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

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Lagrange interpolation for ellipse

Consider the ellipse $$\frac{x^2}{4} + \frac{y^2}{2} =1$$ The line integral $I$ of the ellipse in the first quadrant is $$I=\int^2_0 \Big[ 1+(y'(x))^2 \Big]^{1/2} dx$$ Find the cubic polynomial ...
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40 views

Convergence of iteration scheme of solving matrix equations

Consider the equation $A\mathbf{x}=\mathbf{b}$. It is equivalent to $$S\mathbf{x}=(S-A)\mathbf{x}+\mathbf{b}$$ where $S$ is a splitting matrix. We now consider the iteration scheme ...
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16 views

The Heuristic Gauss-Kronrod Based Error Estimator in Quadpack

I'm trying to understand the local error estimate that Quadpack (and subsequently other libraries such as GSL, quadpack++, cubature, etc.) uses for it's general adaptive quadrature subroutine QAG. The ...
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1answer
29 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
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15 views

How to numerically solve a Green's function using mathematica? [migrated]

Suppose we have a Green's function $$LG(x)=\delta(x),$$ how to numerically solve it by mathematica? Can the mathematica read the delta function directly? For analytical calculation, we usually ...
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1answer
17 views

Numerical Analysis - Upper bound for $|R|$

I am asked to find an upper bound for $|R|$ valid for all $x\in[-1,1]$ that is independent of $x$ and $\xi$. Given that, $$R(x)=\frac{|x|^6}{6!}e^\xi$$ for $x\in[-1,1]$ where $\xi$ is between $x$ ...
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47 views

A difficult question about Runge-Kutta method and Euler's method

I found this question extremely difficult... Especially part b. For part b, I've tried several times. Is it correct that we need to use Taylor expansion to solve it? Since $e'(t) = f(...) + ...
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1answer
18 views

Problem applying Simpson's rule

I am having a problem applying composite Simpson's rule for the integral $$I=\int_0^2\dfrac{1}{x+4}dx$$ with $n=4$. The exact value of the integral is about $0.405$, however, Simpson's is giving ...
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34 views

Minimize/Maximze a function against its approximation.

Let $f \in C^{\infty}[1,2]$ be a function we would like to approximate, let be $g$ such approximation, you can assume $g$ is a spline function (at least quadratic). In literature I have seen that a ...
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18 views

Characteristic of Euler- Tricomi equation

Consider the Euler-Tricomi equation $$u_{xx}-xu_{yy}=0$$ Determine its characteristics. I know the characteristics is $x dx^2=dy^2$ from wikipedia, can anyone show me how to get it?
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25 views

In practice what is (modified) Gram Schmidt used for?

Modified Gram-Schmidt is known to be numerically less stable than methods like Householder orthogonalization and also not quite as fast at approximately $2mn^2$ flops. So in practice do we ever use ...
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26 views

Numerically solve a set of “coupled” ODE

I am looking for some numerical methods to solve a set of coupled ODE, coupled in the sense that $$ \frac{d\alpha}{dt} = F\left( \alpha(t), \beta(t), \frac{d\beta}{dt}, t \right) $$ $$ ...
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11 views

Iterative method to find a solution

Here is the equation i tried to solve and only got one answer : $101.33=x.exp(1.045(1-x)^2).74.218 + (1-x).exp(1.045x^2).101.05$ And $exp(1.045(1-x)^2)=g1$ , $exp(1.045x^2)=g2$ Which they show ...
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19 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
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1answer
9 views

Adding two functions represented by a table of values with a different step size?

Let $f(t)$ be some numerically obtained $T$-periodic function represented by a table of values over one period or a set of points $(t, y)$ with a time step $\Delta t.$ Now let's change the ...
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1answer
51 views

Rounding unit vs Machine precision

I'm not sure if this question should be asked here... For a general floating point system defined using the tuple $(\beta, t, L, U)$, where $\beta$ is the base, $t$ is the number of bits in the ...
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1answer
24 views

on a characterization of convergent matrices

Let $A\in \mathbb R^{n\times n}$ a matrix. It's known that the following statements are equivalent: 1) $A$ is convergent, namely $\lim_{k\to\infty}(A^k)_{ij}=0$ 2) $\lim_{k\to\infty}||A^k||=0$ for ...
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2answers
29 views

Understanding convergence of fixed point iteration

I was reading some slides explaining the convergence of the fixed point iteration, but honestly I'm not seeing or having an intuitive idea of how fixed-point iteration methods converge. Assuming ...
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17 views

Order of convergence Secant method

As we know, the order of convergence of the secant method is 1.618... But, when I try to find the convergence const of the function: f(x)=(1+x^2)^(1/3)-1, it's not convergence... it's going to ...
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1answer
24 views

eigenvalues lesser than 1 implies affine maps are eventually contractive

Consider $(\mathbb R^n,d)$ where $d$ is the Euclidean metric. A map $w:\mathbb R^n\to \mathbb R^n$ is said $\textbf{contractive}$ if there exists $0<s<1$ such that for every $x,y\in \mathbb R^n$ ...
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1answer
26 views

Determining the most appropriate set of eigenmodes for a modal decomposition of an experimental data set

I have a complex vector of the transverse amplitude and phase distribution of a laser beam, derived from experimental data. When modelling these field distributions, ordinarily the eigenmodes of the ...
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1answer
31 views

Understanding this trigonometric identity $\frac{n}{2} (2 \cos \frac{\alpha_n}{2} \sin \frac{\alpha_n}{2}) = \frac{n}{2} \sin \alpha_n$

I was looking an example that motivates rounding errors using the quadrature of a circle (?). And at a certain point there's this identity: $$\frac{n}{2} \left( 2 \cos \frac{\alpha_n}{2} \sin ...
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How to find the ground energy state solution in a quantum harmonic oscillator?

Recently, I came across a question which asks to solve the Schrödinger equation for a harmonic oscillator on $ [a, b] $ : $-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = ...
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24 views

Eigenvalue equation and the diffusion equation

I am running some finite-element software on Matlab that generates solutions to the diffusion equation over a punctured, rectangular domain. This is the same as the $k \times k$ matrix system ...
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2answers
39 views

In practice, what does it mean for the Newton's method to converge quadratically (when it converges)?

I was studying about the Newton's method (and other root-finding methods) and apparently Newton's method converges quadratically (or more) when it does. Suppose that the sequence $\{x_k \}$ ...
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1answer
43 views

Computing square roots with arithmetic-harmonic mean

We know that if we iterate arithmetic and harmonic means of two numbers, we get their geometric mean. So, basically if we need to compute the square root of $x$: $$\sqrt{x}=\sqrt{1 \cdot ...
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24 views

Which method among bisection, Newton's and secant is the best for this function?

I've an exercise where I need to decide which root-finding method to use among bisection, Newton's and secant given a certain function. The function is defined in the interval $[0, 4]$ and looks like ...
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0answers
25 views

Solve the equation by Jacobi's method : $5x + 2y +4z = 12$ ;$x+4y=2z=15$ ;$x+2y+5z=20$

Solve the equation by Jacobi's method : $$5x + 2y +7z = 30$$ $$x+4y+2z=15$$ $$x+2y+5z=20$$ I am getting the values as $$x=4.08,y=3.375,z=4.12.$$ My equations : $$x=\frac{1}{5} \{30-2y-7z\}$$ ...
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1answer
28 views

How are floating-point numbers logarithmically distributed?

From what I remember from a lecture I had of a course I'm attending called "introduction to computational science", floating-point numbers are distributed logarithmically. What does it mean? And how ...
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2answers
23 views

Help with calculating relative error in approximation of x

So i have $$x=\displaystyle\sum_{k=1}^{\infty}2^{-k}+\displaystyle\sum_{k=0}^{\infty}2^{-6k-1}$$ and i need to calculate relative error when approximating above x in $$MARC-32 \ \dots ...
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2answers
48 views

Computing Bezier curve of a high order

I have a set of ten points that much be used to compute a bezier curve. As you are probably aware, computing a bezier curve of order 9 is a very strenuous activity. I need it in polynomial form. I ...
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0answers
23 views

Is this finite difference approach correct? [migrated]

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. and so, the two equations I have are: $$ \frac{\partial u}{\partial x} = 0$$ $$ \frac{\partial P}{\partial x} = ...
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1answer
29 views

How many iterations are required to reduce the convergence error by a factor of 10?

I've the following function $$g(x) = x^2 + \frac{3}{16}$$ for which I found the two fixed points $x_1 = \frac{1}{4}$ and $x_2 = \frac{3}{4}$. I noticed that the fixed-point iteration $$x_{k+1} = ...
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15 views

Bounds on the size of Voronoi cells

I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain $D$ is partitioned according to the Voronoi cells $D_1,\dots,D_n$ with Voronoi ...
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1answer
73 views

Newton's method for square roots 'jumps' through the continued fraction convergents

I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square ...
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2answers
28 views

Find if a fixed-point iteration converges for a certain root

I'm asked to find if the fixed-point iteration $$x_{k+1} = g(x_k)$$ converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$. ...
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61 views

Deriving Milne's predictor of order four from extrapolation polynomial

I would like to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = ...
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47 views

Derivative error for Lagrange interpolation

I was reading a book and I found this (with some context): if $f(x)=L(x)+R(x)$, with $L$ the quadratic interpolation with three points $x_0, x_1$ and $x_2$, then $R(x)=\dfrac{f'''(\epsilon(x))}{6} ...
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15 views

Numerical Multivariate Integration with respect to a Multivariate Gaussian

I have the following likelihood which is basically the convolution between the product of m independent Poisson densities and a multivariate Gaussian density: $L(\mathbf{y}) = ...
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1answer
36 views

Numerical Integration error for simpson's rule through taylor series

I am looking at the derivation of the simpson's rule as well as an error analysis in my textbook and I am slightly confused over two things. Firstly, in the derivations, the text uses a taylor ...
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12 views

Blow-up from PDE or aliasing error? How to know and check?

I have this PDE: $u_{t}= u_{xx}+g(t)\cdot u_{x}^{2}$ (with periodic boundary conditions in the interval $(-\pi,\pi)$) that I don't have an exact solution for or know if it blows up. However, by ...
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1answer
26 views

Calculate what function is approximated with the Lagrange Polynomial

I would like to find out what the sum estimates and prove that it estimates that function. $$\sum_{j=0}^ml_j(x)*x_j^k=?$$ From the Lagrange interpolation polynomials we know that $$l_k(x) = ...
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1answer
32 views

drawbacks of non positive definite matrices

I'm currently writing a report on the Element-free Galerkin method, which is a meshless method to solve PDES. In the method the stiffness matrix K(which is positive definite) has to be modified to ...
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Drawing uniform samples from the *range* of a non-invertible function

I am looking for a Bayesian technique to draw samples from a uniform distribution over the range of a non-invertible (that is, there isn’t even a formula) function $\mathbf{f}: \mathbb{R}^N ...
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1answer
20 views

Numerically solve differential equation with limits of integration

I am trying to solve the following equation, $c$ is a constant and $f(0)$ is known. I've never solved a differential equation that used the limits of integration and I'm unsure how to proceed. It ...
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0answers
12 views

Error of Chebyshev Interpolation

I am trying to find the error when using Chebyshev interpolation on the Runge function, but I am having trouble understanding how to do this. Specifically, I would like to use points for n from 10 to ...
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1answer
46 views

eigenvalues lesser than $1$ implies contractive map

Consider $(\mathbb R^n,d)$ where $d$ is the Euclidean metric. A map $w:\mathbb R^n\to \mathbb R^n$ is said contractive if there exists $0<s<1$ such that for every $x,y\in \mathbb R^n$ we have ...
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0answers
11 views

Numerical scheme for first order partial differential equation.

Does anyone know some numerical methods for solving such equations? $$ \begin{cases} V_t + \langle V_x,\, F(t, x) \rangle = 0\\ V(0, x) = h(x) \end{cases}, \quad 0 \leqslant t \leqslant T, $$ where ...
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13 views

Trapezoidal rule (differential equation) is not symplectic

Trapezoidal rule $y_n = y_{n-1}+\frac12h(f(y_n+y_{n-1}))$ is not symplectic. I have no clue to prove the claim. Can anyone give me some hints? Thanks for your time.
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2answers
105 views

Backward Euler's Method

This question was asked in CSIR. please help me to find out correct choice Let $y(t)$ satisfy the differential equation $$y'=\lambda y;y(0)=1$$. Then the backward Euler method for $n\geq 1$ and ...