Forces: Calculating Them, and Using Them Shobhana Narasimhan JNCASR, Bangalore, India

Size: px
Start display at page:

Download "Forces: Calculating Them, and Using Them Shobhana Narasimhan JNCASR, Bangalore, India"

Transcription

1 Forces: Calculatig Them, ad Usig Them Shobhaa Narasimha JNCASR, Bagalore, Idia Shobhaa Narasimha, JNCASR 1

2 Outlie Forces ad the Hellma-Feyma Theorem Stress Techiques for miimizig a fuctio Geometric optimizatio usig forces Froze Phoo Calculatios Molecular Dyamics with forces Shobhaa Narasimha, JNCASR 2

3 The Bor-Oppeheimer Approimatio Ma Bor ( R. Oppeheimer ( Shobhaa Narasimha, JNCASR 3

4 The Bor-Oppeheimer Approimatio The may-particle wavefuctio ψ is a fuctio of both uclear ad electroic coordiates. Separate out the Hamiltoia ito a uclear ad electroic part. The wavefuctio ca the be writte as: Ψ tot =ψ ψ uc elec OK because electros are much lighter tha uclei ad therefore respod istatly o time-scale of uclear motio. Shobhaa Narasimha, JNCASR 4

5 Forces Need for geometry optimizatio ad molecular dyamics. Ca also use to get phoos. Could get as fiite differeces of total eergy - too epesive! Use force (Hellma-Feyma theorem. Richard Feyma s Seior Thesis! (whe he was 21 Shobhaa Narasimha, JNCASR 5

6 Hellma-Feyma Theorem Wat to calculate force o io I: Get three terms: Whe is a eigestate, -Substitute this Shobhaa Narasimha, JNCASR 6

7 Hellma-Feyma Theorem Wat to calculate force o io I: Get three terms: Whe is a eigestate, -Substitute this Shobhaa Narasimha, JNCASR 7

8 Hellma-Feyma Theorem Wat to calculate force o io I: Get three terms: Whe is a eigestate, -Substitute this Shobhaa Narasimha, JNCASR 8

9 Hellma-Feyma Theorem (cotd. The force is ow give by Note that we ca ow calculate the force from a calculatio at ONE cofiguratio aloe huge savigs i time. If the basis depeds o ioic positios (ot true for plae waves, would have etra terms = Pulay forces. If is ot a eact eigestate (electroic calculatio ot well coverged, may get big errors i forces calculated usig this prescriptio! Shobhaa Narasimha, JNCASR 9

10 Hellma-Feyma Theorem (cotd. The force is ow give by Note that we ca ow calculate the force from a calculatio at ONE cofiguratio aloe huge savigs i time. If the basis depeds o ioic positios (ot true for plae waves, would have etra terms = Pulay forces. If is ot a eact eigestate (electroic calculatio ot well coverged, may get big errors i forces calculated usig this prescriptio! Shobhaa Narasimha, JNCASR 10

11 Hellma-Feyma Theorem (cotd. The force is ow give by Note that we ca ow calculate the force from a calculatio at ONE cofiguratio aloe huge savigs i time. If the basis depeds o ioic positios (ot true for plae waves, would have etra terms = Pulay forces. If is ot a eact eigestate (electroic calculatio ot well coverged, may get big errors i forces calculated usig this prescriptio! Shobhaa Narasimha, JNCASR 11

12 Usig H-F Theorem i a (plae-wave DFT calculatio Force o io I give by: where =(pseudopotetial due to io cores ad = iteractio of ios with each other. Shobhaa Narasimha, JNCASR 12

13 Stress Strai: Stress: Stress Theorem (Nielse & Marti, 1985 as for forces, ca calculate at a sigle cofiguratio. What if the primitive lattice vectors (specifyig uit cell are ot optimal? - Forces o atoms may = 0 (e.g., a FCC crystal with wrog lattice costat - Stress will ot be zero, however. < 0 cell would like to epad. > 0 cell would like to cotract. Shobhaa Narasimha, JNCASR 13

14 Geometric Optimizatio Wat to move the atomic positios aroud util the lowest-eergy equilibrium cofiguratio is obtaied. At equilibrium, for all I. We are searchig for a miimum i a 3N I -dim space. Fidig global miimum is hard! Will usually ed up i earest miimum Shobhaa Narasimha, JNCASR 14

15 Miimizatio Relevat to may parts of DFT calculatios: - Optimizig ioic positios - Miimizig eergy fuctioal - Diagoalizig Hamiltoia - Achievig self-cosistecy (miig A brief foray ito umerical methods. Covetio: subscripts coordiates, superscripts iteratios Shobhaa Narasimha, JNCASR 15

16 Miimizatio i 1-D usig gradiets Cosider a fuctio f(; we wat to fid 0, the value of where the fuctio has its miimum value. Iterative methods: successive approimatios 1, 2, 3, 0 Ca take several small dowhill steps, i the directio opposite the gradiet f (. = β '( + 1 ( f Might take a log time to coverge. Note: eed first derivatives Shobhaa Narasimha, JNCASR 16

17 Fidig zeroes i 1-D: the Newto-Raphso Method Use iformatio from first derivatives to make a series of guesses: f f ( 0 '( = + 1 f( + 1 = f ( f '( Note: will coverge i oe step if f is liear +1 Shobhaa Narasimha, JNCASR 17

18 Fidig miimum i 1-D: the Newto-Raphso Method Lookig for a miimum i f is equivalet to lookig for a zero of f. So replace f by f ad f by f to get: or sometimes: = = f '( f "( f '( β f "( Note: eed first derivatives ad secod derivatives Note: will coverge i oe step if f is liear. So will coverge i oe step if f is quadratic. Shobhaa Narasimha, JNCASR 18

19 Miimizatio i a N-d space Cosider a fuctio f( of N variables = 1, 2,, N The gradiet is f(. Note that at a give, this poits i directio of maimum icrease of f(. Wat to fid 0 s.t. f( has its miimum value at 0, i.e., f( 0 = 0. Will fid iteratively, through a sequece of poits i the N-dimesioal space that are steppig stoes to fidig 0. Covetio: subscripts coordiates, superscripts iteratios. Shobhaa Narasimha, JNCASR 19

20 Steepest Descet Keep goig dowhill i directio opposite local gradiet. Could try takig lots of little dowhill steps: +1 = β f( = + βg Always, g perpedicular to g +1. Better: Oce directio g idetified, do lie miimizatio alog it. Ca be slow (may ot reach miimum! Problem: whe movig alog ew directio, lose some miimizatio alog old oe(s. Shobhaa Narasimha, JNCASR 20

21 Secod derivatives i a N-d space: The Hessia is a N N matri, whose elemets are give by: H ij = 2 f ( / i j It gives iformatio about the curvature of the fuctio. Shobhaa Narasimha, JNCASR 21

22 Newto-Raphso Miimizatio i N-d 1-d N-d f f( 1/f H = f '( β f "( = β [ H( ] f ( Need to compute gradiets ad iverse Hessias Shobhaa Narasimha, JNCASR 22

23 Quasi-Newto-Raphso Methods We had: = β [ H( ] f ( But H -1 may be hard/epesive to compute directly. Istead, build up estimates for H -1 by aalyzig successive gradiet vectors. Various prescriptios for doig this, e.g., Broyde method, BFGS method. Shobhaa Narasimha, JNCASR 23

24 Shobhaa Narasimha, JNCASR 24 BFGS Miimizatio Broyde, Fletcher, Goldfarb, Shao (1970. ( ] [ ( ] [ f f H H = + ρ [ ] [ ] [ ] [ ] [ ] = + T T T T T T T s s H H s s s s s H H H ( ( ( ( ( ( ( Trust radius Iverse Hessia approimated as: where ( ( 1 f f ad ( ] ( [ 1 f H s (No doubt you will agree that this is a good poit at which to ed our foray ito umerical methods Newto-Raphso

25 Back to the Problem of Ioic Relaatio Fuctio f to be miimized is total eergy E tot. Poits i 3N I -d space correspod to set of ioic coordiates ( 1, y 1, z 1, 2, y 2, z 2, NI, y NI, z NI. Gradiets f( correspod to set of 3 compoets of forces o the N I ios. Forces ca be computed usig Hellma-Feyma theorem. Now use a miimizatio scheme to fid the ioic positios that give the lowest value of E tot, which is also whe the forces o all ios are (close to zero. Shobhaa Narasimha, JNCASR 25

26 The Vibratioal Frequecy of a Diatomic Molecule Ca obtai from eergies or from forces: e.g., for CO: E = k + O( 2 F = k + O( 2 Saada Biswas Shobhaa Narasimha, JNCASR 26

27 Froze-Phoo Calculatios Take sapshots of a crystal as it is vibratig i a particular mode. Use Hellma-Feyma theorem to compute forces as a fuctio of displacemet. Force vs. displacemet gives phoo frequecy. Note: Ca istead use desity fuctioal perturbatio theory (later today, Stefao Baroi s lecture! Shobhaa Narasimha, JNCASR 27

28 The Force-Costat Tesor Φ α,β (R i -R j force iduced o atom j i directio β, upo movig atom i i directio α Ca be obtaied easily if forces ca be computed. Shobhaa Narasimha, JNCASR 28

29 Gettig Phoo Frequecies Dyamical Matri: ~ D( k = i k R e R Φ( R Diagoalize to get phoo frequecies: Det ~ ~ 2 [ D( k Mω ( k ] = 0 Shobhaa Narasimha, JNCASR 29

30 What else ca you do with forces? Newto s equatios of motio: F = m a = I I I m I 2 d R 2 dt I If forces ca be computed, ca itegrate these to get ioic positios as a fuctio of time. Molecular Dyamics (tomorrow Shobhaa Narasimha, JNCASR 30

31 Computig Forces & Stress with PWscf tprfor =.TRUE. (Set automatically if calculatio = rela or md or vc-md tstress =.TRUE. (Set automatically if calculatio = vc-md Shobhaa Narasimha, JNCASR 31

32 Forces Obtaied usig PWscf e.g., for a (urelaed poit defect (vacacy i graphee: Kacha Ulma Shobhaa Narasimha, JNCASR 32

33 Ioic Relaatio i PWscf Tell the program to carry out ioic relaatio, ad say which method to use to fid miimum &cotrol calculatio = rela. io_dyamics = bfgs &cotrol calculatio = md. io_dyamics = damp Shobhaa Narasimha, JNCASR 33

34 Ioic Relaatio i PWscf (cotd. Say which atoms are to be moved & i which directios. e.g., for a four-layer Al(001 slab: ATOMIC_POSITIONS Al Al Al Al Allow these atoms to move oly alog z Fi these atoms Shobhaa Narasimha, JNCASR 34

35 The Ed! May the Force Be With You. Shobhaa Narasimha, JNCASR 35

5.76 Lecture #33 5/08/91 Page 1 of 10 pages. Lecture #33: Vibronic Coupling

5.76 Lecture #33 5/08/91 Page 1 of 10 pages. Lecture #33: Vibronic Coupling 5.76 Lecture #33 5/8/9 Page of pages Lecture #33: Vibroic Couplig Last time: H CO A A X A Electroically forbidde if A -state is plaar vibroically allowed to alterate v if A -state is plaar iertial defect

More information

Chapter 5 Vibrational Motion

Chapter 5 Vibrational Motion Fall 4 Chapter 5 Vibratioal Motio... 65 Potetial Eergy Surfaces, Rotatios ad Vibratios... 65 Harmoic Oscillator... 67 Geeral Solutio for H.O.: Operator Techique... 68 Vibratioal Selectio Rules... 7 Polyatomic

More information

HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples:

HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples: 5.6 4 Lecture #3-4 page HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT Do t restrict the wavefuctio to a sigle term! Could be a liear combiatio of several wavefuctios e.g. two terms:

More information

Lecture #18

Lecture #18 18-1 Variatioal Method (See CTDL 1148-1155, [Variatioal Method] 252-263, 295-307[Desity Matrices]) Last time: Quasi-Degeeracy Diagoalize a part of ifiite H * sub-matrix : H (0) + H (1) * correctios for

More information

Notes on iteration and Newton s method. Iteration

Notes on iteration and Newton s method. Iteration Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said

More information

The Born-Oppenheimer approximation

The Born-Oppenheimer approximation The Bor-Oppeheimer approximatio 1 Re-writig the Schrödiger equatio We will begi from the full time-idepedet Schrödiger equatio for the eigestates of a molecular system: [ P 2 + ( Pm 2 + e2 1 1 2m 2m m

More information

1 Adiabatic and diabatic representations

1 Adiabatic and diabatic representations 1 Adiabatic ad diabatic represetatios 1.1 Bor-Oppeheimer approximatio The time-idepedet Schrödiger equatio for both electroic ad uclear degrees of freedom is Ĥ Ψ(r, R) = E Ψ(r, R), (1) where the full molecular

More information

v = -!g(x 0 ) Ûg Ûx 1 Ûx 2 Ú If we work out the details in the partial derivatives, we get a pleasing result. n Ûx k, i x i - 2 b k

v = -!g(x 0 ) Ûg Ûx 1 Ûx 2 Ú If we work out the details in the partial derivatives, we get a pleasing result. n Ûx k, i x i - 2 b k The Method of Steepest Descet This is the quadratic fuctio from to that is costructed to have a miimum at the x that solves the system A x = b: g(x) = - 2 I the method of steepest descet, we

More information

Lecture 25 (Dec. 6, 2017)

Lecture 25 (Dec. 6, 2017) Lecture 5 8.31 Quatum Theory I, Fall 017 106 Lecture 5 (Dec. 6, 017) 5.1 Degeerate Perturbatio Theory Previously, whe discussig perturbatio theory, we restricted ourselves to the case where the uperturbed

More information

ENGI 9420 Engineering Analysis Assignment 3 Solutions

ENGI 9420 Engineering Analysis Assignment 3 Solutions ENGI 9 Egieerig Aalysis Assigmet Solutios Fall [Series solutio of ODEs, matri algebra; umerical methods; Chapters, ad ]. Fid a power series solutio about =, as far as the term i 7, to the ordiary differetial

More information

PC5215 Numerical Recipes with Applications - Review Problems

PC5215 Numerical Recipes with Applications - Review Problems PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: 0 0 05 00 0 00 Note that it has 8 bits for the epoet,

More information

5.80 Small-Molecule Spectroscopy and Dynamics

5.80 Small-Molecule Spectroscopy and Dynamics MIT OpeCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy ad Dyamics Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture # 33 Supplemet

More information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t,

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t, Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio

More information

Physics 7440, Solutions to Problem Set # 8

Physics 7440, Solutions to Problem Set # 8 Physics 7440, Solutios to Problem Set # 8. Ashcroft & Mermi. For both parts of this problem, the costat offset of the eergy, ad also the locatio of the miimum at k 0, have o effect. Therefore we work with

More information

Chapter 2: Numerical Methods

Chapter 2: Numerical Methods Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

CS537. Numerical Analysis and Computing

CS537. Numerical Analysis and Computing CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be

More information

Introduction to Optimization Techniques. How to Solve Equations

Introduction to Optimization Techniques. How to Solve Equations Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually

More information

Statistical Inference Based on Extremum Estimators

Statistical Inference Based on Extremum Estimators T. Rotheberg Fall, 2007 Statistical Iferece Based o Extremum Estimators Itroductio Suppose 0, the true value of a p-dimesioal parameter, is kow to lie i some subset S R p : Ofte we choose to estimate 0

More information

Differentiable Convex Functions

Differentiable Convex Functions Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for

More information

Math 5311 Problem Set #5 Solutions

Math 5311 Problem Set #5 Solutions Math 5311 Problem Set #5 Solutios March 9, 009 Problem 1 O&S 11.1.3 Part (a) Solve with boudary coditios u = 1 0 x < L/ 1 L/ < x L u (0) = u (L) = 0. Let s refer to [0, L/) as regio 1 ad (L/, L] as regio.

More information

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients. Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,

More information

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka) 7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.

More information

Optimization Methods MIT 2.098/6.255/ Final exam

Optimization Methods MIT 2.098/6.255/ Final exam Optimizatio Methods MIT 2.098/6.255/15.093 Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short

More information

For example suppose we divide the interval [0,2] into 5 equal subintervals of length

For example suppose we divide the interval [0,2] into 5 equal subintervals of length Math 1206 Calculus Sec 1: Estimatig with Fiite Sums Abbreviatios: wrt with respect to! for all! there exists! therefore Def defiitio Th m Theorem sol solutio! perpedicular iff or! if ad oly if pt poit

More information

There are 7 crystal systems and 14 Bravais lattices in 3 dimensions.

There are 7 crystal systems and 14 Bravais lattices in 3 dimensions. EXAM IN OURSE TFY40 Solid State Physics Moday 0. May 0 Time: 9.00.00 DRAFT OF SOLUTION Problem (0%) Itroductory Questios a) () Primitive uit cell: The miimum volume cell which will fill all space (without

More information

Root Finding COS 323

Root Finding COS 323 Root Fidig COS 323 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy

More information

CS321. Numerical Analysis and Computing

CS321. Numerical Analysis and Computing CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca

More information

Quantum Annealing for Heisenberg Spin Chains

Quantum Annealing for Heisenberg Spin Chains LA-UR # - Quatum Aealig for Heiseberg Spi Chais G.P. Berma, V.N. Gorshkov,, ad V.I.Tsifriovich Theoretical Divisio, Los Alamos Natioal Laboratory, Los Alamos, NM Istitute of Physics, Natioal Academy of

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5 CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio

More information

Castiel, Supernatural, Season 6, Episode 18

Castiel, Supernatural, Season 6, Episode 18 13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6 Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

Orthogonal transformations

Orthogonal transformations Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear

More information

Linear Support Vector Machines

Linear Support Vector Machines Liear Support Vector Machies David S. Roseberg The Support Vector Machie For a liear support vector machie (SVM), we use the hypothesis space of affie fuctios F = { f(x) = w T x + b w R d, b R } ad evaluate

More information

Assignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.

Assignment 2 Solutions SOLUTION. ϕ 1  = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ. PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio

More information

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

18.657: Mathematics of Machine Learning

18.657: Mathematics of Machine Learning 8.657: Mathematics of Machie Learig Lecturer: Philippe Rigollet Lecture 0 Scribe: Ade Forrow Oct. 3, 05 Recall the followig defiitios from last time: Defiitio: A fuctio K : X X R is called a positive symmetric

More information

ENGI 9420 Lecture Notes 3 - Numerical Methods Page 3.01

ENGI 9420 Lecture Notes 3 - Numerical Methods Page 3.01 ENGI 940 Lecture Notes 3 - Numerical Methods Page 3.0 3. Numerical Methods The majority of equatios of iterest i actual practice do ot admit ay aalytic solutio. Eve equatios as simple as = e ad I = e d

More information

4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1

4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1 4755 Mark Scheme Jue 05 * Attempt to fid M or 08M - M 08 8 4 * Divide by their determiat,, at some stage Correct determiat, (A0 for det M= 08 stated, all other OR 08 8 4 5 8 7 5 x, y,oe 8 7 4xy 8xy dep*

More information

Numerical Methods in Fourier Series Applications

Numerical Methods in Fourier Series Applications Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets

More information

Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =

Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t = Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,

More information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture 9: Pricipal Compoet Aalysis The text i black outlies mai ideas to retai from the lecture. The text i blue give a deeper uderstadig of how we derive or get

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

Physics Supplement to my class. Kinetic Theory

Physics Supplement to my class. Kinetic Theory Physics Supplemet to my class Leaers should ote that I have used symbols for geometrical figures ad abbreviatios through out the documet. Kietic Theory 1 Most Probable, Mea ad RMS Speed of Gas Molecules

More information

PHY4905: Nearly-Free Electron Model (NFE)

PHY4905: Nearly-Free Electron Model (NFE) PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

For example suppose we divide the interval [0,2] into 5 equal subintervals of length

For example suppose we divide the interval [0,2] into 5 equal subintervals of length Math 120c Calculus Sec 1: Estimatig with Fiite Sums I Area A Cosider the problem of fidig the area uder the curve o the fuctio y!x 2 + over the domai [0,2] We ca approximate this area by usig a familiar

More information

METHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS

METHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS Please cite this article as: Staisław Kula, Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 009,

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

MATH 10550, EXAM 3 SOLUTIONS

MATH 10550, EXAM 3 SOLUTIONS MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

More information

Appendix A. Nabla and Friends

Appendix A. Nabla and Friends Appedix A Nabla ad Frieds A1 Notatio for Derivatives The partial derivative u u(x + he i ) u(x) (x) = lim x i h h of a scalar fuctio u : R R is writte i short otatio as Similarly we have for the higher

More information

Matsubara-Green s Functions

Matsubara-Green s Functions Matsubara-Gree s Fuctios Time Orderig : Cosider the followig operator If H = H the we ca trivially factorise this as, E(s = e s(h+ E(s = e sh e s I geeral this is ot true. However for practical applicatio

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

MTH Assignment 1 : Real Numbers, Sequences

MTH Assignment 1 : Real Numbers, Sequences MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a

More information

IP Reference guide for integer programming formulations.

IP Reference guide for integer programming formulations. IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more

More information

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001. Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;

More information

Principle Of Superposition

Principle Of Superposition ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

Machine Learning Regression I Hamid R. Rabiee [Slides are based on Bishop Book] Spring

Machine Learning Regression I Hamid R. Rabiee [Slides are based on Bishop Book] Spring Machie Learig Regressio I Hamid R. Rabiee [Slides are based o Bishop Book] Sprig 015 http://ce.sharif.edu/courses/93-94//ce717-1 Liear Regressio Liear regressio: ivolves a respose variable ad a sigle predictor

More information

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent.

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent. 06 微甲 0-04 06-0 班期中考解答和評分標準. ( poits) Determie whether the series is absolutely coverget, coditioally coverget, or diverget. Please state the tests which you use. (a) ( poits) (b) ( poits) (c) ( poits)

More information

Root Finding COS 323

Root Finding COS 323 Root Fidig COS 33 Why Root Fidig? Solve or i ay equatio: b where? id root o g b 0 Might ot be able to solve or directly e.g., e -0. si3-0.5 Evaluatig might itsel require solvig a dieretial equatio, ruig

More information

Systems of Particles: Angular Momentum and Work Energy Principle

Systems of Particles: Angular Momentum and Work Energy Principle 1 2.003J/1.053J Dyamics ad Cotrol I, Sprig 2007 Professor Thomas Peacock 2/20/2007 Lecture 4 Systems of Particles: Agular Mometum ad Work Eergy Priciple Systems of Particles Agular Mometum (cotiued) τ

More information

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y

More information

Hydrogen (atoms, molecules) in external fields. Static electric and magnetic fields Oscyllating electromagnetic fields

Hydrogen (atoms, molecules) in external fields. Static electric and magnetic fields Oscyllating electromagnetic fields Hydroge (atoms, molecules) i exteral fields Static electric ad magetic fields Oscyllatig electromagetic fields Everythig said up to ow has to be modified more or less strogly if we cosider atoms (ad ios)

More information

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j. Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α

More information

A widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α

A widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows

More information

Chimica Inorganica 3

Chimica Inorganica 3 himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule

More information

Series: Infinite Sums

Series: Infinite Sums Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree

More information

EXPERIMENT OF SIMPLE VIBRATION

EXPERIMENT OF SIMPLE VIBRATION EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

Math 312 Lecture Notes One Dimensional Maps

Math 312 Lecture Notes One Dimensional Maps Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,

More information

Kinetics of Complex Reactions

Kinetics of Complex Reactions Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet

More information

Lecture 2 October 11

Lecture 2 October 11 Itroductio to probabilistic graphical models 203/204 Lecture 2 October Lecturer: Guillaume Oboziski Scribes: Aymeric Reshef, Claire Verade Course webpage: http://www.di.es.fr/~fbach/courses/fall203/ 2.

More information

Markov Decision Processes

Markov Decision Processes Markov Decisio Processes Defiitios; Statioary policies; Value improvemet algorithm, Policy improvemet algorithm, ad liear programmig for discouted cost ad average cost criteria. Markov Decisio Processes

More information

1. Szabo & Ostlund: 2.1, 2.2, 2.4, 2.5, 2.7. These problems are fairly straightforward and I will not discuss them here.

1. Szabo & Ostlund: 2.1, 2.2, 2.4, 2.5, 2.7. These problems are fairly straightforward and I will not discuss them here. Solutio set III.. Szabo & Ostlud:.,.,.,.5,.7. These problems are fairly straightforward ad I will ot discuss them here.. N! N! i= k= N! N! N! N! p p i j pi+ pj i j i j i= j= i= j= AA ˆˆ= ( ) Pˆ ( ) Pˆ

More information

SECTION 2 Electrostatics

SECTION 2 Electrostatics SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

Assignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1

Assignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1 Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate

More information

1. Hydrogen Atom: 3p State

1. Hydrogen Atom: 3p State 7633A QUANTUM MECHANICS I - solutio set - autum. Hydroge Atom: 3p State Let us assume that a hydroge atom is i a 3p state. Show that the radial part of its wave fuctio is r u 3(r) = 4 8 6 e r 3 r(6 r).

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

Lyapunov Stability Analysis for Feedback Control Design

Lyapunov Stability Analysis for Feedback Control Design Copyright F.L. Lewis 008 All rights reserved Updated: uesday, November, 008 Lyapuov Stability Aalysis for Feedbac Cotrol Desig Lyapuov heorems Lyapuov Aalysis allows oe to aalyze the stability of cotiuous-time

More information

State Space Representation

State Space Representation Optimal Cotrol, Guidace ad Estimatio Lecture 2 Overview of SS Approach ad Matrix heory Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore State Space Represetatio Prof.

More information

Using Spreadsheets as a Computational Tool in Teaching Mechanical. Engineering

Using Spreadsheets as a Computational Tool in Teaching Mechanical. Engineering Proceedigs of the th WSEAS Iteratioal Coferece o COMPUTERS, Vouliagmei, Athes, Greece, July 3-5, 6 (pp35-3) Usig Spreadsheets as a Computatioal Tool i Teachig Mechaical Egieerig AHMADI-BROOGHANI, ZAHRA

More information

We will conclude the chapter with the study a few methods and techniques which are useful

We will conclude the chapter with the study a few methods and techniques which are useful Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs

More information

Lecture 6: Integration and the Mean Value Theorem

Lecture 6: Integration and the Mean Value Theorem Math 8 Istructor: Padraic Bartlett Lecture 6: Itegratio ad the Mea Value Theorem Week 6 Caltech - Fall, 2011 1 Radom Questios Questio 1.1. Show that ay positive ratioal umber ca be writte as the sum of

More information

Axis Aligned Ellipsoid

Axis Aligned Ellipsoid Machie Learig for Data Sciece CS 4786) Lecture 6,7 & 8: Ellipsoidal Clusterig, Gaussia Mixture Models ad Geeral Mixture Models The text i black outlies high level ideas. The text i blue provides simple

More information

Ray-triangle intersection

Ray-triangle intersection Ray-triagle itersectio ria urless October 2006 I this hadout, we explore the steps eeded to compute the itersectio of a ray with a triagle, ad the to compute the barycetric coordiates of that itersectio.

More information

Machine Learning Theory Tübingen University, WS 2016/2017 Lecture 12

Machine Learning Theory Tübingen University, WS 2016/2017 Lecture 12 Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract I this lecture we derive risk bouds for kerel methods. We will start by showig that Soft Margi kerel SVM correspods to miimizig

More information

Math 257: Finite difference methods

Math 257: Finite difference methods Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...

More information

is also known as the general term of the sequence

is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

More information