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

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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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30 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
39 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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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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20 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
68 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
47 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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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} &= ...
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1answer
43 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
70 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: ...
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1answer
65 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 ...
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1answer
32 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)\ ...
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2answers
32 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 ...
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22 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 ...
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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 ...
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1answer
117 views

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 ...
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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 ...
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2answers
33 views

Taylor Series approximation

Let $f(x) = (1-x)^{-1}$ and $x_0=0$. Find the $n$-th Taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$. Find a value of $n$ necessary to approximate $f(x)$ within $10^{-6}$ on $[0,0.5]$. I am ...
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1answer
14 views

Selecting denominator for relative error margins

When looking at this page: http://floating-point-gui.de/errors/comparison/ there are values a, and b that are being compared ...
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10 views

Subset of density one

A subset S of positive integers will be said of density 1 if the following expression D_(N )(S) :=1/N #(S∩[1,N]) tends to 1 as N tends to infinity. The question is: Prove the equivalence of: 1) S ...
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1answer
42 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
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14 views

Embedding an Implicit Runge-Kutta Algorithm

I am interested in implementing a fourth-order RK method with variable step-size. However, in order to test for accuracy, the solver needs to periodically utilize a higher-order method to see if the ...
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0answers
51 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
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1answer
59 views

Looking for fast computation method of $Ax=b$ ($A$ is sparse matrix)

I am looking for fast method to solve linear equation $$Ax=b$$ In which A is sparse matrix. Could you suggest to me some current method for this task. Thank in advance
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31 views

Generalize an average to a sum

Any help would be appreciated! Thanks so much!
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42 views

High dimensional Differential Evolution

I want to minimize a cost function with Differential Evolution (DE) algorithm and I have 55 unknown parameters as an input for DE algorithm. Therefore, the DE should search in high-dimensional space ...
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1answer
28 views

Runge Kutta Method

Here,$$y'(x)=x^2+y^2,y(0.9)=14.3$$ I calculated the value of $y(1.0)$ using step sizes of h=0.1 and then h=0.05. However,my result for the different step sizes are very different. I got ...
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1answer
16 views

minimum order of bspline curve for C2 continuity

Given a control polygon with five pairwise different points $d_0,...,d_4$ what is the minimum order of B-Spline curve for this polygon such that it is $C^2$ continuous ?
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32 views

Question about forward slash and set theory with matrices and vectors …

Hi People I'm new to this forum and this is my first question,(Please forgive me for my improper posting method.) In the Below equation which is where we start with a linear equation such as ...
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61 views

Error estimation absurd for Simpson's rule

If $f:[a,b]\to \mathbb R$ is a $\mathrm C^{3}([a,b])$, Simpson's Rule (or Newton-Côtes for $a$, $(a+b)/2$ and $b$) gives that if $P$ interpolates $f$ in those points, $$ \int_a^b f \approx \int_a^b P ...
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1answer
21 views

Bisection method

I know how to use the Bisection Method to find the roots, however I have never used it to find the point of interconnection on two graphs. I looked it up and found that you can just subtract the two, ...
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16 views

Stability of Lax-Wendroff Approach for Advection Equation

The Problem: I am attempting to solve the following problem in 1D over a periodic region: "In one dimension, the mass density $\rho$ is advected with velocity $v$, so that it follows the equation: ...
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20 views

integration rule for singular function

It is well known that for sufficiently smooth function $f(t)$, error bounds for midpoint are $$ \int_{t_i}^{t_{i+1}} f(s)ds=hf(t_{i+1/2})+\frac{h^3}{24}\frac{d^2 f(s)}{ds^2}|_{s=t_i} +O(h^5). $$ where ...
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1answer
128 views

Prove $1! + 2! + 3! + \ldots + n! =y^3$ has only one solution in the set of natural numbers?

I actually know that the above equation is true for $n=1$ and $y=1$ but am unable to prove it for the entire set of natural numbers. Can anyone please help me solve this in a simple way?
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1answer
55 views

How does Matlab, Maple, etc…solve algebraic and differential equations internally?

I would be very interested finding out how does Matlab, Maple, etc…solve algebraic and differential equations internally? Anyone know how they do it?
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24 views

Finite Difference Approach for the 1D Conservative Advection Equation with Spacially Varying Velocity

I am attempting to numerically solve the following conservative advection equation in 1D, using a finite difference method. $\frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial ...
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1answer
43 views

Solving for many points in a curve at the same time

Suppose there is a well-behaving monotonic function $f(x)$ where do not have analytical form of $f'(x)$, and we need to solve for many points on this function at once, that is, we need to know the set ...
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20 views

Discretisation of a product of two functions

Suppose I have two functions, $f(x,t)$ and $g(x,t)$, and for an upwind scheme I want to use the quantity $\partial_x (fg)$ to solve the advection equation $$ \frac{\partial f}{\partial t} + ...
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1answer
33 views

Machine limit analysis of $\sqrt {x^2-a^2}-(x-a)$

Let $L(x)=\sqrt {x^2-a^2}-(x-a)$. I've been messing around with this equation on the calculator and found out that for certain values of $x$, the equations behave as $x \gg a$. Considering only for $x ...
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21 views

numerical errors near at the borders

I use some kind of partitioning on my data and then I do some interpolation and some other mathematical operations using chebyshev points. I have noticed that in the borders of each partition, It ...
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20 views

BDF2 and TR-BDF2: what is better?

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? The BDF2 method requires the values of $y_{n-1}$ and $y_n$ for computing $y_{n+1}$ ...
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1answer
24 views

Solving boundary value problem, put up linear equation system

For $\Omega = (0,1)^2 \subseteq \mathbb R^2, f \in C(\Omega)$ consider the boundary value problem: $-\Delta u(x,y) + u(x,y) = f(x,y)~ \forall (x,y) \in \Omega \\u(0,y) = u(1,y)~ \forall y \in (0,1) ...
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2answers
41 views

How to solve $Ax=b$ via backward and forward substitution on Matlab

How can I solve $Ax=b$ in Matlab code via LU factorization. I know that the command [L,U]=LU(A) stores the ...
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28 views

Help on Lagrange Error Calculation [duplicate]

Here is an example in Burden's Numerical Analysis book. My problem is in bold In example 2 we found the second Lagrange polynomial for $f(x)=1/x$ on $[2,4]$ using the nodes $x_0=2$, $x_1=2.75$, ...
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13 views

Solving a sum-exponential equation

I was wondering if someone could point me to the right resource towards numerically solving an equation of the form: $c = \dfrac{\sum_i a_i^{2x}}{\left( \sum_i a_i^x \right)^2}$ $c$ and $a_i$ are ...
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11 views

choice of iterative linear system method

while implementing an unconstrained optimization problem, using Newton's method, I am faced with a Hessian matrix that is very large (10^8 by 10^8) but very, very sparse - Non zero elements along the ...
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1answer
52 views

Why easier to numerically minimize than to maximize a function

Is it easier, in terms of coputational complexity or speed, to numerically minimize a function $f$ than to maximize $-f$? Why is that so? I have noticed that most optimization algorithms in Matlab are ...