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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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If and only if for Gaussian quadrature.

Consider a quadrature rule of the form $$ Q(f) = \sum_{i=0}^{n} A_{i}f(x_{i})$$ to approximate the integral $$\int_{a}^{b}w(x)f(x)dx,$$ where $w:[a,b] \to \mathbb{R}$ is a positive function. Prove that $Q(f)$ is exact for all polynomials $f$ of…
esavaleo8
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Solving a homogenous recurrence relation with variable coefficient

How can I solve a recurrence relation with variable coefficient like the one below: $$J_{n+1}(x) = \frac{2n}{x}J_{n}(x) - J_{n-1}(x)$$ for simplicity assume $x=1$. I need to solve this in order to analyse the error trend of $\in_{n+1}(x) = |J_{n+1}…
Upperwal
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Jacobi Gauss lobatto quadrature nodes and weights

According to my calculations, For $N = 5$ i.e., $n=(0,1,2,3,4)$ and $\alpha = \beta = 0$ in the interval [-1, 1] the Jacobi Gauss Lobattto nodes are: x = [-1, -0.655, 0, 0.655, 1] and its corresponding weights are: w = [0.1, 0.544, 0.711, 0.544,…
Abbeha
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Solving a PDE system with Neumann and Robin boundary conditions numerically

I've tried to solve the following PDE system with Neumann and Robin boundary conditions numerically for a long time, but it just doesn't work: $\frac{\partial c_+}{\partial t} = -v \frac{\partial c_+}{\partial x} + D\frac{\partial^2 c_+}{\partial…
Peter123
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merging two power equations into one

Let, we have two power equations: $$y=k_1x^{a_1}$$ $$y=k_2x^{a_2}$$ Is there any way (analytical or numerical) to combine these two equations into one i.e. into the form: $$y=kx^a$$ Infact,what I am trying to do here is: I have got around 20…
fnafis
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Des chiffres et des lettres: proof that a random "right count" has a 94% success, how to prove it? (rules inside)

I am not sure whether this is the correct forum. Anyway, here goes... The game "Des chiffres et des lettres" is the most long living television game in French history; in its current form, it dates back as far as 1972. There are two separate…
fge
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Local truncation error calculation

The other day, I tried to answer someone's question, you can read the question and the answer here: Numerical Solutions of ordinary differential equations The problem is, I got stuck towards the end of the calculation. Now my question is: does…
Rudy the Reindeer
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Square root of x : $\sqrt{x}$ (Numerical Method)

$$f(x) = \sqrt{x}$$ has to be approximated by polynomial interpolation $p(x_n) = f(x_n)$ with the positions $\{x_n\} = \{1,4\}$. For such problem which method is the fastest? And find $p(2)$. My attempt: Newton's Method: $p(x) = x_{n+1} = x_n +…
rndflas
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Bisection Method and midpoint

If $f(a) < 0$ and $f(b) > 0$, then prove that the point $c$ computed in the bisection method is the point where the line through $(a, \operatorname{sign}(f(a)))$ and $(b, \operatorname{sign(f(b))})$ intersects the $x$-axis. Can anyone give me hints…
k7dy
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Approximate solution of a trigonometric equation using only pen and paper

I found an exam question that I managed to solve via calculator but not by using only pen and paper. Is there a solution to this? Prove that there is an $x$ satisfying $10x-9 = 9\sin x-10\cos x$ and $0.654
studying
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Discretisation of Euler's method

I don't quite understand the discretisation error of Euler's method $$L(t,h)=\frac{x(t+h)-x(t)}{h}-f(t,x(t))$$ What I don't understand is: $\frac{x(t+h)-x(t)}{h}$ is the "gradient" and $f(t,x(t)$ is the value of the function. How can we get the…
user71346
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Determining whether function is logarithmic or logarithmic with addition of variable

Suppose that I have a function: $f(x,U) = x + \log(U)$ where $x$ and $U$ can be complex numbers. I take $f(x,U)$ and evaluate it a number of times for different $x$ and $U$. Since $U$ can be complex, I take $\log(U)$ as the complex logarithm…
Nicholas Kinar
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How to find the number of iterations needed within a certain degree of accuracy in the bisection method

I know how to find a zero of a function by the bisection method. But I am not sure how to find the number of iterations needed within a certain degree of accuracy. Let's say, when we use the bisection method to find the zero $x^*$ of the function…
user71346
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Please explain the solution

Use a Taylor series expansion to compute an error estimate in approximating the derivative of the function $f:\mathbb R\to\mathbb R$ using the formula $$ f'(x_0) \approx \frac{f(x_0-2h)-4f(x_0-h)+3f(x_0)}{2h} $$ So $\left|f…
stackdsewew
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Numerically stable version of calculation with cancellation

What's a numerically stable way to compute $$ \frac{2^{1/n}}{2^{1/n}-1} $$ for large (integer) $n$?
Charles
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