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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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Numerical estimation of second order derivative

Given that y" = [y(x+h)-2y(x)+y(x-h)]/h^2 (central difference) with error = O(h^2). Find an expression for y" that has a smaller error than O(h^2). Note: h is small (say 0 < h < 1). Thanks in advance
William Kubin
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Derive the Newton-cotes formula for $\int f(x) dx$ using nodes $o, \frac13, \frac23, \1$.

Can someone please let me know if I am on the right track here. Obviously, I still need to perform the integrations as well as evaluate the integrations(from 0 to 1) to determine the coefficients. What I gather from my notes, and the text book, this…
user551155
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Why are Machin-like formulas with more terms "better"

As in the title, when looking at the Wikipedia page, it seems that the more terms the Machin-like formula the better. Could someone explain this to me? I think it is for two reasons: It is easier to "split up" the computation. I can compute one…
2132123
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Convergence Iteration matrix

Let $u_0:=v \in R^d$ and $u_{k+1}:=Au_k, A \in M_d(\mathbb{R})$ being a symmetrical matrix with non-negative coefficients such that the sum of its row elements is $1$ for each row. Moreover, one eigenvalue is $1$, the other eigenvalues are…
user249018
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Approximate ODE by discretization

I consider the the ODE $ -u''(x)=f(x),\quad x\in \left]0,1\right[ $ with $ u(0)=u(1)=0 $. So I search a function $ u:\ [0,1]\rightarrow \mathbb{R} $ which solves this problem. First step is to discretize the domain of $ u $ with $…
hallo007
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How do I show a function has a unique positive root?

I have to show that the function: $f(x) = x^3 + x^2 - 1$ has a unique positive root. It’s for numerical methods, would anyone have any tips? I’ve been trying to use fixed point methods but I’m not getting anywhere.
user731762
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"Explicitization" of Implicit Euler

Consider the ODE $\dot y=f(t,y)$. Explicit Euler says: $$y_{n+1}=y_n+hf(t_n,y_n)\tag{1}$$ Implicit Euler says: $$y_{n+1}=y_n+hf(t_{n+1},y_{n+1})\tag{2}$$ The above schemes are well-known. In a way similar to what is done in Heun's method, we could…
pluton
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Having problem to understand the expression $∆x = ∆x_1 + ∆x_2 + · · · + \Delta x_n$

I am studying this pdf file. in the file they say: $2.1.1$ Errors in sum and difference. Let us consider the exact numbers $X_1, X_2, . . . , X_n$ and their corresponding approximate number be respectively $x_1, x_2, . . . , x_n$. Assumed that…
Etemon
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Let be a root of $f ∈ [, ]$ and $f$ is differentiable and $_$ be a sequence of iterates obtained in bisection method. Then show that ...

Then show that $ − _n = −f(_) /f'(_)$ where $ < _ < _n$ I know this statement represents the mean value theorem, I think It can be proved using proof of the mean value theorem but Are there any other methods to prove it using the bisection method?
Galaxylokka
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Subtractive cancellation (same sign)

I know that subtractive cancellation occurs when subtracting two numbers close to each other. But if you have two numbers close to each other, one being negative and the other postive. You subtract the postive from the negative. Will there be…
Sorry
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Iterative method proof

We choose the iterative method $$\vec{u}^{i+1} = Q\vec{u}^i + \vec{s}$$ with $\vec{u}^0 = 0$ to solve $A\vec{u} = \vec{f}$. I want to proof that for $\vec{u}^i$ the following holds: $$ \vec{u}^i = (I - Q^i)A^{-1}\vec{f} $$ Under certain assumptions.…
Phoenix_10
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Newton-Cotes for $n=2$ and $[a,b]=[0,1]$

Using Newton-Cotes formula: $$ \int_{a}^{b} f(x) d x \approx \int_{a}^{b} p(x) d x=\sum_{i=0}^{n} f\left(x_{i}\right) \int_{a}^{b} \ell_{i}(x) d x $$ I want show that: $$ \int_{0}^{1} f(x) d x \approx \frac{1}{6} f(0)+\frac{2}{3}…
Sorry
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Maximum error of $\int_{1}^{2 m} \frac{1}{y} d y$ using the Trapezoid Rule

Consider $\int_{1}^{2m} \frac{1}{y} d y$. Show that the integral can be calculated using Trapezoid Rule with maximum error $$\frac{\left(2^{m}-1\right) h^{2}}{12}$$ for a given step length $h$. Answer The error is given by $E = -f^{\prime…
Sorry
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Finding a polynomial that interpolates data (Newton's method with Horner's method)

Problem Find the polynomial that interpolates the data: $$ \begin{array}{|c||c|c|} \hline x & 3 & 7 \\ \hline y & 5 & -1 \end{array} $$ So I have Newton's polynomials given by $$ p_{0}(x)=c_{0}, \quad p_{i}(x)=p_{i-1}(x)+c_{i}…
Sorry
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Prove that the rate of convergence of a sequence is unique

I know that the rate of convergence of a sequence is unique. But I do not know why. Please prove that the rate of convergence of a sequence is unique. Rate of Convergence
Behrooz Fard
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