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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 the approximations to within $10^{-4 }$ to all the real zeros of the following polynomial using Newton's method.

We have $P(x)=x^3-2x^2-5$. I know the formula of Newthon's method. That is given here. The problem is, how do I find the approximations to within $10^{-4}$ to all the real zeros of the following polynomials using Newton's method?
user3543012
  • 139
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Three point formula exercise question

Using the table in the figure and the three point formula find the approximate values of the derivative required f'(1.2).Also calculate Ea and Ev ( Actual error and error bound) We notice that h=0.1 so f'(1.2)=[1/(2*0.1)]* [f(1.3)-f(1.1)] Is this…
user3543012
  • 139
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Solving equations in R

I have a question if a method is true and the proof of it: So, let us have a real function f which satisfies that f(a) < 0 and f(b) > 0, f is continuous in [a,b], and f'(x)>0 in [a,b]. Then we know that there is an unique point c from (a,b) such…
nikola
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Is there a formula of coefficients of Newton-Cotes Method in numerical intergation?

We know the coefficients of Newton-Cotes method in numerical integration are: 2-points $ 0.5$ , $0.5$ 3-points $ 1/6$, $2/3$, $ 1/6$ 4-points $1/8$, $3/8$, $3/8$, $1/8$ and so on I ask if there is a formula…
user187581
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What is the best method to calculate the square root when I know that the root is always an integer?

I have been through the wikipedia page, but wanted to know if there was a preferred (most efficient) method when there is an exact solution to find?
KenInLondon
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The formula of the order of multistep methods

How can I derive this $$(1+\xi) \left(1+\frac{1}{2}\xi-\frac{1}{12}\xi^{2}\right)+O(\xi^3)$$ from $$\frac{1+\xi}{1-\frac{1}{2}\xi+\frac{1}{3}\xi^{2}}+O(\xi^3)$$ ? The whole formula is below. This is from "A first course in the numerical analysis of…
user26767
  • 299
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condition number for a matrix with a variable

How would I go about calculating $cond(A)$ for A= $\begin{bmatrix}1 & c\\c & 1\end{bmatrix}$, $|c|\neq 1$
user141745
  • 410
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Finding the function when the Newton-Rapson formula is give.

The question is, "Show that the Newton-Raphson method of the form $$x_{n+1}= \frac{12x_n-5x_n^3}{8}$$ can be used to estimate $\sqrt{0.8}$. Show that this method will converge if the initial estimate $x_1$ satisfies $\sqrt{4/15}\lt…
user46044
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Gaussian Quadrature - Construction

Suppose you have $w(x)= 1/\sqrt{x}$ as your weight function, and the integration of the form $\int_0^1 f(x) w(x) dx$. I am tasked with creating a quadrature of exactness 3. So I know I need a polynomial with degree $n=2$, and that I need to map the…
Michael Senter
  • 47
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Quadratic interpolation.

Quadratic interpolation means the following: If we are given a table of values $y_i=f(x_i), 0\leq i\leq 2n$, then on each interval $[x_{2j},x_{2j+2}], 0\leq j\leq n-1$,the quadratic interpolating polynomial is used on that interval to interpolate…
Flip
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integral of the sum

I am trying to integrate the following sum. I need to get at least first 5 terms (using math or computer). I've tryed wolfram alpha online-did not work. I should find $$ \int_0^\infty \sum_{j=0}^\infty c_jx^{2j}e^{-\frac{x^2}{6}} \;dx, $$ where…
David
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RK4 stability problem.

I'm trying to solve the following problem using RK4: $$y'=y^{\frac{1}{3}}, \quad y(0)=0, \quad t\in[0,6].$$ The exact solution is $y(t)=\sqrt{\left( \frac{2t}{3} \right)^{3}}$. I wrote the following code: int main(){ double a = 0; double b =…
Spine Feast
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Newton's Method Convergence

Let r be the zero of multiplicity 2 of the polynomial p(x), how do I prove that $x_{n}$ converges quadratically to r? I only know the basic Newton's Method which is $x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$, I can't think other way to prove…
Yagami
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For what values of $x$ is the assignment $y=1-\cos x$ problematic, and why?

So I'm kind of stuck on this question and I don't exactly know how to describe this on the title header and I apologize... For some values of $x$, the assignment statement $y := 1-\cos(x)$ involves a difficulty. What is the difficulty? What values…
user3390252
  • 43
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name of function $x_{n+1} = f(x_n)$ and the sequence it generates

this is a very trivial question, but I couldn't find what the proper name a function of the form $x_{n+1} = f(x_n)$, where $f: X \rightarrow Y$ for some initial $x_0$ is, and what the the sequence $(x_n)_{n=0}^{\infty}$ which generates is…
sarah
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