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

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11 views

Good derivation of Romberg's method?

I'm confused about the Romberg's method even after viewing numerous explanations of it. I understand the "high-level" view of what's happening, but not how it's derived/proven. Can anyone recommend a ...
1
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3answers
39 views

Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
2
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0answers
273 views

Universal Approximation Theorem — Neural Networks

Universal approximation theorem states that "the standard multilayer feed-forward network with a single hidden layer, which contains finite number of hidden neurons, is a universal approximator among ...
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0answers
17 views

Nonlinear system and Optimization: Unconstrained Optimization

Let $f(x) = \frac{1}{2} x^T Q x + b^T x + c.$ Prove that Newton's method finds a critical point after a single iteration. Here $Q$ is positive definite. For this: $\nabla f(x) = $ If $x^*$ is a ...
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2answers
16 views

Substituting $t=\tan(\theta)$ to $\int_0^{\infty} \frac{\cosh[(1+t^2)]^{-1/2}]}{1+t^2} dt $

How does one substitute $t=\tan(\theta)$ to $$\int_0^{\infty} \frac{\cosh[(1+t^2)]^{-1/2}]}{1+t^2} \,dt \, ? $$
5
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1answer
78 views
+50

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

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, ...
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2answers
33 views

Is my solution correct ?

Determine a fixed-point function $g$ in the interval $[0,1]$ that produces an approximation to a positive solution of $$3x^2-e^x=0$$ So I would rearrange and make $x=ln(3x$^2$)$ and then go on to do ...
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0answers
12 views

Gauss quadrature on arbitrary interval

I'm trying to solve the following exercise, unfortunately I'm not really sure, how to solve it: Derive for two nodes on $[a,b]$ the Gauss quadrature formula from the formula ...
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1answer
11 views

Why does Euler-Maruyama method use a square root of the time step

Euler-Maruyama method is supposed to be an extension of the Euler method for ODE, but applied to SDE. This means that if we have an equation: $$ dY_t = Y_t dW_t $$ where $W_t$ is the Wiener process, ...
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0answers
11 views

Matlab - Solve PDE (method of lines) with an ODE boundary condition depending on a different compartment

I am modelling simple onedirectional 1D diffusion in one compartment using method of lines in Matlab. However, at x=n, I'd like to impose a boundary condition that depends on the exchange of the ...
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0answers
20 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
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26 views

Is my approach correct

Since there are 6 degrees of freedom, then $$\array{f(x)&=&ax^6&+&bx^5&+&cx^4&+&dx^3+ex^2+fx+g \\ f'(x)&=& &&6ax^5&+&&&\cdots \\ ...
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0answers
14 views

Find the best line [on hold]

Given the equation: $$[(k, a + exp(b k) sin (c k) + d/k )]; k = 1 : 10. where$$ $$a=7, b=1, c=14, d=27$$. How do i find the best line and also write the points $(Xk, Yk)$ to be fit? Any little ...
1
vote
1answer
38 views

Please explain the last step of this newton method for system of equations

The step of working out x$^1$. I know the above is the formula but do they actually work out the inverse of the derivative matrix, is there a quicker way to do this?
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0answers
19 views

Computing a Green's function - where did I go wrong?

This is from a homework problem that was recently returned to me in a numerical analysis course. The grader even noted that he didn't know where I went wrong but the solution was marked as incorrect. ...
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1answer
11 views

Where am I going wrong in my cubic spline, work out a,b and c

So I know $f^{'}_1(x)=6x+3ax^2$ and $f'_2$(x)=6x+3bx^2$ and $f^{"}_1(x)=6+6ax$ and $f^{"}_2(x)=6+6bx$ Now, $f_1(0)=0$ and $f_2(0)=c$, therefore $c=0$ And $f^{"}_1(0)=0$ and $f^{"}_2(0)=0$ But ...
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0answers
30 views

Multistep Method: Gear's Formula Interpolation

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.
3
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2answers
214 views

Looking for finite difference approximations past the fourth derivative

I scanned the internet and could not find further representations of the central difference approximations past the fourth derivative. Are there published results past the fourth derivative? Ideally ...
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0answers
8 views

Can I use Euler approximation for backwards computation?

I want to solve the following Ricatti Equation: in Matlab. I'm given the final value of P and as far as I understand I have to go backwards solving this. My question is, can I use the Euler ...
0
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1answer
42 views

Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...
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1answer
20 views

How to get the exact solution $x^*=x_n-f(x_n)\frac{x^*-x_n}{f(x^*)-f(x_n)}$ from Newton's method? [on hold]

In Newton's method of finding solutions of nonlinear equations: $f(x^*)=f(x_n)+\frac{f(x^*)-f(x_n)}{x^*-x_n}(x^*-x_n)$ how can we get the exact solution: ...
0
votes
1answer
29 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 ...
1
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1answer
30 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 ...
2
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1answer
22 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 ...
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2answers
34 views

Numerically stable version of calculation with cancellation [on hold]

What's a numerically stable way to compute $$ \frac{2^{1/n}}{2^{1/n}-1} $$ for large (integer) $n$?
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
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| ...
0
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0answers
14 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
171 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
48 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
72 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
votes
1answer
179 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 ...
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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 ...
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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
397 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
31 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 ...
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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 ...
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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 ...
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2answers
17k 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
15 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 ...
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0answers
55 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 ...
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0answers
18 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&\\ ...
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0answers
35 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
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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 ...
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0answers
20 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
16 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 ...