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

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How to show that the one step method can't have consistency $p=3$?

I was looking at some exercises from last years' of my Intro to numerical math class, and found this: Consider the following explicit one step method: $$\psi^h x=x+h \gamma_1 f(x)+ h \gamma_2 ...
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31 views

Fastest way of finding eigenvectors from eigenvalues

Given the eigenvalue of a matrix of large dimensions, I want to know if there is a fast way of finding the corresponded eigenvectors?
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31 views

Convergence of the newton method

If f(x) has m'th root at r. consider the newton method $x_{i+1}=x_i-\frac{mf(x_i)}{f'(x_i)}$. why it has quadratic convergence?
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2answers
55 views

Newton Raphson Method for double roots

I am currently working on Newton Raphson Method. I am kind of facing a problem how Newton Raphson Method work on more than second order quadratic functions with double roots. I have googled and ...
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1answer
38 views

Initial value of Newton Raphson Method

I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it. The questions that I have are ...
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1answer
40 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
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1answer
48 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
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1answer
66 views

How do I construct such a numerical method for solving ODE?

I am asked to expand $x(t+h)$ and $x(t+2h)$ around $t$ up to the rest term of the third order, find $A, B, C \in \mathbb R$ such that $$x'(t)=\frac{Ax(t)+Bx(t+h)+Cx(t+2h)}{h} + O(h^2)$$ and based on ...
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1answer
33 views

Anyone recognize this pattern? Plotting relationship between two parameters and their response.

First time asking a question here so hopefully I can provide enough information to you guys without explaining more than necessary. I'm doing some amateurish numerical analysis in MATLAB on some ...
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2answers
52 views

Integral over implicit function

Suppose the equation is given: $$y^5+2y^4-7y^3+y-x=0$$ (or any other equation that cannot be expressed explicitly). Let the solutions be implicitly given as $y=g(x)$. Is there an approach to solve ...
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0answers
25 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
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0answers
29 views

condition number of orthogonal matrix

Let $A\in M_n(\mathbb R)$ be an orthogonal matrix. Then: $cond (A) =1$. I tryed to use facts about the eigenvalues but is did not help. In 2-norm it is easy to prove it since $||A||_2 = \sqrt{\rho ...
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2answers
72 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
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1answer
49 views

Which numerical method to use for ODE?

In practice what is the most common way to numerically estimate $y(t)$ (possibly using a series expansion) in the ODE with initial conditions, $$ y'(t) = f(t,y(t)), \qquad y(t_0)=y_0 $$ Wikipedia has ...
2
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1answer
32 views

Where did I go wrong: B-Spline recursion and B-Spline using determinants

For $ B^2_1(t) $ with knot values $ t_1 = 1, ..., t_4 = 4 $ Using the determinant method $ B^d_i(t) = (-1)^{d+1} (t_{i+d+1}-t_i) \frac1D A $ where D is the determinant of $\begin{bmatrix} 1 \ t_i \ ...
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1answer
39 views

3D surface fitting

I am attempting to find the mathematical representation of a surface given a set of (x,y,z) data points. I recently tried using the method of least-squares which worked well for most of my situations. ...
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1answer
19 views

Numerical evaluation of a complex integral

I have to evaluate numerically $f(z)$ via the Cauchy representation (so via a complex integral), in other words, I have to calculare $f(z)$ performing a complex integral: $\dfrac{1}{2\pi ...
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8 views

Choleski's Algorithm Query

In Choleski's algorithm, I wonder how can one be sure that the diagonals elements of L (except for the first one) to be all positive?
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1answer
58 views

What method are there for “numerically” computing arclengths!

I know the originals formula for arc-length is: $$\int_{a}^b \sqrt{1+{f'(x)}^2}$$ However most of the formulas don't have closed formed solutions, and are unsolvable in terms of this equation. So ...
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1answer
27 views

What numerical quadrature algoritm can be use to handle $\int_b^c K_0 (x-a)-K_1(x-a) dx$?

I am curious what numerical algorithm can be used to handle $$\int_b^c [K_0 (x-a)-K_1(x-a)] dx$$, where $a\lt b\lt c$ and K is the modified bessel function of the second kind. From plotting the ...
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2answers
37 views

Solution of $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$

I'm trying to prove that $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$,given that $A$ is symmetric positive-definite has only one minimum. I've found the derivative is $f'(x)=Ax-b$, and in order to find ...
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1answer
93 views

How to find the integer part of big number?

How to calculate the integer part $$\left \lfloor10^{10^{10^{10^{10^{-10^{10}}}}}} \right \rfloor ?$$ Does this equal $$10^{10^{10}}? $$ Both Maple and Mathematica fail with it. PS. Unmotivated votes ...
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2answers
52 views

Power iteration

If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration Especially, I often see that we ...
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0answers
25 views

Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we ...
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1answer
28 views

Two point Gaussian Quadrature rule

I want to use the two point Gaussian Quadrature rule to approximate (evaluate) $\int_0^1 \! 6x^2-2x+1 \, \mathrm{d}x $ Since, with the two point Gaussian Quadrature rule, n=2 and the integral of ...
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1answer
47 views

Korteweg–de Vries equation: why is there a substantial literature on their numerical solutions if they are analitically integrable?

Given the initial value problems for the Korteweg-de Vries equation $u_t + u_{xxx} = u u_x; \quad u(0,x) = u_0(x)$ I have read that they can be solved exactly by the inverse scattering method, but ...
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1answer
24 views

Find Iterative Method Convergence Rate

Given $f(x)\in C^2[a,b]$ s.t there is a point $x_0$ s.t $f(x_0)=0,f'(x_0)\ne 0 $, and the iterative method is defined as follows : $$ x_{k+1} = x_k - f(x_k)/g(x_k) ,\qquad g(x_k) = \frac{f(x_k ...
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1answer
35 views
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2answers
37 views

Solving $f(x) = e^{(-sin4x)} - 3/4$ with 3 digits after the decimal point correction

I needd to solve $f(x) = e^{(-sin4x)} - 3/4$ with 3 digits after the decimal point correction, but cannot find out how. I'd really appreciate it if anyone could point me to the solution. I think I ...
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4answers
66 views

Numerical integration - $\int_{-1}^{1} f(x)dx$

I'm currently studying numerical integration, and ive come across a question i'd like help answering. We are given an integration rule as follows: $I(f)=\int_{-1}^{1}f(x)dx = \frac{2}{3} ...
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1answer
28 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
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0answers
15 views

Kernel density estimation of a divergent probability density function

I'm working with a 2D probability distribution function (pdf) that will be something like $$P\left(r,\theta\right)\approx\frac{3}{\pi^3}\frac{1}{e^{r}-1},$$ when written in polar coordinates (i.e. ...
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1answer
24 views

numerical algorithms for determining least common multiple of polynomials

I have a pair of rational polynomial fractions $\frac{A(x)}{B(x)} + \frac{C(x)}{D(x)}$ where A, B, C, and D are all polynomials in x, and I have their coefficients as an array of numbers. I would ...
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1answer
28 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
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35 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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1answer
28 views

Integration Rule Exact Degree

Given the integration rule $Q(x) = \alpha_1f(0)+\alpha_2f(1)+\alpha_3f'(0)$ for interpolating the integral $\int_0^1f(x) dx$ , I need to find $\alpha_1,\alpha_2,\alpha_3$ values s.t Q has exact degree ...
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57 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
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0answers
31 views

solving partial differentiation using finite difference method

I have been trying to solve right hand side (RHS) of the following one-dimensional partial derivative equation: $\frac {\partial p} {\partial t}=\frac {\partial} {\partial x} ({D(x)}e^{-\beta V(x)} ...
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1answer
22 views

Proof that Jacobi method will converge to the solution of a system Ax=b [closed]

Can anyone show me a statement that this works and a proof? Thanks
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2answers
67 views

Why does newtons method converge to the root of an equation?

I'm trying to understand why the Newton Raphson method converges to the root of a given equation? Can someone explain it to me theoretically. Thanks
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0answers
25 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
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0answers
96 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
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2answers
25 views

What are the possible limits of the iteration?

Consider the function $f(x) = \sqrt{2 + x}$ for $x \geq -2$ and the iteration $x_{n+1} = f(x_n) ; n \geq 0$ for $x_0 = 1$. What are the possible limits of the iterations ? $\sqrt{2 + \sqrt{2 ...
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1answer
49 views

How to numerically handle a double integral with a singular endpoint on the outer integral

I am trying to numerically integrate $$\int_0^a f(x) \int_{\sqrt{x}}^\infty \frac{\exp(-u^2)}{\sqrt{u^2-x}}du dx$$ where a is some positive real number and f(x) is some well behaved function. The ...
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1answer
28 views

Can an iterative method converge for some initial approximation?

Studying iterative methods for solving(or approximating) linear equation systems, I came accross the following theorem$^1$: Let the following be an iterative method: $$x^{(0)},\qquad known\\ ...
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34 views

Solving solely continuous system of ode's with matlab

I'm working with the numerical integration of the system of differential equations, $\dot{x}=f(x)$ with the vectorfield, $f(x)$ being solely continuous. Examples of the systems which I'm working on ...
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1answer
202 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
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23 views

Inverse Iteration to Find Eigenvalues - Question about Method

So I'm doing Inverse Iteration in Excel to find the dominant eigenvalue and eigevector of a matrix. This particular method involves estimating an eigenvalue, multiplying the identity matrix by it, ...
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1answer
33 views

Linear interpolation by hand - Any quick ways to do this?

I have to calculate the roots of the equation $x^3 + x^2 -3x -3 = 0$ in the interval $[1,2]$ using linear interpolation to six decimal places, by hand. Now I know this is trivial in excel, but when ...
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0answers
15 views

Numerical solution of ODE

I have a general question about numerical solution of ODE. I want to solve a ODE on an interval where two solutions can exist and intersect. As far as I understand a numerical solution will give the ...