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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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How is "Condition Number" different from derivative? (Numerical analysis)

From Wiki: "the condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument." I just can't see why do we need this when we already have a great…
so.very.tired
  • 2,065
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Calculate the degree of convergence

I was studying about numerical methods to find root of functions. In the False Position Method, I came to the following claim: $$\text{Degree of convergence } = P = \frac{1+\sqrt5}{2}$$ I cannot find any proof about it. Please help.
far
  • 21
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Banach fixed point theorem for an equation

I am asked to show that the set of equations $$x^2+4xy=1$$ $$x^2+3y^2=9$$ Has exactly one solution for $(x,y)\in [0,1]\times[1,2]$. Also, I shell give an iteration converging to this solution. I know I will have to use Banach fixed-point theorem and…
Hagadol
  • 401
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Is the following scheme for generating $p_n=(1/3)^n$ stable or not. $p_n=(5/6)p_{n-1}-(1/6)p_{n-2}$.

Is the scheme for generating $p_n=(1/3)^n$ stable or not? $$p_n= \frac{5}{6} p_{n-1} - \frac{1}{6}p_{n-2}.$$
user128024
  • 21
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Bound on Number of Iterations for Fixed Point Method

Given the fixed point iteration $$ p_n = \frac{p_{n - 1}^2 + 3}{5}, $$ which converges for any initial $p_0 \in [0, 1]$, estimate how many iterations $n$ are required to obtain an absolute error $\left| p_n - p \right|$ less than $10^{-4}$ when $p_0…
michael straws
  • 515
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Numerically stable Lanczos process? I need to compute Elements of inverse in sparsity pattern of A

I have a large sparse symmetric positive definite matrix NxN matrix $A$. Let $s$ be the average number of non-zeros per row (i.e. $sN$ total non-zero elements). I would like to compute the elements of the inverse of A only at locations where A is…
user126071
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Plotting $\log^{1/n}(1/(1 - u^n))$

The function: $$f(x)=\log^{1/\alpha}\left(\frac{1}{1 - x^\alpha}\right),\quad x\in[0,1),\ \alpha\in\mathbb{R^+}$$ Looks timid enough. To my understanding, the function is well defined over the entire interval, and moreover should be almost identical…
Nathaniel Bubis
  • 33,018
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We are learning about LU Decomposition .. because?

I know what LU Decomposition is but I don't know why we have have to learn about it. What are we using it for? (What's the point to know about it?) Thanks.
UnknownW
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Different termination criterion (bisection method)

I follow why the error at the nth step of the bisection method has $\epsilon_n \le\frac{1}{2^{n+1}}|b_0-a_0|$. Where $\epsilon_n=|\alpha-c_n|$: $c_n=\frac{a_n+b_n}{2} $, $\alpha$ is the root. My question is: Suppose $f$ is a continuous function and…
user68293
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Equivalent formulations of Runge Kutta Methods?

I have given two formulations of a general Runge Kutta Method with $s$ stages $$ y(x+h) = y_0 + h\sum_{i=1}^{s}b_ik_i$$ $$ k_i = f\left(x_0+c_ih;\,y_0+h\sum_{i=1}^{s}a_{ij}k_j\right)$$ $$ y(x+h) = y_0 +…
Osch Bert
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How do I combine "error of order" terms in numerical analysis?

Looking at a 2nd-order Taylor series approximation of the function $f$, I have this: $$f(t_1) = f(t_0) + hf'(t_0) + {h^2\over 2}f''(t_0) + O(h^3)$$ Now say I approximate $f''(t0)$ with a $2$nd-order central difference method: $$f''(t) = {1\over…
jjkparker
  • 123
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Finding $5^{1/3}$ with Newton's method

I have this exercise in my book: Let $f(x)=x^3-5$ and we are looking for the solution $f(x^*)=0$, that is $x^*=5^{1/3}$, with Newton's method. So $x_{n+1}=x_n-\frac{x_n^3-5}{3x_n^2}$ and let $e_n=x_n-x^*$ be the $n$-th error. If we take $x_0>x^*$,…
xan
  • 2,053
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Is scalar product a well-conditioned operation?

I'm reading a course and one of the exercises is about establishing whether scalar product is a well-conditioned operation. Here's their solution. They disturb each element of the vector by multiplying it by $(1+\alpha_{i})$ where $\alpha$ is really…
Arek Krawczyk
  • 2,114
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Writing Lagrange form of an interpolating polynomial

Write the Lagrange form of the interpolating polynomial of degree at most 2 that interpolates $f(x)$ at $x_0, x_1,$ and $x_2$, where $x_0
Drake
  • 851
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Finding rates of convergence

I'm doing some textbook problems in Burden and Faires Numerical Analysis, when I encountered this question that I did not understand how to do. How would I go about finding the rate of convergence for $$\lim_{n\to\infty}\sin\dfrac{3}{n}=0$$ I know…
Sujaan Kunalan
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