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
4
votes
3 answers

Newton's Method - Slow Convergence

I'm using Newton's method to find the root of the equation $\frac{1}{2}x^2+x+1-e^x=0$ with $x_0=1$. Clearly the root is $x=0$, but it takes many iterations to reach this root. What is the reason for the slow convergence? Thanks for any help :)
Drake
  • 851
4
votes
3 answers

Newton's Method - Why is there slow convergence with a high multiplicity

I'm using a calculator to observe the sluggishness with which Newton's method converges with $f(x) = (x-1)^8$. I let $x_0 = 1.1$. Clearly it's taking forever to get to the root $x=0$. I'm not completely sure why this is. I know it has to deal with…
Drake
  • 851
4
votes
1 answer

Can you calculate the accuracy of a calculator?

I have a phone with an inbuilt calculator. I love to play with calculators. So I did the following and the following was shown by the calculator. When I went in the scientific tab and wrote $\pi$ it returned $3.1415926535$. When I wrote $\pi \times…
Rohinb97
  • 1,702
4
votes
2 answers

Why are we choosing the Krylov projection method like this?

I am attending a course on numerical linear algebra, where we talk about Krylov-Methods right now. We want to construct a sequence which converges to a solution of a system of linear equations $Ax = b$. Lets say that $x_0$ is our starting point. In…
Josh.K
  • 105
4
votes
1 answer

Explanation needed: $g \,\colon [0,2] \to \Bbb R \,$ be a twice continuously differentiable function

I am stuck on the following problem: Let $\,\,g \, \colon [0,2] \to \Bbb R \,$ be a twice continuously differentiable function. If $\displaystyle \int_{0}^{2}g(x)dx \approx 2 g(1),$ then the error in the approximation is : $\frac{g'(\xi)}{12}$ for…
learner
  • 6,726
4
votes
2 answers

Fixed point iteration problem of $f(u)=u^3-u-1$

I was thinking about the following problem: Let $f(u)=u^3-u-1$. Then I have to verify whether the following statements are true/false? 1.Starting with the initial guess $u^{(0)}=1.5,$ the fixed point iterates of the equation $u=g(u)$,where…
user53386
4
votes
2 answers

Why does forward-difference give a different value than backward-difference?

We are given that for $x_0 = 0.5, x_1 = 0.6$ and $x_2 = 0.7$, we have $f(x_0) = 0.4794, f(x_1) = 0.5646$ and $f(x_2) = 0.6442$. Also $h = 1$. The forward-difference formula is $f'(x) = \dfrac{f(x_i + h) - f(x_i)}{h}$ and the backward-difference…
Boognish
  • 43
  • 4
4
votes
3 answers

What's the largest number?

I have problems with the problem: What's the largest number? \begin{align} &60^{\frac{1}{3}} \mbox{ or } 2 + 7^{\frac{1}{3}}& \end{align} I tried using factorization, but I didn't get a good result and I dislike using an estimate. Thanks.
Daniel Brandão
  • 41
4
votes
2 answers

Question about newton's interpolation and rootfinding

Suppose we want to solve $f(x) = 0$. We start from the newtons interpolation formula on two points, say $x_n$ and $x_{n-1}$. ${\bf Attempt.}$ Since we are given two points say $(x_n, y_n)$ and $(x_{n-1},y_{n-1})$, the Newton interpolant is given…
James
  • 3,997
4
votes
1 answer

Newton-Cotes error term homework

The two-point open Newton-Cotes formula provides an estimate of the integral $$\int_{-1}^{2} (x^3 - x^2 + x)dx$$ The error term is expressed in terms of $f''(z)$. The problem asks to find $z$. I know that the two-point open Newton-Cotes formula…
natn2323
  • 157
4
votes
2 answers

Loss Of Significance Error For: $\tan(x) - \tan(y)$

For $x \approx y$, how may we avoid loss of significance errors in computing $\tan(x) - \tan(y)$? I don't think Taylor polynomials will be useful here, so I am thinking the answer lies in some sort of trig identity trick. I have managed to get the…
tylerc0816
  • 4,123
4
votes
4 answers

How to Determine Interval $[a,b]$ for a Fixed-Point Iteration?

Determine an interval $[a,b]$ on which the fixed-point ITERATION will converge. $x = g(x) = (2 - e^x + x^2)/3$ I've determined that $g'(x) = (2x -e^x)/3$, but I don't know how to determine the interval without the guess-and-check method (which…
user59808
  • 41
4
votes
1 answer

Simple linear interpolation in $\mathbb{R}^k$?

When doing simple linear interpolation on a data set in $\mathbb{R}^2$, I just sort the points by one co-ordinate and then do piecewise interpolation between a point and its successor in the sorted data set. How would I do this in $\mathbb{R}^3$,…
user8534
4
votes
1 answer

Error propagation in numerical analysis.

my professor is using the following slides to teach error propagation in numerical analysis: I am finding hard times understanding the material because of its extreme formal notation and brevity. Can anyone explain this discussion? I am mainly…
user366312
  • 1,641
4
votes
1 answer

Use finites differences for the BVP

Use finite differences for the BVP $u(x)''=\tan^{-1}(u(x))+2u(x)+\cos(x)$ with $u(0)=u(1)=0$. I don't know what I have to do with $\tan^{-1}(u(x))$. When I applied finite differences, I need to have the term $u(x)$ without depending on the…
clm
  • 225
1 2 3
…
99 100