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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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Find $n\in \mathbb N$ such that $n \gt 1$ and the binary expansion of $\frac{1}{n}$ terminates

Question: Find $n\in \mathbb N$ such that $n \gt 1$ and the binary expansion of $\frac{1}{n}$ terminates. My try: I know an algorithm for writing the binary expansion. Step 1: Consider $n \in \mathbb N$ and $A \in (0,1)$ Step 2: For $i$…
Arman Malekzadeh
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Chebyshev polynomials of first kind

I know the chebyshev polynomials of the first kind can be approximated using the cosine function, where $T_n(\cos \theta)=\cos(n \theta)$ and I know that chebyshev polynomials are a family of orthogonal polynomials. How would I prove that…
A A
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Find the **Local Truncation error** in the form of $ \ o(h^k) \ $ of the multi-step method

Find the Local Truncation error in the form of $ \ o(h^k) \ $ of the multi-step method $ 2u_{i+3} = −3u_{i+2} + 6u_{i+1} − u_i + 6hf(t_{i+2}, u_{i+2}) $. Also determine it is convergent or not. Answer: The scheme is $ \ 2u_{i+3} = −3u_{i+2} +…
MAS
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Convergence of the BDF-algorithm

I have the following problem: Show that the BDF-algorithm for a constant stepsize $h$ and $r\in{1,2}$ is convergent for all enough smooth right sides $f$. I already know, that for $r=1$ the BDF-algorithm equals the implicite Euler-algorithm, which…
Tobi92sr
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Newton Iteration method derivation

How is Newton's Iteration achieved? I mean, can you please explain where does Newton's iterative formula $x_{k+1}=\frac{1}{2}(x_k+\frac{N}{x_k})$ come from?
John Smith
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Interpolation of a function

Given the function $$f (x) = x\bigg(x − {1\over4}\bigg)\bigg(x − {1\over2}\bigg)$$ How can I interpolate $f(x)$ with $p(x) = a_0T_0(x) + a_1T_1(x) + a_2T_2(x) + a_3T_3(x)$ to show that $$a_0 = -{3\over8},\ \ a_1 = {7\over8},\ \ a_2 = −{3\over8},\ \…
StealzHelium
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Forward and Backward Euler.

I want to consider this differential system: $$ \ \frac{dx}{dt} = -y(t)\\ \frac{dy}{dt} = \ x(t) $$ where $t>0$ with initial condition$ (x(0),y(0))=(1,0).$ First I want to show that this differential equation admits an invariant of $I = x(t)^2 +…
Jackson Hart
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Deriving Chebyshev Polynomials

I'm trying to show that the derivative of Chebyshev polynomials at $x = 1$ satisfy $$T_k'(1) = k^2$$ for each $k ≥ 0$. I can get the derivative to come out as $$T'_k(x) = \frac{k \sin(k\theta)}{\sin(\theta)} $$ but after that it always ends up as…
StealzHelium
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Numerical Analysis ODE's

I am trying to solve this problem, but am having some trouble. The ODE $u''=\cfrac{u'}t - 4t^2u$ has the solution $u(t)=\sin(t^2)+\cos(t^2)$. I want to plot the exact solution over the interval $[1,20]$. Can someone help me figure out this exact…
Jackson Hart
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What is a formal definition of "algebraic degree of accuracy" regarding numerical integration?

I am given one example that deals with numeric integration. It is a classic example of determining coefficients $A_1,...A_n$ such that the given quadratic formula is valid for all algebraic polynomials of some degree. The next part of this problem…
user300045
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When to use forward or central difference approximations?

While trying to approximate derivatives in my numerical methods class, we were taught about forward and central difference approximations, however apart from questions when the method to be used is stated i have no idea which is to be used in…
Sonny Da Silva-Peters
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Curve fitting challenge: Find a better fit

I have tried to use the method of deepest descent to find the $4$ parameter variables $v_1, \ldots, v_4$ of a chosen function $y(x,v_1,v_2,v_3,v_4)$ that most closely matches the $y$-values in the table below. The best fit values of the variables,…
Jens
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All roots of a non-polynomial function

Say I have a function $f(x)$ which crosses the zero several times. Suppose I want to find all the $x$ in a range $[a,b]$ where the function $f(x)$ would be zero. I have got quite some methods like Newton Raphson and bisection but those enables us to…
Mitradip Das
  • 183
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cg-method: Convergence of $A=I+B$

I have the following situation: If $A=I+B$ is a symmetric positive definite (spd) matrix and $rank(B)=r$, then the cg-method converges at most in $r+1$ steps. So first of all, I hope I do understand correctly that I have to show that the cg-method…
Yasuduck
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Writing expressions as column matrix...B-splines?

The question: Evaluate $$\sum_{k=0}^4 c_kx^k$$ for $x=0,1,2,3,4$. Write these five expressions as a matrix product $Mc$, where $M$ is a 5x5 matrix, and $c$ is a column matrix with components $c_0,c_1,....,c_4$. Would I just plug in the different $x$…
Alti
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