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.

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  1. Philip551

    Looking for resources to help me understand the basics of PDEs for physics

    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...
  2. C

    Number of bits required for floating point representation

    Relevant equations ( a quote from the book A Concise Introduction to Numerical Analysis by A. C. Faul explaining what is floating point representation ): __________________________________________________________________________________________________________ I've been reading A Concise...
  3. C

    How does one convert a decimal float to binary using the multiply by 2 method?

    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...
  4. CandidFlakes

    I Why is (a+b)/2 not recommended in the Bisection method?

    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...
  5. Graham87

    Numerical Analysis - Richardson Extrapolation on Riemann Sum

    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...
  6. S

    Interested in Renewable Energy and Numerical Analysis? Join our Forum Community!

    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...
  7. Theia

    MHB Would you read a blog about numerical analysis?

    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...
  8. The Coronavirus Curve Simulation - Numberphile

    The Coronavirus Curve Simulation - Numberphile

    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.
  9. K

    Determining 'realmax' and 'realmin'

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  10. W

    I Number of Solutions of a Nonlinear System

    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...
  11. T

    A Lecture notes on Finite Difference Methods

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  12. L

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    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?
  13. M

    I How can you know if a numerical solution is correct?

    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!
  14. N

    Your thoughts on numerical analysis teaching

    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...
  15. E

    A Simulation from a process given by "complicated" SDE

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  16. M

    A Successive iteration problem in quantum dissipation article

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  17. T

    Finite difference method; finding the 'Stencil'

    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...
  18. J

    Applied Numerical Analysis book question

    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...
  19. T

    Intro to analysis, intro to real analysis I, numerical analysis

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  20. PhysicsKid0123

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    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...
  21. R

    A How can I formulate spectral methods for numerical computation?

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  22. awholenumber

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  23. B

    Diagonalization of Gigantic Dense Hermitian Matrices

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  24. C

    MHB Numerical Analysis problem Newton's method

    Can anyone help in the solution of this problem? how can i determine the zero x*??
  25. K

    Prove One but not both of these systems is consistent.

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  26. T

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  27. M

    Get equation that describes set of measured values

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  28. J

    MATLAB Solving Orbital Motion Equations with RK4 Using MATLAB

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  29. MidgetDwarf

    Courses Good Books for Numerical Analysis & Mathematica for Learning

    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...
  30. E

    A Measuring the degree of convergence of a stochastic process

    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...
  31. J

    A What is the best Quantum Model to solve many body problems?

    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...
  32. E

    Project idea in Numeric Analysis

    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
  33. J

    Solving the 3D Diffusion Equation with Fourier Spectral Techniques

    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...
  34. J

    MATLAB 3D Diffusion Equation in MATLAB

    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...
  35. H

    How to check a numerical simulation is energy conservative?

    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.
  36. S

    How Does IEEE Single Precision Format Store 7 Decimal Digits?

    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}##?
  37. M

    MHB Numerical Analysis: Project 01 | CPP

    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.
  38. PinkGeologist

    Numerical modeling of hot magma

    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...
  39. O

    Numerical Analysis Programming Language

    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...
  40. K

    How to improve stabilities of numerical solutions of PDEs

    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...
  41. ognik

    Tricky Intregral for numerical quadrature

    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...
  42. S

    Mathematica Mathematica: Speed up NIntegrate

    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...
  43. ch3cooh

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  44. Q

    Best Numerical Analysis Software for Physics

    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?
  45. J

    Intuitive understanding of the reference solution (numerical analysis)

    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...
  46. M

    Taylor Polynomials and Numerical Analysis

    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)...
  47. T

    MHB Numerical Stability of Difference Schemes for Differential Operators

    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...
  48. T

    Numerical analysis homework

    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...
  49. A

    Advanced numerical analysis - numerical integration

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  50. B

    Evaluating expression (numerical analysis)

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