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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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Why do we impose |ρ(ξ)|^2 ≤ 1 in the Crank-Nicholson scheme?

I'm being introduced to numerical methods for PDE's. I've learned that in the Crank Nicholson scheme, we can map the discrete Fourier transform of the current space lattice to the FT of the same lattice for the next time interval: $$ \hat U^n = \rho…
BallisticThomist39
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Approximating polynomial roots without complex arithmetics

Suppose that we have polynomial of degree greater than two $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+_\cdots+a_{1}x+a_{0}$ with real coefficients and we want to approximate all roots of this polynomial without complex arithmetic It seems to be good idea to get…
J Doe
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Euler method fails to approximate the exact solution $y(x)=(2x/3)^{3/2}$

How can I show that the Euler method fails to approximate the exact solution $$y(x)=(2x/3)^{3/2}$$ to the IVP $$y'=y^\frac{1}{3}$$ $$y(0)=0$$ Here we have $f(t,y)= y^\frac{1}{3}$, $y_0=0$ and so $f(t_0,y_0)=f(0,0)=0$ and $$y_{n+1}=y_n +h…
ali
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Compare the rate of convergence of Jacobiand Gauss-Seidel method for the matrices

Compare the rate of convergence of Jacobi and Gauss-Seidel method for the matrices $$ A=\left(\begin{array}{rrr} 1 & 2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}}…
Tariro Manyika
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Logarithm computation of a multiple precision floating point number

I am working on a library for multiple precision floating point numbers. I implemented computation of natural logarithm as follows: Factoring of decimal powers first: $$ln(m*10^e) = \ln(m*10^{(1-n)}) + (e - 1 + n)*\ln(10)$$ here m is mantissa, n…
stencillogic
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accurancy of Tertiary bisection method

The question is: Assume we change the bisection method into "tertiary bisection" which divides the interval into 3 parts and chooses the one from the left which is minimal and changes sign. e.g in the following sketch if the first interval is…
user65985
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analysing a numerical method

I am learning numerical methods and even though I read several articles I still could not have a summarized idea of how and what exactly the following means. What is the idea/definition of accuracy, stability, consistency and convergence of a…
user46044
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Numerical Methods - order of accuracy

I am implementing a program which solves differential equations - 1d diffusion. I am using the Crank - Nicolson method whose accuracy should be second-order for time and second-order for space. Unfortunately my results are second-order for time and…
GenLancelot
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$t(x)=\prod \limits _{k=1}^{2n}\sin \left (\dfrac{x-x_k}{2}\right )$ a trigonometric polynomial proof

Show that, for real $x_1,\ldots ,x_{2n}$,the function:$$t(x)=\prod \limits _{k=1}^{2n}\sin \left (\frac{x-x_k}{2}\right )$$is a trigonometric polynomial$$\frac{1}{2}a_0+\sum \limits _{j=1}^na_j\cos (jx)+b_j\sin (jx)$$with real coefficients…
a dysthemic student
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Numerical evaluation of Student's t density

The density of Student's t distribution is given by $$f_n(x) = \frac{1}{\sqrt{n\pi}}\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac n2\right)}\left(1+\frac{x^2}n\right)^{-\frac{n+1}{2}},$$ where $n$ is a positive real (the degrees of…
Syd Amerikaner
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If the points $x_1,x_2,\ldots,x_n$ are distinct,then...

I am stuck on the following problem that says: If the points $x_1,x_2,\ldots,x_n$ are distinct,then for arbitrary real values $y_1,y_2,\ldots,y_n$, prove that the degree of the unique interpolating polynomial $p(x)$ such that $p(x_i)=y_i,\,\,(1…
learner
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Bisection and Fixed-Point Iteration Method algorithm for finding the root of $f(x) = \ln(x) - \cos(x)$.

I am trying to find the root of $f(x)=ln(x)-cos(x)$ by writing an algorithm for bisection and fixed-point iteration method. I am currently using python but whenever I'm running it using either of the two methods, it prints out "math domain error". I…
mathnewbie
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$\int_{\Omega} \nabla u_{h} \cdot \nabla v_{h}=\int_{\Omega} f v_{h} \equiv \int_{\Omega} \nabla u_{h} \cdot \nabla \phi_{i}=\int_{\Omega} f \phi_{i}$

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$, with $d=2,3$. Let $V_{h}$ be a finite dimensional space of dimension $n$ : $$ V_{h}=\operatorname{Span}\left\{\phi_{1}, \ldots, \phi_{n}\right\} $$ i.e, the functions $\phi_{i}$ form a basis for…
user1023562
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What are equivalent expressions for the following functions that avoid a loss of significance?

$\ln ( x + 1 ) - \ln ( x ) $ for large $x$ $\cos^2 x - \sin^2 x $ for $x \approx \frac{\pi}{4}$ I know in general that when we have subtraction of two very near amounts, it may lead to catastrophic cancelation, therefore I need to change the…
JenniferBagly
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Mysterious term in semi-implicit Euler scheme

In the paper i'm currently working with I don't understand the role of the term $C^m$ in the following semi-implicit numerical Euler scheme they use, which consists of following two recurrence relations, where $A^m = A^0 + B^m$ ($A^0$ is a constant)…
Andrusch
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