathbb{Z}$, then $x\mid N$.
3
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1answer
79 views

solution of multidimensional PDE

I'm looking for a way to find a solution 'f' to the following PDE. $$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + ...
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2answers
68 views

Is there a proper way to prove that $f:[a,b] \to[a,b]$

Is there any proper way to know whether a function has the same domain and range $[a,b]$ where $a,b<\infty$ i.e. $f:[a,b] \to [a,b]$ ? For example: $$ f(x) = e^{−x} ,\qquad [\ln(1.1), \ln(3)] $$
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2answers
60 views

prove that $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R

Can anybody prove $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R using the fixed point iteration theorem?
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1answer
29 views

Show that$ |e_n| \leq 2^{-(n+1)}(b_0 - a_0)$

I would like to know if someone can shed some light on it.I'm not sure but I think Lipschitz or contraction mapping theorem is involved. Let $x_n = \frac{a_n + b_n}{2} , r=\lim_{n \to \infty }x_n$ ...
2
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1answer
48 views

matlab program help

Wanting to write a matlab program to solve the following iteration: $x^{(k+1)}=b+\alpha\begin{bmatrix}2&1\\1&2\end{bmatrix}x^k,k=0,1,2,\cdots$ where alpha is a real constant. Find the values ...
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1answer
57 views

How do I solve for the zeros of a Chebyshev polynomical? (on a computer)

I am working on a computer program and have a method that returns a number for a given $x$, $y$. So $f(x, y) = z$, where $f$ is my method. if I know $y$ and $z$, can I find what $x$ will be, without ...
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0answers
50 views

How to solve one differential equation with two independent variables in heat transfer.

$$A\frac{ \partial T_a}{\partial t}=B(T_p-T_a)+C(D-T_a)-E\frac{\partial T_a}{\partial x}$$ Where $A, B, C, D, E$ are constants, $t$ is time and $x$ is $x$-axis of the box in which heat transfer is ...
2
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1answer
43 views

condition number for a matrix with a variable

How would I go about calculating $cond(A)$ for A= $\begin{bmatrix}1 & c\\c & 1\end{bmatrix}$, $|c|\neq 1$
2
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1answer
194 views

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
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0answers
30 views

polymonial of $n+1$ degree inequality

Let: $$p(x) = x^{n+1} + a_{n}x_{n} + a_{n-1}x^{n-1} + \cdots + a_0 $$ Show $\lVert p \rVert_{\infty}^{[-1,1]} \geq \left(\frac{1}{2}\right)^{n}$ Obviously my first naive approach is induction. ...
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0answers
48 views

Show that Newton’s Method is well-defined for all k and converges to 0 for $x_0>0$

Let $f : R → R$ with $f$ twice continuously differentiable, $\gamma > f''(x)>\delta, f(0)=0,f'(x)>\rho $ for $x ≥ 0$. Show that for any $x_0 > 0$ that Newton’s Method is well-defined for ...
0
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1answer
172 views

Incomplete Cholesky decomposition conjugate gradient method in Matlab

I have a problem in finding the numerical material that describing in detail for incomplete Cholesky combined with conjugate gradient method by using Matlab. Someone can help me? Many thank in ...
2
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0answers
125 views

Calculating gradient from finite difference results

I am solving the steady-state heat equation in two dimensions: $$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial ...
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1answer
29 views

Functions for interpolation

Do we always need to be given points to do interpolation? Or can we be given only a function? For lagrangian interpolation we require points, and does it apply for others also?
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1answer
316 views

Need some facts about Newton-Schulz iterative method and its application to sparse matrices

I am studying Newton-Schulz iterative method for obtaining an approximate inverse , which is given by $V_{k+1}=V_{k}(2I-AV_{k})$, wherein $I$ is the identity matrix and it converges, when the ...
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0answers
54 views

Numerical integration formula

I'm looking for a numerical method/scheme which can be used to solve the following equation $$ \int_{t_n}^{t_n+h} \sin\left((t_n+h-s)\omega\right) g\left(x(s)\right)ds = \frac{1}{2\omega} \sin^2(h ...
3
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2answers
92 views

Examples of orthonormal bases for $L^2[0,1]$ that are not trigonometric?

What are examples of orthonormal bases for $L^2([0,1],dx)$? For instance, the following trigonometric polynomials are orthonormal basis $$\left\{1, \sqrt{2}\sin(2\pi jx),\sqrt{2}\cos(2\pi j ...
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2answers
498 views

minimum number of iteration in Bisection method

One root of the equation $e^{x}-3x^{2}=0$ lies in the interval $(3,4)$, the least number of iterations of the bisection method, so that $|Error|<10^{-3}$ is (a) 10 (b) 6 (c) 8 (d) 4
2
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1answer
58 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
26 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 ...
0
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1answer
17 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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0answers
13 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
31 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 ...
1
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0answers
60 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
84 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?
0
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1answer
40 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 ...
1
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0answers
165 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 ...
1
vote
1answer
377 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 ...
0
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1answer
61 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 - ...
2
votes
2answers
290 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
45 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 ...
1
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1answer
192 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. ...
0
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1answer
272 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 ...
1
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2answers
51 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
14 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
27 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 ...
0
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1answer
82 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. ...
0
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1answer
73 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
24 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 ...
2
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2answers
45 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
332 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 ...
0
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1answer
212 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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0answers
25 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 ...
1
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1answer
29 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 ...
1
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1answer
131 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 ...