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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 Chebyshev's method have cubic convergence?

Chebyshev's method: $$ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}-\frac{f''(x_n)}{2f'(x_n)}\left(\frac{f(x_n)}{f'(x_n)}\right)^2$$ The question might seem simple, but I can't seem to find an answer to it. I have scanned through several numerical…
pls_halp
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Initial Value Problem into Euler and Runge-Kutta scheme

Having a bit of difficulty with this question: Convert the initial value problem, $$\frac{d^2y}{dx^2}-\frac{dy}{dx}+2y=x+1$$ where $$y(0)=2,\ \frac{dy}{dx}(0)=-1,$$ into a set of two coupled first-order initial value problems. Firstly I let…
Reety
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Numerical stability of algorithms

Let be $f(x) = x_1 - 2x_2 +x_3, x_1 \approx x_2 \approx x_3 $ Find out (by using forward stability analysis) which of those following algorithms is the most stabil one: a) $ \tilde{f} (x) := ( x_1 \ominus 2x_2) \oplus x_3$ b) $ \tilde{g} (x) := (…
wondering1123
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Newton's method and derivative is 0 at root?

Newton's method of root-finding uses the derivative at $\tau$ to find the value of $x_i$ on the next iteration. But I have read that a function can still have a root if $f'(\tau) = 0$..is there something special about the root in this case? I can't…
dukenukem
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Find the error bound

Hey guys I am unsure how to find the error bound. Use the langrange interpolating polynomial of degree 3 or less and four digit chopping arithmetic to approximate $\cos(.750)$ using the following values. Find an error bound for the approximation.…
math101
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Numerical analysis:: meaning of "half a unit in the nth place"

What exactly does it mean by "half a unit in the nth place"? I am reading numerical analysis and this term is popping out very frequent in the first chapter itself,my first encounter with this is while reading the rounding-off error rules. From few…
Quixotic
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Accuracy of Lax-Wendroff scheme

Consider the Lax-Wendroff scheme $$\frac{u_j^{n+1}-u_j^n}{\Delta t}+\frac{u_{j+1}^n-u_{j-1}^n}{2\Delta x}+\frac{\Delta t}{2}\frac{2u_j^n-u_{j+1}^n-u_{j-1}^n}{\Delta x^2}=0$$ for differential equation $$\frac{\partial u}{\partial t}+\frac{\partial…
Sukan
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Fixed Point Iteration doesn't converge, how to find its convergence?

$g(x)=(2/3)(\cos x-\sin x)$ $x_n=g(x_{n-1})$ with initial guess $x_0=1$ I need to compute for n= 0,1,...,8 When I try it, my sequence diverges: $x_1=-0.20078$, $x_2=0.78623$, $x_3=-0.00079$, etc... In a following question, I'm meant to compare it to…
user314580
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The accuracy from left to right and that from right to left of the floating point arithmetic sums

Question 1 Show that floating point arithmetic sums $$s_n=\sum_{k=1}^n\frac{1}{k^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\dotsb+\frac{1}{n^2}$$ with accuracy $\mathcal O(n)\epsilon$ from left to right, while summing from right to left gives accuracy…
J.doe
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Maxwell-Area Construction Problem

I am solving a problem which asks to find the equation of a horizontal line which crosses the graph of $$y=x^3-3x+1$$ at three distinct points, such that the two areas bounded by such curves are equal. I am creating a program for this problem. What…
James
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Bairstow's method: Rate of convergence

In numerical analysis, I was asked whether Bairstow's algorithm convergence rate is quadratic. My initial feeling was that it does, since it is essentially Newton's method for a system of non-linear equations, and Newton's method converges…
Ahia Cohen
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What types of polynomials cause root finders to fail?

I am under the impression there are certain types of polynomials that root finders have trouble with. In other words, multiple real roots, complex roots very near to each other, etc. I am not interested in polynomials with order greater than 100. I…
user5108_Dan
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Semi-discrete time discretisation of PDEs, stability and convergence.

I'm having a lot of difficulty and I was hoping someone could help me out. I'm looking at a variety of PDEs but for the sake of this post, lets just look at the advection equation $$u_{t} = -u_{x}$$ If I apply the Backward Euler formula (implicit)…
Matthew Cassell
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system of First-Order ODES

I am looking at the following exercise: Consider the initial value problem $\left\{\begin{matrix} x''(t)=x(t)\\ x(0)=a\\ x'(0)=b \end{matrix}\right.$ Write it as a system of First-Order ODES with suitable initial values and show that Euler method…
evinda
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What is the logic behind Jacobi iterative method?

The book I follow and on net also, all that I can find is the algorithm to find the solution, but I don't quite understand the physical significance or logic behind the algorithm. Can someone please help.
Manish
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