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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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relationship between forward and backward error?

Let $\tilde{f} $ be some algorithm: we have: $$ \| f(x) - \tilde{f}(x) \| = \| f(x) - f(\tilde{x}) \| \leq \|f'(x) \| \|x - \tilde{x} \|$$ I'm curious on the last step, how did they get the inequality? It kind of looks like the mean value theorem…
DH.
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Order of convergence of nonlinear iterative solver

I'm given a sequence $x_n \rightarrow \alpha$ of a nonlinear solver such that $$\lim_{n\rightarrow\infty}\frac{x_{n+1}-\alpha}{x_n-\alpha}=c$$ converges linearly (i.e. $c\in(0,1)$). Now, I need to determine the asymptotic rate of convergence $p^*$…
MrT
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Numerical Methods for ODE

I have been working through Iserles books A first course in Numerical Analysis of Ordinary Differential Equations. I am trying to figure out how to prove that the error constant for an s-step BDF is -beta/(s+1) I have tried using the…
ADA
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O(h) operator over uniform grid

For a uniform grid $$x_n = -1 + nh$$ where $h = \frac{2}{N}$ and the integration rule $$I_N(f) = h\sum_{n=0}^{N-1}f(x_n)$$ which corresponds to a left hand Riemann sum or to integrating an interpolant in $\mathscr{S}_0^{-1}$, how would I show that…
John Jay
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A proof on forward difference operator

Show that $u_0-u_1+u_2-u_3+...=\frac{1}{2}u_0-\frac{1}{4}\Delta u_0+\frac{1}{8}\Delta^2 u_0-\frac{1}{16}\Delta^3 u_0+...$, where $\Delta$ is the forward difference operator. My…
user88923
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Lipschitz inequality required for Broyden convergence proof

I'm trying to understand the proof of the convergence of the Broyden method through the book Numerical Methods for Unconstrained Optimization and Nonlinear Equations, and at some point the proof relies on the following lemma (page 77): Lemma 4.1.15…
Luís Pureza
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numerically solving differential equations

$\frac{d^2 \theta}{dx^2} (1 + \beta \theta) + \beta \left(\frac{d \theta}{d x}\right)^2 - m^2 \theta = 0$ Boundary Conditions $\theta=100$ at $x = 0$, $\frac{d\theta}{dx} = 0$ at $x = 2$ $\beta$ and $m$ are constants. Please help me solve this…
fagbenro olayiwola
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Euler's method setup

How can we use Euler's method to approximate the solutions for the following IVP below: $$y' = -y + ty^{1/2},\text{ with }1 \leq t \leq 2,\ y(1) = 2,$$ and with $h = 0.5$ The main concern is the organization, i.e., set up of it for this particular…
mary
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If root is very near to the max/min, what happenns with Newton Raphson method? Does it diverge?

If root is very near to the max/min, what happens with Newton Raphson method? Does it diverge? Or converges slowly? I know if some iteration involves a stationery point then we can not go further. But is there a possiblity that even in such a case…
Jacob
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How many iterations of interval bisection would be requited to obtain a given level of accuracy?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)=x^{5}+x^{3}+1$. Given that $f$ has a root between -1 and 0, how many iterations of interval bisection would be required to obtain the root to an accuracy of $5\times10^{-11}$? Thanks
M Smith
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Interpolation question

I have a polynomial such that for i = 1, 2,..., N distinct nodes. How would I show that this possesses a unique solution and find a way of constructing it?
John Jay
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Find a bound for the error in this interpolation (Interpolation inside)

Consider the piecewise constant interpolation of the function $f(x) = ln(x)$ , $10 ≤ x ≤ 11$ , at points $x_i = 10+ih$, where $h = 0.1$. Thus, our interpolant satisfies $v(x) = ln(10+ih)$, $10 + ih − 0.5h ≤ x < 10 + ih + 0.5h$, for $i = 0, 1, . . .…
user129509
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Numerical Analysis Question That Deals with the Mclaurin Series

The Maclaurin series for the arctangent function converges for $−1 < x ≤ 1$ and is given by, $\arctan x=\lim P_{n}(x)$=$\lim \sum_{i=1}^{n}(-1)^{i+1}$$\frac{x^{2i-1}}{2i-1}$ Use the fact that $\tan π/4 = 1$ to determine the number of n terms of the…
Cody S
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Rearranging equation to prevent denominator blowup in C

I know the title mentions C, because this is a programming related problem, but I think this specific issue is more pure mathematics so I figured here would make more sense! For a homework assignment, we have to implement an equation in C and…
elykl33t
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Does the following limit exist as a result of the bisection method?

Does the following limit exist as a result of the bisection method? $$\lim_{n\rightarrow\infty}\dfrac{|r-c_{n+1}|}{|r-c_{n}|}$$ where $r$ is the root as a result of the method, and $c_n=\dfrac{a_n+b_n}{2}$ My professor has told me that the limit…
unholypride25
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