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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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Implementing Simpson's rule in Matlab

I am trying to implement Simpson's rule, but for one reason or another, this works only for constant functions, but for every other kind of function, my code doesn't give a decent approximation. The code itself is rather simple, but I just don't see…
Julian
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solve equation with R

I want to solve the equation with R: $$\frac{c_1^{n_1+1}}{[(n_2-1)+(n_1-1)c_1]^{n_1+n_2-2}}=\frac{c_2^{n_1+1}}{[(n_2-1)+(n_1-1)c_1]^{n_1+n_2-2}};$$ $$[F(c_2;n_1-1,n_2-1)]-[F(c_2;n_1-1,n_2-1)]=1-\alpha.$$ I want to solve out the numerical of…
Jayden
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The cutting method for finding the solution of equation $16x^2+3-9/x=0,x\neq 0$

Find one real and positive solution of equation $16x^2+3-9/x=0,x\neq 0$ using the cutting method with precision up to $\epsilon=0.01$. I think that I am getting a mistake by using the secant method. Let $f$ is continuous function such that $f(a)>0$…
user7920621
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How can I use mathematical software to solve this matrix?

The previous post I made was not well explained. I hope this new image would pass the information required across. So I'm trying to solve for the a's in terms of the other variables. The softwares I have available are; Matlab,Maple,Scientific work…
OdunayoO
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Write in normalized form, using 4 decimal digit IEEE rounding.

The numbers are: a = 124.68 a = -1.2345 a = 0.55555 a = -0.0054321 I'm not sure what the format is for rounding numbers in this way. Explanations would be great!
Phoenixdeath
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Calculating Best Approximation of $f(x)=e^x$ on $P_0[-1,1],P_1[-1,1],P_2[-1,1]$ with L2-Norm

I want to calculate the best approximation $f(x) = e^x$ on the function spaces $P_0[-1,1],P_1[-1,1],P_2[-1,1]$ with the L2 Norm. In the lecture we derived that given a orthonormal system of the function space of $P_n[-1,1]$, e.g. the Legendre…
eager2learn
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What is the Quadrature rule under these conditions?

all Is there any quadrature rule with error of $O (h^6)$ when using 5 nodal points? The five nodal points are uniformly distributed. For instance, [-1,0.5,0,0.5,1] or [a, ....,b] I am trying to find the one which satisfies those condition. If so,…
bluejyellow
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Method of least squares

Consider the non-linear equation $x^3 = 3375$. The real solution is given by $x^* = 15$. The iteration sequence $\{x^{(k)}\}$ has the convergence order $p$, if $$\lim_{k \rightarrow \infty} {||x^{(k+1)} - x^*||_2 \over ||x^{(k)} - x^*||_2^p} =…
Julian
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Numerical Tests of Convergence

I was once told that, since numerical integration out to infinity is not possible, that one method which is used to test whether an integral converges, say over ($0, \infty$), is to take a large interval, eg ($0, 10^6$). The method that the…
T-Ray
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What are collocation and interpolation points?

I'm studying numerical integration for solution of second order ivp. But I don't seem to get the concept of collocation and interpolation points. How are they being assigned? What are some things to consider? I need a layman intuition on this topic.
OdunayoO
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How to find values for which an iterative sequence converges?

I'm trying to figure out this question, or rather, part a of this question from a practice final. I know that if you consider the fixed points of $g(x) = (α+1)x-x^2$ (which are $ x = 0, x= α $), but I don't know how to find the interval for which…
klamont15
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Quadrature Formulas

I'm doing a exam problem in the subject numerical methods with the book "Numerical Mathematics and Computing" 7th edition by Ward Cheney and David Kincaid The problem is in the picture below the relevant part of the solution which I have question…
Postskjerm
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Function corresponding to Newton Iteration

Given the following Newton iteration: $x_{n+1}=2x_n-x^{2}_{n}y$ Identify the function being iterated. The solution was: $$x_{n+1}=2x_n-x^{2}_{n}y= x_n-(x^{2}_{n}y-x_n)=x_n-\frac{y -\frac{1}{x_n}}{\frac{1}{x^{2}_{n}}}$$ which gives $f(x)=y -…
user137481
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Fixed point iteration for function $\sin x^{100}$

Suppose that the fixed point iteration for the function $\sin x^{100}$ converges to the fixed point 0. Determine the order of convergence with justification. I may have misunderstandings for this question. 1) Is it right to let $f(x) = \sin x^{100}$…
Little Rookie
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finding parameters for measured values

From a physical system I got the following relation: $$ c_M = \sqrt{ \frac{c_s}{\sin{\alpha_s}} } \sqrt{ \frac{L}{\Delta t - 2 t_s} } $$ From measurements multiple pairs of $(L,\Delta t)$ are known, $c_s$, $\alpha_s$ are given constants. It is…
Mario
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