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

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4
votes
3answers
149 views

Robust Numerical ODE Solver?

I made a little explicit Runge-Kutta 4th order solver a few days ago, but when testing it against various 1st and 2nd order ODEs chosen at random (for example $d^{2}y/dt^{2} = -y \sin(y)$, ...
1
vote
0answers
56 views

Sequence and intermediate value theorem

For part (a), By intermediate value theorem, there exist c between 0 and 1 such that $f(c) = 0$ Now, I supposed that there also exist d between 0 and 1 such that $ f(d) = 0 $ and $ c \neq d $ I am ...
1
vote
0answers
32 views

Find the real roots

I have the following equation: \begin{equation} (y-1)^a - C~~ y~~ \exp(b x)=0 \end{equation} where $a, b$ are real constants, $C$ may be a complex number. I need to find the real solution of the ...
3
votes
1answer
318 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
0
votes
0answers
44 views

Condition under which newton raphson converges

I see in a book that under the following condition newton-raphson method (for finding the zero of a function) converges: 1) The function is continuously differentiable 2) The function is positive in ...
0
votes
1answer
104 views

Numerical computation of the logarithm of the generalized incomplete Beta function

Define $\mathrm{B}(x,y;\alpha,\beta)$ as the generalized incomplete beta function: $$\mathrm{B}(x,y;\alpha,\beta) = \int_x^y t^{\alpha-1}(1-t)^{\beta-1}\mathrm{d}t$$ You can assume that ...
2
votes
1answer
62 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
3
votes
1answer
233 views

Numerical verification of solution.

I have the non-linear equation \begin{align} & \left( -\frac{1}{4}\left({\frac { \left( 4\,{x}^{3}+2\, ex \right) ^{2}}{ \left( {x}^{4}+e{x}^{2}+f \right) ^{3/2}}}\right)+\frac{1}{2}\left({ \frac ...
1
vote
0answers
32 views

Taking the limit of a derived function

I need to evaluate an expression of the form $$ \int_0 ^ a dx \left[\frac{\partial}{\partial \alpha} \left[ \frac{\partial^n}{\partial \beta^n} \left[\frac{\partial}{\partial \gamma} ...
1
vote
1answer
2k views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
3
votes
1answer
340 views

Fourier Transformation: an Animated GIF

Here I found the animated GIF below. I don't get it! Would someone explain it please?
0
votes
1answer
362 views

Proof for multivariate Newton-Raphson method

How can the proof for Newton's method for a single variable be extended to the multivariate version? Forgive me if this is trivial, but I don't seem to get it. Any links or proofs would be greatly ...
2
votes
1answer
113 views

Can I compute this integral analytically?

I will give a small background and explain the variables and the system first. I have two images which are observed and are constant and we can treat them as continuous functions and I will call them ...
1
vote
0answers
96 views

Optimal numerial method for optimization of “Rosenbrock Banana”-like function

Which numerical methods would be optimal to find an extremum of a function with an almost flat "valley" (but a single minimum in the middle of the valley)? In this context optimal means the least ...
1
vote
0answers
51 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
1
vote
0answers
69 views

Numerical algorithm to solve quadratic eigenvalue problem.

Given the equation $$-4 \left(a^2+a (n-1) (2 t^2-1)\right) \left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2 \\ +\frac12 \left(\sum _{i=0}^n \alpha_i t^{2 i}\right) \left(t \left(8 a (t^2-1)+1\right) \sum ...
1
vote
0answers
28 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
1
vote
1answer
268 views

Integrating a discrete 3D surface, in spherical coordinates

I have an matrix which contains height information for a sheet suspended in air. Like a checkerboard, each value in the matrix represents a sampled height. Here's the hard parts: the data in the ...
1
vote
2answers
73 views

Euler method application: step size

Suppose we have a system of ODE's: $a' = -a - 2b$ and $b' = 2a-b$ with initial conditions $a(0)=1$ and $b(0)=-1$. How can we find the maximum value of the step size such that the norm a solution of ...
1
vote
0answers
32 views

How to establish a lower bound on this difference operator?

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
1
vote
2answers
982 views

Is there a general formula for estimating the step size h in numerical differentiation formulas?

Using three-point central-difference formula $$ f^{\prime}(x_0)\approx \frac{f(x_0+h)-f(x_0-h)}{2h} $$ and for $f(x)=\exp(x)$ at $x_0=0$ we have $$ \begin{array}{c, l, r} h & f^{\prime}(0) ...
0
votes
1answer
59 views

How to verify the gradient of a symbolic function using numerical gradient?

I have a function $f$, which takes as inputs a three arrays and returns an array. I have written a symbolic function $g$ to calculate the gradient of this function and I want to verify that it ...
3
votes
1answer
98 views

Numerical integration fails

I am doing something wrong. This is my algorithm to evaluate the integral $$\int_0^1 \frac{1}{1+x}dx= \log(2).$$ with the Newton Cotes algorithm (Simpson and 3/8). Both give me that for large n ...
0
votes
1answer
150 views

Computational complexity of numerical integration of gaussian function

$ \int^{b}_{a} \exp(-x^2)\,dx$. I have the following two questions regarding the above integral expression of the Gaussian function: Is there a numerical method we can use to solve the above ...
1
vote
1answer
207 views

What is a tensor-product Chebyshev grid?

What is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?
0
votes
1answer
345 views

Using Finite Difference to compute derivative in the Newton-Raphson root finding Algorithm

In the Newton-Raphson method we come across the following equation: $$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}$$ Can you please let me know if we can calculate the derivative term like this - $$f'(x_n) = ...
1
vote
1answer
34 views

Difference between derivative and its approximation

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
7
votes
1answer
593 views

Fast algorithm for approximating Eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16}$x$2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather ...
1
vote
1answer
34 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
1
vote
1answer
50 views

Differentiating the $QR$ decomposition?

Let $A(t)$ be a smooth family of invertible $n \times n$ matrices with $A(0) = I$, and let $A(t) = Q(t) R(t)$ be the $QR$ decomposition. Given $\dot{A}(0)$, what is an algorithm to compute ...
1
vote
1answer
140 views

Euler's method vs midpoint method

Are the following methods equally accurate and if not, why? Using Euler's method with a step size of $h$. Using the midpoint method with a step size of $2h$. Even though Euler's method has a ...
6
votes
3answers
195 views

optimal way to approximate second derivative

Suppose there is a function $f: \mathbb R\to \mathbb R$ and that we only know $f(0),f(h),f'(h),f(2h)$ for some $h>0$. and we can't know the value of $f$ with $100$% accuracy at any other point. ...
2
votes
1answer
78 views

Change of variable

I have to approximate the following integral, using Simpson's Composite $1/3$ Rule: $\displaystyle \int\limits_{0}^1 \mathrm{\frac{e^{2x}}{\sqrt[5]{x^2}}}\,\mathrm{d}x$. The only problem is that ...
10
votes
2answers
477 views

Why is the numerical solution of this equation unstable? Is this equation stiff?

I am trying to solve the following equation with an explicit fourth-order using the Runge-Kutta method: $$y' = t(y - t \sin t)$$ with initial conditions $y(0) = 1$ over the interval $[0, 10]$. The ...
1
vote
0answers
30 views

characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
1
vote
0answers
55 views

Continuation fixed points of parameter dependent Newton

Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in ...
1
vote
1answer
97 views

Numerical evaluation of the first (K) and second (E) complete elliptic integrals

To get a numerical evaluation of the first (K) and second (E) complete elliptic integrals: $$K(k)=\int_0^1\frac{dt}{(1-t^2)^{1/2}(1-k^2t^2)^{1/2}}, \ \ \ \ \ ...
0
votes
1answer
33 views

Interpolation based on $n$ uniformly distributed points

We are given $n+1$ uniformly distributed points in the segment $[0,1]$: $x_i=\frac{i}{n}$, $i=0,1,...,n$ and a function $f(x)=e^{-x}$ $P(x)$ is the interpolation polynomial of $f(x)$ where ...
0
votes
2answers
177 views

Numerical precision of product of probabilities (normal CDF)

I'm trying to calculate $\prod_k{p_k}$ where $p_k$ are (potentially) very high probabilities of independent, zero-mean, standard normal random variables and $k>100$. However, I'm running into ...
2
votes
1answer
43 views

Numerical computation of power differences: $x^a - y^a$

I want to calculate a power difference, $x^a - y^a$, where $a$ can be large, and the numbers $x,y$ are of similar magnitude. What's a sound numerical way to approach this? Note: $x,y,a$ are all ...
1
vote
1answer
34 views

Computing $\mathrm{B}_{x,y}(\alpha+1,\beta) / \mathrm{B}_{x,y}(\alpha,\beta)$ numerically

I need to compute numerically ratios of the form: $$\frac{\mathrm{B}_{x,y}(\alpha+1,\beta)}{\mathrm{B}_{x,y}(\alpha,\beta)} \tag{1}$$ where $\mathrm{B}_{z_1,z_2}(\alpha,\beta)$ is the incomplete ...
1
vote
1answer
122 views

Finite Element method Implementation

I have written a program for the finite element method for an elliptic one dimensional problem.Initially I assumed a mesh that had only 10 points , but since the error was far above my tolerance ...
1
vote
1answer
344 views

Solving Differential Equations theoretically and using matlab

i am trying to solve the initial value and elliptic boundary value problems below. but now i need some help solving them using matlab. for the elliptic problem, any method is ok, but for the initial ...
1
vote
1answer
799 views

Newton's method linear convergence proof

How would you show that if f'(a)=0 then the Newton's Method is linear convergent when 1. $f''(a)\neq 0$ 2. $f''(a)=0, f'''(a) \neq 0$? I am having some trouble getting it to the point where you can ...
0
votes
1answer
119 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
0
votes
0answers
45 views

Interpolation using four nodes

Suppose there are four points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ my target is to interpolate any point $x_I$ between $x_2$ and $x_3$. Is there any Interpolation method which gives linear ...
1
vote
0answers
50 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
0
votes
1answer
60 views

Normalizing a dataset from the interval [0,1] with fixed properties.

So I have a rather large dataset where values are from the interval $[0,1] \in \mathbb{R}$. But the problem is that a big portion of the values are extremely close to $0$. So firstly I'm looking for ...
0
votes
1answer
157 views

Plot as you go in MATLAB

I'm self studying some numerical analysis and I'd like to get a feel for how an ODE solver speed varies as you move forward in time. Is it possible to use matlab to numerically solve an equation ...
3
votes
1answer
56 views

Where comes the +1 from in this formula?

I'm working on two papers ([1] equation 8, [2] equation 2.3) and I can't figure out why there is an identic formula I can't explain on both. $p(z) \in \mathbb{C}[z]$ is a monic polynomial with simple ...