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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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Newton's method fractals generator?

So I am trying to find the relationship between fractals and newton's method, and I am trying to use software to generate the fractals pattern. which software can I use to do that and how? Also i am trying to compare two different types of…
Tommy Lassa
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How should I determine the number of quadrature points when using Gaussian quadrature integration?

For integration of polynomials, the number of qradrature points can be determined easily according to the $(2N-1)$ rule. While in most cases, it's not necessary to use Gaussian quadrature for simple polynomials since they can be calculated…
user123
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Writing difference equation for $\nabla \cdot (u \nabla v)$ [Finite differences- Numerical Analysis]

Assume I try to work in 1D, and solve an equation which contains expressions of the form: $u''$, but also an expression of the form : $(u v' ) ' $. Assume I discretize my region of solution, and denote $h$ to be my spacing. I am trying to work…
georgia
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Romberg Method (quadrature)

Romberg Method says: $$\int_a^bf(x)\,dx\;{\color{red}=}\;T_h^0+c_2 h^2+c_4 h^4+\dots$$ where $T_h^0$ is the Trapezoidal Rule: $$T_h^0=(b-a)\left[\dfrac{f(b)+f(a)}{2}\right]$$ My question is: why is true the "$\color{red}=$" in the Romberg Method?
yemino
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Determine Hermite interpolation polynomial and interpolation polynomial using linear splines

y:=f(x) xi || -1 0 1 2 yi || 4 6 2 10 a) Determine Hermite interpolation polynomial and evaluate the appropriate…
TheGruela
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Find $F$ which interpolates $f$ at all points $x_0,x_1,\cdots ,x_n$

Suppose the function $u(x)$ interpolates $f(x)$ at $x_0,x_1,\cdots ,x_{n-1}$ and the function $v(x)$ interpolates $f(x)$ at $x_1,x_2,\cdots ,x_n$. Then find a function $F(x)$ which interpolates $f(x)$ at all the points $x_0,x_1,\cdots…
Empty
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More accurate way to compute $f(x)=\sqrt {x-1} - \sqrt{x-2}$

The formula: $f(x)=\sqrt {x-1} - \sqrt{x-2}$ is used to compute a function $f$. Suggest a more accurate way to compute the same function. I simply do not know how to start with this question... I made the attempt employing the fixed-point iteration,…
QmmmmLiu
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How do we get this equation?

Given the problem $$-u''(x)+q(x)u(x)=f(x), 0 \leq x \leq 1, \\ u'(0)=u(0), \ \ u(1)=0$$ where $f,g$ are continuous functions on $[0,1]$ with $q(x) \geq q_0>0, x \in [0,1]$. Let $U_j$ be the approximations of $u(x_j)$ at the points $x_j=jh, j=0, 1,…
evinda
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How to compute equation that includes a^n / n! for large a and/or n?

I'm trying to compute $$e^{-a} \times \sum_{n=0}^N \frac{a^n}{n!}$$ where $e^{1}=2.7183$, N is an integer > 0, and a is a real value > 0. For example, N=a=1500. But for large n, the n! goes to infinity, and for large a, the numerator a^n goes to…
user46688
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Weighted quadrature formula

I need to calculate a quadrature rule with maximum degree of accuracy that looks like this: $$ \int_0^\infty e^{-x}f(x)dx = A_1f(x_1) + A_2f(x_2) + R(f) $$ where $f(x) = cox(x)$, presumably. Questions: 1) Can this be solved in a general manner, for…
scribu
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Modified Euler explanation

I've implemented the modified Euler's method to solve ODEs as described in my lecture (here only important parts): F = @(x,Y) [Y(2),-sin(Y(1))]; ... for i=1:N Yh=Y+h/2*F(x,Y); Y=Y+h*F(x+h/2,Yh); x=x+h; ... end What I don't get…
Hobo
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2 Stage, Order 3 IRK

Looking for a program for this method. Any suggestions on where I can find one? The tableau for this method is, 0| 1/4 -1/4 2/3| 1/4 5/12 -------------- …
sjf2468
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Relative error in numerical computations doubt

I wonder why in numerical computations, if, in order to estimate the error it's done with respect to a higher precision calculus than the one used, why don't take the higher precision result as the final solution to take into account? Because of the…
Felix L.
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Homotopy Methods: Show there is no path connecting roots

This is a question from Cheney and Kincaid's Numerical Analysis (3rd ed, pg 138). Consider the homotopy $h(t,x)=tf(X)+(1-t)g(x)$, in which $f(x)=x^2-5x+6$ and $g(x)=x^2-1$. Show that there is no path connecting a root of $g$ to a root of $f$. I'm a…
Georgia S
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Why is the adjoint of d/dx -d/dx with zero boundary condition?

Weird observation about discretization of $\nabla^T\nabla$ In the previous question page, I saw "the adjoint of d/dx is -d/dx" I think that it is related to integration by parts. But I don't know understand what it mean, clearly.
jakeoung
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