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1. Finite Differences
Point Group Symmetry Up Vibrational Spectra Previous Prerequisites Contents Index Finite Differences The simplest way to calculate the vibrational spectrum is by finite
http://www.cpmd.org/manual/node95.html

Extractions: Next: Point Group Symmetry Up: Vibrational Spectra Previous: Prerequisites Contents Index The simplest way to calculate the vibrational spectrum is by finite differences via the keyword VIBRATIONAL ANALYSIS . This type of calculation supports that largest variety of system setups. The corresponding section can look like this. The flag GAUSS tells CPMD to produce a fake Gaussian type output file, VIB1.log , which can be used for visualization with programs like Molden, Molekel, or gOpenMol. After the initial wavefunction optimization (which will take only one step since we start here from an already optimization wavefunction), every atom is displaced in positive and negative x-, y-, and z-direction by a small distance and the gradient computed. At the end the resulting dynamical matrix is diagonalized and the resulting vibrational spectrum in harmonic approximation is computed. Unless a very tight geometry optimization was performed, a few very low frequency modes will appear. Since in our case we didn't optimize, these frequencies are fairly large, and there are several imaginary (=negative) frequencies as indication of the geometry not being fully optimized. For higher accuracy of the results a **************************************************************** HARMONIC FREQUENCIES [cm**-1]: -209.8467 -102.0840 -58.0525 -37.5732 42.6296 214.9993 1581.2523 3712.7275 3821.3442 PURIFICATION OF DYNAMICAL MATRIX **************************************************************** HARMONIC FREQUENCIES [cm**-1]: -0.0001 -0.0001 0.0000 0.0000 0.0000 0.0001 1621.5287 3721.6537 3814.9509

2. Epic Consulting Services, A Division Of Baker Hughes Canada Company
Worldwide reservoir engineering and characterization. Both finite difference and streamline simulation.
http://www.epiccs.com/

3. Finite Difference -- From Wolfram MathWorld
Spiegel, M. Calculus of Finite Differences and Differential Equations. New York McGrawHill, 1971. Stirling, J. Methodus differentialis, sive tractatus de summation et
http://mathworld.wolfram.com/FiniteDifference.html

Extractions: Finite Difference The finite difference is the discrete analog of the derivative . The finite forward difference of a function is defined as and the finite backward difference as The forward finite difference is implemented in Mathematica as DifferenceDelta f i If the values are tabulated at spacings , then the notation is used. The th forward difference would then be written as , and similarly, the th backward difference as However, when is viewed as a discretization of the continuous function , then the finite difference is sometimes written where denotes convolution and is the odd impulse pair . The finite difference operator can therefore be written An th power has a constant th finite difference. For example, take and make a difference table The column is the constant 6. Finite difference formulas can be very useful for extrapolating a finite amount of data in an attempt to find the general term. Specifically, if a function is known at only a few discrete values , 1, 2, ... and it is desired to determine the analytical form of

4. Finite Differences In Inhomogeneous Media
Finite Differences in Inhomogeneous Media Atomic, Solid State, Comp. Physics discussion

5. FISHPAK
A library for the direct solution of finite difference approximations to two-dimensional Helmholz equations. Includes documentation and source code.
http://www.cisl.ucar.edu/softlib/FISHPAK.html

Extractions: FISHPAK is a free, portable library for the direct solution of finite difference approximations to two dimensional Helmholz equations in Cartesian, polar, cylindrical, interior spherical coordinates, and surface spherical coordinates, with various combinations of periodicity, normal derivative, or solution of the boundaries of a regular domain. Versions of these two dimensional codes are provided for both standard and staggered grids. Additionally, FISHPAK provides two routines for solving more general two dimensional separable elliptic equations, and one routine for solving a system of linear equations resulting from the discretization of a three dimensional separable elliptic equation. Recommendation: Use FISHPAK if your application requires direct solvers and free software. Otherwise use CRAYFISH . If you need an iterative solver, consider using MUDPACK Usage documentation is available in SCD's FISHPAK source directory , as comment lines in the Fortran source files. Overview documentation is available in the README file in the FISHPAK source directory. You may obtain Fortran source code from the FISHPAK source directory.

6. Finite Difference Method - Wikipedia, The Free Encyclopedia
One way to numerically solve this equation is to approximate all the derivatives by finite differences. We partition the domain in space using a mesh x 0, , x J and in time using a
http://en.wikipedia.org/wiki/Finite_Difference_Method

Extractions: From Wikipedia, the free encyclopedia   (Redirected from Finite Difference Method Jump to: navigation search Not to be confused with "finite difference method based on variation principle" , the first name of finite element method In mathematics finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives. Finite-difference methods approximate the solutions to differential equations by replacing derivative expressions with approximately equivalent difference quotients . That is, because the first derivative of a function f is, by definition, then a reasonable approximation for that derivative would be to take for some small value of h. In fact, this is the forward difference equation for the first derivative. Using this and similar formulae to replace derivative expressions in differential equations, one can approximate their solutions without the need for calculus. Assuming the function whose derivatives are to be approximated is properly-behaved, by

7. Mathematical Modeling – Finite Differences
Mathematical Modeling – Finite Differences . Section 2.2 . Charles Babbage (England, 1821) created a forerunner of the computer called the Difference Engine.
http://www.math.wvu.edu/~rmayes/2.2 Finite Differences.ppt

8. Meep - AbInitio
A free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems, along with the MPB eigenmode package.
http://ab-initio.mit.edu/wiki/index.php/Meep

Extractions: var skin = 'monobook';var stylepath = '/wiki/skins'; Jump to: navigation search Meep (or MEEP ) is a free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems, along with our MPB eigenmode package. Its features include: Meep Download Release notes FAQ Meep manual ... Acknowledgements Free software under the GNU GPL Simulation in , and cylindrical coordinates. Distributed memory parallelism on any system supporting the MPI standard. Portable to any Unix-like system ( GNU/Linux is fine). Arbitrary anisotropic dispersive nonlinear conductivities PML absorbing boundaries and/or perfect conductor and/or Bloch-periodic boundary conditions. Exploitation of symmetries Complete scriptability Scheme scripting front-end (as in libctl and MPB ), or callable as a C++ library; a Python interface is also available. Field output in the standard scientific data format, supported by many visualization tools. Arbitrary material and source distributions. Field analyses including flux spectra, frequency extraction, and energy integrals; completely programmable. Multi-parameter optimization, root-finding, integration, etcetera (via

9. Finite Difference
I suppose I should find the 3rd degree polynomial (y=ax 3 +bx 2 +cx+d) that fits the above data. There are numerous methods. I will just add and subtract multiples of equations
http://www.jimloy.com/algebra/finite.htm

Extractions: x=6, y=305 And you would like to know which function fits those values. One possibility is an n-degree polynomial: y=ax +bx +cx +dx +ex +fx+g, for example. You could actually plug the above x and y values into this equation. Then you would have seven linear equations (like 1=64a+32b+16c+8d+4e+2f+g) with seven unknowns. And there are a few fairly easy ways to solve them, to get a, b, c... A valuable short-cut is called the Finite Difference method. We take the numbers in the table, and find their differences (between consecutive elements), then we find the differences between the differences, etc: x y diff1 diff2 diff3 diff4 5 -5 1 6 1 12 2 1 18 19 12 3 20 30 49 12 4 69 42 91 12 5 160 54 145 6 305 It can be shown that for an n-degree polynomial, the nth difference is constant (and the (n+1)th difference is 0). So our function is

10. Levenberg-Marquardt In C/C++
Package containing double and single precision flavors of the Levenberg-Marquardt algorithm. Included are versions with analytic and finite difference approximated jacobians. Open source, GPL
http://www.ics.forth.gr/~lourakis/levmar/

11. Finite Differences Tutorial
A Limited Tutorial on Using Finite Differences in Soil Physics Problems written by Donald L. Baker reviewed by H. Don Scott. home This and the soil physics tutorial section
http://www.aquarien.com/findif/Findifa4.html

Extractions: A Limited Tutorial on Using home This and the soil physics tutorial section have been the most popular sections on this site. This is a brief and limited tutorial in the use of finite difference methods to solve problems in soil physics. It is meant for students at the graduate and undergraduate level who have at least some understanding of ordinary and partial differential equations. After an explanation of how to use finite differences in cook-book fashion, the equations, computer code and graphic results are given for three examples: heat flow, infiltration and redistribution, and contaminant transport in a steady-state flow field. Often, for problems of heat flow, or unsaturated water flow or contaminant transport in soil, there may be no analytic solutions or neat equations describing the result. In such cases, we use numerical methods on a computer. Perhaps the simplest of the numerical methods to understand and to program are finite differences, derived from Taylor series expansions (DuChateau and Zachmann, 1989). Some methods are so simple, they can even be done in a spreadsheet. But in the interests of accuracy, we will only discuss the methods that require some ability to program in a computer language such as C, BASIC or FORTRAN. The examples here given will be in FORTRAN, but can be converted to other languages. Because they are often confusing to the neophyte, we will not discuss Taylor series derivations. Finite differences can be explained and used in cook-book manner, if one is careful. If the reader has no other experience in these methods, he or she should keep in mind that this is a limited discussion. Such issues as stability, convergence, iteration methods, implicitness, discretization errors and non-Darcian flow will not be covered. So the reader should be careful to understand that a great deal more study is necessary to use these methods successfully in many cases. This is only a brief synopsis.

12. Migration By Finite Differences
WAVEEXTRAPOLATION EQUATIONS Up Table of Contents Migration by finite differences In the last chapter we learned how to extrapolate wavefields down into the earth.
http://sepwww.stanford.edu/sep/prof/iei/omx/paper_html/node1.html

Extractions: Next: WAVE-EXTRAPOLATION EQUATIONS Up: Table of Contents Migration by finite differences In the last chapter we learned how to extrapolate wavefields down into the earth. The process proceeded simply, since it is just a multiplication in the frequency domain by .Finite-difference techniques will be seen to be complicated. They will involve new approximations and new pitfalls. Why should we trouble ourselves to learn them? To begin with, many people find finite-difference methods more comprehensible. In ( t x z )-space, there are no complex numbers, no complex exponentials, and no ``magic'' box called FFT. The situation is analogous to the one encountered in ordinary frequency filtering. Frequency filtering can be done as a product in the frequency domain or a convolution in the time domain. With wave extrapolation there are products in both the temporal frequency -domain and the spatial frequency k x -domain. The new ingredient is the two-dimensional -space, which replaces the old one-dimensional -space. Our question, why bother with finite differences?, is a two-dimensional form of an old question: After the discovery of the fast Fourier transform, why should anyone bother with time-domain filtering operations? Our question will be asked many times and under many circumstances. Later we will have the axis of offset between the shot and geophone and the axis of midpoints between them. There again we will need to choose whether to work on these axes with finite differences or to use Fourier transformation. It is not an all-or-nothing proposition: for each axis separately either Fourier transform or convolution (finite difference) must be chosen.

13. Package FD
Fortran 77 package by Jiri Zahradnik for 2-D P-SV elastic second-order finite differences.
http://seis.karlov.mff.cuni.cz/software/sw3dcd5/fd/fd.htm

Extractions: klimes@seis.karlov.mff.cuni.cz Package FD contains programs for 2-D P-SV elastic second-order finite differences. See file fdver.htm for the list of released versions and changes made in this version. Package FD employs package FORMS for unified memory management and unified compilation All Fortran 77 source code and include files of the FD package are assumed to be located in a single directory together with all source code and include files of the FORMS and MODEL packages when being compiled and linked. The files with main programs contain, at their ends, Fortran 90 INCLUDE command for all subroutine files required. In this way, each program may simply be compiled and linked as a single file. All filenames are assumed to be expressed in lowercase.

14. Finite Difference: Definition From Answers.com
Boole, George, A Treatise On The Calculus of Finite Differences, 2 nd ed., Macmillan and Company, 1872. See also Dover edition 1960. Levy, H.; Lessman, F. (1992).

15. Theory Of Finite Differences
Theory of Finite Differences Given a sequence that follows a distinctive pattern (the pattern does not have to be known, in fact, the pattern, or formula, is what we are
http://jwilson.coe.uga.edu/EMT668/EMAT4680.2000/Massey.Walt/emat4690/Problems/Fi

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16. FISHPACK
Collection of Fortran subprograms which utilize cyclic reduction to directly solve second- and fourth-order finite difference approximations to separable elliptic PDEs in a variety of forms.
http://www.cisl.ucar.edu/css/software/fishpack/

Extractions: Roland Sweet A modernization of FISHPACK 4.0 is available at Details of the changes FISHPACK is incompatible with FISHPACK90. (figure provided by Jacque Marshall) This document consists of three parts: FISHPACK Introduction FISHPACK Solver Description Obtaining FISHPACK Software FISHPACK is a collection of FORTRAN subprograms that use cyclic reduction to directly solve second- and fourth-order finite difference approximations to separable elliptic Partial Differential Equations (PDEs) in a variety of forms. (See MUDPACK for software that uses multigrid iterative techniques to approximate separable and nonseparable elliptic PDEs). FISHPACK includes solvers for the Helmholtz equation in Cartesian, Polar, Cylindrical, and Spherical coordinates as well as solvers for more general separable elliptic equations. Provisions are made to handle coordinate-system-induced singularities (e.g., at the origin r=0 in cylindrical coordinates and at the poles in spherical coordinates). Solutions obtained in the least-squares sense are computed for singular problems. The following table summarizes the contents of FISHPACK. Descriptions can be obtained by clicking on the solver name. Descriptions of the PDEs solved are also included after the table.

17. Finite Differences — The Endeavour
If f(x) is a function on integers, the forward difference operator is defined by. For example, say f(x) = x 2. The forward difference of the sequence of squares 1, 4, 9, 16
http://www.johndcook.com/blog/2009/02/01/finite-differences/

Extractions: The Endeavour The blog of John D. Cook by John on February 1, 2009 If f(x) is a function on integers, the forward difference operator is defined by For example, say f(x) = x There are many identities for the forward difference operator that resemble analogous formulas for derivatives. For example, the forward difference operator has its own product rule, quotient rule, etc. These rules are called the calculus of finite differences. The finite results are often much easier to prove than their continuous counterparts. integration by parts for integrals. This is quite useful technique and I intend to write a separate post on it. The product rule for forward differences looks a little odd: The left hand side is symmetric in f and g though the right side is not. There is also a symmetric version: Here is the quotient rule for forward differences. One of the first things you learn in calculus is how to take the derivative of powers of x. If f(x) = x n n-1 . There is an analogous formula in the calculus of finite differences, but with a different kind of power of x. For positive integers n, define the n th falling power of x by Then Falling powers can be generalized to non-integer exponents by defining The formula for finite difference of falling powers given above remains valid when using the more general definition of falling powers. Falling powers arise in many areas: generating functions, power series solutions to differential equations

18. Note On Nonsymmetric Finite Differences*for Maxwell’s Equations
Journal of Computational Physics 161, 723727 (2000) doi10.1006/jcph.2000.6524, available online at http//www.idealibrary.comon NOTE Note on Nonsymmetric Finite Differences
http://www.math.udel.edu/~driscoll/pubs/nonsymmetricFD.pdf

19. Airfoil Software
Windows software for analyzing airfoils using CFD and analytical techniques. VisualFoil can export an O-Mesh for finite difference/volume CFD analysis. On-line purchase.
http://www.hanleyinnovations.com/airfoils.html