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

learn more… | top users | synonyms (2)

0
votes
0answers
15 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 ...
0
votes
0answers
11 views

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

$E=g(p,v)$ $\frac{dp}{dv}=F$ $\frac{dE}{dv}$=$g_pF+g_v$ $\begin{align}\frac{d^2E}{dv^2}&=(g_pF+g_v)_pF+(g_pF+g_v)_v \\ &=g_{pp}FF+g_pF_pF+g_{vp}F+g_{pv}F+F_vg_p+g_{vv} \end{align}$ ...
0
votes
0answers
20 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 ...
2
votes
0answers
13 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 $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
2
votes
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) \ ...
0
votes
0answers
17 views

plotting equilibrium conditions

According to an equilibrium model I am trying to understand: The stock market is in equilibrium when $$D_L(p,p_s)+D_S(p,p_s)=N$$ and lending market is in equilibrium when ...
-2
votes
0answers
63 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)= ...
2
votes
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 $k>0$ ...
1
vote
0answers
26 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: $$ x^2 − 2x + 3y = − 1 \\ 2x^2 - 3.9999x + 6.0001y = - 1.9999 $$
0
votes
0answers
30 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 $f(x) = 0$ ...
0
votes
1answer
31 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 ...
3
votes
3answers
126 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 ...
0
votes
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 ...
2
votes
2answers
27 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
1answer
7 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) $-1$ ...
0
votes
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 ...
0
votes
0answers
24 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 ...
0
votes
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 $1/8$, $3/8$, ...
0
votes
0answers
17 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: ...
1
vote
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 ...
1
vote
0answers
14 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 ...
1
vote
0answers
18 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. ...
2
votes
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?
0
votes
0answers
25 views

Power method convergence explanation. [on hold]

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 ...
1
vote
0answers
94 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 ...
2
votes
2answers
16 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 ...
0
votes
0answers
17 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 ...
1
vote
0answers
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 ...
0
votes
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)_{+}:= ...
1
vote
0answers
24 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 ...
1
vote
1answer
36 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) ...
1
vote
0answers
14 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 ...
0
votes
0answers
17 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 ...
1
vote
1answer
66 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 ...
1
vote
1answer
40 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 ...
0
votes
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 ...
0
votes
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} &= ...
0
votes
0answers
22 views

Numerical analysis and localization of the roots of a polynomial [closed]

I want a digital Matlab application of root location of a polynomial theorem: Enestrom and Kakeya theorem.
1
vote
0answers
30 views

Euler's method in python [closed]

I'm trying to implement euler's method to approximate the value of e in python. This is what I have so far: ...
1
vote
1answer
41 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)] ...
0
votes
1answer
68 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: ...
1
vote
1answer
60 views

Numerical convergence depending on summation order

I'm looking for an example of convergent series such that the numerical convergence depends on the order of summation? Or perhaps a series of positive terms where the partial sums value depend on the ...
0
votes
1answer
31 views

Heat equation in 1D with collocation method

I want to use the collocation method to solve $u_t=u_{xx}$. I impose the PDE pointwise and expand the solution in Fourier Series: $$ \partial_{t}\sum_{k=-K}^{K}\hat{u}_{k}(t)\ ...
1
vote
2answers
31 views

How should terms be scaled by finite dx and dt in numerical integration of 1D diffusion?

I am familiar with numerically integrating systems of ordinary-differential equations, but I feel that I am missing something important in terms of how numerically integrating ODEs differs from ...
0
votes
0answers
21 views

Chebyshev polynomials approximation - Is there a way to generalize this

In an exam I was given this question: let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
1
vote
1answer
27 views

A problem about lub and glb of matrix

For any matrix $A\in \mathbb{C}^{n\times n}$, define $$lub_K(A):= \inf\{\alpha\geq 0: AK\subset \alpha K\},$$ and $$glb_K(A):= \sup\{\alpha\geq 0: \alpha K\subset AK\},$$ where $K$ is a equilibrated ...
4
votes
1answer
80 views
+50

How can one find intermediate digits of a root of an algebraic equation?

I was wondering whether there is a way to find intermediate digits of an algebraic equation. For example, if I have $$234x^{\frac{1}{12345}}-24621x^{\frac{1}{3456}}=1$$ And I want to find the ...
1
vote
0answers
41 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...