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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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A function for which the Newton-Raphson method slowly converges?

I'm doing a MATLAB assignment in which you work out and implement a better version of Newton-Raphson using a second degree Taylor polynomial instead of a first degree one. I have the algorithm worked out and it is working good. The second part of…
Zeta Two
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Why do implicit numeric methods have superior stability properties than explicit ones?

I've been taking classes in numerical methods and we were told that implicit methods are more stable than explicit ones. This is very easy to show with some differential equations, but I dont understand why this would be a general phenomenon. I was…
juggernugger
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How to obtain the logarithm from this numerical integral?

Evaluate $\int_0^1\frac{dx}{1+x}$ using the trapezoidal rule for integration and hence find the value of $\log(2)$. I solved the first part with an interval of 0.125 and obtained: $\int_0^1\frac{dx}{1+x} ≈ 0.694075$. However, how do I find the…
Dipendra Shrestha
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Proving error bound on Simpson's Rule, Numerical Integration

The approximation from "Simpson's Rule" for $\int_a^b f(x)\, dx$ is, \begin{equation} S_{[a,b]}f = \bigg[\frac{2}{3}f\Big(\frac{a+b}{2}\Big) + \frac{1}{3}\Big(\frac{f(a) + f(b)}{2}\Big)\bigg](b-a). \end{equation} If $f$ has continuous derivatives…
Merkh
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what is "kink"?

Pleas tell me that what a "Kink" is and what this sentence means: Distance functions have a kink at the interface where $d = 0$ is a local minimum.
narges
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Is there a meaningful distinction between "direct" and "iterative" methods for solving equations?

I'll motivate this question with an example. The Abel-Ruffini theorem states that there is no general "formula" for the roots of polynomials of degree greater than 4. (Specifically it states that there is no solution that can be expressed in terms…
user541686
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What is the convergence rate of Brent's method (root-finding algorithm)?

As far as I know, Brent's method for root finding is said to have superlinear convergence, but I haven't been able to find any more concrete information. Is its convergence rate known to be at least bounded between some known values? What is a good…
Vicent
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Jacobi method convergence for a symmetric positive definite matrix in $\mathbb{R^{2 \times 2}}$

I have to prove that a symmetric positive definite matrix $A \in \mathbb{R^{2 \times 2}}$ converges for the Jacobi method. Any ideas? The matrix $A$ is said to be positive definite if $x^t A x > 0 \ (\forall x \neq 0$ ) Thus, we can consider matrix…
marlanbar
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Fixed Point Iteration is Not Converging to the Desired Root

What is the fixed point for the following function? $f(x) = 2sin(\pi x) + x = 0$ between $[1,2]$ I expressed $f(x)$ as $x=g(x)$ such that $g(x)$ can be $- 2sin(\pi x)$ or $\frac{1}{\pi}sin^{-1}(\frac{-x}{2})$. However, in both cases, the method is…
Ziya
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Rate of convergence for sequences

I have numerically determined that the sequence $\{f_x\} = \frac{\sin(x^2)}{x^2}$ approaches $1$ (as $x$ approaches $0$) faster than the sequence $\{g_x\} = \frac{\sin^2(x)}{x^2}$. However, I am stuck for determing the rate of convergence of each…
user85362
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Understanding the Runge pheonomena and Chebyshev nodes.

I am studying the Runge's phenomenon and there are a couple things I would like to understand better. Suppose we interpolate the Runge's function with use equally spaced nodes to interpolate the function on the interval $[-1,1]$. From what I…
John Keeper
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Using Newton's Divided Difference formula, prove that

Using Newton's Divided Difference formula, prove that $$f(x) = f(0) + x\,\Delta f(-1) +\frac{x(x+1)}{2!} \, \Delta^2f(-1) + \frac{(x+1)(x)(x-1)}{3!} \, \Delta^3f(-2)+ \cdots$$ where $\Delta$ is forward difference operator Now i know that Newton's…
Taylor Ted
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Which of the following sums is a closer approximate to $e^{-5}$?

Here are two sums given: $$e^{-5}\approx \sum_{i=0}^9 \frac{(-1)^i 5^i}{i!}$$ $$e^{-5}=1/e^5\approx \frac{1}{\sum_{i=0}^9 \frac{5^i}{i!}}$$ Using Matlab ;) I've got (2) to be closer. But computer-aside, is there anyway to say which one is closer…
Mill
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Numerical integration of a data set with uncertainties

I have a 1D data set {xi, yi} with no uncertainties in xi and with uncertainties dyi in yi. The resulting discrete function is monotonic and relatively smooth and I would like to integrate the function. If there were no uncertainties, I…
user26718
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Numerical analysis textbooks and floating point numbers

What are some recommended numerical analysis books on floating point numbers? I'd like the book to have the following In depth coverage on the representation of floating point numbers on modern hardware (the IEEE standard). How to do arbitrary…
user782220
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