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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

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Problem in understanding the derivation of relationship between relative error and the number of significant numbers.

The following is excerpted from Numerical Analysis by K. Mukherjee where he discusses a theorem relating the relative error with number of significant figures: Theorem: If the first significant figure of a number is $k$ and the number is correct to…
user142971
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Practical defn of "Order of Convergence"

Given an algorithm producing the sequence of iterates $\{x_n\}$ converging to some value $x$, it is customary to express the manner in which the sequence converges using the concept of the order of convergence. Theoretically, this definition is…
Matt Brenneman
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Fourier-Legendre series vs Lagrange interpolating polynomial

Let's say we want to approximate a regular function f(x) within closed interval [-1,1], Using Lagrange interpolating series we can write: f(x) = Lj(x)*fj http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html Using Fourier-Legendre series…
cylee
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Solving nonlinear ODE system with continuous Newton method

Is anyone knows how to apply continuous Newton method for solving nonlinear ODE systems? For example let the system is: $$\left|\begin{array}{cc} y'= \frac{z}{x} \\ z'= \frac{z(y+2z-1)}{x(y-1)}\end{array}\right.$$
vladislavn
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To find a real root

find real root of given equation $$5x - 2 \cos x -1=0 $$ I only know that I should use $$x_n = \frac{ x_{n-2} f_{n-1} - x_{n-1} f_{n-2} } { f_{n-1} - f_{n-2} }$$ And I applied it, but didn't get desired answer Can anyone explain me about some…
akash
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Variation of Trapezoidal Rule found by chance?

Even though this question is DSP related, I am interested in the math behind the algorhitm discussed below. To give some background: I'm working with power frequency (50/60Hz) AC signals and need to compute the integral of an input signal in real…
faguerre
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Some question about definition of splines and periodic splines

I have some questions about the definition of splines and in particular periodic splines. So in non periodic case splines are piecewise polynomials of degree $<=m$ which are $m-1$ times differentiable.(I took $m-1$ time differentiability for…
Katja
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Prove uniqueness of linear spline

Let $f: \mathbb{R} \to \mathbb{R}$ be linear spline (continuous, piecewise polynomial of degree $\le 1 $) at knots $x_0 < x_1 < … < x_n$. Prove $f$ can be uniquely represented at form: $$\displaystyle f(x) = a+bx+ \sum_{j=0}^n c_j|x-x_j|$$ for some…
Jessy
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Prove limit of a sequence in Newton's method

Given the $ f(x)=x^3+x-1 $, I have shown so far that $ f$ has a unique root $r\in(0,1)$ and that for the sequence $(x_{n}), n>=0$ produced by Newton's method we have $$\lim_{n\to\infty} x_{n}=r$$ for every $x_{0}\in\mathbb{R}$. How do I prove that…
asdf
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Numerical computation of $\sqrt{x} - \sqrt{y}$ where $x$ and $y$ are almost equal

A problem asks the following Compute $\sqrt{4321} - \sqrt{4318}$ using 4 decimal digit arithmetic. This comes out to $0.02,$ when the exact result should be $0.02282...$. It then asks the following, Reformulate the expression $\sqrt{x} -…
Edward Evans
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Derive $f''(x) = \frac{f(x+2h) -2f(x+h) +f(x)}{h^2} + E(h) $ using taylor series expansions.

We have: $f(x+2h) = f(x) +2hf'(x) + 2h^2f''(x) + \frac{4h^3}{3}f'''(\zeta_1)$, where $\zeta_1 \in (x, x+2h) $ $f(x+h) = f(x) +hf'(x) + \frac{h^2}{2}f''(x) + \frac{h^3}{6}f'''(\zeta_2)$, where $\zeta_2 \in (x, x+h) $ Substituting into the formula…
Btzzzz
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Convergence of quadrature for non-smooth integrands

Standard convergence estimates for quadrature formulas $(w_i,x_x)_{i = 1}^n$ are of the form $$ \left|\int_a^b f(x) \, dx - \sum_{i = 1}^n w_i \, f(x_i)\right| \leq C (b - a)^{p+1} \max_{x \in (a,b)} |f^{(p)}(x)|, $$ see…
gTcV
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Sum of Gauss Quadrature Weights

The sum of weights for Gaussian quadrature depends on the dimension of the interval. For example, the sum of the weights over the interval [-1, 1] is 2. My question is simply why?
AEW
  • 171
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Is there any formula for computing the product of $(n+1)(n-1)(n-3)(n-5)...(n+1-2k)$

I need to compute the product of $(n+1)(n-1)(n-3)(n-5)...(n+1-2n)$ in order to use the Stirling's formula to calculate, but I am stuck. Thanks! I am a little bit confused now... should the last term of $w(n/2+1/2)$ be $(n+1-2n)$ or $(n+1-2k)$?
J.doe
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make the Exponential interpolation vanish and show it has a unique solution

I have several questions concerning different parts of the question: a) Is it sufficient to show that ${1, e^x,...,e^{nx}}$ are linearly independent over the vector space of differentiable functions on R? b) For part b, I don't quite understand…
J.doe
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