# What is Numerical approximation: Definition and 24 Discussions

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics (predicting the motions of planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Before the advent of modern computers, numerical methods often depended on hand interpolation formulas applied to data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas nevertheless continue to be used as part of the software algorithms.The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.
Numerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to real-world measurements by translation into digits, it gives approximate solutions within specified error bounds.

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1. ### Minimizing the friction of the end of a lever against the top of a moving piston

I'm trying to design a simple mechanism in CAD and got stuck on this problem. Consider a lever (blue shape) that pivots at one end (blue circle). The other end rests on a piston (pink rectangle). In my application the lever is pushing the piston (the lever pushes through the full travel of a...
2. ### I Finding a Rational Function with data (Pade approximation)

Dear Everybody, I need some help understanding how to use pade approximations with a given data points (See the attachment for the data). Here is the basic derivation of pade approximation read the Derivation of Pade Approximate. I am confused on how to find a f(x) to the data or is there a...
3. ### Numerical approximation of the 2nd derivative across a diffuse interface

Imagine you create a diffuse interface in space and determine which side of the interface you are on by a local scalar value that can be between 0 and 1. We could create a circle, centered in a rectangular ynum-by-xnum grid, with such a diffuse interface with the following MATLAB code: xnum =...
4. ### Numerical integration - Gauss Lobatto

Homework Statement I need calculate the points (##x_i##) and weights (##w_i##) with Gauss Lobatto seven points on the interval [a,b]. With the points and the weights I am going to approximate any integral at this interval.Homework Equations I have found the relevant points and weights at the...
5. ### MATLAB Numerical approximation of the area under curve

I am very new too Matlab and how it all works but I am having trouble understanding at what axis the numerical integration is occurring from on the graph that I plotted. So I am currently doing an experiment in gamma ray spectroscopy and due to issue with the software we found it hard to...
6. ### Coding a numerical approximation for a damped pendulum

Hi there. I have a question about the damped pendulum. I am working on an exercise where I have already numerically approximated the solution for a simple pendulum without dampening. Now, the excercise says that I can simply change the code of this simple situation to describe a pendulum with...
7. ### Numerically find the energy of the delta-well's bound state

I'm working on an assignment where I'm required to numerically find the energy of a delta-potential's bound state. To do this, we've converted the time-independent schrödinger equation to an eigenvalue problem with E the eigen value, ψ the eigen vector and H a matrix as follows: with ##t =...
8. ### Help with interpreting an interpolation problem

Homework Statement I don't know if this is the appropriate place to ask this, but I really do need some help. I am doing a homework problem and I don't understand what is being asked. It goes as follows: > Write a MATLAB function to evaluate the trigonometric interpolant ##p_n(x)## for a given...
9. ### Secant Method Minimum Estimation

Homework Statement I am having some trouble understanding the following solved problem: Use two iterations of the Secant method to estimate where ##x^2 -x## has a local minimum. Start with ##x^{(0)} =1##, ##x^{(1)} =1/2.## Answer: Homework Equations Secant iteration: x_k = x_{k-1} -...
10. ### Richardson Extrapolation with 3 steps?

Homework Statement [/B] use richardson extrapolation to estimate the first derivative y=ln(x), x=5 using steps of 2, 1, 0.5. Four decimal points. obtain true relative error for the last estimate and comment on its value. Homework Equations [/B] deriv ln(x)=1/xThe Attempt at a Solution I know...
11. ### Is there an analytic solution to this system of equations?

I have the following system of equations with variables ##a,m##, and I'm wondering—can this system be solved symbolically/analytically? \begin{align} m &= 100 + \frac{ \left( 200 \frac{\ln{\frac{1}{2}}}{26.8} \right) }{\left(\dfrac{\ln{\frac{1}{2}}}{26.8} + a \right)} \\ \\ 50 &= me^{-a\left(...
12. ### Is this correct second order approximation?

I have a second order differential equation of the form (theta is a function of time): \theta ''=F\left(\theta ,\theta '\right) Turning them to two first order equations I get: \begin{cases} \theta '\:=\omega \\ \omega '=F\left(\theta ,\omega \right) \end{cases} And here's the algorithm...
13. ### Algorithm for Numerical approximation to add data points

Hi, I am working on TDR (Time Domain Reflectometry). I send a 7GHz bandwidth fast rising edge (14ns) square wave into a coax. I get a return Signal. I have an ADC with 10Msamples/sec. I am using MPLAB IDE for coding the microcontroller. Now I would like to increase the Points on the...
14. ### Numerical Approximation and addition of new data points

Hi I am new member and I am new to the Signal processing so I hope I could get some help from the members to able to understand the concepts. I have a Signal. I have a 10Msaples/sec ADC. I view the Signal on an Oscilloscope which has 20Gsamples/sec sampling rate. The Point where I am...
15. ### Numerical Approximation of a Rocket's motion

So, I ve been trying to add orientation to my model of the flight dynamics of a rocket but I ve been running into a lot of problems. I didn't bother actually doing the math for the moments of inertia and everything because I guess it really doesn't have that much of an effect on the general...
16. ### Numerical Approximation of a 4D System of ODE's

What is the most general method of approximating arbitrary systems of ODEs of 4 variables(x,y,z,t) that fit these conditions? The conditions that are assumed true of the ODEs are: 1) that I require differentials to be explicitly defined (but they can be defined in terms of other...
17. ### MHB How can I optimize numerical approximation with fewer samples?

Hi all, i have a problem to solve that i want maybe to solve with MATLAB o excel. I have a numerical samples and with linear approsimation i have a function, but now i want to use less samples for example only 20 and i want to find the best set of samples to approsimate in the best way the...
18. ### Error in numerical approximation of an integration

Homework Statement a,b\in R, a<b, n\in N\\ h=\frac{b-a}{n} , x_i = a+ih , i=0..n \\ f\in C^1[a,b] we approximate the integral of f in a,b with Q_n(f) = h\left[f(x_1) + f(x_1) + ... + f(x_n)\right] Find the error R_n(f) = \int_a^bf(x)dx - Q_n(f), as function of the first derivative of f...
19. ### Numerical approximation of the eigenvalues and the eigenvector

Homework Statement This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example. We will define two sequences of vectors{vk} and {uk} (a) Choose any vector u \in R2 as u0 (b) Once uk has been determined, the...
20. ### Euler's method for numerical approximation

y' = 3 + t - y, y(0) = 1 A) Find the approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, 0.4 using the Euler method with h = 0.1. B) Repeat part A with h = 0.05. Compare the results found in A. I did part A correctly, but cannot get the right...
21. ### Numerical Approximation to Roots

Homework Statement I'm trying to find a root-finding method for a function [tex] f: \mathbb R^n \to \mathbb R [/itex] Homework Equations x is a root of f(x) if f(x) = 0 The Attempt at a Solution There is lots of work done for this problem when n=1, and also lots of work done when...
22. ### Numerical approximation of the solution

u''(x)=f(x), boundary conditions u(a)=0,u(b)=0. (u(x+h)-2u(x)+u(x-h))/h^2=f(x); maltab code: clear all a=0; b=1; n=10; h=(b-a)/(n+1); x_with_boundary=linspace(a,b,n+2)'; x=x_with_boundary(2:n+1); A=h^(-2).*(diag(ones(1,n-1),-1)+diag(-2.*ones(1,n),0)+diag(ones(1,n-1),1))...
23. ### Numerical approximation: Forward diffrerence method

Homework Statement [U][t]=-U+k[U][xx] u(x,0)=U(L,0)=0 u(x,0)sin(pix/L) Write down difference equations for the approximate solution of this problem using the following methods: 1)forward difference 2)backward difference 3)crank nicholson Homework Equations I can do...
24. ### Third order differential equation numerical approximation

Homework Statement There is a fluid flowing over a hot plate. We non-dimensionalized the problem from three partial diff eq's to two ode's. I am modeling I have two coupled differential equations that are a system of initial value problems. I am supposed to numerically integrate the two...