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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

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Runge's phenomenon-question

I am looking at the Runge's phenomenon and I have a question. We have the interval $[a,b]=[-1,1]$ and $f\in C^{n+1}[-1,1]$. We know that $\forall x \in [-1,1] $ $\exists$ $\xi\in[-1,1]$ so…
evinda
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Is the iterative method suitable??

We have the equation $g(x)=x^{2}-3x-4=0$ that has the roots $-1$ and $4$ and we are looking for a suitable iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges to the root $4 \forall x_{0} \in [3,5] $.Is this…
evinda
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How do you find $x$ from $y=\operatorname{sinc}(x)$?

I have an equation: $$\operatorname{sinc}(x) = \frac{\sin(x)}{x} = 0.5$$ How do I find $x$ from this? I realise there's probably not a simple equation to describe the inverse, but is there a numerical method I can use to solve this?
Edd
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how to show that K is positive-definite,knowing that A is positive-definite

Given a positive-definite and symmetric matrix $A$,which can be written as: $A=\begin{bmatrix} d & u^{T}\\ u & H \end{bmatrix}=\begin{bmatrix} \sqrt d & 0\\ \frac{u}{\sqrt d} & I_{n-1} \end{bmatrix}\begin{bmatrix} 1 & 0\\ 0 & K…
evinda
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Quadratic convergence with Newton's method.

Let $g(x)=x^2 \ln(x)$ We have the following equation $g(x)=a$ and let $\alpha(a)$ be the solution for $a>0$ Suppose $a>0$ and $1<\alpha(a)<1+a$. Show that it converges, except for the first step, to $\alpha(a)$ while using Newton's method, used on…
DeanTheMachine
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Gaussian Quadrature

I have a question about when determining the weights in Gaussian quadrature for integrating a polynomial of degree 3: $c_0 + c_1x + c_2x^2 + c_3x^3$ up to the point when: \begin{align*} &c_0(\omega_0 + \omega_1 - \int_{-1}^1dx) + c_1(\omega_0x_0 +…
Yue Harriet Huang
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Conjugate Gradient Methods problem

I have one proble to solve. I have to determine $x$ that minimizes $d$. $C$ is the centroid of the figure. This is the problem given problem my numerical analysis class which is something that I did not learn in class. However I do not even…
eChung00
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Finding roots/zeros for collision detection in a video game

For the most simple of 2D games, I have implemented a posteriori collision detection (overlapping rectangles) on the $xy$ Cartesian plane, but am now interested in understanding the basics of a priori collision detection... On Wikipedia's entry on…
JackOfAll
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Quadrature errors of midpoint and trapezoid formulas

I have this question in my homework in Numerical Analysis but I can't figure it out. Can someone have a look and help me? Question: Let $E_0(f)$ and $E_1(f)$ be the quadrature errors of the midpoint and the trapezoid formula, respectively. Prove…
Albanian_EAGLE
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Simpson Rule special case of Gauss quadrature

Prove that the Simpson Rule for integrating the function $f$ on the interval $[a,b]$: $$I_2(f)=\frac{b-a}{6}(f(a)+4f(\frac{a+b}{2})+f(b))$$ is actually the Gauss quadrature for the weight $g=1$ and $3$ nodes. I can not get why when we have $3$ nodes…
user53969
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Numerical plot of function which cant be integrated.

How to plot numerically function $F(x)=\int e^{-x^2}dx$ ?
user109447
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dissipative function and stable fixed points

I have to prove that if $ f:\mathbb{R^d}\rightarrow\mathbb{R^d} $ is dissipative with respect to the scalar product < . , . > then every fixed point of $x\prime =f(x)$ is stable. I wanted to use $ \forall~\epsilon>0 ~\exists~\delta>0~ :|dx_{0}…
Xi Tong
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Numerical integration of $\int_0^2 \frac{1}{x+4}dx.$

I have homework problem. Determine the number of intervals required to approximate $$\int_0^2 \frac{1}{x+4}dx$$ to within $10^{-5}$ and computer the approximation using (a) Trapezoidal rule, (b) Simpson's rule, (c) Gaussian quadrature rule. I think…
eChung00
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Estimating error linear interpolation

Let $p(x)$ be the linear interpolation polynomial of $\sin(x)$ at the points $x_0$ and $x_0 + h$. We know that $$ \sin(x) - p(x) = \frac{-\sin(\xi)}{2}(x - x_0)(x - x_0 - h)$$ for some $\xi \in (x_0, x_0 + h)$. Because of $\sin(\xi) \approx \sin(x)…
user109536
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Upper bound for Newton-Raphson first approximation

I'm trying to determine why the upper bound for the first approximation $p_0$ of the root $p$ of a function $f \in C^2$ must satisfy the condition: $$|p-p_0|\leq \frac{2|f'(p)|}{|f''(p)|}$$ when $p$ is a simple root, i.e.: $f(p)=0$, but $f'(p)\neq…
MonsieurCandie
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