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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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Numerical Method - Computing relative error

So I am supposed to: Consider a decimal machine in which floating point numbers are represented with a precision of 12 decimal places and compute the relative error for numbers (assume numbers are rounded correctly). [ | x - fl(x) | ] / |x| <--…
Amai
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Second Order Numerical Differentiation

If we take the central difference twice we get $$\frac{f(x+2h)+f(x-2h)-2f(x)}{4h^2}$$ It is said that taking the forward difference and then the central difference is equal, but I do not get the same…
newhere
  • 3,115
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How to derive the convergence condition for $x_{n+1}=g(x_n)$ is $|g'(x_0)<1|$

In the method of fixed point iteration we are looking for solutions to $$f(x)=0,$$ by rearranging to give: $$x=g(x)$$ we then iterate using $$x_{n+1}=g(x_n)$$ using an intial guess of $x_0$. I'm struggling to understand how the taylor expansion…
Freeman
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In Verlet integration, why is $a*dt*dt$ considered "force vector"?

In Verlet integration, why is $a*dt*dt$ considered "force vector"? Intuitively, since Verlet integration is: $$x_{i+1}=x_i+(x_i - x_{i-1}) + a * dt * dt$$ then this doesn't have other "force" terms than $a * dt *dt$. However I'm unsure as to what…
mavavilj
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Find the stability function of $y_{n+1}-y_{n-1} = 2h\lambda y_n$

I'm asked to find the stability function $R(z)$ in the expression $y_{n+1} = R(z)y_n$ given the method $$y_{n+1}-y_{n-1} = 2h\lambda y_n.$$ How do I do this? I can't do it when three stages is present in the equation.
IdiotWithNoShame
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Barycentric form of polynomial interpolation undefined at $P(x_{j})$

For example, $P(0)$, $P(1)$, $P(2)$, $P(3)$ will be undefined because the denominator will be $0$. Does that mean we can't use barycentric form at $P(x_{j})$?
Joshua Leung
  • 555
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I don't know Extrapolation Method for initial value ODE

my text book burden $ y'(t)=f(t,y), a \leq t \leq b ,y(a)= \alpha$ apply for the extrapolation technique we let $ h_0 = \frac{h}{2}$ and use Euler method with $w_0=\alpha$ $ w_1 = w_0+h_0f(a,w_0)$ and then midpoint method $ t_{i-1}=a…
seyunkim
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How to calculate $q_{1}(t)$, $q_{2}(t)$, $p_{1}(t)$, $p_{2}(t)$ using Euler's method?

From my understanding, this assignment wants me to calculate $q_{1}(t)$, $q_{2}(t)$, $p_{1}(t)$, $p_{2}(t)$ using Euler's method. I am a quite confused as to how to proceed. For example, at $t_{0}$, $q_{1}(t)$ is $1-e$. What about at $t_{1}$?…
Joshua Leung
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What is the $f(t,y)$ in this question?

Euler's method requires $f(t,y)$. What's the $f(t,y)$ function in this assignment? $q(t)$ and $p(t)$ are vectors. How am I supposed to apply them in Euler's method?
Joshua Leung
  • 555
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How to discretize the following equation?

How do I discretize $-\frac{d^2}{dx^2}u = .0000001(300^4-u^4)$ using the finite difference method where $u(0)=900=u(L)$?
user63575
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text book definition of local truncation differ from other

in my text book [burden faires] ODE inital value problem $ y' = f(t,y) , a \leq t \leq b, y(a)= \alpha $ and the difference method $ w_0=\alpha, w_{i+1} = w_i + h\phi(t_i,w_i)$ then local truncation error $\tau_{i+1}(h)= \frac{y_{i+1}-y_i}{h} -…
seyunkim
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the relation of Local Truncation error and approximation error

I'm students at undergraduate school 2year I don't have much background knowledge I study nummerical analysis Runge - Kutta Method not a few my text book induce Runge - Kutta midpoint method by 2 order Taylor series $T^2$ and it's approximating…
seyunkim
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In Runge Kutta Methods Why Midpont Method is accurate than Modified Euler Method?

Both Midpoints Method and Modified Euler Method is local truncation error $ O(h^2) $ but Midpoint Method is accurate than Modified Euler Method why?
seyunkim
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the numer p* is said to approximate p to t significant digits if t is the largest nonneative integer

|p-p*|/|p|<=5*10^(-t) said by numerical analysis textbook burden then how we define significant digits? do we have to choose any integer t? example of this text book p=0.54617, p*=0.5462 how we get significant digits by definition? this text book…
seyunkim
  • 51
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How Many Digits of an Integral can be Accurately Computed?

Assuming $I$ is an integral, and we want to approximate $I$ with numerical integration, if the question asks you 'how many digits of $I$ can you accurately compute', how can you calculate that? Is the result of $|\text{abs}_{\text{err}} -…
Joshua Leung
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