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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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Using an already calculated QR factorization to solve a different system

The exersice requested to find a QR factorization of $A$ $A=\begin{bmatrix} 1&1&0\\1&1&1\\0&1&-1 \end{bmatrix}$ Wich we did and obtainted $Q=\begin{bmatrix} \frac{-\sqrt{2}}{2}&0&\frac{-\sqrt{2}}{2}\\\frac{-\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\\0&-1&0…
Chalkling
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Numerical analysis - boundary value problem

So I need a push in the right direction with this question, so any help is much appreciated :) Show that the solution to the boundary value problem $-u''(x) + r(x)u(x) = f(x)$, $u(a)=\alpha$, $u(b)=\beta$ where $r(x) \gt 0$ for all $x \in [a, b]$,…
Swayy
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Solve $x^x = a$ for known $a$?

For example if you have $x^x = 2$, can you express $x$ as a numerical expression containing only the addition, multiplication and exponentiation operators?
Alraxite
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Induction proof of polynomial interpolation theorem

show if $\phi (x) = f(x)g(x)$, this is valid: $\phi [x_0,x_1,...,x_n]=\sum\limits_{r=0}^n f[x_0,x_1,..,x_r]g[x_r,x_{r+1},...,x_n]$ by induction. I have tried to prove it by the divided differences formula but things are standing still at the…
pjoltergeist
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Richardson extrapolation integral

Use Richardson extrapolation to compute $\displaystyle \int_{-4}^4\frac{dx}{1+x^2}$ with the number of subintervals $[n=1,2,4,8,\ldots$]. Compare the results $\displaystyle I_1^{(0)},I_2^{(1)},I_4^{(2)},I_8^{(3)},\ldots$ to trapezoidal and …
user60514
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What is the idea behind expm1 to avoid cancellation error?

It is well known that when x is close to 0, computing exp(x) - 1 introduces significant cancelation errors. As such, we have expm1 implemented in c99 and python. My question is how expm1 avoids cancellation error? Can anyone give me a general idea…
user1559897
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Is there periodic sequence generated by secant method?

Is there $f(x)$, $x_0, x_1$ such that $\{x_n\}$ is periodic sequence? $$x_n=x_{n-1}-f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}$$
Sabrina
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Finding the global error of a numerical method

For the IVP $u' = f(t,u), u(0) = u_0$, I have the following numerical method: $$U_0 = u_0 \\ U_{n+1} = U_n + hf \left(t_n + \frac h2, U_n + \frac h2f(t_n,U_n) \right)$$ I am asked to show the global error $e_n = u(t_n) - U_n$ satisfies $$e_{n+1} =…
ant
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Newton's Method, Analytical Formula

Currently I am learning about Netwon's Method. Given the function f: $\frac{1}{5} x^5 - \frac{2}{3}x^3 + x$ and $x^{(0)} = \sqrt{\frac{ 25+2\sqrt{55} }{27} }$, I want to analytically determine the sequence $x^{(n+1)} = \Psi(x^{(n)})$ with $\Psi(x)…
TestGuest
  • 1,053
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Gradient Descent: Optimal fixed step size for a quadratic objective?

All of the literature I'm reading immediately skips to the idea of adjusting the step size as you iterate (as far as I can tell) to maximize the rate of convergence. In the context of neural network modeling, I'm building up from fixed-step…
PatternMatching
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Evaluate $\cos(x)$ at a particular point

I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point. Currently, I am thinking first write $x=\pi+x'$ where…
Qomo
  • 285
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Euler's method for second order differential

Given the differential: $y'' + y' - y = x$ , $y(0) = 2$ , $y'(0)=1$ I am asked to calculate $y'(2)$ for (a) $h=2$ and (b) $h=1$ I have used Euler's method for a second order differential so I am unsure if how I answered the question is correct and…
katara
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Newton polynomial divided differences

When trying to interpolate data points $(t_i,y_i)$, one can use Newton's polynomials which are spanned by the basis of $P_{n-1}$ given by polynomials of the form $$ t-t_1, (t-t_1)(t-t_2), ..., \prod_{j=1}^{n-1} (t-t_j) $$ So, in the newton basis, an…
James
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Best Linear Approximation that minimizes the following L1 norm

Best linear approximation that minimizes the following $L^1$ norm: The $L^1$ norm is defined as $$ \| f\|_1 = \int_a^b |f(x)|{\rm d}x $$ Best Linear approximation $l(x) = a_0 + a_1 x$ that minimizes $\|e^ x - l(x) \|_1$ on the interval $[-1,…
user565684
  • 167
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Forward Euler stability

I am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability. $$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$ $$y(0) = 1$$ then the book shows this: $$u_0 = 1 \qquad \text{does this come…
BRabbit27
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