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

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3
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48 views

Comparison of Newton-Cotes Quadrature and Gaussian Quadrature formulas

Newton-Cotes quadrature formulas are a generalization of trapezoidal and Simpson's rule. The trapezoidal rule involves $2$ points, Simpson's rule involves $3$, and in general Newton-Cotes formulas ...
2
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1answer
28 views

Cubic spline interpolation realization

dear mathematicians! I was trying to code cubic spline interpolation algorythm. So I found one here. But I was confused. Let's see why. So let say I got 2 vectors - vector $X$ and $Y$ (with ...
0
votes
1answer
14 views

The 'order of error'

If I do a trapezoidal rule estimate and get 0.6386 and the true value of the integral is 0.636294, then the error is 0.002306. If I was asked to find the order of error, does it just mean the error ...
1
vote
1answer
42 views

Why does secant method converge

Assume $f$ is continuous and twice differentiable on $[a,b]$ such that $f'(x)>0$ and $f''(x)>0$, $x \in [a,b]$. If $f(b)>0$ and $f(a)<0$ and I choose $x_0=a$,why are we gauraunteed ...
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0answers
39 views

Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...
1
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1answer
28 views

Finite Difference Method Stability with diffusion equation

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
1
vote
1answer
17 views

Looking for a approximation/solution to my mortgage calculator function

I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
0
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1answer
48 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
1
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1answer
30 views

How can we apply the forward Euler method to $x''=x^2$?

If we want to apply the forward Euler method to $x''=x$ with $x(0)=0, x'(0)=1$, we can introduce a new function $$u:=\begin{bmatrix}x'\\x \end{bmatrix}$$ then $$u'=\begin{bmatrix}0&1\\1&0\\ ...
2
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0answers
35 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
1
vote
1answer
29 views

Estimating the absolute error of the function $f(x)=4x^2$

I have to estimate the value of $f(x)=4x^2$ for $x\in [1,2]$, and $x$ is unknown. the approximated value for $x$ is $\tilde x$, which is also in $[1,2]$. What is the maximum absolute error of $x$, ...
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0answers
35 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
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0answers
12 views

Absolute and relative error of a binary number?

Given is the following system $A:={ \pm 1.a_1 a_2 a_3 a_4 \cdot 2^e}$, $a_{i}\in \left \{ 0,1 \right \}$, $i\in \left \{ 1,2,3,4 \right \}$, $e\in \left \{ -8,\ldots,8 \right \}$. Find the absolute ...
1
vote
1answer
28 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
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0answers
34 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
2
votes
2answers
24 views

How to implement a numerically stable solution of a quadratic equation?

Solving $a x^2 + bx +c=0$ for $x$ gives $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ for $a \ne 0$. But for $a = 0$ we get $x=-\frac{c}{b}$. How to implement a numerically stable algorithm computing ...
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0answers
18 views

Quadrature methods: even order?

I noticed that all quadrature methods I know (Newton-Cotes and Gaussian quadrature) have always even order in the sense that a quadrature method is of order $n$, if all polynomials of degree $n-1$ are ...
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0answers
20 views

Finding root by mean of Bisection Method

I have a problem with this function f(x) = e^-x - x^2 i try to find The roots by Bisection method , I'am using Maple 16 this the command i used for calling Bisection method : Bisection(f(x), x = ...
0
votes
1answer
31 views

Jacobi Method for the linear systems (for first two iterations)

I have the following 3 linear systems: $$4x_1 + x_2 - x_3 = 5$$ $$-x_1 + 3x_2 + x_3 = -4$$ $$2x_1 + 2x_2 + 5x_3 = 1$$ for x^(0) = 0 Then, I write it as following: $$x_1 = -1/4x_2 + 1/4x_3 + ...
3
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1answer
30 views

Newton Rhapson Algorithm Accuracy

I read somewhere that the NR algorithm in general (given an appropriate initial value) increases in accuracy by roughly two decimal places per iteration. Is this something that can be proven, or is ...
3
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1answer
68 views

I want to study Numerical linear algebra [closed]

Would you like to recommend a book to me? the proof is explicit and easy to understand is preferred.
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1answer
36 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
3
votes
1answer
109 views

Numerical integration of $\sin(p_{m})$ and $\cos(p_{m})$ for a polynomial $p_{m}$

I was wondering if anyone knew about any numerical methods specifically designed for integrating functions of the form $\sin(p_{m})$ and $\cos(p_{m})$ where $p_{m}$ is a polynomial of degree $m$. I ...
1
vote
1answer
52 views

Computing a solution of the Laplace-Eigenvalueproblem with Neumann-b.c.

Good day! I was considering the Laplace-Eigenvalueproblem with Neumann b.c., i.e. find $u \in H^1(\Omega) \setminus \{0\}$ and $\lambda \in \mathbb{R}$, such that: \begin{eqnarray} -\Delta u \ ...
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0answers
19 views

Runge-Kutta for newton's law with dependency

I'm trying to determine the changes (position and velocity) on a mechanical system during a step of time. I have a mobile mass whose position (everything is 1D-only) is denoted $x(t)$, velocity $v(t)$ ...
2
votes
0answers
10 views

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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vote
0answers
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 ...
0
votes
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 ...
0
votes
1answer
37 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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votes
3answers
87 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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0answers
19 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 $k_1=f(t,y)$, $k_2=f(t+\lambda_2,y+\mu_{21}k_1)$, ...
0
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0answers
19 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 ...
1
vote
1answer
64 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 ...
1
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0answers
36 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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votes
1answer
13 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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0answers
16 views
1
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2answers
131 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 ...
0
votes
0answers
27 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 ...
0
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0answers
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 ...
0
votes
1answer
41 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
32 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 ...
2
votes
2answers
77 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 ...
2
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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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0answers
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
32 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 ...
2
votes
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} ...
6
votes
1answer
70 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 ...