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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

14158 questions
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Without using a calculator, find the cube root of 2, correct to two decimal places

This is a variation from my personal favorite 'A Concise Introduction to Pure Mathematics' by M. Liebeck (Chapter 3, Exercise 4). Without using a calculator, find the cube root of 2, correct to two decimal places. This is how I work. (Only odd…
Dimitris
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How to find how many terms in a sum you need to get a given amount of precision

Context: I recently discovered the formula for $\pi$ by Machin, $$\frac{\pi}{4} = 4\arctan \left(\frac15 \right) - \arctan{\left(\frac1{239} \right)}$$ In order to apply this formula, I used the Taylor series expansion for $\arctan x $ to create the…
Mailbox
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How would I apply the nonlinear shooting method to a problem with "mixed" boundary conditions?

I have the BVP: $$ \frac{d^2y}{dx^2}=e^{2x}+y\frac{dy}{dx} $$ With the following mixed boundary conditions: $y(0)=1$ and $y'(1)=0$ How would I apply the shooting method to this BVP? It's throwing me off that the right hand BC is a derivative, so I'm…
kitty123
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calculate the 4th order of Newton's (divided diff ) interpolating polynomial by hand

I wanted to calculate the 4th order of Newton's interpolating polynomial by hand but the results weren't that good, so I wanted to make sure that I got it right is $$a_4 =\frac{f(x_4)-a_0 - a_1(x_4-x_0) -a_2(x_4-x_0)(x_4-x_1) -a_3…
hn_gara
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Backward Euler method problem

Let $\operatorname{g}\left(x,t\right)=\alpha t + x$ and $\alpha < 0$ is a parameter s.t. $\left\vert\alpha\right\vert$ is large. To numerically solve the equation $Y'= \operatorname{g}\left(x,Y\right)$, the Backward Euler Method is used. Find how…
rudytheduck
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Numerical solution to a Legendre differential equation - approximating using tridiagonal matrix

My task is to solve this differential equation: $$(1−x^2)\frac{d^2y}{dx^2}−2x\frac{dy}{dx}+p(p+1)2y=0,y(0) = y₀, y(a)=yǎ$$ I need to solve it numerically using linear equations with a tridiagonal matrix. I am struggling with the first step -…
Jacos
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What is the Mehrestellenverfahren of finite difference scheme?

When I read the book A First Course in the Numerical Analysis of Differential Equations by A. Iserles, in Part II, p.163, there is one method called Mehrestellenverfahren, which is a modified nine-point formula. However, this method is not well…
Mingdong HE
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Finding a numerical stable way to evaluate different functions

For each of the following functions, suggest a numerical stable way to evaluate $y$ for small $x$ in order to avoid loss of precision: $y=x^2-\sin(x)^2$ - ignore terms of order $o(x^7$) $y=\frac{x^2-\sin(x)^2}{2-\sqrt{x+4}}$ - ignore terms of order…
Jens
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How to use finite difference method to solve second order ODE with image eigenvalue?

$-\phi'' +c(x) \phi(x) = f(x), \phi (0) = \alpha $ and $\phi (1) = \beta $ If we use central difference method to discretize $\phi ''$, we know when $c(x) \geq 0 $ the matrix is invertible. Thus, we can solve the linear algebra problem to find the…
user823572
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Trusting numerical methods

So I'm currently learning numerical methods and I'm curious in which scenarios you can blindly trust the numerical methods? Are there scenarios where you should show suspicion towards the numerical result? As of right now I'm focused on numerical…
Avina
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Proving $L^2$ error estimate for piecewise linear error bounds

Let $f(x)$ be a $C^2$ function on $[0,1]$, split uniformly into intervals of length $h=1 / n$. Let $p$ be the piecewise linear polynomial interpolating $f$ on each interval. I am trying to prove the following $L^2$ error…
Questioner5453
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Divergence preservation and change of variable

I'm currently studying a numerical scheme which preserves the divergence through time, i.e $div(A^{n+1}) = div(A^{n})$ where $A$ any vector field in $\mathbb{R}^n$ and $n, n+1$ time stations. This is true in Cartesian coordinates but I was wondering…
Amzocks
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How can I show that the l2 norm holds true for triangle inequality

I need to show that the l2 norm : $\|x\|_2 = \sqrt{ x_1^2 + x_2^2 + ... + x_n^2}$ has the following properties: i) $\|x\| > 0$ if $x\neq 0$ ii) $\|\lambda x\| = |\lambda|\cdot\|x\|$ for $\lambda \in \mathbb{R}$ iii) $\|x+y\| \leq \|x\| + \|y\|$ I…
user87803
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Differentiating terms involving evaluation operator and Wronski matrix

We have the initial value problem $$\dot{\mathbf{y}} = f(\mathbf{y}), \mathbf{y}(0) = \mathbf{y_0},$$ with $f(\mathbf{y})$ continuously differentiable. There exists a $T > 0$ such that $\mathbf{y_0} = \mathbf{\Phi}^T \mathbf{y_0}$ and $\mathbf{y_0}…
Huy
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Exercise 1.12 - Mark H. Holmes Introduction to Numerical Methods in Differential Equations

I'd like to prove that the order of the two-steps Backward Differentiation Formula is $O(k^2)$. For this, I write $$y'_i\approx \dfrac{y_{i}-{\tfrac {4}{3}}y_{i-1}+{\tfrac…
Quiet_waters
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