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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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What are the possible limits of the iteration?

Consider the function $f(x) = \sqrt{2 + x}$ for $x \geq -2$ and the iteration $x_{n+1} = f(x_n) ; n \geq 0$ for $x_0 = 1$. What are the possible limits of the iterations ? $\sqrt{2 + \sqrt{2 +\sqrt{2 + ...}}}$ -1 2 1 I think $x_1 = \sqrt3$, $x_2…
Struggler
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Linear interpolation by hand - Any quick ways to do this?

I have to calculate the roots of the equation $x^3 + x^2 -3x -3 = 0$ in the interval $[1,2]$ using linear interpolation to six decimal places, by hand. Now I know this is trivial in excel, but when doing it by hand it becomes very tedious. Is there…
NMH
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Convergence of Gauss-Seidel.

I am preparing for an exam and I have an exercise type, without any example. Give the following system : $$ \begin{bmatrix} 2 & 1\\ -1 & 2\\ \end{bmatrix} *x = \begin{bmatrix} 3\\ 1\\ …
user137209
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Numerical computation of the logarithm of the generalized incomplete Beta function

Define $\mathrm{B}(x,y;\alpha,\beta)$ as the generalized incomplete beta function: $$\mathrm{B}(x,y;\alpha,\beta) = \int_x^y t^{\alpha-1}(1-t)^{\beta-1}\mathrm{d}t$$ You can assume that $x,y\in[0,1]$ and that $\alpha,\beta > 0$. The ordinary…
a06e
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Using Finite Difference to compute derivative in the Newton-Raphson root finding Algorithm

In the Newton-Raphson method we come across the following equation: $$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}$$ Can you please let me know if we can calculate the derivative term like this - $$f'(x_n) = \frac{f(x_n) - f(x_{n-1})}{x_n-x_{n-1}}$$ Will…
Discretizer
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Numerical one-step method: initial value and non consistent method

We had to code a program solving a starting point problem. (Runge-Kutta 6th Order) The ordinary differential equation (first order) is: $y'*y= \cos x$ with $f(0)= 2$; $[0,10]$ I have 2 questions for that since Im not that familiar with differential…
Jln
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We have x=0.2234 and Δx =0.2*10^-3.Find how many accurate digits does the number have?

So we know that $Δx≤(1/2)*10^{-n}$ where n is the number of the accurate digits. Now I just have to replace $Δx =0.2*10^-3$=$(1/2)*10^{-n}$ and find n. But why have I been given x=0,2234 since I dont need it?
fsdd
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Use bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on [0,1]

I have to use the bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on $[0,1]$. However I simplify I get, $$x^{2x(1-x^{2}+x)}-e^{x^{2}-1}=0$$ Which is undefined at $0$. How would I go about approaching this problem?
Gamecocks99
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Least Squares Problem - Show that $F(x) = (b - Ax)^T(b - Ax) + \alpha x^T x $...

Consider the function $$F(x) = (b - Ax)^T(b - Ax) + \alpha x^T x $$ where $A$ is a real $ m \times n$ matrix and $\alpha$ is a positive real number. We want the minimum point of $F$ for given $A$, $b$, and $\alpha$. Show that $$F(x + h) - F(x) =…
Ozera
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What is the recomputation of residuals in the deepest descent method?

From Jonathan Richard Shewchuk: By using equation (13), we have lost track of $x_(i)$ in the process of iteration. How is it possible to recompute the correct residual using equation (10) ever again?
qed
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Gaussian Quadrature with polynomial of order one

Let $w(x) > 0$ be a weight function on the interval $[0, 1]$ and let $P_n(x), n = 0,1,2\dots$ be a sequence of orthonormal polynomials of exact degree $n$ which satisfy $$\int_0^1P_n(x)P_m(x)w(x)dx = \delta_{nm}$$ Where $\delta_{nm}$ is the…
user117682
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Prove the trapezoidal rule approximation

$\displaystyle L^i(x)= f(x_{i-1}) + \frac{f(x_i) - f(x_{i-1})}{x_i - x_{i-1}} (x-x_{i-1})$ Show that this linear approximation gives the trapezoidal rule. I know the formula of the trapezoid rule, but not how and where to use this L at all.
kiwifruit
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Find $t$ which minimizes $\|A(x+ty)-b\|^2_2$

Let \begin{align}f(t) &= \|A(x+ty)-b\|^2_2 = (A(x+ty)-b)^T(A(x+ty)-b))\\ &= x^TA^TAx + 2tx^TA^TAy+t^2y^TA^TAy-2b^TA(x+ty)\end{align} Then, letting $f'(t) = 0$, we have $$ t = \frac{(b^TAy)-x^TA^TAy}{y^TA^TAy}$$ Ok I am confused by the whole thing.…
user136422
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Polynomial Interpolation question

I do not understand the part that I have underlined in green. I thought that if you fit a polynomial $P_n$to given data for say $n+1$ distinct points you got an approximation of $f(x)$, where there is an error $f-P_n$.
usainlightning
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calculate an approximate value of integral

Calculate an approximate value of integral : $$\int_1^{3.4}\frac {2}{\sqrt{x}+x}$$ Take 8-subintervals $n=8$ by using trapezoidal rule How can I calculate this?
user143507
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