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

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Remainder of the minimax approximation polynomial - number of extrema

Recall some definitions. Let $f \in C [a,b]$. The minimax polynomial $p_n$ is the polynomial $p_n (x) $ of degree $\leq n$ that minimizes $||f-p_n||_\infty $. It can be proved that this polynomial ...
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12 views

Order of Romberg's method

We call a method(numerical integration) of $n-$th order, if it can integrate any polynomial of degree $n-1$ without any error. In this sense: The simpson rule is of $4$-th order and the trapezium ...
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1answer
9 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
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1answer
35 views

Show that Newton method converges for every choice of $x_0$

Let $f\in\mathcal C^{2}(\mathbb R)$ with $f'(x)>0$ and $f''(x)<0$. If $f(x^*)=0$, show that the Newton method converges to $x^*$ for every choice of $x_0$ Wlog, choose a start point $x_0$, ...
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3answers
83 views

Intuitive Numerical Analysis Texts

Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement ...
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18 views

runge kutta 4th order

i need detailed/exact Taylor expansions used in 4th order Runge - Kutta method. To be precise; $y(t+h)=y(t)+a_1k_1+a_2k_2+a_3k_3+a_4k_4$ where $k_1=f(t,y)$, $k_2=f(t+\lambda_2,y+\mu_{21}k_1)$, ...
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17 views

Difference Between Prescaling and Preconditioning

I am confused about prescaling and preconditioning. Are they in fact the same thing? If prescaling is done, then the numerical stability will also improve. And thereby, it will have the effect of a ...
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1answer
58 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
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35 views

Error bound by the Simpson's rule

My lecture notes have a little exercise. Two functions are given: $$ f(x) = \cos(x) \ \text{and} \ g(x)=\sqrt{x+1} $$ And we're asked about the error bound of the Simpson's rule to estimate the ...
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1answer
12 views

Converting x number of petaFLOPS into a base 2 number

I would like a few different formulas or methods for doing a couple of conversions and calculations: 1) How can I convert petaFLOPS into a base $2$ number representing how many operations per second ...
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16 views
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2answers
128 views

What is the sum that the square root button on calculator does so I can do it without the calculator button [duplicate]

I am not very good when it comes to Maths but the current work I am doing means I need to get better and quick. I have been teaching myself about areas, diagonals and square roots. However I am ...
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24 views

Lagrange interpolation polynomial and error estimation

Given is a function $f(x)$ with $f(0)=1$, $f(\frac{1}{2})=2$ and $f(1)=-1$. Additionally is given that $max_{x\in \left [ 0,1 \right ]}f''(x)=1$. Find its Lagrange interpolation polynomial $P$ and ...
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16 views

How to design Boundary condition for Euler equations (CFD)?

I'm developing on the calculation of the euler equations using the finte volume method. As you may know each cell is calculated by the incoming and outgoing flux. That means I need in a 1D System the ...
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1answer
39 views

An integration formula using Gauss-Laguerre method

Using Gauss-Laguerre method show that: $ \int_{0}^{\infty}\frac{e^{-x}}{x+a}dx=\frac{a+3}{a^2+4a+2}+\frac{4}{\theta^5} $ where: $ 0<\theta<1, a>0 $
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1answer
31 views

Degree of Precision Effect on Quadrature Accuracy

For an $n$ point Gaussian quadrature, one can show that it has degree of precision $2n - 1$ meaning it will exactly integrate polynomials of that degree or lower. Is it always true that a quadrature ...
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2answers
74 views

An analytical proof that the sequence from the chord's iterational method is monotonic

Assume $f:[a,b]\to\mathbb R$ is twice differentiable and $f'f''\not=0$ on the interval. Assume $f(a)<0<f(b)$. Let $x_0$ be the point of intersection of the $x$-axis with the line through the ...
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1answer
38 views

How to compute the pade approximation?

Like $\log(1+x)$? Is there any algorithms? I have read many materials but doesn't have an idea
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13 views

Solving a nonlinear volterra integral equation with two integrals each with a non separable kernel

I am trying to solve the nonlinear volterra integral equation ...
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0answers
16 views

Why do we get oscillations in Euler's method of integration and what is the period?

When using Euler's method of integration, applied on a stochastic differential eq. : For example - given $$\dot v = -\gamma v \Delta t + \sqrt{\epsilon \cdot \Delta t }\Gamma (t) $$ we loop over $$ ...
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0answers
29 views

Calculating log and trigonometric functions using only +,-,/,*

How to calculate logarithm and trigonometric functions (sin, cos etc.) on base n with using +,-,/,* ? Is there any way to do it?
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0answers
29 views

implicit non-linear equations with complex variables

I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex ...
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2answers
57 views

Determinant of an ill conditioned matrix

I have the following ill conditioned matrix. I want to find its determinant. How is it possible to calculate it without much error \begin{equation} \left[\begin{array}{cccccc} ...
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1answer
69 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
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0answers
28 views

Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...
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8 views

How to determine the upper bound of the global error when I calculate a ODE using Euler method

I have an ODE say, $u^\prime(t) = tan^{-1}u(t)$ $(0 \le t \le T)$, $u(0) = a$. Now I would like to decide the upper bound of the global error when using Euler method to solve this ODE. I know that the ...
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1answer
16 views

Help inverting a non-linear polynomial system of equations

I have a set of two equations like this $$ \gamma_3=\left(\frac{1}{\sqrt{1+2c_3^2+6c_4^2}}\right)^3 \left( \alpha_1\,c_3^3 + \alpha_2\,c_3c_4^2 + \alpha_3\,c_3c_4 + \alpha_4\,c_4\right)\\ ...
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1answer
17 views

Quadratic Optimization Problem with Box Constraints

I want to solve a problem of form $$\min_x x'Ax + b'x \;\;\mbox{ s.t. } l\leq x \leq u$$ where $A$ is a positive semidefinite matrix, thus the function I'm optimizing should be convex. However the ...
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33 views

System of linrar equations and condition number

The relative error of the solution of a system of linear equation $Ax=b$, for any natural norm $\|\cdot\|$ is bounded by $$ \frac{1}{\| A\| \|A^{-1} \|} \frac{\|r\|}{\|b\|} \le \frac{\|e\|}{\|x\|} \le ...
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0answers
23 views

Binary search (bisection method) - is it worth checking continuity

I am implementing a rather simple matlab code, that gets a function $f$ and 3 real numbers $a,b, \epsilon$ where $\epsilon >0$ is a very small positive number (for instance, no larger than ...
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35 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
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1answer
25 views

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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2answers
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
37 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
45 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 ...
2
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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
23 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
28 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
70 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
48 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 ...
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1answer
31 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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0answers
7 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 ...