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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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Evaluate a differential equation at a given point

I have a second order differential equation $x^2\cdot y''(x)+x\cdot y'(x)+(x^2-10)=0$ with the initial condition $y(100)=1$ and $y'(100)=0$. I want to evaluate $y(x)$ when $x=103$ with 8 digit accuracy. The first step is rewrite this equation to a…
Qomo
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What does the term $c_x$ mean in the theorem of Taylor's remainder?

Theorem (Taylor's remainder) Assume that $f(x)$ has $n+1$ continuous derivatives on an interval $\alpha\leq x\leq\beta$, and let the point $a$ belong to that interval. For the Taylor polynomial $p_n(x)$, let $R_n(x) \equiv f(x)-p_n(x)$ denote the…
Schole
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Discretisation of 2D Poisson equation in Sylvester matrix form

I am working on fast Poisson solvers and I have to understand some basic concepts with the discretisation of the 2D Poisson equation ($U_{xx}+U_{yy}=f$) in the Sylvester equation form ($KX+XK=F$). I have been able to discretise the 2D Poisson…
steve mike
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skinned surface

In the representation of a skinned surface using $B$-Spline, I have $K+1$ given curves of degree $p$ on a common partition $U$ and I want to construct the surface $S(u,v)$ with these curves as isoparametric curves. Assuming that $p$ is the degree in…
fsfdfdsfsdfsdf
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Writing a program using the trapezoidal rule

Write a program to evaluate $I=\int_a^bf(x)dx$ using the trapezoidal rule with $n$ subdivisions, calling the result $I_n$. Use the program to calculate the following integrals with $n=2,4,8,16,32,64,128,256$. Analyze empirically the rate of…
user60514
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Hermite problem formula

Consider the Hermite problem $$p^{(r)}(x_i)=y_i^{(r)},\ i=1,2 ;\ r=0,1,2$$ with $p(x)$ a polynomial of degree $\leq 5$. a - Give a Lagrange type of formula for $p(x)$. b - Give a Newton divided difference formula and also derive an error…
user60514
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Question about numerical calculation

Quasi-newton method： If the second derivative of the target function doesn't exist at all, can we use Quasi-newton method? If the second derivative of the target function at some point doesn't exist, can we use Quasi-newton method? My question comes…
abraxas
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Computing the square root function with Newton's method

Show that Newton's method can be used to compute the square root function $\sqrt a$ using the formula $$x_{n+1} = \frac{1}{2}\left(x_{n} + \frac{a}{x_{n}}\right)$$ show that the error is $$\sqrt a - x_{n+1} = -\frac{1}{2x_{n}}\left(\sqrt a -…
BMEdwards37
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efficient algorithm for solving equation $\sum_n a_n/(x - x_n ) = 1$

Here $x_n $ are real and $a_n$ are positive, and we have a finite summation. The picture is very clear. But what numerical algorithm is stable and efficient? Supposed $x_n $ are ordered in the increasing order, then between $x_n $ and $x_{n+1}$…
S. Kohn
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Gauss Quadrature in numerical methods

In the following code I have implemented gauss quadrature. It is working correctly for my first function but for my second function I am getting an error. So, Is it possible to do this code without sing p_roots? and if not how do I fix it so my…
fr14
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Properties of Bernoulli Polynomials

I have pored over the internet and can't seem to find any proof of a particular property of Bernoulli Polynomials. The property is: $P'_{n}(x) = nP_{n-1}(x)$ I have not found anything conclusive in any of my textbooks that lay out a clear proof or…
djohnson
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Curve Fitting Atmosphere

I'm trying to build a sounding rocket to reach the 100km as my senior project next year. To do this I need an equation to approximate the atmospheric density at any given altitude in the rocket's ascent, so that I can better understand the engine…
WhisperingShiba
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What is the definition of spectral convergence?

I don't understand what is spectral convergence. The definitions I found in google are very physics. Is there any definition in mathematics ?
Haiyi Tan
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Find the interpolating polynomial of degree $3$ that interpolates $f(x) = x^3$

Find the interpolating polynomial of degree $3$ that interpolates $f(x) = x^3$ at the nodes $x_0=0, x_1 = 1, x_2=2, x_3 = 3$. Here are my workings below The basic Lagrange polynomials are: $$L_0(x) =…
mt12345
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If $A$ is invertible, then $A^{-1} = V \sum ^{-1}U^*$

The singular value decomposition of a complex-values $n\times m$ matrix $A$ is defined as $$A = U \sum V^*$$ where $U$ and $V$ are unitary $n \times n$ matrices and $m \times m$ matrices respectively, and $\sum $ is an $n \times m$ rectangular…
fr14
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