Mathematics Stack Exchange
Mathematics Stack Exchange

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

14158 questions
1
vote
1 answer

Constructing piecewise quadratic polynomial

The question asks to construct a piecewise quadratic polynomial defined on the interval $\mathbb{R}$ of the form $$ B_0= \begin{cases} p(x)=x^2,\qquad\qquad\quad\; 0\leq x<1,\\ q(x)=ax^2+bx+c,\quad\, 1\leq x<2,\\ r(x)=(3-x)^2,\qquad\quad 2\leq…
Alti
  • 2,458
1
vote
1 answer

Number of iterations required for $x_n$ to be correct to 100dp

Given $a \in (0.5, 1)$, one seeks to compute $\frac 1a$ using Newton–Raphson method with $$f(x) = a − \frac 1x$$ Taking $x_0 = 1.5$, compute $x_2$ and estimate which $x_n$ is first correct to $100$ decimal places, taking the error as $|ax_k −…
Gragbow
  • 661
1
vote
1 answer

BDF2: How to Implement

While I had a course a couple years ago concerning the implementation of linear time marching methods, I'm pretty rusty and am getting stumped here. I'm attempting to implement the BDF2 time marching method in order to convect Lagrangian points in…
Marius
  • 111
1
vote
1 answer

Give an example to show that computing $ \frac{x - y}{\sqrt x + \sqrt y}$ is not always more accurate for computing $fl(\sqrt x - \sqrt y)$?

I assume that we are working on a machine with rounded four-digit decimal arithmetic, I know that we have this formula: $\sqrt x - \sqrt y = \frac{x - y}{\sqrt x + \sqrt y}$ Since there is cancelation error in machine so it is better to use this…
haleh
  • 133
  • 9
1
vote
3 answers

Finding N roots of an oscillating function with infinite roots on interval [0,1]

I am attempting to numerically approximate $$\int_0^1 \frac{1}{x} \cos\left(\frac{\log(x)}{x^2}\right) \, dx,$$ whose graph is shown below. To that end, I am attempting to calculate roots of the function to serve as bounds for subintervals. My plan…
user425340
  • 23
1
vote
1 answer

The error in lagrange interpolation

The picture contains the formula of the error which occurs when we interpolate a function $f(x)$ to a polynomial $P(x)$, my questions is: what is the difference between this error and the absolute error $|f(x)-P(x)|$? and what does $f\in…
soso sos
  • 349
1
vote
1 answer

Euler's Method Error Term (Big O Notation)

I have just been looking at the error in Euler's Method, and I noticed something strange. I understand that the error is proportional to $O(h)$ via the argument that the local truncation error is proportional to $O(h^2)$, and there are n of these…
Resquiens
  • 373
1
vote
1 answer

A method to evaluate rounding errors

I'd like to learn to evaluate how much there is error if I compute algebraic expressions and round my intermediate steps. For example, I had data of length of couples height as (167,183), (165,165), (167,178), (163,173), (178,180), (165,173). I…
statstudent
  • 35
1
vote
1 answer

Least Squares approximation of the first degree to the function $f(x) = \sin(x)$.

I was solving a problem of approximation by Least Square Method,it asks the least squares approximation of first degree to the function $f(x) = \sin(x)$ over the interval $[-\pi/2 , \pi/2]$, I got answers varying from textbook answer , $I_{0} =…
BAYMAX
  • 4,972
1
vote
1 answer

Quadrature Error for Trapezoidal Rule

Question: If the trapezoidal rule is used to approximate the integral of the function $f$ over the interval $[a,b]$ with 201 evenly spaced dissection points, estimate the quadrature error for $$\int_0^1log(1+x^2)dx$$ For this style of question…
Nicky
  • 671
1
vote
3 answers

Taylor Series Based Numerical Method for Non-Linear Equations

There is a numerical method for solving non-linear equations which uses the Taylor Series: $$f(x_n+e_n)=f(x_n)+f'(x_n)e_n+\cdots+\frac{f^{(k)}(x_n)}{k!} e_n^k+\cdots+=0$$ The solution is then found through the iteration: $$x_{n+1}=x_n+e_n$$ For…
Adam
  • 351
1
vote
1 answer

Can we approximate complex roots of nonlinear equations by numerical methods

Can we approximate the complex roots of nonlinear equations by using Bisection or Newton Raphson mehods? Is there any numerical example where these methods were applied to approximate complex roots of nonlinear equations? Thank you very much for…
Srijan
  • 12,518
  • 10
  • 73
  • 115
1
vote
1 answer

How to prove $\bigtriangledown = \bigtriangleup E^{-1}$?

How can we prove $\bigtriangledown = \bigtriangleup E^{-1}$? where,$\bigtriangleup \rightarrow \text{ Forward difference operator }$ $\bigtriangledown \rightarrow \text{ Backward difference operator }$ $E \rightarrow \text{ Shift operator }$ I tried…
Quixotic
  • 22,431
1
vote
1 answer

Loss of significance $\frac{1-x}{1+x}-\frac{1}{3x+1}$

find a way to overcome $\frac{1-x}{1+x}-\frac{1}{3x+1}$ loss of significance for which values is there loss of significance? is it 0 to solve it I get need to multiply it by…
gbox
  • 12,867
1
vote
1 answer

Use the Mean Value Theorem to estimate absolute error

Let $f \in C[a,b]$ be a function whose derivative exists on $(a,b)$. Suppose $f$ is to be evaluated at $x_0$ in $(a,b)$, but instead of computing the actual value $f(x_0)$, the approximate value, $\hat f(x_0)$, is the actual value of $f$ at $x_0 +…
Gjekaks
  • 1,133
1 2 3
…
99 100