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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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Numerical solution to ${d\over dt} \nabla^2 p = {dp\over dx}$

I need help numerically solving $$\frac{\partial \nabla^2 p} {\partial t} = \frac{\partial p} {\partial x}$$ I know that to solve \begin{equation} \frac{\partial p} {\partial t} = \frac{\partial p} {\partial x} \end{equation} the finite…
user3100212
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Use Newton's method to approximate a unique root in [-1,1] for $f(x)=arccos(x)-2x-3$ to within $10^{-6}$

Edited for readability I'm having some trouble with this because I've tried solving this myself through the iterations, and I've also tried Matlab and an online calculator. The first time I tried the iterations myself I began with x=1. On the second…
Mary Martinez
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Loss of Significance

I understand that when subtracting two values that are nearly identical there is a loss of significance. Aside from this however I'm not quite sure how to tell what results in the next largest loss of precision. For example, here is a problem in my…
Valrok
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How would I add the following numbers?

I have got two floating point numbers and need to calculate them in the following way: -1.724444389 (and a ton more digits) * 2^(126) + -1.342222094535 (and a ton more digits) * 2^(-108) My question is how would I go about doing this in a 'clean'…
Cooliostasis
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Show that the number 2.46^(1/64) is known within less than one unit in the place of its fifth significant digit

This question (from Hildebrand's Introduction to Analysis) states: Show that the number $2.46^{\frac{1}{64}}$ is known within less than one unit in the place of its fifth significant digit if $2.46$ is known only to be corrected to three…
Stoof
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Give an estimate for the error.

Use the first three nonzero terms of Taylor’s formula for $\sin x$ to find an approximate value for the integral $\int_0^1 \frac{\sin x}{x}$ and give an estimate for the error.(It is understood that the quotient of $\frac{\sin x}{x}$ is equal to…
Flip
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Error evaluation of consecutive terms at fixed point iteration methods.

Q: Give an evaluation of error between two consecutive terms for methods of type $p_{n+1}=g(p_{n})$. I tried solving it, but I think that my solution is correct only when the method converges to a fixed point, or when the function that we consider…
davd
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Newton's method convergence criteria

To use Newton's method on interval $[a,b]$ we need to guarantee that $f(a)f(b)<0$ on the interval which is true for $[0,1]$. $f'(x)$ and $f''(x)$ are continuous on the interval $[a,b]$ (which they are) $f'(x) \neq 0$ on $[a,b]$ and $f''(x)$ does…
pseudomarvin
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How to find a conservation scheme for a second order ODE.

Give a second order ODE: $y''(t)+y(t)+g(y)=0, t>0$, with initial data $y(0)=0,y'(0)=1.$ Define $$E(t)=\frac{[y'(t)]^2}{2}+\frac{y^2(t)}{2}+\int_0^{y(t)}g(s)ds.$$ How to find a second order finite difference scheme such that $E(t)$ is conserved in…
Shine
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Method of False Position

I'm trying to use the Method of false position on the functions below to find solutions to within $10^{-5}$ (1) $f(x) = \ln (x-1) + \cos(x-1) = 0$, for $1.3 \leq x \leq2$ and (2) $2x \cos(2x) -(x-2)^2 = 0$, for $[2,3]$ and $[3,4]$ I am trying to…
James R.
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Secant Method for $f(x) = \ln(x-1) +\cos(x-1)$, interval $1.3 \leq x \leq 2$

I am trying to see if I got the right answer so far for: Using the Secant Method for $f(x) = \ln(x-1) +\cos(x-1)$, interval $1.3 \leq x \leq 2$ $p_0 = 1.3$ $p_1 = 2$ $p_2 = 1.520607$ $p_3 = 1.2043557751$
James R.
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Two dimensional Numerical integration

If I am numerically integrating my function $f(x,y)$ on a two dimensional cartesian grid, say $[0,1]\times[0,1]$ with $\Delta x=\Delta y$ using the values at the center of each cell, what is the order of accuracy, and how do I see this?
mononono
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Understanding what exactly an upper bound on an error is in numerical analysis

I think the hardest part of numerical analysis for me is understand what constitutes an "upper bound", and this has caused me alot of strife because often times my answer differs from the book, but I'm still "correct", which would imply to me that…
Steve
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Numerical analysis - Showing fixed point exists

Let $g(x) = \frac{1}{2}(e^{−x})\cos x$. Prove that $g(x)$ converges to a fixed point. The answer provided by my lecturer is: $g(x)$ is continuous on $[0,1]$, and we can easily verify that $0 ≤ g(x) ≤ 1$ for $x ∈ [0,1]$. Furthermore, since $g'(x) =…
epsi
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maximum error of a near-minimax approximation

I'm trying to find the maximum error in the degree $n$ near-minimax approximation to $g(t)=\tan^{-1}(t)$, $0 \leq t \leq 1$ for $n=1,\ldots,6$. Does anyone know of a formula to find this? The book doesn't give a formula and it kind of vague about…
Kailee
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