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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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Fixed Point Iterations - Root Finding

Given any function $f(x)$, how can you come up with the corresponding $g(x)$ such that $g(p)=p$ (where p is the root)? Say, $$f(x)= sinx -\frac{x}{1.4}$$ my professor told me to simply isolate for $x$ and get: $$g(x)=1.4sinx$$ Why does this work and…
CivilSigma
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Showing that $\sin(x) + x = 1$ has one, and only one, solution

Problem: Prove that the equation $$\sin(x) + x = 1$$ has one, and only one solution. Additionally, show that this solution exists on the interval $[0, \frac\pi2$]. Then solve the equation for x with an accuracy of 4 digits. My progress: I have no…
Alec
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Interpolating $n$ points by piecewise quadratic polynomial

Given $n$ data points. Is it possible to interpolate them by piecewise quadratic polynomials with knots at the given data such that the quadratic interpolant is: (a) Once continuously differentiable? (b) Twice continuously differentiable? If it's…
ghjk
  • 2,859
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Second order to first order equations

I need to write $$\frac{d^2\theta}{dt^2} + 4\sqrt{k}\,\frac{d\theta}{dt}+g\sin(\theta)=0$$ as a first order equation. What I have done so far is: Let $z = \frac{d\theta}{dt}$ Then $z' = \frac{d^2\theta}{dt^2}$ Therefore the second order equation…
user233606
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When does "successive substitution" not work?

Successive substitution is a technique, we learned, used to find the roots of a polynomial $f(x)=x^2-2$ for example. We must construct some function $g(x)$ so that $g(x)=x$ iff $f(x)=0$, for example $g(x)=f(x)+x$. It is done by taking some start…
steedsnisps
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Maximum timestep for RK4

I'm trying to find the maximum timestep that can be used when applying a RK4 numerical method to solve this system. I know how I would do this for single equation but have no idea how to solve it for a system of equations. Please can someone…
Tim
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Numerical method to calculate sum of infinite series?

for example: I have a series is there numerical computation method to find it ? thanks
T.Nhan
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Least squares problem, using derivative to find the normal equation?

for a matrix $A \in \mathbb{R^{m\times n}}$ and for $x,\epsilon \in \mathbb{R^n}$ and $\epsilon$ small we have…
blueonken2
  • 85
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1 answer

The order of accuracy of the implicit Euler method is equal to $1$

I want to show that the order of accuracy of the implicit Euler method is equal to $1$. That's what I have tried: We have the initial value problem $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ Using…
evinda
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Backward Euler method- How do we get the approximation?

Approximating $y'(t^n)$ at the relation $y'(t^n)=f(t^n,y(t^n))$ with the difference quotient $\left[\frac{y(t^{n+1})-y(t^n)}{h} \right]$ we get to the Euler method. Approximating the same derivative with the quotient…
evinda
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$A(\theta)-$ stable method, region of absolute stability

We have to look for numerical methods for the numerical solution of $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ that have 'great' regions of absolute stability. Methods of which the region of…
evinda
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psuedo inverse of a matrix counter-example

I have proved $A^{\dagger}A = I$ for a $m$ by $n$ matrix with $m\geq n$ and $\text{rank}(A) = n$ I am trying to find a counter example which shows that $AA^{\dagger} \not= I$ but to no avail. The only one I have is the 0 matrix, but I guess that's…
psuedoguy2
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Cobweb plot for secant method

For a numerical methods class I have to implement the secant method to find the roots of a 2nd order polynomial and create a cobweb diagram. I'm a bit confused on the cobweb diagram. I read here that a cobweb diagram is used to "Investigate the…
void life
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Bisection method guessing interval

I know that generally the bisection method is used given a certain function and an interval where we know a root exists within it. What if we don't know the interval? Is there a way of "guessing" the interval to use?
user219426
  • 21
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Finite element method - basis functions for triangular mesh

In FEM, with a triangular mesh over $R^2$, could $\phi\left(x\right)=x_1\cdot\left[x\in T\right]$ be a basis function for the triangle $T$ with vertices in $\left(0,0\right), \left(0,1\right), \left(1,0\right)$? My doubts come from the fact it is…
Adrian Panasiuk
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