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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

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Accuracy of $e^x$ approximation using remainder of Taylor Series Approximation.

The Problem: If we consider approximating $e^x$ on $[-1,1]$, the Taylor Theorem for $x_0 =0$ says we can represent $e^x$ using a polynomial with a (known) remainder: $e^x =\{1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \dots +\frac{1}{n!}x^n \} +…
Sharky
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Numerical Analysis - Upper bound for $|R|$

I am asked to find an upper bound for $|R|$ valid for all $x\in[-1,1]$ that is independent of $x$ and $\xi$. Given that, $$R(x)=\frac{|x|^6}{6!}e^\xi$$ for $x\in[-1,1]$ where $\xi$ is between $x$ and $0$. I began with stating…
cryptomath
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Problem applying Simpson's rule

I am having a problem applying composite Simpson's rule for the integral $$I=\int_0^2\dfrac{1}{x+4}dx$$ with $n=4$. The exact value of the integral is about $0.405$, however, Simpson's is giving $0.8$, and by increasing the number $n$ up to $8$,…
Tofi
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If $f(x)=e^{ax}$, show that $\Delta^n f(x)=(e^{ax}-1)^ne^{ax}$

Given as an assignment for Interpolation. I, first of all, doubt whether the question is correct or not, because $$\Delta f(x)=e^{ax+ah}-e^{ax}=e^{ax} (e^{ah}-1)$$ $$\Delta^2 f(x)=[e^{ax+ah} (e^{ah}-1)]-[e^{ax} (e^{ah}-1)]=e^{ax} (e^{ah}-1)^2$$ So,…
Jignesh Shah
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Showing Lagrange polynomials form a basis for $\Pi_3$.

Here's the question: Let $x_0=-2, x_1=0, x_2=1, x_3=4$ and let $L_j$ be the Lagrange polynomial for $j=0,1,2,3$. Show that $L_0, L_1, L_2, L_3$ form a basis for $\Pi_3$. So I've calculated $L_0,...,L_3$. I'll supply $L_3$ and $L_0$ just to see if I…
user314580
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Is there a way to measure how close a matrix is to being rank-deficient?

I'm working on a least-squares problem with an overdetermined matrix, and I've noticed that changing the data very slightly leads to a huge change in my solution (in this case, plotting an elliptical path). I suspect that this is because the changes…
NNN
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Mesh points natural ordering - Finite Difference Method

Imagine that you have a descritized matrix A which the points are $A_{i,j}$ with $i=1,...,n$ and $j=1,...,m$ Then we can label the points as $A_{k}$, where $k=i+n(m-j)$. I had already seen some examples but I have never seen the proof of this. Can…
pipita
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Use Romberg integration to estimate the integral

I was asked to use Romberg integration to evaluate the integral$$\int_0^1x^{-x}dx=\sum_{n=1}^\infty n^{-n}$$ and compare the result with the result I get from the sum. And I also need to estimate how many function evaluation Romberg integration will…
J.doe
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Can Steffensen's Method be applied if there is a possibility of a singularity?

I realize that the initial nonlinear equation needs to be linearly convergent and that we then use Steffensen's Method (a modification of Aitken's $\Delta ^2$ Method) to speed up the convergence, but I'm unsure if having the possibility of a…
Richard Slabbert
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Convergence, supremum and limit

Hey guys I am kind of confused about these three terms. How to prove if a sequence is convergent? How to prove the sup S or inf S exists? How to prove the limit of a sequence exists? How to show that a sequence is bounded? To show that the limit…
J.doe
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Bisection Method Numerical analysis Problem.

I am trying to solve the following $$(\frac{x}{2})^2-\sin x = 0$$ with initial starting points $a_o = 1.5, b_0 = 2$ and $n = 1(1)5$ using Bisection Method. From the little I have studied, I went about solving it like so; Iteration 1 $f(x_1) =…
Diamond
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Newton's Method Solving an Equation

What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^2-2bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? Additionally, I need to show that $|g'(x)|\le 1/2$ whenever $|x-b|\ge d/\sqrt{2}$ and also that…
user310546
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The fixed point iteration part b

I've solved part a. And for part b, I solved $g'\left(x\right)^2$ and when c=0 or $x=-b/c$, we have the minimum, but according to the problem, we can't reach it. So the minimum occurs when $x^2=2|c|$ and $b^2=9/2|c|$. To finish the proof, I need to…
J.doe
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How to find the value of 'c' using Trapezoidal rule

If the Trapezoidal rule with single interval [0,1] is exact for approximating the integral $\displaystyle\int_0^1(x^3-c\ x^2) \ dx $, then what is the value of c ? I am confused with the word "exact". What does it mean ? and give me some hints to…
user308807
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Using Stirling's Approximation to Find Maximum

How can we use Stirling's approximation, $$n!\approx\sqrt{2\pi n}(n/e)^n$$ to find the size/location of the max term in, $$\sum_{n=0}^\infty\frac{x^n}{n! }$$ for any $x>0$. I started off by writing the series in terms of Stirling's…
user308046
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