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

Calculating the partial derivative and plugging in interval given.

I have dervied:…
Cherry
  • 11
  • 1
1
vote
1 answer

Modern book on error analysis in numerical calculations

There is a classical book by Wilkinson ( "Rounding Errors in Algebraic Processes"), but reviewer in the Amazon writes: Unfortunately, the book discusses these critical topics in terms of 1963 technology. A modern eye finds it jarring to see computer…
Rustam Mamin
  • 69
1
vote
1 answer

Is there any formula for the infinite surd $\sqrt{k+\sqrt{k^2+\sqrt{k^3+\sqrt{k^4+...}}}}$?

Is there any related studies about the topic?
Cleo Tejano
  • 11
1
vote
2 answers

What does round-off error in double arithmetic mean?

Let's say I have a function $f$ and furthermore let there also be a recursion formula for $f$. So I can evaluate $f(x)$ directly and I can evaluate $f(x)$ using the recursion formula. Now, in the recursion formula when $x$ is close to $0$ we…
user711482
  • 325
1
vote
0 answers

Verify the error estimate

$(a)$ Derive adaptive quadrature formula to evalute $\int_a^bf(x)dx$. $(b)$ Given, $I=\int_0^{\frac{\pi}{4}}cos^2xdx$ compute $S(0,\frac{\pi}{4}),S(0,\frac{\pi}{8})$ & $S(\frac{\pi}{8},\frac{\pi}{4})$. Also verify the error estimate…
Dr.Antidode
  • 169
1
vote
1 answer

Function approximation from list of numbers in linear time

so what I need is a bit hard to describe for me, so I start with giving a context: I have a list of numbers that is not sorted (yet). One part of a more complex algorithm is that I need a rough approximation of a list of arbitrary numbers so that I…
Tia
  • 111
  • 3
1
vote
2 answers

How to find the correct iterative function for iterative method

We want to compute the root of $$f(x) = x^3-2x-5$$ Since $f(1.5)<0, f(2.5)>0$ the root must be within the interval $[1.5,2.5]$. 1) An intuitive iterative function would be $\phi(x) = 0.5 \cdot (x^3-5)$, that defines the iterative…
JB-Franco
  • 852
1
vote
0 answers

Nth Degree Bernstein Polynomial Properties

I am told to prove for any $n$, $p_n(0)=p_0$ and $p_n(1)=p_n$. Where $p_0,p_1,...,p_n$ are a set of points with $p_i=(x_i,y_i)$ in the plane and the n-th Bernstein polynomial is given by
Sherien Hassan
  • 21
1
vote
0 answers

Prove that polynomials L(x) have degree n, satisfying L(x)=1 and L(x sub j)=0 whenever i is not equal to one.

For this question I basically need to prove the properties of Lagrange interpolation using basis functions. I understand the exsistence of an identiy matrix causes for us to have these properties but I am unsure how to go about proving it.
Sherien Hassan
  • 21
1
vote
1 answer

Floating Point system with base 10

It is a long time ago I used floating points, so I struggle with the following question and hope that you can help me out. I need to list all numbers that can be represented exactly in a normalised floating- point number system with base 10, two…
KingDingeling
  • 1,642
1
vote
1 answer

Interval Transformation for Numerical Integration

I want to write a python program that evaluates the following integral using: $$\int_{-1}^1 f(x)dx \approx \sum_{i=0}^n w(x_i)f(x_i) $$ In addition, I want to implement a linear transform of the interval with $\ x = mt +c$, such that I can…
ABCCHEM
  • 99
1
vote
1 answer

Fast scheme for the evaluation $x^k$ for $k\in[0,1]$

In numerical analysis, we have the scheme to evaluate $x^{1/2}$ or more generally the scheme to evaluate $x^{1/n}$. For $x^k$ with $k\in[0,1]$, is there a fast scheme to evaluate it? My naive implement is writting $k$ into fration like…
Langma
  • 321
1
vote
2 answers

Complex Numerical analysis

I have been studying numerical analysis in depth and I am wondering if there is a subject like complex Numerical analysis in which we study the concept of numerically solving complex variable equations, complex integration etc.?
kam
  • 151
1
vote
1 answer

Absolute error versus relative error — more suitable?

I understand the differences between relative and absolute error. I can think of examples when relative error is a more suitable error measure but not when absolute error is more suitable than relative error. So when is absolute error a more…
Moo
  • 43
1
vote
2 answers

behavior of the solution of a polynomial equation

Given a positive integer $m\ge 3$, consider the function $f:[0,1]\longrightarrow \mathbb{R}$ given by \begin{equation*} f(x)=(1-x)^{m-2}-\frac{3x}{2-x} \end{equation*} It is easy to see that $f(x)=0$ has unique solution in $[0,1]$, $\phi_m$. What is…
user84909
  • 412
1 2 3
…
99 100