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

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0
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2answers
26 views

Numerically stable version of calculation with cancellation

What's a numerically stable way to compute $$ \frac{2^{1/n}}{2^{1/n}-1} $$ for large (integer) $n$?
2
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0answers
16 views

Prove multidimensional Newton's method converge at least quadratically

Newton's method for root finding is simply $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. The following is a theorem from my textbook. where 6.1.22 is shown below Now I want to prove a similar ...
0
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1answer
996 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
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0answers
23 views

Find LU decomposition of a matrix using partial pivoting

I've the following matrix: $$ A= \begin{bmatrix} 0& 7& 5& 1 \\ 4& 3& 2& 1 \\0 &0& 0& 1 \\ 0& 0& -1& -2 ...
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0answers
10 views

Intermediate value theorem for fixed point convergence error

Hi I am trying to understand the convergence analysis for fixed point, and this is what I am not getting. So Let r be a root i.e r=g(r) Iteration Xk+1=g(Xk) Error=Ek=|xk-r| Ek+1=|xk+1-r|=|g(Xk)-r| ...
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0answers
13 views

How to numerically evaluate a integral whose limits are functions of x (using Gauss quadrature rule)?

I am trying to numerically evaluate an integral $\int_q^1 \ln (\sum_i \alpha_ix_i) dq$, in which $\ln (\sum_i \alpha_i x_i)$ is related to $q$ via the following: $z_i=(1-q)\frac{\alpha_ix_i}{\ln ...
12
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2answers
162 views

Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation?

In the biopic ``infinity'' about Feynman. (11:48~15:50) Feynman compute $\sqrt[3]{1729.03}$ by a mental calculation. I guess that he use the linear approximation. That is, he observe that ...
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0answers
43 views

Intergration $\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]$

I need to calculate the integral: $$\int^{\infty}_{0}\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]dx$$ I guess complex analysis is suitable for this integral, but I still have no ideas which kinds of ...
0
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5answers
66 views

Why, if a matrix $Q$ is orthogonal, then $Q^T Q = I$?

I was looking at the definition of an orthogonal matrix, which is as follows: Square matrix $Q$ is orthogonal if its columns are pairwise orthonormal, i.e., $$Q^TQ = I$$ Hence also ...
0
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1answer
24 views

If $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many solutions

I was reading this pdf: https://www.math.ohiou.edu/courses/math3600/lecture10.pdf and it tells you that if $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many ...
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3answers
3k views

When does the Newton Raphson method fail?

Can someone please tell me the conditions under which the Newton Raphson method will not converge? I looked around online, and couldn't find a general way to determine. For example, for the Fixed ...
0
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1answer
178 views

Upper bound for the error magnitude

For the function $f(x) = \mathrm{e}^x$ on the interval $[0,1]$, by using polynomial interpolation with $x_0 = 0$, $x_1 = 1/2$, and $x_2 = 1$, find the upper bound for the magnitude $$ \max_{0 ...
0
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0answers
19 views

Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
0
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0answers
11 views

Can we solve linear equations via EM algorithm?

I have coupled linear equations, written as $\vec{y}={\bf M}~\vec{x}$ in the matrix form, where $\vec{y}$ and ${\bf M}$ are completely known, so I would like to find an unknown vector $\vec{x}$. Of ...
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0answers
12 views

Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
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2answers
395 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
0
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0answers
29 views

method of undetermined coefficients and come up with a new quadrature.

I'm trying to solve some problems which is related method of undetermined coefficients to determine some weights and to come up with a new quadrature. the interval x∈[0,1]. given values of a function ...
1
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1answer
21 views

Can gauss quadrature integrate this function exactly $f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}}$?

Suppose I had a function $$f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}},$$ that I wanted to integrate on the interval $[\pi, 2\pi]$. Can Gauss quadrature of order $2$ (ie. with two points ...
0
votes
1answer
12 views

Admissibility condition for wavelets

The admissibility condition for a wavelet $\psi$ is: $\int \frac{|\hat\psi(x)|^2}{|x|} dx < \infty$, with $\hat\psi$ the Fourier transform of $\psi$. A necessary and sufficient condition should ...
1
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1answer
24 views

Numerical Method for KdV travelling waves

Can someone please direct me to the best numerical method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV ...
0
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0answers
9 views

Can we build a list of terms that indicate the numerical stability of problems?

Can we build a list of terms that indicate the numerical stability of problems? I read (but not fully understood why) that condition number gives an indication of numerical stability of linear systems ...
7
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2answers
16k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
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0answers
14 views

Significance of the complex conjugation symmetry of the DFT for real-valued input

For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that $$X_{N-k} = X_k^*$$ where * denotes complex ...
2
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0answers
52 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
0
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0answers
17 views

Compute the condition number of the matrix and show for what $\Delta x$ it is singular

Given the laplacian $N \times N$ matrix \begin{align*} A=\frac{1}{(\Delta x)^2}\begin{pmatrix} 2&-1& & &\\ -1&2&-1& &\\ &\ddots&\ddots&\ddots&\\ ...
0
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0answers
34 views

Mathematical conjectures for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
2
votes
2answers
49 views

Evaluating integral with a singularity.

I want to evaluate an integral numerically that contains one singularity. The software I use for this is Python. The actual integral I want to evaluate is quite long with a lot of other constants so I ...
0
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0answers
18 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that ...
1
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1answer
15 views

Accurate summation of mixed-sign floating-point values

Due to the finite representation, floating-point addition loses significant bits. This is particularly noticeable when there is catastrophic cancellation, such that all the significant bits can ...
0
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3answers
38 views

What does the function domain with letter C stand for?

I am reading a mathematics textbook on the subject of numerical analysis. In one theory the author says let us assume $f$ to be a function in $C^{n+1}[a,b]$. I understand that $[a, b]$ is the ...
2
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2answers
29 views

Reciprocal of a quadratic form

I am working with an expression of the form $$ \frac{x^TAx}{{x^TBx}}$$ and would like to simplify it. I understand that vectors do not have inverses, but viewing the bottom number as a 1 by 1 matrix, ...
0
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0answers
16 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
3
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2answers
98 views

Prove that $f:\mathbb{R}^2\to\mathbb{R}^2$ has a fixed point in a subset of $\mathbb{R}^2$

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ by defined by $f(x_1, x_2) = \begin{pmatrix} \frac{1}{3}x^2_2 + \frac{1}{8} \\\ \frac{1}{4}x^2_1 - \frac{1}{6} \end{pmatrix}$ and let $D = \{ x \in \mathbb{R}^2 ...
1
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2answers
59 views

Fixed point iteration question

This image was taken from this youtube video https://www.youtube.com/watch?v=OLqdJMjzib8 Since only one of them converges, how do we know in advance which formula to work with?
4
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1answer
64 views

Any math competitions dedicated to calculations by hand (college level math)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
0
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1answer
12 views

Error and residue in linear system

Consider the linear system $A x = b$ with a computed solution, what will be the relation between error and residue?
0
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0answers
10 views

Find analytically the sequence of iterations $x_n$ for Newton's method applied to the function $f(x) = x^2$ with the starting point $x_0 = 1$.

Find analytically the sequence of iterations $x_n$ for Newton's method applied to the function $f(x) = x^2$ with the starting point $x_0 = 1$? How would you do this, we can't use the recursive ...
0
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2answers
12 views

Implementing Adams-Bashforth of order 2 (AB2) algoirthm

Assuming we are given the initial condition for an ODE such that: $$ \begin{cases} x' = f(x,t) \\ x(t_0) = x_0 \end{cases} $$ We are going to solve it numerically using AB2. We know that the ...
0
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0answers
22 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
0
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0answers
18 views

reformulating a BVP into a system of first order ODEs

I need to convert the following bounded value problem $y'''+3y''+2y'^2-5y^2=1$ with conditions $ y(1)=1, y'(0)=1 $ and $y''(1)=0$ Into a system of first order Initial value problems in order to ...
0
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0answers
14 views

Connection between power iterations and QR Algorithm

I am seeking an intuitive understanding of why the QR Algorithm solves the symmetric eigenvalue problem. In class, and also in Golub and Van Loan, it has been suggested that there is somehow deep ...
4
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1answer
311 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
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0answers
10 views

$L_2$ norm of trigonometric interpolation

Let $I_N f(x)$ be the trigonometric interpolation of discrete function $f(x)$ with Fourier coefficient $g_l$. How can I prove this relation: $$\|f||_2=\|I_N f(x)||_2$$
1
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2answers
71 views

Solving equations like $xe^x = c$ via functional iteration

Yesterday I randomly thought of solving $xe^x = c$ via functional iteration (FI) after manipulating the equation into a form "$x = \cdots$" that gives the 'fastest' convergence rate regardless of the ...
15
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6answers
9k views

Calculate logarithms by hand

I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits. By pen and paper that is. I'm doing this old school. What first came to mind was to use $\log(ab) = ...
0
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0answers
6 views

CG as an orthogonal projection

I have heard that the Conjugate Gradient method can be viewed as an orthogonal projection onto the Krylov subspace $K(A,r_0)$, but I can't find a reference that deal with it in this way. Could you ...
0
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0answers
10 views

Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
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0answers
20 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
2
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1answer
46 views

Numerically Solve a Second Order ODE with singular coefficients

I need to solve the following numerically: $$xy''+y'+xy=x$$ with initial conditions $y(0)=0$ and $y'(0)=1$. I need the solution for $x:[0, 10]$. I've written the ode as a system of first order odes ...
0
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0answers
16 views

Help understanding a homework problem (Preconditioning matrices, numerical methods)

Below is a link to the problem (because I didn't want to have to go through the pain of TeXing it all out myself), the basic idea is we are supposed to be first showing that a specific matrix has a ...