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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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Condition number of matrix

Could you tell me what is equal to $K_{\infty}(A)$ (condition number of matrix $A$, using the infinity norm) if $A$ is is the Hilbert matrix with dimension $n=100$ ??
evinda
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Solution consisting scheme (?)

I need to prove that this implicit and multisteps scheme is consistent or not: $$y_{k+1}=\frac{1}{8}[9y_k-y_{k-2}+3hf(x_{k+1},y_{k+1})+6hf(x_k,y_k)-3hf(x_{k-1},y_{ k-1}) ]$$ but I'm a little lost because this is a multistep method and I have no…
Hiperion
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numerical analysis-biection method

I have to write a program that implements the bisection method.The program should end if the number of iterations surpass the maximum number of iterations,or if one or both of these conditions : $\left | x_{k}-x_{k-1} \right | $ < ε and $ \left |…
evinda
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Brownian Motion and MonteCarlo Simulation

While studying Random Numbers, I have come across brownian motion. In the text I am using (Numerical Analysis by Sauer); there is an example (chapter 9, example 9.9) where the author illustrates how a monte carlo simulation is used to estimate the…
user76020
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Finding if two sequences can be Big o and problem in definition of Big O

I want to know if this is a correct assertion. e$^n$/n$^5$=O(1/n) What I did was, n <= 4 ln(n) whenever n>=3 e^n <= n^4 e^n <= n^5/n e^n/n^5 <= 1/n e^n/n^5 <= (1)(1/n) Thus there exists c=1, and N=3 such that |e$^n$/n$^5$|<=c|1/n| whenever…
clarkson
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Numerical iterative method, estimating error

Given iterative method: $x_{n+1}=0.7\sin x_n +5 = \phi(x_n)$ for finding solution for $x=0.7\sin x +5$, I want to estimate $|e_6|=|x_6-r|$ as good as possible, with $x_0=5$, where $r$ is exact solution. This method obviously converges, because…
xan
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Numerical Analysis: Bisection -- Accuracy

Suppose we have a computer program that estimates the root of an equation $f(x) = 0 $ by bisection. Given that its truncation error $\leq$ a & rounding error for evaluating $f(x)$ is $\leq$ b (for a given range of x), what is the estimated accuracy…
Hitchhiker
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Least Squares & Normal Equations

I encountered the problem below, and I know how to do the least squares for a system of equations with no solutions (inconsistent system) where the number of equations (rows) is greater than the number of variables (columns). In this problem, $m \ge…
user76020
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Help with numerical analysis proof

Let $u$ be a nonzero vector in $\mathbb{R}^n$, and define $\gamma=\frac{2}{||u||_2^2}$ and $Q=I-\gamma uu^T$. Prove Q is a reflector satisfying A) $ Qu=-u$ B) $Qv=v$ if $=0$ My approach: I'm letting some $$\hat u=\frac{u}{||u||_2}$$ so that…
User69127
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Gaussian Quadrature - derivation problem

Derive a formula of the form $$\int_{a}^{b}f(x)dx=w_0f(a)+w_1f(b)+w_2f'(a)+w_3f'(b)$$ that is exact for polynomials of the highest degree possible. Not sure how to derive this formula. Possibly method of undetermined coefficients, but I'm not sure.…
Drake
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Determine the spacing $h$ of the x-coordinates so that the error of linear interpolation for this data is $10^{-3}$.

We want to construct a table of equispaced values to $f(x)=sin(x)cos(x)$. Determine the spacing $h$ of the x-coordinates so that the error of linear interpolation for this data is $10^{-3}$. I know the formula for the error of linear approximation…
PooperScooper
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Numerically solving first order system of ODES

I am currently trying to numerically solve a first order system of ODES, of the…
Freeman
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Derive the Simpon's Rule for numerical integration

I think I'm over thinking this because I'm coming up blank. Any help would be appreciated. Here is the question: Derive the Simpon's Rule for numerical integration in a interval $[x_{0}, x_{2}]$ $$\int_{x_{0}}^{x_{2}} f(x) dx =…
primer
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Numerical iterative method for equation with $\cos(x)$

I am practising for the test of numerical methods and here I stumbled on the exercise I don't know how to solve: Show that equation: $x-0,4 \cos(x)=7$ has only one solution $x^*\in\mathbb{R}$ and that given iterative method: $x_n=0,4\cos(x_{n-1})…
xan
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How can I apply Runge-Kutta to evaluate integral?

I would like to evaluate cumulative normal (0,1)-distribution values using Runge-Kutta method but the problem is that I don't know how to apply the method. Namely, if I have that $y'(x)=e^{-x^2/2}, y(0)=1/2$ and $y(3)$ is the value I would like to…
Beginnerprogrammer
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