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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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Why does Newton's method diverge if the root is an inflection point?

Why does Newton's method not converge to a root $\alpha$ of $f(x) = 0$ if $\alpha$ is an inflection point (i.e. $f^{\prime\prime}(\alpha) = 0$)?
Prasiortle
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Errors of approximate function values

I'm studying for a course called numerical mathematics and let me apologize in advance if some of the terminology I use is unknown to you. It's in another language and I'm having a hard time finding English equivalents to any of this (makes it hard…
Koy
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Find $x$ for a given derivative value.

Given a function $ y=f(x)$. I know that the function is continuous, now for a given $y^{'}_0$, I have to find $x$ such that $\dfrac{\mathrm df(x)}{\mathrm dx} = \dfrac{dy}{dx}=y_0^{'}$ at that $x$. To solve this problem numerically i tried $g(x) =…
Pramod Azad
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Finding intervals using the Contraction Mapping Theorem

I need to identify an interval $[a, b]$ for which the Contraction Mapping Theorem guarantees convergence to the positive fixed point (or verify that there is no such interval) for the function $g(x) = x^3+3x^2-3$. Clearly, the only positive fixed…
Monya Feldman
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Prove that $\frac{d}{dx}f[x_0,x_1,\ldots,x_{i-1},x]=f[x_0,x_1,\ldots,x_{i-1},x,x]$

I have tried to prove it by induction on $i$- For $i=1$, $f[x_0,x]=\frac{f(x)-f(x_0)}{x-x_0}=\frac{f(x)}{x-x_0}-\frac{f(x_0)}{x-x_0}$ So,…
MathBS
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Richardson extrapolation and improvement in error

I am using the Simpson's composite rule for evaluating integrals, this is $O(h^4)$, where $h$ is the step size. We can use the Richardson's trick to improve things (I denote the integral to be evaluated by…
Tony
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Numerical Methods, Neville's Method Missing Values

I tried to solve question below for over 1 hour, and can't figure it out. Can somebody help me? Question
Mehmet Usta
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Is the composition of backward stable algorithms again backward stable?

I have heard the claim that a composition of two backward stable problems is again backward stable, but I have trouble proving it. Let $f: X \to Y, g: Y \to Z$ be problems and $\tilde f, \tilde g$ their algorithms. Take input data $x$. Since $\tilde…
marmistrz
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Nonlinear shooting example calculation

How can we use the nonlinear shooting method with $h = 0.25$ to approx the solution to $y'' = 2y^3$, $-1 \leq x\leq 0$, $y(-1) = 1/2$, and $y(0) = 1/3$. I tried to convert this to a first order system but the RK $4$ didnt look good. Also, I needed…
mary
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embedded models using simpson's rule

We are looking at embedded models with Simpon's rule (order $s=4$) as primary method. We take $\hat{s}=2$. Which methods are suited as secondary models? And why? I do not understand how to choose such secondary models. And why is it not possible to…
Pieter
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Accuracy order of different numerical methods

I don't understand how to determine the accuracy order of different numerical methods I have two examples: (1) (2) For the (1) I get that the quotient for the errors are $16$ and for (2) I get that the quotient is $4$. And from what my teacher…
katara
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Prove that if $f$ is a polynomial of degree $k$, then for $n > k$, $f[x_0,x_1,...,x_n] = 0$.

I've been struggling with divided difference problems a lot. I'm able to program an algorithm to compute them easily, but I have no idea how to even start this proof. Any ideas?
Cameron Potter
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approximation of differential equation through chebyshev differentiation matrices

All, i'm trying to solve a linear differential equation, through discretizing the domain into the chebyshev grid and using spectral method ( approximating first and second derivative with differentiation matrices- polynomial approximation),and…
kaak
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What is the graph of this linear function used for functional iteration excercise?

I'm given the equation: $$3x-5=0$$ I can rewrite it as: $$(2+1)x-5=0 \implies x=-\frac{1}{2}x+\frac{5}{2}$$ The obvious root is $\bar{x}=\frac{5}{3}$ and I can define $g(x)=-\frac{1}{2}x+\frac{5}{2}$.…
zcb
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Is this 'simple' analysis of the Euler Method Error valid?

This question is very similar to this but the answer there does not quite answer my question. I understand that there are 'state of the art' bounds for the global error when using Euler's Method. The important conclusion, as far as I am concerned,…
JP McCarthy
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