Mathematics Stack Exchange
Mathematics Stack Exchange

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
1
vote
1 answer

Can the numerical method for a set of ordinary differential equations be non-conservative?

I have a set of highly coupled ODE that I am solving. I'm just not understanding whether the quantities will be inherently conserved because it's an ODE. Everything I'm seeing on Google is for PDE's.
user1543042
  • 486
1
vote
0 answers

Difference operator corresponding to a given difference scheme

Given is the following difference scheme: $L_h u = \frac{u_{i-1,j} - 2 u_{i,j} + u_{i+1,j}}{h^2} + \frac{u_{i,j-1} - 2 u_{i,j} + u_{i,j+1}}{h^2}$. This can be written in the form $L_h u = \sum_{k,l=1}^{3} \sigma_{l,k} u_{i+2-k,j+2-l}$ with $\sigma =…
Peter123
  • 453
1
vote
0 answers

To find order of convergence

Given $x=\sqrt {a}$ is a fixed point of the function $g(x)= \frac{x^3+3xa}{3x^2+a}$. Determine the order of convergence and asymptotic error constant of sequence $P_n =g(P_{n-1})$ towards $x=\sqrt{a}$. I know the definition of order of convergence…
Kavita
  • 728
1
vote
1 answer

What properties does a numerical approximation method have when it converges regardless of initial value?

What properties does a numerical approximation method have when it converges regardless of initial value? For example, sometimes the function does not allow Newton's method to work if we choose an initial value too far from value we want to go to. …
Simply Beautiful Art
  • 74,685
1
vote
1 answer

To determine the rate of convergence of given sequences.

Determine the rate of convergence of each sequence and numerically determine which of the following sequences approaches 1 faster. $$\lim_{x\to0} \frac{ sinx^2}{x^2} versus \lim_{x\to0} \frac{(sinx)^2}{x^2}$$ ** I just found out the rate of…
Kavita
  • 728
1
vote
1 answer

Weights for Gauss-Radau Quadrature

Gauss-Radau-(Legendre) quadrature rules are defined by having one of the abscissa prescribed at $\pm 1$. Thus the abscissa can be expressed as the roots of $$ \Psi_m(x) = (x_0 - x)r_{m-1}(x). $$ WLOG we take the prescribed abscissa to be at $x_0 =…
Freddie Witherden
  • 148
1
vote
2 answers

How does this error depend on h?

An approximating formula dr(h) is used to calculate the derivative of a given function. With different steplengths these were the result ( in exact arithemtic and no rounding errors) h drh 0.1 0.50801203103 0.05 0.50100687738 0.001 …
Niklas Rosencrantz
  • 2,265
1
vote
1 answer

Determining parameters for a cubic spline

I have $f(x)=\begin{cases} a(x-2)^2 + b(x-1)^3 & x \in (-\infty,1] \\ c(x-2)^2 & x\in[1, 3] \\ d(x-2)^2 + e(x-3)^2 & x\in[3,\infty) \end{cases}$ and wish to find the parameters $a,b,c,d,e$ for which $f(x)$ is a cubic spline.…
Taln
  • 615
  • 4
  • 14
1
vote
1 answer

What is net cost, net sales, and net revenue

If I have two products. Those products are shirts and shoes. Please see the image. Shirts and shoes If I want to calculate the net cost then do I add the all the costs for the shirts for every year, which will give me 60 dollars or do I add it by…
Ted
  • 13
1
vote
1 answer

Invertible Matrix and matrix norms

I am studying numerical analysis and I have a problem with these questions: $1.$ Prove that if a square matrix satisfies and inequality $||Ax||\geq\theta||x||$ for all $x$ with $\theta>0$, the $A$ is nonsingular, and $||A^{-1}||\le\theta^{-1}$. This…
Laura Cadavid
  • 13
1
vote
1 answer

Showing that the normalized power method converges to eigenvector

I'm trying to show that if $ \lambda_1 >0 \Rightarrow $ the vectors $ x^{(k)} $ converges to an eigenvector. Where k is the number of iteration, and k $ \rightarrow \infty $ I started by writing $ x^{(k)}$ $$ x^{(k)}= \frac{A x^{(k-1)} }{ ||A…
Viktor Jeppesen
  • 394
1
vote
1 answer

finding the significant digits for relative error

How exactly do you go about finding the number of significant digits? From what I've found I am suppose to find t where relative error (Re) $ \le$ 5*10^-(t) But I don't understand how you find t. For example, let pi be the exact value, and 3 the…
rex
  • 135
1
vote
1 answer

Normalise a uniformly sampled function

I have a uniform 2D grid of points which represents some unknown function. I wish to normalise the function s.t. it integrates to 1. Is there some method which is best suited for this problem? I had been numerically integrating using the trapezium…
Mike Miller
  • 2,453
1
vote
1 answer

Show that $\nabla\Delta = E$.

My assignment asks me to prove that $$\nabla\Delta = E.$$ I was taught that \begin{align*} \nabla f(x) &= f(x) - f(x-h) \\ \Delta f(x) &= f(x+h) - f(x) \\ E f(x) &= f(x+h) \end{align*} I was taught to show equality assuming we have a function…
Arctus
  • 33
1
vote
2 answers

Condition number of a given linear system $Ax=b$ varying $A$

How would you solve this excersize? I have tried it but I am not posting my calculations since it is wrong and I would spend a lot of time writing it in latex. The problem. Find the condition number with respect to $\Vert \cdot \Vert_{\infty}$…
Nisba
  • 777
1 2 3
…
99 100