Tagged Questions

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

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0
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2answers
35 views

Evaluate the following function using as many significant figures as required to get a final result of 4 digits accuracy

I need to evaluate $$ f_5(0.2) = 5! \left[ e^{0.2} - \left( 1 + (0.2) +\frac{(0.2)^2}{2!}+\frac{(0.2)^3}{3!} + \frac{(0.2)^4}{4!} +\frac{(0.2)^5}{5!} \right) \right] $$ using as many as required ...
0
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0answers
23 views

modifying plot in matlab

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0
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0answers
36 views

MATLAB linear interpolation

I'm trying to write a MATLAB program to do linear interpolation and to check its accuracy. I have to input $x_0$ and $x_1$ and then generate the data values using $y=e^x$. Then, for a variety of ...
0
votes
1answer
28 views

Lagrange interpolation: Getting a bound and finding the error

I am struggling to understand this: The problem asks me to find the lagrange error of the polynomial approximation given the nodes $x_0 = 1, x_1 = 1.25, x_2 = 1.6$ with $x = 1.4$ The function I am ...
3
votes
0answers
478 views

Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2 $. Picking the symmetric case with $0$ mean ...
1
vote
2answers
34 views

Literature and web sources for Computer Aided Geometric Design (CAGD)

Through the Numerical Analysis lecture I came across Bèzier curves, B-Splines and Spline Interpolation and found it very interesting. The title of the chapter was Computer Aided Geometric Design and I ...
1
vote
1answer
34 views

Fixed-point theorem restriction in numerical analysis

The Banach fixed-point theorem states that if $f:[a,b]\to [a,b]$ is $\lambda$-Lipschitz where $\lambda\in[0,1)$ is such that satisfies $|f(x)-f(y)|\leq \lambda |x-y|$ for every $x,y\in [a,b]$ (I'm ...
1
vote
1answer
44 views

Very confused with interpolating polynomials

I have a problem from my homework that I completely botched, and no matter what I do I end up with the wrong answer. Here's the problem: For a given function $f(x)$ let $x_0 = 0, x_1=0.6, x_2 = ...
1
vote
2answers
33 views

newton's formula

How is the number of iterations found using the Newton's formula? I tried $|P-P_n|<k^n\max\{P_0-a_1, b-P_0\}<TOL$ Can anyone help me with another formula in finding $N$ (the number of ...
0
votes
1answer
20 views

Iterative Scheme-Programming Matlab

I don't know if this is going to seem like a dumb question, I am new to this and to matlab, but I'm trying to construct an iterative scheme in MATLAB to compute $\sqrt(b)$ for a given b>0, and program ...
0
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0answers
15 views

for loop standard deviation

I'm very new to MATLAB programming and thus I doubt myself when doing things with matlab. I just wanted to confirm I am doing this correctly. I am supposed to complete this program: function ...
1
vote
1answer
43 views

function program for $e^x$

I'm Using this approximaion: $$e^x\approx \text{myFunc}(x)=\sum_{i=0}^{10}\frac{x^i}{i!}$$ I'm trying to write a function program to evaluate $e^x$ and have an error that is $\leq$ $10^{-7}$ which ...
0
votes
0answers
17 views

How to compare the speed of convergence of infinitesimal of the same order?

Given two iterated function $f(x)$ and $g(x)$. Both of them converge to A, that is, both $\{p_n=f(p_{n-1})\}_{n=1}^\infty$ and $\{q_n=g(q_{n-1})\}_{n=1}^\infty$ converge to A. And the order of ...
1
vote
0answers
59 views

Euler method inequality

Given the problem for $t\neq0$ and $t\le1$ $y'(t)=y^2(t)$ $y(0)=1$ Let $\mu>0$, and $\epsilon_n=\frac12(f(t_{n+1},y_{n+1})-f(t_n,y_n))$, such that $|\epsilon_n|\le\mu|y_n|$ is ...
1
vote
0answers
49 views

Elastica - numerical check

Following on from rmhleo's fantastic answer here, where he states that the deformation of an ideally elastic circle is a problem of the calculus of variations which may be solved with an ODE of the ...
0
votes
2answers
87 views

Quicker way to compare numbers without calculator

Question: Find the order of $(1/2)^{1/2}$, $(1/e)^{1/e}$, $(1/3)^{1/4}$ without using calculator. Extra constraint: You only have about 150 seconds to do it, failing to do so will eh... make you run ...
1
vote
1answer
30 views

A proof of Newton's Iteration for Finding Square Roots

I find this theorem in my textbook: Assume that $A>0$ is a real number and let $p_0>0$ be an initial approximation to $\sqrt{A}$. Define the sequence $\{p_k\}_{k=0}^\infty$ using the recursive ...
0
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0answers
17 views

How to estimate the error of a numerical multiple integration

I'm integrating over the wholes space the function $$f(\vec{r_1},\vec{r_2})=\exp{\bigg[-(r_{1\alpha}+r_{1\beta}+r_{2\alpha}+r_{2\beta})\bigg]} \cdot 1/r_{12}$$ where $\vec{r_\alpha}=(-R/2,0,0), ...
0
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0answers
30 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation by multistep methods?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep methods for the second order ...
0
votes
1answer
16 views

Bound on numerical integral

I'm running a numerical integral over a function which I can only know on specific points as the problem is defined on a lattice. I've been using a trapezoidal method which has for known error: ...
-1
votes
0answers
18 views

Eigenvalues of tridiagonal matrix.

I know about the eigenvalues of a common tridiagonal matrix of order n by n having principle diagonal entries 'a' and the sub diagonal entries as $b$ and $c$. I just wanted to ask if this will hold ...
0
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0answers
18 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep method for the second order ...
-2
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0answers
26 views

The rate of convergence of four iterated functions

The four iterated functions below are used to calculate the approximate value of $21^{\frac13}$. Assume that ...
2
votes
1answer
25 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
0
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0answers
20 views

Numerical solution of SDEs with fractional Brownian motion

I am trying to numerically solve some SDEs representing a nonlinear circuit (possibly chaotic) driven by noise: $$ dX = f(X) dT + \sqrt{P_{w}} dW + \sqrt{P_{f}} dC $$ where $X$ is my circuit state, ...
2
votes
0answers
13 views

stationary stokes problem - inf-sup

I want to show that $\inf_v \sup_{p} \int_{\Omega} \vert \nabla v \vert^2 + p \nabla \cdot v \, dx$ (where $v \in W_2^1(\Omega)$ and $p \in L^2(\Omega)$) is equivalent to the minimization of the ...
2
votes
1answer
99 views

Jacobian Matrix Requirement for Linear Approximation

It is my understanding that when searching for a linear approximation of a nonlinear function, using the Jacobian (matrix) could help. I did some reading, and read that there is a condition: the ...
8
votes
2answers
268 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
0
votes
2answers
37 views

Taylor Approximation

For $f(x)=e^x$, find a Taylor approximation that is in error by at most $10^-7$ on [-1,1]. Using this approximation, write a function program to evaluate $e^x$. Compare it to the standard value of ...
1
vote
1answer
31 views

Approximating a real from some other reals.

Given a list of $n$ real numbers: $R=(r_1,r_2,\ldots,r_n)$ with $r_i < r_{i+1}$, and a target real number $t$, How can we find the subset of $R$ of size $k$ with a sum that best approximates $t$? I ...
0
votes
1answer
70 views

First-degree spline interpolation problem

I have been spending hours working on a problem from Cheney & Kincaid on first-degree spline interpolation problem but am heading nowhere. Any explanation, pointers or hints from you would be bery ...
1
vote
0answers
29 views

Strictly diagonal matrix

Suppose that matrix $A$ is strictly diagonally dominant, show that $||A^{-1}||_{\infty}\leq[min(|a_{ii}|-|\sum_{i\neq j}^n a_{ij}|)]^{-1}$.
0
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0answers
24 views

Proof the described sequence obey the formula

The sequence start by solving algebraic equation $ P_1V_1^{1.4}=P_2(V_1-\frac{a}{n})^{1.4} $ for P2 and substitute value of $ P_2$ into next equation $P_2(V_1-\frac{a}{n})=P_3(V_1+\frac{a}{n})$. ...
0
votes
1answer
26 views

Bounding an approximation error of Bernstein polynomials

I have to show that for $ 0 \leq x \leq 1 $ \begin{align*}\sum|f(x)-f(k/n)|p_{nk}(x)\leq(2M/\delta^2)\sum_{k=0}^n (x-k/n)^2p_{nk}(x)\end{align*} Important to know is that $p_n(x)$ and $p_{nk}(x)$ are ...
0
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0answers
19 views

The weak formulation of Navier Stokes Equation

I come across this problem in the weak formulation in Navier-Stokes equation. In the book, it let $D(\Omega_T)=\{\vec{\phi}\in C_0^{\infty}(\Omega_T),div\vec{\phi}=0 $, where ...
1
vote
1answer
34 views

Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
0
votes
2answers
30 views

Euler method(path s1s2=s2s1)

Given a differential equation $\frac{dy}{dx}=f(x,y(x)), y(x_0)=y_0$. What is the condition for function of f(x,y) such that the result of $y(x_0+S_1+S_2)$ by using Euler forward method, a step size ...
1
vote
2answers
49 views

The Mean Value Theorem

I am trying to understand why the following question is correct. Regarding to the MVT. Given: Find a number c satisfying the conclusion of the Mean Value Theorem $f(x)= x^{1/3}$, on $[-1,2]$ $f(x)= ...
3
votes
2answers
46 views

Is there a problem for which it is known that the only solution is “iterative”?

By "iterative solution" I mean specifically the following type of iteration: given a problem whose solution is $x$, first you compute some approximate solution $x_n$, and then make use of $x_n$ to ...
6
votes
1answer
217 views

Recursion relation of fourth order Runge-Kutta method applied on system

I'm trying to apply the Gauss-Legendre method of fourth order (as Runge-Kutta method) on the following system of equations $$\left\{ \begin{matrix} \dot{a} =& -b \\ ...
3
votes
2answers
39 views

How to numerically solve for a contour curve?

Supple you have a 2D surface z = f(x,y). Given a value z, how to numerically find all the values of x and y that satisfies z = f(x,y)? I know that my surface is well behaving to the extend that the ...
0
votes
0answers
52 views

Real world applications of numerical methods, for a mathematics project

I'm doing a mathematics project and I have been given 3 areas to have a look at and choose from. There's plenty of information on the academic side but not a lot of information on there real world ...
0
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0answers
24 views

Derive Runge-Kutta matrix with known weights and nodes

Can I derive a Runge-Kutta method by choosing freely the weights and the nodes? what are my constraints? So, if this is the general form of the explicit RK method: $$ y_{n+1} = y_n + \sum_{i=1}^s b_i ...
1
vote
1answer
30 views

Relative error in differential equation

Consider the following problem with; \begin{cases} y'(t)=3y(t)-3t & \\ y(0)=\frac13 \end{cases} If the initial value is replaced by $y(0)=\frac13+\epsilon$, compute the relative error of ...
0
votes
0answers
5 views

Degrees of freedom of differential algebraic equations (DAE)

I have a set of DAEs that seem to be giving the solver difficulty (the solver is APMonitor, a web service), and I suspect I haven't formulated them correctly. The physical system is a pair of rigid ...
0
votes
0answers
20 views

derivative of function equal to zero in newton raphson method

In Newton’s Method, if derivative of the function is made equal to zero instead of the function itself at a particular x. What would you call that value of x?
0
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2answers
35 views

Eulers method for a non-linear boundary value problem.

As part of an assignment, I have been asked to numerically solve the following 2nd-order differential equation. For those wondering, it is a model of groundwater flow through an aquifer beset on both ...
1
vote
2answers
78 views

How to prove the eigenvalues of tridiagonal matrix?

Assume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & &\\ b & a & c & ...
1
vote
0answers
25 views

how to prove the eigenvalues of tridiagonal matrix [duplicate]

Assume the tridiagonal matrix $T$ in this form: $$ T = \begin{bmatrix} a & c & & & \\ b & a & c & ...
0
votes
1answer
27 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...