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

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32 views

Show that $\phi'(x)=0$

Let $ f \in C^2([a,b], R)$ and $$\phi(x)=x-\frac{f(x)}{f'(x)}$$ Such that 1.$ f'(x)\neq 0, \forall x \in [a,b]$ $\exists \alpha \in [a,b]: f(\alpha)=0$ Show that $\phi'(x) = 0$ My attempt: ...
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28 views

Is the Hessenberg form of a matrix unique?

I have to calculate the Hessenberg form of an matrix using householder reflectors. For real Matrices I get the same result as the 'hess()' function in Matlab, but for Complex Matrices I keep getting ...
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23 views

A sequence of intervals- Trying to find a fixed point -

This might be a trivial question but I couldn't come up with a clever trick,theorems or whatnot. Suppose $I_0=\left[\frac{1}{h_0},\frac{1}{l_0}\right]$ where $h_0=1$ and $l_0=\frac{1}{2}$. Given ...
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63 views

Alternative to the Gram-Schmidt Procedure for Orthogonalization

I was wondering if there is an alternative to the Gram-Schmidt procedure, which instead of being a successive orthogonalization scheme, would be non-successive (simultaneous)? In other words, is there ...
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30 views

Definition of local truncation error

In the first reference of Wikipedia the local truncation error defined as $$ \tau_n = y(t_n)-y(t_{n-1})-hA(t_{n-1},y(t_{n-1}),h,f) $$ But in the second reference mentioned that $$ \frac{\tau_n}{h} ...
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13 views

quadrature trapezoidal rule to solving integral

I want to solve an integral. when I used $n=50$ for quadrature trapezoidal rule, my answer is better than $n=100$. Is it true, or you think I have a mistake in my program?
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15 views

Approximation of the ito SDE using backward Euler approximation

I have the stochastic SDE $ dX_{t}=a X_{t} dt+ b X_{t} dW_{t}$ I succeeded to formulate a forward Euler approximation to approximate it but I have some problems to derive the right backward Euler ...
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57 views

Order of implicit trapezoidal (Heun's) method

What is the order of local truncation error (LTE) of implicit trapezoidal method? Is it $O(h^2)$ or $O(h^3)$? By integrating the $y^\prime(x)=f(x,y(x)),\, x \in [a,b]$, from $x_i$ to $x_{i+1}$ we can ...
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16 views

Stability of a system of ode

I have the ordinary differential equation $dX_{t}=A X_{t} dt $ where where $ X_{t} \in \mathcal{R}^{2}$ and the matrix A has two real eigenvalues $\lambda_{1} = 1 , \lambda_{2} = -10^{5}$. The ...
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35 views

How to find a separating hyperplane?

I know about support vector machine, and it's quadratic programming approach which delivers the best separating hyperplane. My question is: is there a relatively simple algorithm to find a ...
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25 views

Numerically doing trapezoid rule and richardson extrapolation using haskell.

so im trying to numerically to solve some integrals using trapezoid and richardson methods. Ive implemented the code as followed in haskell. ...
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14 views

The influence of time interval in the Newmark-beta method on the results

I have tried to solve an ODE using the Newmark-beta method. The results changed a lot when the time step changed and the results well exceeded the expected value. How these results came into being? ...
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18 views

Show that A is positive-definite matrix if and only if ${x^{(k)}}$ converge

Let A be invertible Hermitian matrix and $b \in \mathbb{C}$ Also N is invertible matrix such that $N+N^*-A$ is positive definite. In order to solve $Ax=b$ consider the iteration ...
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46 views

Solving a non-linear set of equations with non-exact constant values

I have a set of nonlinear equations which are related to a physical system. The constants of these equations (light hand side values-LHS) are determined through some measurement methods. It means that ...
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17 views

How to express the quantity using significant figures to imply the stated error?

Express the quantity using significant figures to imply the stated error. $$1.77 \pm 0.06$$ I tried to factor out a $6$ and got $6(.295\pm0.01)$ but then when I try to make $.295$ have an error of ...
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33 views

“Big-O” notation with a Taylor Series Expansion

Use a Taylor's expansion to rid the expression $1-\cos x$ of subtractive cancellation for $x$ small. Use a $\mathcal{O}(x^5)$ approximate. I understand Taylor series and I know that the expansion of ...
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66 views

Proof of theorem about iterative methods

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is ...
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26 views

Gap between numbers in fixed-point vs. floating-point arithmetic.

If $r$ is a machine-representable number and $f(r)$ is the next larger machine representable number, are the following true or false? (1) In fixed-point arithmetic, the distance between $r$ and ...
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50 views

Solving implicit theta-method function numerically faster than using fixed point iteration

When we are using the theta-method to solve an IVP. We have the equation: $$y(x_{n+1}) \approx y(x_n) + h[(1-\theta) f(x_n, y(x_n)) + \theta f(x_{n+1}, y(x_{n+1})]$$ where $f(x,y)=y'(x)$ The only ...
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27 views

Zero-stability for numerical methods, only applies to LMM's?

I'm trying to get a better grasp of the notion of zero-stability. Mainly I'm using a book by Leveque (Finite Difference Methods for Ordinary and Partial Differential Equations). Anywho, Leveque ...
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13 views

Why does QUADPACK only enforce the least strict error boundary?

According to this reference (which is in agreement with my own numerical experiments), QUADPACK tries to fulfill the following accuracy requirement on the approximation error: |RESULT - I| $\le$ ...
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94 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
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11 views

fehlberg mthod and control step size

want to use Fehlberg method to solve a system of ode. But I don't how can I control step size? for example I use h=.5 and then approximate $y_{n+1}$ by RK4 and RK5 then what am I doing?
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14 views

Mapping from $\left(-\infty,\infty\right)$ to $[a,b]$ to reduce numerical error

Suppose $A = \left[\begin{array}{cc} \exp\left(x_1\right)&\exp\left(x_2\right)\\ \exp\left(x_3\right)&\exp\left(x_4\right) \end{array} \right]$, where each of $x_i\in\left(-1000,1000\right),$ ...
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14 views

Is there a way to characterize the range of a chebyshev series through its coefficients?

Let $f$ be a Chebyshev series of order $n$ $$ f(x) = \sum_{i=0}^n a_i \cos\left( i \arccos\left(x\right)\right), x \in \left[-1, 1\right]. $$ Is it possible to characterize all the $\lbrace a_i ...
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22 views

numerical (script) fit of function with 2 arguments

I would like to find the least-square fit for a 1D-function that takes two arguments. m(x,y) = d * (x-x0)^2 / (y-y0)^2 I would like to write a c++ routine to ...
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17 views

How to use Legendre Quadrature for Multiple Unbounded Integrals?

Legendre Quadrature as most other methods is designed for $[-1,1]$ interval and some variable change methods are used for extending them to $[a,b]$ interval. I found some method in a textbook for ...
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29 views

fifth and sixth order Runge-Kutta method

I want to solve one system of ODE problem with numerical methods. I used $RK_2$, $RK_3$, $RK_4$ and Adams_Bashforth method.Now I want to solve it with high order Runge Kutta methods. Do you have these ...
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41 views

numerical calculation of an integral

I am having trouble finding the solution of this numerically and wondered if I could get some tips so that I can: $$ \int\limits^1_{0}\left[\min(ax, b) - \min(a x, c)\right] dF(x; p, \rho)$$ (1) ...
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41 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
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50 views

Complexity of the power method

I'd like to find out what the complexity of the power method is depending on the size of the matrix $A \in \mathbb{R}^{n\times n}$ given that the algorithm runs until a certain stop criterion. I.e. ...
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29 views

Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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19 views

Correctly rounding an error to one significant figure

I'm running a process which produces several possible values for a given parameter and returns its mean and standard deviation, both given with an arbitrary precision. I find myself with the ...
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16 views

Initial value problem test cases

I am working on some materials about numerical solution of initial value problem for ODEs. Are there any state of art test cases used to test properties of methods? I have found one in Wikipedia and ...
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20 views

Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
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23 views

Find Lipschitz constant for the equation

I am still confused about the theorem relating to find the Lipschitz constant. I have the following equation: $$y' = \frac{1+t}{1+y},\quad 1\le t\le2,$$ where $ y(1) = 2$ and $h = 0.5$. If I were to ...
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50 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
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35 views

Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
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14 views

Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
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18 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
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42 views

Weierstrass substitution for solving trigonometric eqations

I'm trying to solve a set of equations of a parallel robot . The equations can be writen as $x(\cos(\theta),\sin(\theta))$ $y(\cos(\theta),\sin(\theta))$ so to solve the equation I used Weierstrass ...
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21 views

finding eigenvector with its eigenvalue known

suppose I have a square matrix $A$ of large dimension(>100) with eigenvalue $\lambda$, what are the numerical methods to find its corresponding eigenvector without using the inverse of A?
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27 views

Finite difference method for elliptic pde stability condition

I am trying to implement a finite difference scheme for the elliptic pde $\nabla(a(x_1,x_2)\nabla\phi(x_1,x_2))=0$ where $(x_1,x_2) \in (0,l_1)\times(0,l_2)$ with $l_1 \neq l_2$ necessarily. I know ...
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27 views

4th order method

I am asked to solve a ODE using the 4th order Runge-Kutta method, and then given the analytical answer, 'show the method is 4th order numerically' . What does the question 'show the method is 4th ...
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17 views

preconditioning techniques for Householder transformation

I am using Householder transformation to get similar matrices for the original symmetric positive definite matrices of large dimensions. However the eigenvalues of the similar matrices are quite ...
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46 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 ...
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51 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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40 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 ...
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41 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 = ...
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25 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 $v(t)$ ...