What is Numerical analysis: Definition and 119 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.
TL;DR Summary: I am taking a math methods course this semester of which a large part are PDEs. I don't understand the context behind the order in which we are solving PDEs. I am interested in learning how other people were taught PDEs and any book recommendations you might have.
I am taking a...
Relevant equations ( a quote from the book A Concise Introduction to Numerical Analysis by A. C. Faul explaining what is floating point representation ):
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I've been reading A Concise...
I've started reading A concise introduction to numerical analysis, A.C. Foul and on the first page there's the following paragraph about how a floating point in fixed point precision can be represented:
I don't understand the example where it says " ##\beta=2## and ##p=20##, the decimal number...
The image attached above from a textbook, explains that we should refrain from using (a+b)/2 while applying Bisection method but I am unable to get the reason why it is asking to do so?
While the image above is from another textbook. This book uses (a+b)/2.
I am really confused by two different...
I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?).
For example sin(x) for [0,1], n with trapezoidal rule = 0.420735...
With my own...
Hello !
My name is Amine, Renewable Energies Student (Thermal Conversion Masters degree) .
interested in numerical analysis and applied mathematics .
Currently focusing on Heat and Mass Transfer , Fluid Mechanics and Matlab programming. ready to efficiently help on those subjects
please...
Hey! I have been thinking about different topics on mathematics, and for some reason I feel, that applied mathematics and doing numerical computations is often overlooked.
When I was studying in university, more than 95 % of all examples which would require numerical methods were either omitted...
Ben Sparks uses https://www.GeoGebra.org to explain (and code) the so-called SIR Model being used to predict the spread of cornavirus (COVID-19). A pretty cool way to visualize a set of differential equations.
Summary:: figuring the 'realmax' and 'realmin' from a hypothetical distribution of bits
I am attempting to figure the 'realmax' and the 'realmin' from this hypothetical distribution of bits.
I know the largest the mantissa can be is (1-eps), and the smallest is just 1/2. Also I note the...
I developed a program to solve the nonlinear system below through the method of successive approximations and was only able to find one solution, namely ##x_1 = 0.93377## and ##x_2 =0.88417##, even though I tried many different starting points. I was wondering if there's a way to determine if...
I have lately been working with Numerical Analysis and I am using Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque. It was recommended to me by a friend of mine (physicist)
https://epubs.siam.org/doi/book/10.1137/1.9780898717839?mobileUi=0&...
I am doing DMRG (in traditional formalism, not MPS) for Hubbard model
H = -t ∑i ∑σci,σ ci+1,σ + U∑ini,σni,σ-
In every iteration we add two sites to the system, but how do we set that how many particles are allowed in the system?
Hi PF,
Suppose I numerically solve a nonlinear system of differential equations. How can I know if my solution is correct (if there is no known analytic solution)? What are the standard practices people do? I have a couple of ideas, but I want to know what people are already doing.
Danke!
Hello all,
I may get a contract to teach numerical analysis. I did quite a lot of numerical work during my PhD but that was a while ago. Now when I look at most books on the topic, I get the feeling that a lot is outdated, and I also feel that a lot of what I knew is outdated as well...
Actually this is more of a simulation question but since PF doesn't have Simulation category I ask here.
I need to simulate a path from a proces given by this Stochastic DE:
$$ dX_t = -a(X_t-1)dt+b\sqrt{X_t}dB_t $$ where ##B_t## is wiener process/brownian motion and a and b are just some...
Hey Guys, Trying to figure out how to replicate the following from an article, but can not understand their notations;
The main points are:
The bounce action can be written as the equation
$$\left( n^2 \Theta^2 + 2\alpha n\Theta -1\right)R_n = 2 \sum_{m=1}^\infty R_{n+m}R_m + \sum_{m=1}^n...
Homework Statement
Determine the Finite Difference Method stencil for approximating a second derivative u''(x) at a discrete set of nodes with maximum accuracy for stencil of sizes (0,4) (off-centered).
My questions:
I think I am able to answer the question I am just not sure about what is...
So in my future numerical analysis class the recommended book i liked the most was Numerical mathematics by Quarteroni :
https://books.google.pt/books/about/Numerical_Mathematics.html?id=m-bHBAAAQBAJ&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false
Since i like this...
Hello,
Is there a difference from these courses, or are they the same course with different names? I need to know which one to choose for the upcoming semester...
Intro to Analysis, Intro to Real Analysis I, and Numerical Analysis
Thank you,
Tracie
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...
Hi,
I am new to spectral methods, does anyone have a good reference which shows some worked examples of formulating the method to be computed?
E.g. go from the general governing equations through transformation to the form which will be passed to a computer.
Cheers
https://www.amazon.com/dp/1305253663/?tag=pfamazon01-20
Are all these called root finding algorithms? All of it ?
<Moderator note: Link changed because of undetermined copyright situation. fresh_42>
Hi there,
This is a question about numerical analysis used particularly in the computational condensed matter or anywhere where one needs to DIAGONALIZE GIGANTIC DENSE HERMITIAN MATRICES.
In order to diagonalize dense Hermitian matrices size of 25k-by-25k and more (e.g. 1e6-by-1e6) it is not...
Given the two systems below for an ##m \times n## matrix ##A##:
(Sy 1): ##Ax = 0, x \geq 0, x \neq 0##
(Sy 2): ##A^Ty > 0 ##
I seek to prove: (Sy 1) is consistent ##\Leftrightarrow## (Sy 2) is inconsistent.
I figured out how to prove Q ##\Rightarrow ## P by proving the contrapositive...
A question for those who are computational physicist:
Dear Computational physicist,
I am struggling between computational physics course or numerical analysis. They are both in graduate level (so very intensive), one in physics department and another in math. Both are taught by leading experts...
Hello.
A whole decade passed since I graduated mathematics and shifted to other profession, so my knowledge is very rusty.
There is an important problem for a scientific work that I need help for.
Let's say factor t is being calculated from factors x, y and z, all some parameters from living...
I'm trying to plot the solutions of the second order differential equation d^2R/dt^2 = GM/R^2 + Lz^2/R^3. I'm reducing this to a system of first order ODEs and then using RK4 to solve this system.
My code is given by
function RK4system()
Tsim = 10...
I am currently working through Spivak, and have finished Geometry by Moise. I wanted to learn numerical analysis on my own. I could take it at the university, but I rather use the elective for differential geometry.
What are good books on Numerical Analysis?
I have completed 2 books in Linear...
Consider a sample consisting of {y1,y2,...,yk} realisations of a random variable Y, and let S(k) denote the variance of the sample as a function of its size; that is
S(k)=1/k( ∑ki=1(yi−y¯)2)
for y¯=1/k( ∑ki=1 yi)
I do not know the distribution of Y, but I do know that S(k) tends to zero as k...
I know to describe Quantum Mechanical systems we can use:
-Schrodinger equation
-Feynman Path Integral method
-Heisenberg Matrix formulation
Well my question is, if you want to calculate molecular properties, and want to understand biochemistry (protein), you have a system with several...
hey,
need an idea for a paper on differential equations, which can be implemented in two or three days work.
also it will be nice if that paper has graphs on it.
any help would be most welcome
thx
Hi guys,
I've distilled the 3D Diffusion Equation into the following PDE using Fourier spectral techniques:
∂C(m,n,p,t)/∂t + k(p^2+m^2+n^2)C(m,n,p,t)=0,
where C is the Fourier coefficient of the 3D Fourier transform, {m,n,p} are the spatial frequencies, and t is time. I've tried using a...
Hi guys,
I have functioning MATLAB code for my solution of the 3D Diffusion equation (using a 3D Fourier transform and Crank-Nicolsen) that runs just from the command window and automatically plots the results. However, it seems like my solution just decays to zero regardless of what initial...
I am modelling a 1D fluid wave propagation problem and I needed to know how I can check that my results are energy conservative. please reply, urgent. thank you.
Could someone please show me some examples how IEEE single precision format can used to store approximately 7 decimal digits of a number when the number is written in decimal format? How this is related to the statement ##2^{-23}=1.19\times10^{-7}##?
http://www.cpp.edu/~jlcannons/mat_201_2015_spring/Project_01.pdf
I don't know if this counts as cheating, but I hope someone could look this over and see what my professor is ask.
Ok, I've built a numerical model to show the cooling of hot magma sills entered into the crust over time. The results show that the volume of the "hot" zone when the emplacement of a constant volume of hot sills is all done will vary as a matter of two things: the overall rate at which the magma...
In two semesters or so, i will be taking a numerical analysis course as part of an Applied Mathematics major. In the course description it says that you should have knowledge of at least one programming language, but i cannot find any more information about this. What do you all think would be...
This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction t and one spatial direction z and the numerical method is formulated by stepping forward in time. The problem is that I obtain instabilities, either at...
Hi - I have just started 'Computational Physics' by Koonin & Meredith, - through distance learning.
Exercise 1.3 needs a program to evaluate an integral - I'm finding myself kinda rusty on integrals. The hint says - split range of integration into parts, use different change of variable in each...
I have simplified my problem here, in my actual problem the matrix is much bigger which makes it impossible to find eigenvalues analytically. So, I chose standard BCS problem (2x2 matrix) to demonstrate my problem with Mathematica. But, with even this simple case, it takes forever to integrate...
Chebyshev polynomials and Legendre polynomials are both orthogonal polynomials for determining the least square approximation of a function. Aren't they supposed to give the same result for a given function?
I tried mathematica but the I didn't get the same answer :( Is this precision problem or...
Two criteria
1. Freely Available
2. Large support for additional scientific libraries/packages/modules for reuse
http://en.wikipedia.org/wiki/Comparison_of_numerical_analysis_software
Which one would you recommend and why?
Hello
I am having some trouble understanding what the difference is between the reference solution and the "true" solution of a set of certain differential equation. The book I'm working with is Numerical Analysis by Walter Gautschi. Equation 5.16 from this book is a set of first order...
Homework Statement
Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6.
*To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use.
Homework Equations
x = 45 or pi/4, x0 = 42 or 7pi/30
cos(x) = Pn(x) + Rn(x)...
Sorry, but this is the only subject I could not pass even if I gave it my all every day and night of the semester. And I will still surely fail this subject, but as a last resort I will try to post my problem here, hoping to get solution and maybe an explanation. Sorry if some of the phrasing...
Sorry, but this is the only subject I could not pass even if I gave it my all every day and night of the semester. And I will still surely fail this subject, but as a last resort I will try to post my problem here, hoping to get solution and maybe an explanation. Sorry if some of the phrasing...
If I have a x,y table of discrete datapoints with a discrete dataset, such that delta x is not a constant, what are some of the more advanced techiques that I can use to integrate this?
I remember that there were Trapezoidal rules and Simpson's rule where delta x IS a constant (and there are...
Hello, using octave when I evaluate this expression (1+1/x)*x-x, with x = 10^(15) i get as result 1.125, I didn't undesrtood why,
I know that octave show 15 digits in format long so when i evaluate (1+1/x)*x i get 1e15, the last digit is not visible and it is 1, so when i do substraction i get...