Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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54 views

Galerkin Method: Why Set the Residuals to Zero?

I don't understand why the Galerkin method weighs the residual by the shape functions and sets it equal to zero. I'd like to know the reason why. Any intuitive explanation would be greatly appreciated....
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72 views

Conjugate Gradient

Why sometime our conjugate gradient routine never reaches the stop condition even if the result is correct? As stop condition we use the following: $$\delta > \epsilon^2 \delta_0$$ To avoid ...
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145 views

extension of trigonometric functions as basis functions to higher dimensions

Trigonometric functions forms an orthonormal basis functions for $L^2[a,b]$, with corresponding normalization coefficients. I want to know if this result can be extended to higher dimensions. For ...
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29 views

Best way to fit an equation for the given graph

I have 450 pair $(x,y)$ of data. The plot is like this: I need to fit an equation: $y=f(x)$ for the given data, and to find out values of $y$ when $x=500$. Now, my question is: What kind of ...
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56 views

Composition of two functions in normed spaces

Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||DF(x)^{-1}||...
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36 views

Properties of carry in base $b$ multiplication

Consider $n$ bit numbers $A$ and $B$. Let they be represented in base $b$. When you multiply $A$ and $B$ using school multiplication: $(1)$ how many carry propagations can one expect? $(2)$ what ...
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362 views

Nearest-neighbor interpolation

I read in a book that the nearest-neighbor interpolation results in a function whose derivative is either zero or undefined. Can anyone explain what does it mean when the derivative of a function is ...
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70 views

How to solve an inverse of derivative ode

How can I solve $$(\phi'(y))^{-1}=y^{-c_1}+y^{-c_2},$$ where $c_1,c_2$ are constants and $(\cdot)^{-1}$ is inverse? Since I have inverse of derivative and it's nonlinear I think it has to be done ...
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287 views

Finite difference numerical differentiation

I needed to find an O(h2) method to find f'''(x). Using Taylor expansions, I found: $$f'''(x)=\frac{f(x+2h)-2f(x+h)-2f(x-h)+f(x-2h))}{2h^3} + O(h^2)$$. However, I have also found that: $$f'''(x)=\frac{...
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474 views

Relative Error with Respect to Frobenius Norm

I'm look at this tiny book called "Deblurring Images: Matrices, Spectra, and Filtering" by Hansen, Nagy, O'Leary. This is a self study, but I believe my question is broad enough so that it can be of ...
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29 views

Calculating bounds interpolation and approximation

If I have a function $f(x)=e^x$ and nodes $x_0=a-h/\sqrt(3)$ and $x_1=a+h/\sqrt(3)$ to linearly interpolate $f(x)$ on the interval $[a-h,a+h]$ for some real numbers $a$ and $h$, $h>0$. How do I ...
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66 views

Having trouble grasping curve fitting

I have an exam coming up next week in my Applied Numerical Methods class. Our professor gave us a list of about 12 things that we need to be able to do for the exam, all of which are pretty ...
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101 views

When can definite integration be numerically computable?

under what condition,can the integration $$\int_{\Delta}f(x_1,x_2,\dots,x_n)dx_1dx_2\dots dx_n, \text{where } \Delta \text{ is integration domain defined by function},f(x_1,x_2,\dots,x_n) \text{ ...
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43 views

Is the optimization problem right?

If we want optimize the following problem $$ \min_x \{a(x)+c(x)\} $$ and we have $$ a = \min_y b(y) $$ then, could we directly optimize the following problem? $$ \min_x \{b(x)+c(x)\} $$
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304 views

Interpolation polynomial types

I was wondering if both the Maclaurin and Taylor series are two types of interpolation polynomials? I was under the impression that they were not because they only go though one point in an interval ...
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109 views

Expression of the exponential integral $E_1$ using standard functions with real arguments

In standard numeric packages (for C++) the function $$E_1(z)=\int_1^\infty \frac{e^{-zt}}{t}dt$$ is only implemented for real arguments. For a specific calculation I need to be able to evaluate this ...
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241 views

V-shape of error function of numerical derivative vs. analytical derivative

I'm given the following function: $$f(x) = \frac{x^2}{\sin(x)}$$ and I'm supposed to derive the derivative numerically at the point $x=1$ with the following central difference quotient: $$\frac{f(...
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621 views

Cubic convergence of itearative method

thank you for your time at first! It's my homework, so I don't expect answer with result, only some hint. With given iteration method $$x_{n+1} = \frac{x_n(x_n^2 + 3U)}{3x_n^2 + U} $$ show cubic ...
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94 views

Explanation of the Leibniz formula

I am reading the book Solving Ordinary Differential Equations I - Nonstiff Problems (1987) by Hairer et al. My question is from Section II, chapter 2 (Order conditions for RK methods), equation 2.4. ...
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600 views

Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $$y''(x)+ e^{y(x)} = 0, \quad \lambda = 1, \quad x \in(0,1),$$ $$y(0) = y(1) = 0$$ I have to solve this using the tridiagonal ...
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812 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
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47 views

Compute the following norm derivatives

I was wondering if anyone can explain me how to compute the derivatives of the following norms: $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$ $\bigtriangledown ||u(x)||^...
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69 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
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227 views

How to solve this using gauss jordan method?

I am trying to solve the following equation using gauss jordan method but unable to solve due to the type of equations.At the end i am getting unwanted zeros in 2nd and 3rd row.Here is my work... <...
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154 views

Rewriting partial differential equation

I have some trouble rewriting a partial differential equation, more specifically the heat equation in one dimension: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t)\\ $ ...
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620 views

Evaluate the following function using as many significant figures as required to get a final result of 4 digits accuracy

I need to evaluate $$ f_5(0.2) = 5! \left[ e^{0.2} - \left( 1 + (0.2) +\frac{(0.2)^2}{2!}+\frac{(0.2)^3}{3!} + \frac{(0.2)^4}{4!} +\frac{(0.2)^5}{5!} \right) \right] $$ using as many as required ...
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1answer
68 views

Iterative Scheme-Programming Matlab

I don't know if this is going to seem like a dumb question, I am new to this and to matlab, but I'm trying to construct an iterative scheme in MATLAB to compute $\sqrt(b)$ for a given b>0, and program ...
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1answer
109 views

Euler method inequality

Given the problem for $t\neq0$ and $t\le1$ $y'(t)=y^2(t)$ $y(0)=1$ Let $\mu>0$, and $\epsilon_n=\frac12(f(t_{n+1},y_{n+1})-f(t_n,y_n))$, such that $|\epsilon_n|\le\mu|y_n|$ is ...
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48 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep method for the second order ...
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53 views

How to estimate the error of a numerical multiple integration

I'm integrating over the wholes space the function $$f(\vec{r_1},\vec{r_2})=\exp{\bigg[-(r_{1\alpha}+r_{1\beta}+r_{2\alpha}+r_{2\beta})\bigg]} \cdot 1/r_{12}$$ where $\vec{r_\alpha}=(-R/2,0,0), \vec{...
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357 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. Example:...
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144 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 $|\alpha|&...
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126 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 ...
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2k 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 $x=b$....
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42 views

Alternate expression for finite summation

"How many arithmetic operations are required to directly compute $$y=1+x+x^2+...+x^{1023}$$ Use a formula for the sum to come up with an alternate expression for $y$, and show that only 10 ...
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232 views

Stuck on condition number derivation of the perturbed equation $(A + \Delta)\tilde{x} = b + \delta_b$

I've almost got what I want. We start with $Ax = b $ and $(A + \Delta)\tilde{x} = b + \delta_b$. What I have then is \begin{align*} \tilde{x} - x &= -A^{-1}\Delta\tilde{x} + A^{-1}\delta_b \\ \|\...
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1answer
127 views

How can I determine the base of the following numbers for the operations to be correct?

Given: 24)A + 17)A = 40)A How can I find the base of the following number (A) so the operations are correct? NOTE: I am not sure what topic this would fall under. Hence sorry for any misplaced ...
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57 views

Numerical analysis Taylor $1/(1-x)$

How does accuracy depend on the degree of the Taylor Polynomial and the distance from the point its being expanded about (say $x=0$). So I'm considering the function $f(x) =\frac{1}{1-x}$ centered at $...
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1answer
3k views

Graphical estimate of convergence rates?

I am studying some numerical optimization methods, but I am not an expert in numerical maths. Question: If the convergence rate is linear, then the logarithm log(x_n) of the distance x_n to the ...
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51 views

Show that $\phi'(x)=0$

Let $ f \in C^2([a,b], R)$ and $$\phi(x)=x-\frac{f(x)}{f'(x)}$$ Such that 1.$ f'(x)\neq 0, \forall x \in [a,b]$ $\exists \alpha \in [a,b]: f(\alpha)=0$ Show that $\phi'(x) = 0$ My attempt: $$\...
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87 views

relation between euler function and divisors $n$

Please hint me. Let $S$ be the number non trivial divisors $n$. prove that $ S<\phi(n)+1$. $\phi$ is euler function. hint: we know $ \sum_{d|n}\phi(d)=n$ so $ \phi(1)+\phi(n)+\sum_{1,n\not=d|n}\...
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37 views

A sequence of intervals- Trying to find a fixed point -

This might be a trivial question but I couldn't come up with a clever trick,theorems or whatnot. Suppose $I_0=\left[\frac{1}{h_0},\frac{1}{l_0}\right]$ where $h_0=1$ and $l_0=\frac{1}{2}$. Given $k_0\...
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747 views

Alternative to the Gram-Schmidt Procedure for Orthogonalization

I was wondering if there is an alternative to the Gram-Schmidt procedure, which instead of being a successive orthogonalization scheme, would be non-successive (simultaneous)? In other words, is there ...
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1answer
43 views

Question about “Magnifying” an inequality for big-oh analysis

Here's an example directly from my numerical analysis text book: $$ a_n = \frac {n+1}{n^2} $$ The goal is to find the convergence rate. So we know, $\lim\limits_{n \to \infty} \frac{n+1}{n^2} = 0$ ...
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240 views

Solving a non-linear set of equations with non-exact constant values

I have a set of nonlinear equations which are related to a physical system. The constants of these equations (light hand side values-LHS) are determined through some measurement methods. It means that ...
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1answer
419 views

Ordinary Differential Equations divergence of successive approximations

I searched this and couldn't find my question on here so here it goes: This is an example from a text I'm reading and I was hoping someone could shed some light on my misunderstanding. Let $y'=2x-2\...
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1answer
674 views

min/max in front of a sigma sign (Numerical Analysis)

I have an assignment in my Numerical Analysis class that involves an approximation of an integral with a sum. The sum looks to me like a Riemann sum, but I don't know what the "min" in front of the ...
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477 views

Proof of theorem about iterative methods

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is ...
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1answer
44 views

Initial value problems with known solutions?

I'm trying to find a list of IVPs with known solutions to test my implementation of some numerical techniques. The only one I know of is: $$f(x,y)= y' =-\lambda y\;,\;\;\; y(0)=1$$ with the ...
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2answers
199 views

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & a_{...

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