Lecture #18
|
|
- Doris Peters
- 6 years ago
- Views:
Transcription
1 18-1 Variatioal Method (See CTDL , [Variatioal Method] , [Desity Matrices]) Last time: Quasi-Degeeracy Diagoalize a part of ifiite H * sub-matrix : H (0) + H (1) * correctios for effects of out-of-block elemets: H (2) (the Va Vleck trasformatio) *diagoalize H eff =H (0) + H (1) + H (2) coupled H-O s 2 : 1 (ω 1 2ω 2 ) Fermi resoace example: polyads 1. Perturbatio Theory vs. Variatioal Method 2. Variatioal Theorem 3. Stupid oliear variatio 4. Liear Variatio ew kid of secular Equatio 5. Liear combied with oliear variatio 6. Strategies for criteria of goodess various kids of variatioal calculatios 1. Perturbatio Theory vs. Variatioal Method Perturbatio Theory i effect uses basis set goals: parametrically parsimoious fit model, H eff fit parameters (molecular costats) parameters that defie V(x) order - sortig (1) H k (0) E (0) <1 Ek errors less tha this mixig agle times the previous order o zero correctio term ( is i-block, k is out-of block) because diagoalizatio is order (withi block). Variatioal Method best possible estimate for lowest few E, ψ (ad properties derivable from these) usig fiite basis set ad exact form of H.
2 18-2 Vast majority of computer time i Chemistry is spet i variatioal calculatios Goal is umbers. Isight is secodary. Ab Iitio vs. semi-empirical or fittig [itetioally bad basis set: Hückel, tight bidig qualitative behavior obtaied by a fit to a few microscopic like cotrol parameters] 2. Variatioal Theorem ot ecessarily ormalized If φ is approximatio to eigefuctio of  belogig to lowest eigevalue a 0, the α φ A φ φφ a 0 ay observable PROOF: eigebasis (which we do ot kow but kow it must exist) A = a φ = φ A φ = φφ = φ φ A φ = φ 2, φ φ = φ 2 α φ A φ φφ = completeess a δ eigebasis a φ 2 φ 2 subtract a 0 from both sides α a 0 = a a 0 ( ) φ 2 φ 2 the variatioal Theorem expad φ i eigebasis of A, exploitig completeess 0 a all terms i both sums are 0 agai, all terms i both sums are 0
3 18-3 because, by defiitio of a 0, a a 0 for all ad all terms i sum are 0. α a 0. QED but useless because we ca' t kow a or φ It is possible to perform a variatioal calculatio for ay A, ot limited to H. 3. Stupid Noliear Variatio Use the wrog fuctioal form or the wrog variatioal criterio to get poor results illustrates that the variatioal fuctio must have sufficiet flexibility ad the variatioal criterio must be as it is specified i the variatioal theorem, as opposed to a clever shortcut. The H atom Schr. Eq. (l = 0) H = r 2 r r2 r T 1 r V ad we kow 12 / r ψ1s () r = r 1s = π e E1s = 1 / 2 au 1 au = cm 1 [ ] but try r 3 12 / [ ] ( ) φ = ξ 2π ξr e ξr ormalized for all ξ ξ is a scale factor that cotrols overall size of φ(r) [actually this is the form of ψ 2p (r)] which is ecessarily orthogoal to ψ 1s! STUPID! 12 / ( φ( 0) = 0 but ψ1 s( 0) = π ) ε = φ H φ φφ = 4 ξ 2 3ξ 3 8 skipped a lot of algebra miimize ε: dε dξ = 0 ξ = 3/ 2 ε = 3/ 8 mi mi au FAILURE! 1 c.. f the true values: E1s = 1/ 2 au, E2s = au 8
4 Try somethg clever (but lazy): What is the value of ξ that maximizes φ 1s? 18-4 ( ) = for the best variatioal ξ = 3 / 2, C1s φ ξ = 3 / 2 1s if we maximize C1s wrt. ξ : ξ = 5 / 3 C1s = better? but ε = results, a poorer boud tha ξ = 3/2 ε = * eed flexibility i φ * ca t improve o d ε by employig a alterative variatioal strategy dξ This was stupid ayway because we would ever use the variatioal method whe we already kow the aswer! 4. Liear Variatio Secular Equatio φ = N =1 c χ KEY TOPIC for this lecture χ H χ = H χ χ = S overlap itegrals (o-orthogoal basis sets are ofte coveiet) φh φ ε = = φφ, mm, cc H c c S m m mm rearrage this equatio ε c m c m = c c m, m m S m, ε = 0 for each j c j H to fid miimum value of ε, take c liear variatio! for each j ad require that because we are seekig to miimize ε with respect to each c j. Fid the global miimum of the ε(c 1,c 2, c N ) hypersurface. j the oly terms that survive c j are those that iclude c j.
5 εc m ( S mj + S jm )= c H j + H j m if χ ( ) { } are real S ij = S ji, H ij = H ji ( ) N 0= c H j εs j =1 oe such equatio for each j (same set of ukow {c }) 18-5 These are all of the survivig terms (i.e. those that iclude j). Each j term appears twice i both sums, oce as a bra ad oce as a ket. N liear homogeeous equatios i N ukow c s No trivial {c } oly if H εs = 0 (Not same form as H 1E = 0) The result is N special values of ε that satisfy this equatio. CTDL show: all N ε values are upper bouds to the lowest N E s ad all {φ } s are othogoal! (provided that they belog to differet How to solve H εs = 0 values of E ) 1. Diagoalize S USU = S S ij = siδij (orthogoalize {χ} basis) 2. Normalize S ( S ) 1/2 S ( S ) 1/2 =1 S = T ST 3 diagoal matrices where T = US 1/2 s ƒ 12 / ƒ 12 / ( S ) = S = 0 0 / 0 s O / 1 12 uitary This is ot a orthogoal trasformatio, but it does ot destroy orthogoality because each fuctio is oly beig multiplied by a costat.
6 Trasform H to orthoormalized basis set H Sƒ U HU Sƒ 12 / 12 / = ( ) T T U diagoalizes S ot H ew secular equatio H ε S =0 but S =1 H ε1 =0 usual H diagoalized by usual procedure! 5. Combie Liear ad Noliear Variatio typically doe i ab iitio electroic structure calculatios Basis set: χ ( ξ r) ψ c χ ξ r S = ( ) liear variatio where ε is a radial scale factor ( ξ, ξ ), H ( ξ, ξ ) oliear variatio 0. pick arbitrary set of { ξ i } 1. calculate all H ij ξ i,ξ j 2. Solve H - εs =0 ( ) & S ij ( ξ i,ξ j ) a. S S diagoalize S (orthogoalize) b. 12 / S ( ) (ormalize) c. d. H H diagoalize H oliear variatio begis fid global miimum of ε lowest with respect to each ξ i
7 3. chage ξ from ξ ξ = ξ + δ Solve H- εs 0 to obtai a ew set of ε. Pick lowest ε. 6. calculate 1 () 1 0 () 1 1 () 1 0 recalculate all itegrals i Had Sivolvig χ ε = { } i old ew lowest εlowest εlowest = ξ () () ξ ξ1 1 i repeat #3 6 for each ξ i (always lookig oly at lowest ε i ) This defies a gradiet o a multidimesioal ε(ξ 1, ξ N ) surface. We seek the miimum of this hypersurface. Take a step i directio of steepest descet by a amout determied by ε/ ξ steepest (small slope, small step; large slope, large step). This completes 1st iteratio. All values of {ξ i }are improved. 8. Retur to #3, iterate #3-7 util covergece is obtaied. Noliear variatios are much slower tha liear variatios. Typically use ENORMOUS LINEAR {χ} basis set. Cotract this basis set by optimizig oliear parameters (expoetial scale factors) i a SMALL BASIS SET to match the lowest {φ} s that had iitially bee expressed i large basis set.
8 Alterative Strategies * rigorous variatioal miimizatio of E lowest : ab iitio * costrai variatioal fuctio to be orthogoal to specific subset of fuctios e.g. orthogoal to groud state to get variatioal covergece. Applies oly to higher members of specific symmetry class or orthogoal to core: froze-core approximatio. Pseudopotetials (use some observed eergy levels to determie Z eff (r) of froze core) * least squares fittig miimize differeces betwee a set of measured eergy levels (or other properties) ad a set of computed variatioal eige-eergies (or other properties computed from variatioal wavefuctios). { } molecular costats { observed E } parameters i H eff experimetal ψ s i fiite variatioal basis set * semi-empirical model replace exact Ĥ by a grossly simplified form ad restrict basis set to a simple form too. The adjust parameters i H to match some observed patter of eergy splittigs. Use parameters to predict uobserved properties or use values of fit parameters to build isight.
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 informationPerturbation Theory I (See CTDL , ) Last time: derivation of all matrix elements for Harmonic-Oscillator: x, p, H. n ij n.
Perturbatio Theory I (See CTDL 1095-1107, 1110-1119) 14-1 Last time: derivatio of all matrix elemets for Harmoic-Oscillator: x, p, H selectio rules scalig ij x i j i steps of 2 e.g. x : = ± 3, ± 1 xii
More informationApply 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 informationAssignment 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.
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 informationBasic Iterative Methods. Basic Iterative Methods
Abel s heorem: he roots of a polyomial with degree greater tha or equal to 5 ad arbitrary coefficiets caot be foud with a fiite umber of operatios usig additio, subtractio, multiplicatio, divisio, ad extractio
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationPhysics 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 informationLecture 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 informationREGRESSION (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 informationIntroduction 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 information5.76 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 information1 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 informationForces: Calculating Them, and Using Them Shobhana Narasimhan JNCASR, Bangalore, India
Forces: Calculatig Them, ad Usig Them Shobhaa Narasimha JNCASR, Bagalore, Idia shobhaa@jcasr.ac.i Shobhaa Narasimha, JNCASR 1 Outlie Forces ad the Hellma-Feyma Theorem Stress Techiques for miimizig a fuctio
More informationTHE 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 informationTMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
More informationThe time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): ( ) ( ) 2m "2 + V ( r,t) (1.
Adrei Tokmakoff, MIT Departmet of Chemistry, 2/13/2007 1-1 574 TIME-DEPENDENT QUANTUM MECHANICS 1 INTRODUCTION 11 Time-evolutio for time-idepedet Hamiltoias The time evolutio of the state of a quatum system
More informationis 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 informationNBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?
NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationThe Phi Power Series
The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.
More information1. 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 informationLanczos-Haydock Recursion
Laczos-Haydock Recursio Bor: Feb 893 i Székesehérvár, Hugary- Died: 5 Jue 974 i Budapest, Hugary Corelius Laczos From a abstract mathematical viewpoit, the method for puttig a symmetric matrix i three-diagoal
More informationPAPER : 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 informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationChapter 8 Approximation Methods, Hueckel Theory
Witer 3 Chem 356: Itroductory Quatum Mechaics Chapter 8 Approximatio Methods, Huecel Theory... 8 Approximatio Methods... 8 The Liear Variatioal Priciple... Chapter 8 Approximatio Methods, Huecel Theory
More informationChapter 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 informationChimica 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 informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More informationLecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box
561 Fall 013 Lecture #5 page 1 Last time: Lecture #5: Begi Quatum Mechaics: Free Particle ad Particle i a 1D Box u 1 u 1-D Wave equatio = x v t * u(x,t): displacemets as fuctio of x,t * d -order: solutio
More informationn 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 informationA Challenging Test For Convergence Accelerators: Summation Of A Series With A Special Sign Pattern
Applied Mathematics E-Notes, 6(006), 5-34 c ISSN 1607-510 Available free at mirror sites of http://www.math.thu.edu.tw/ ame/ A Challegig Test For Covergece Accelerators: Summatio Of A Series With A Special
More informationUnbiased Estimation. February 7-12, 2008
Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationLinear Programming and the Simplex Method
Liear Programmig ad the Simplex ethod Abstract This article is a itroductio to Liear Programmig ad usig Simplex method for solvig LP problems i primal form. What is Liear Programmig? Liear Programmig is
More information5.61 Fall 2013 Problem Set #3
5.61 Fall 013 Problem Set #3 1. A. McQuarrie, page 10, #3-3. B. McQuarrie, page 10, #3-4. C. McQuarrie, page 18, #4-11.. McQuarrie, pages 11-1, #3-11. 3. A. McQuarrie, page 13, #3-17. B. McQuarrie, page
More information10/2/ , 5.9, Jacob Hays Amit Pillay James DeFelice
0//008 Liear Discrimiat Fuctios Jacob Hays Amit Pillay James DeFelice 5.8, 5.9, 5. Miimum Squared Error Previous methods oly worked o liear separable cases, by lookig at misclassified samples to correct
More informationPhysics 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 informationPHY4905: 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 informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationMath 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 informationMachine 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 informationIP 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 informationOptimization 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 informationGrouping 2: Spectral and Agglomerative Clustering. CS 510 Lecture #16 April 2 nd, 2014
Groupig 2: Spectral ad Agglomerative Clusterig CS 510 Lecture #16 April 2 d, 2014 Groupig (review) Goal: Detect local image features (SIFT) Describe image patches aroud features SIFT, SURF, HoG, LBP, Group
More informationAdmin REGULARIZATION. Schedule. Midterm 9/29/16. Assignment 5. Midterm next week, due Friday (more on this in 1 min)
Admi Assigmet 5! Starter REGULARIZATION David Kauchak CS 158 Fall 2016 Schedule Midterm ext week, due Friday (more o this i 1 mi Assigmet 6 due Friday before fall break Midterm Dowload from course web
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More information5.74 TIME-DEPENDENT QUANTUM MECHANICS
p. 1 5.74 TIME-DEPENDENT QUANTUM MECHANICS The time evolutio of the state of a system is described by the time-depedet Schrödiger equatio (TDSE): i t ψ( r, t)= H ˆ ψ( r, t) Most of what you have previously
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationLinear Regression Models
Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More informationMATHEMATICAL SCIENCES PAPER-II
MATHEMATICAL SCIENCES PAPER-II. Let {x } ad {y } be two sequeces of real umbers. Prove or disprove each of the statemets :. If {x y } coverges, ad if {y } is coverget, the {x } is coverget.. {x + y } coverges
More information10.6 ALTERNATING SERIES
0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More informationΩ ). Then the following inequality takes place:
Lecture 8 Lemma 5. Let f : R R be a cotiuously differetiable covex fuctio. Choose a costat δ > ad cosider the subset Ωδ = { R f δ } R. Let Ωδ ad assume that f < δ, i.e., is ot o the boudary of f = δ, i.e.,
More informationState 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 informationLecture 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 informationLecture 6. Bonds to Bands. But for most problems we use same approximation methods: 6. The Tight-Binding Approximation
6. The Tight-Bidig Approximatio or from Bods to Bads Basic cocepts i quatum chemistry LCAO ad molecular orbital theory The tight bidig model of solids bads i 1,, ad 3 dimesios Refereces: 1. Marder, Chapters
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More informationAME 513. " Lecture 3 Chemical thermodynamics I 2 nd Law
AME 513 Priciples of Combustio " Lecture 3 Chemical thermodyamics I 2 d Law Outlie" Why do we eed to ivoke chemical equilibrium? Degrees Of Reactio Freedom (DORFs) Coservatio of atoms Secod Law of Thermodyamics
More informationFactor Analysis. Lecture 10: Factor Analysis and Principal Component Analysis. Sam Roweis
Lecture 10: Factor Aalysis ad Pricipal Compoet Aalysis Sam Roweis February 9, 2004 Whe we assume that the subspace is liear ad that the uderlyig latet variable has a Gaussia distributio we get a model
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationLECTURE 17: Linear Discriminant Functions
LECURE 7: Liear Discrimiat Fuctios Perceptro leari Miimum squared error (MSE) solutio Least-mea squares (LMS) rule Ho-Kashyap procedure Itroductio to Patter Aalysis Ricardo Gutierrez-Osua exas A&M Uiversity
More informationLecture 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 informationINTEGRATION BY PARTS (TABLE METHOD)
INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationCalculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)
Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationLecture 15: Learning Theory: Concentration Inequalities
STAT 425: Itroductio to Noparametric Statistics Witer 208 Lecture 5: Learig Theory: Cocetratio Iequalities Istructor: Ye-Chi Che 5. Itroductio Recall that i the lecture o classificatio, we have see that
More informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationv = -!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 information8.3 Perturbation theory
8.3 Perturbatio theory Slides: Video 8.3.1 Costructig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (u to First order erturbatio theory ) Perturbatio theory Costructig
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationLinear 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 informationModern Discrete Probability Spectral Techniques
Moder Discrete Probability VI - Spectral Techiques Backgroud Sébastie Roch UW Madiso Mathematics December 1, 2014 1 Review 2 3 4 Mixig time I Theorem (Covergece to statioarity) Cosider a fiite state space
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 11
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract We will itroduce the otio of reproducig kerels ad associated Reproducig Kerel Hilbert Spaces (RKHS). We will cosider couple
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationMarkov 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 informationDETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.
ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics
More informationFor a 3 3 diagonal matrix we find. Thus e 1 is a eigenvector corresponding to eigenvalue λ = a 11. Thus matrix A has eigenvalues 2 and 3.
Closed Leotief Model Chapter 6 Eigevalues I a closed Leotief iput-output-model cosumptio ad productio coicide, i.e. V x = x = x Is this possible for the give techology matrix V? This is a special case
More informationLesson 03 Heat Equation with Different BCs
PDE & Complex Variables P3- esso 3 Heat Equatio with Differet BCs ( ) Physical meaig (SJF ) et u(x, represet the temperature of a thi rod govered by the (coductio) heat equatio: u t =α u xx (3.) where
More informationAxis 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 informationRevision Topic 1: Number and algebra
Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol
More informationDisjoint set (Union-Find)
CS124 Lecture 7 Fall 2018 Disjoit set (Uio-Fid) For Kruskal s algorithm for the miimum spaig tree problem, we foud that we eeded a data structure for maitaiig a collectio of disjoit sets. That is, we eed
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationNew Version of the Rayleigh Schrödinger Perturbation Theory: Examples
New Versio of the Rayleigh Schrödiger Perturbatio Theory: Examples MILOŠ KALHOUS, 1 L. SKÁLA, 1 J. ZAMASTIL, 1 J. ČÍŽEK 2 1 Charles Uiversity, Faculty of Mathematics Physics, Ke Karlovu 3, 12116 Prague
More informationA) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.
M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks
More informationLecture 20. Brief Review of Gram-Schmidt and Gauss s Algorithm
8.409 A Algorithmist s Toolkit Nov. 9, 2009 Lecturer: Joatha Keler Lecture 20 Brief Review of Gram-Schmidt ad Gauss s Algorithm Our mai task of this lecture is to show a polyomial time algorithm which
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More information