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

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Gauss-Jordan elimination in the form of (A|I)

So Gauss-Jordan elimination can be performed through the form of $(A|I)$ where $I$ is the identity matrix. We carry out row elementary operations as usual until the matrix becomes the form $(I|B)$, ...
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0answers
12 views

Diffusion of a chemical species inside a Y-shaped tube

I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration ...
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4answers
51 views

How do I find the error of nth iteration in Newton's Raphson's method without knowing the exact root

In our calculus class, we were introduced to the numerical approximation of root by Newton Raphson method. The question was to calculate the root of a function up to nth decimal places. Assuming that ...
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16 views

numerical solution of drift diffusion equation

0 down vote favorite in this link (in semiconductor physics section) you can see four coupled equations. do you know that finite element method is more accurate for discretization and numerical ...
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3 views

How to know the rate of convergence of a majorization - minimization algorithm?

The basic idea of majorization-minimization (MM) principlein optimization is to convert a hard problem (for example, non-smooth) into a sequence of simpler ones (for example smooth). To minimize ...
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1answer
24 views

Reduce this third order ordinary differential equation to first order to use Runge Kutta

The ODE I'm workin with is $$\dddot{x} + t^2\ddot{x} + 4x = 0$$ with $$x(0)=1, \dot{x}(0)=0, \ddot{x}=-1$$ I've written a very basic program in C++ to use the RK4 method to approximate a solution to ...
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0answers
27 views

Sparse Matrices and Tridiagonalization.

Assume that we are given a sparse matrix,let it be 90*90(1000*1000), would you say that a vector with lots of zeros(let it be 90*1(1*1000),and 65(500) zeros are there),is a smart option to initialize ...
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0answers
13 views

Stick breaking point (discretized ODE)

I cannot find nontrivial solutions to the following problem. Let $x\in[0,1]$ and $y(x)$ be the deflection of the stick. Then this is described by the diff.eq.: $$\alpha^{-1} P y(x)+y(x)''=0 $$ where ...
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2answers
227 views

Deriving formula for derivative

I have a formula in my book for differentiating numerically. $$f'(x_0)=\frac{1}{12h}[-25f(x_0)+48f(x_0+h)-36f(x_0+2h)+16f(x_0+3h)-3f(x_0+4h)]+\frac{4}{5}f^{(5)}(\xi)$$ I was wondering if anyone ...
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0answers
17 views

Convergence of quadrature formulas and interpolating polynomials

There is a theorem of Polya (1933), which says: 1) If a interpolatory quadrature formula converges for all continuous functions on [a, b] and quadrature weights are all positive, then the formula ...
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0answers
15 views

Finding 1st,2nd and 3rd derivative for funtion of 2 variable

$E=g(p,v)$ $\frac{dp}{dv}=F$ $\frac{dE}{dv}$=$g_pF+g_v$ $\begin{align}\frac{d^2E}{dv^2}&=(g_pF+g_v)_pF+(g_pF+g_v)_v \\ &=g_{pp}FF+g_pF_pF+g_{vp}F+g_{pv}F+F_vg_p+g_{vv} \end{align}$ ...
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0answers
21 views

What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...
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0answers
14 views

Mean value theorem for sequences

This is a problem I am trying to solve. Given a sequence $x_n$ defined $x_{n+1}=F(x_n)$. Assume $\lim_{n \to \infty}x_n=x$ and $F'(x)=0$. Need to show that $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
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0answers
27 views

Trapezoidal rule - Multivariable

If I wanted to integrate the function $f(x,y)$ over the region $[a,b]\times[c,d]$ with two segments, am I going about this the right way? $$I(f) = \int_a^b \int_c^d f(x,y)\ dy\,dx = \int_a^b g(x) \ ...
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3answers
159 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1(x)_n dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1) dx$$ to five significant digits. I've used Mathematica, ...
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0answers
75 views

Evaluating a product (and sum) of polynomials in MATLAB [on hold]

I'm new to MATLAB (and programming in general) and there's something I've been having a lot of trouble with. I want to evaluate the Lebesgue function with MATLAB. The function is as follows: $ L(x)= ...
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0answers
31 views

Definite integral of a hypergeometric function of an imaginary argument

How would one deal with such an integral? $$\int_0^\infty\frac{e^{-n r}}{r}{}_1F_1(i/k+1;2;2i kr) \, \mathrm{d} r$$ Here $F$ is the confluent hypergeometric function, $n\in\mathbb{N}$ and $k>0$ ...
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0answers
27 views

What is the best method to solve the ill-conditioned non-linear systems? [on hold]

What is the best method to solve the ill-conditioned non-linear systems? for example: $$ x^2 − 2x + 3y = − 1 \\ 2x^2 - 3.9999x + 6.0001y = - 1.9999 $$
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31 views

If a function is well defined and continuous can it have singularities?

In home work I was given this question: Consider the function $f(x) = xe^x - 2$; We want to study the properties of $f(x)$ so that we can apply numerical methods to solve the equation $f(x) = 0$ ...
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1answer
34 views

Simpson's 3/8 Rule

When deriving Simpson's 1/3 Rule, I used a second order polynomial $P(x) = Ax^2 + Bx + C$, and integrated over the region $[-h,h]$ Integrating gave me: $ \ \dfrac{h}{3}(2Ah^2 +6C)$ I evaluated ...
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3answers
128 views

How can I solve this equation $x^{x^{x^{x^{.^{.^{.}}}}}}-a=0$

I always use the Newton-Raphson Method if I want to find the roots of any equation as follow $$x_{1}=x_{0}-\frac{y_{0}}{y'_{0}}$$ But I don't know how to use this method if the equation takes the ...
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1answer
28 views

What is the relation between analytical Fourier transform and DFT?

First of all let me state that I searched for this topic before asking. My question is as follows we have the Analytical Fourier Transform represented with an ...
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2answers
29 views

Evaluating differential entropies with Matlab: NaN issue

With Matlab I am trying to evaluate differential entropies. These are integrals like $$\int_\mathbb{R} p(x) \log (p(x)) \mathrm{d}x$$ where $p(x)$ is a probability density function. My $p(x)$ is ...
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1answer
13 views

Derivation of $f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$

I have the following function: $$f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$$ I would like to calulate the numeric root of: $n\pi, n\ge0.$ In order to do that, I want to use ...
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0answers
25 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
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1answer
8 views

Using divide difference formula find the value of $f\left[x_{0},x_{1},x_{2},…,x_{10}\right]$

Consider the polynomial $f(x)=x^{10}+x-1$ , $x\in \mathbb R$ & let $x_{k}=k$ for $k=0,1,2,...,10$. Then the value of the divide difference $f\left[x_{0},x_{1},x_{2},...,x_{10}\right]=$ (a) $-1$ ...
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2answers
22 views

Give an estimate for the error.

Use the first three nonzero terms of Taylor’s formula for $\sin x$ to find an approximate value for the integral $\int_0^1 \frac{\sin x}{x}$ and give an estimate for the error.(It is understood that ...
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0answers
26 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...
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0answers
11 views

Is there a formula of coefficients of Newton-Cotes Method in numerical intergation?

We know the coefficients of Newton-Cotes method in numerical integration are: 2-points $ 0.5$ , $0.5$ 3-points $ 1/6$, $2/3$, $ 1/6$ 4-points $1/8$, $3/8$, ...
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0answers
21 views

Lipschitz Constant (Burden and Faires Exercise)

There's an exercise in Burden & Faires Numerical Analysis book, Section 5.1 #2a, where they appear to want the reader to verify that a Lipschitz constant exists for the following ODE: ...
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1answer
16 views

polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
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0answers
18 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
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0answers
20 views

Name of method which includes Taylor linearization inside fixed point iteration

I read paper about Horn-Schunck multiscale method for computing optical flow Core part of this algorithm is minimizing some functional. One part of functional contains nonlinear term inside L2 norm. ...
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0answers
26 views

What is the best method to calculate the square root when I know that the root is always an integer?

I have been through the wikipedia page, but wanted to know if there was a preferred (most efficient) method when there is an exact solution to find?
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0answers
29 views

Power method convergence explanation. [closed]

A sufficient condition for the power method to converge for a given diagonalizable matrix A is that the eigenvalues of A satisfy: $|\lambda_{1}|>|\lambda_{2}|\geq...\geq|\lambda_{n}|$ If this ...
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0answers
113 views

Can gradient descent solve this problem $\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2$?

How can I find the (approximate) solution to the following problem: $$\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2,$$ where $Var(.)$ denotes the variance? $A$ is matrix and $b$ and $x$ are ...
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2answers
18 views

The formula of the order of multistep methods

How can I derive this $$(1+\xi) \left(1+\frac{1}{2}\xi-\frac{1}{12}\xi^{2}\right)+O(\xi^3)$$ from $$\frac{1+\xi}{1-\frac{1}{2}\xi+\frac{1}{3}\xi^{2}}+O(\xi^3)$$ ? The whole formula is below. This is ...
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19 views

With the secant method, how can we ensure the constraint to prove super=linearity?

I know that as long as the first derivation does not equal to 0, then the secant method is super-linear. However, we're not typically given the derivative in things such as MATLab. How are we ...
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23 views

Efficient method for refining parameters in nonlinear curve fitting

I have time-series electrical current data $i(t)$ with transient steps in it which are convoluted with the hardware filter used in data acquisition. As a result, the real steps in current, which would ...
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1answer
26 views

spline derivation

Assume the following representation for cubic splines with $T$ interior knots is given. Let $g(Y)=\sum_{j=0}^3 \alpha_j Y_j+\sum_{t=1}^T \gamma_t (Y-\zeta_t)_{+}^{3}$ where $(Y-\zeta_t)_{+}:= ...
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29 views

Numerical Integration of $\int^{t_i}_{t_i-\Delta t}\frac{e^{-\frac{a}{t_n-\tau}}}{\sqrt{t_n-\tau}}d\tau $ for heat conduction problem

I am looking for a quadrature method to accurately evaluate the integral: $$I=\int^{t_i}_{t_i-\Delta t}\frac{e^{-\frac{a}{t_n-\tau}}}{\sqrt{t_n-\tau}}d\tau $$ Where $a$ is a positive constant of the ...
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1answer
38 views

Calculate the divide difference $f[1,2,3,4]$

Let, $f:[0,4]\to \mathbb R$ be a three times continuously differentiable function. Then the value of the divide difference $f[1,2,3,4]$ is (a) $\frac{f'(\xi)}{3}$ , for some $\xi \in (0,4)$ (b) ...
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0answers
15 views

LU-factorisation of a square matrix

I need to show that the following matrix cannot be factor into the product LU. \begin{equation} A=\begin{bmatrix}1&2&-1\\2&4&0\\ 0&1&-1\end{bmatrix} \end{equation} I did the ...
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0answers
18 views

Saturation Modeling in ODE45

I have a machine with an arm that can move in a linear one dimensional way. There are 3 limits on the arm: The arm has boundary for its location $(x_{min},x_{max})$ The arm has limit on its velocity ...
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1answer
67 views

How to solve the equation $\int_0^{t}\frac{1}{200+4(x+1)\arctan{\left(\frac{x+1}{100}\right)}}dx=1$

Let $l(x)=200+4(x+1)\arctan{\left(\frac{x+1}{100}\right)}$. I want to find real number $t>0$ such that $s(t)=l(t)$, where $s'(x)=\dfrac{l'(x)}{l(x)}s(x)+1$, $s(0)=0.$ It is a first order linear ...
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1answer
46 views

Matlab numerical integration involving Bessel functions returns NaN

I need to numerically compute integrals such as this (some parameters omitted for simplicity): $$ \int_{0}^{\infty} e^{-x^2} I_{0}(x) K_{0}(x) \mathrm{d}x $$ where $I_{0}$ and $K_{0}$ denote the ...
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0answers
24 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve $Ax = b$, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous ...
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1answer
26 views

Proving $\Delta^nx^n=n!h^n$.

How can I prove $\Delta^nx^n=n!h^n$. Here $\Delta$ is forward difference and h is the step size. I used induction . When $n=k$ assume the result is true. $$\begin{align}\Delta^{k+1}x^{k+1} &= ...
1
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1answer
42 views

Use Newton's method to find root for the following equations

I have to use Newton's method to find the roots with accuracy $10^{-5}$ of the following equation : $e^{x} + 2^{-x} +2\cos x -6 =0$ in the interval $(1,2)$ So $f'(x)= e^x - [2^{-x}]*[\log(2)] ...
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1answer
69 views

Help for Integral and evaluating - Eikonal equation

Hy guys I'm reading a paper of "Finding Exact Solutions to the Two- Dimensional Eikonal Equation" - E.D. Moskalensky. link for the paper: ...