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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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Showing an LDU of a LU matrix is unique.

How do I show that every LU matrix A∈K$_n$$_×$$_n$ admits a unique LDU factorisation, that is a triple L∈L$_1$(Kn),D∈D(Kn) and U∈U$_1$(Kn) such that A=LDU? I'm new to these factorisations and I'm just trying to get to grips with the composition of…
wizardofoz
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A question about floating point representation

This question is simple but I don't know the exact answer: My Professors' lecture notes say $(0.01)_2 = 0.1 \times 2^{-1}$ and he insists that all numbers are written in base 2. However, I think it's wrong and I interpret this expression in this…
Emad
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Explain why it is not necessary to use numerical method to find the two solutions of the equation.

This question I havent been able to understand. I am self studying and have no teacher to ask so I have been stuck for the past hour and don't get this is the full question the part I don't get is 14 c Any help is much appreciated. Thank you!
mary james
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Interpolation at Tschebyschff nodes

We are given a function $f:[-1,1]\rightarrow \mathbb{R}$ and the Tschebyscheff nodes, $x_j= $ cos$(\frac{2j+1}{2n}\pi), \,j=0, 1, ....n-1.$ We require that lim$_{n\to \infty} {\left\lVert f- p_n\right\rVert}_{C[-1,1]}=0,\,\,p_n$ being the…
user249018
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Machine epsilon definition

Assume representation of floating point in decimal, in the form $0.q_1q_2...q_n\cdot 10^{exp}$. By definition $1+\epsilon_{\text{mach}} = 1$, but a textbook claims that with this definition, $\epsilon_{mach} = 10^{-n}$. The question I have is,…
nabu1227
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prove this is not true. $e^x-1 = O(x^2)$ as $x \rightarrow 0$

Hello I need to show that this assertion is not true and I ran into an issue. I was taught that I could prove this by doing $\lim_{x\to0} \frac{(e^x-1)}{x^2}=c\neq0$. I used l'hoptial rule once but then was stuck at $\frac{1}{0}$. When I plugged the…
johnsdgh
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Numerical differentiation - find bound for the error

I have be given a set of points $(-3.0, -2.8, -2.6, -2.4, -2.2, -2.0)$ and a function $f(x)=e^{x/3} + x^2$ and asked to find the bound for the error in each case. I already find $f'(x)$ for each point but I don't know how we should find the bound…
GO VEGAN
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Estimation update for finding complex roots using Müller's method

Let there be a polynomial whose roots to be found and let us start with initial guesses $x_0\ x_1\ x_2$. Let the quadratic $a(x-x_2)^2+b(x-x_2)+c$ developed from these values have complex roots. Denote them as $\alpha+\beta i $ and $\alpha-\beta…
Ali Kıral
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is there a tool for figuring out set combinations with limited numerical entries?

i'm sitting here with a pin-locked pad, trying to figure out which combination clicks, i have been given the hint that the code can only contain the numbers: "0" "9" "8" and that it's a 4-digit Pin. Now i'm stressing here to figure out how exactly…
Clay
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Help - construct Hermite interpolation polynomial

Construct Hermite interpolation polynomial if: $ f(1)=0, f'(1)=3, f''(1)=4, f(3)=1, f''(3)=4 $. Can you give me some tips how to do this task?
olikalady
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Numerical analysis objective question.

The equation $e^x-4x^2=0$ has a root between $4$ and $5$. Fixed point iteration with iteration function $\frac{1}{2}e^{\frac{x}{2}}$ is $1.$ diverges . $2.$ converges. $3.$ oscillates. $4.$ converges montotonically. The same question is asked before…
neelkanth
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What does this image mean exactly?

This is an image of the cover of my numerical analysis book. I can see there is a curve and a tangent line to the curve at a point, of course slope of this line is equal to derivative of function of the curve at that point. but we can see also a…
Etemon
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Solving the intersection of two functions with fixed point iteration

I have to find the intersection(s) of two functions $\frac{1}{\sqrt{\ln(x)}}$ and $5-x^2+\sqrt[3]{x}$ with the fixed point iteration method. I suppose that I have to find the roots of the equation f1-f2 = 0, but when I try to iterate this equation…
B3ne
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Divergence of the secant method

Those ten iterations are obtained using the secant method for the function $\tan(x\pi)-6$ starting with an interval of $[0\,\,\,0.48].$ The actual root is about $0.447431543$. Can you explain why the method exhibited poor performance?
Ali Kıral
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Crank-Nicolson Method affected by odd-numbered iterations

For an undergraduate project I am analyzing the performance of a couple of Crank-Nicolson schemes on a time-dependent problem. For some reason, the order of convergence/general performance of one scheme is noticeably worse when I use an odd number…
D.Bar
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