Entropy of Markov Information Sources and Capacity of Discrete Input Constrained Channels (from Immink, Coding Techniques for Digital Recorders)

Size: px
Start display at page:

Download "Entropy of Markov Information Sources and Capacity of Discrete Input Constrained Channels (from Immink, Coding Techniques for Digital Recorders)"

Transcription

1 Entropy of Marov Informaton Sources and Capacty of Dscrete Input Constraned Channels (from Immn, Codng Technques for Dgtal Recorders). Entropy of Marov Chans We have already ntroduced the noton of entropy n a conceptually smple stuaton: t was assumed that the symbols are ndependent and occur wth fxed probabltes. That s, the occurrence of a specfc symbol at a certan nstant does not alter the probablty of occurrences of symbols durng any other symbol ntervals. We need to extend the concept of entropy for more complcated structures where symbols are not chosen ndependently but ther probabltes of occurrng depend on precedng symbols. It s to be emphaszed that nearly all practcal sources emt sequences of symbols that are statstcally dependent. Sequences formed by the Englsh language are an excellent example. Occurrence of the letter Q mples that the letter to follow s probably a U. Regardless of the form of the statstcal dependence, or structure, among the successve source outputs, the effect s that the amount of nformaton comng out of such a source s smaller than from a source emttng the same set of characters n ndependent sequences. The development of a model for sources wth memory s the focus of the ensung dscusson. In probablty theory the notaton P r (A B) means the probablty of occurrence of event A gven that ( ) event B has occurred. Many of the structures that wll be encountered n the subsequent chapters can usefully be modeled n terms of a Marov chan. A Marov chan s a specal type of stochastc process dstngushed by a certan Marov property A (dscrete) Marov chan s defned as a dscrete random process of the form where the varables Z t are dependent dscrete random varables tang values n the state alphabet S= {s,, s N }, and the dependence satsfes the Marov condton In words, the varable Z t s ndependent of past samples Z t-2,z t-3... f the value of Z t- s nown. A (homogeneous) Marov chan can be descrbed by a transton probablty matrx Q wth elements The transton probablty matrx Q s a stochastc matrx, that s, ts entres are nonnegatve, and the entres of each row sum to one. Any stochastc matrx consttutes a vald transton probablty matrx. Imagne the process starts at tme t = by choosng an ntal state n accordance wth a specfed probablty dstrbuton. If we are n state s at tme t =, then the

2 process moves at t = 2 to a possbly new state, the quantty q s the probablty that the process wll move to state s at tme t = 2. If we are n state s at nstant t = 2, we move to s at nstant t = 3 wth probablty q. Ths procedure s repeated ad nfntum. The state-transton dagram of a Marov chan, portrayed n the followng fgure (a) represents a Marov chan as a drected graph where the states are emboded by the nodes or vertces of the graph; the transton between states s represented by a drected lne, an edge, from the ntal to the fnal state, The transton probabltes q correspondng to varous transtons are shown mared along the lnes of the graph. Another useful representaton of a Marov chan s provded by a trells (or lattce) dagram (see (b)). Alternatve representatons of a three-state Marov chan. (a) state-transton dagram, (b) trells dagram for the same chan. Permssble transtons from one state to the other are depcted by lnes. Ths s a state dagram augmented by a tme axs so that t provdes for easy vsualzaton of how the states change wth tme. In theory, there are many types of Marov chans; here we restrct our attenton to chans that are ergodc and regular. Roughly speang, ergodcty means that from any state the chan can eventually reach any other state; regularty means that the Marov chan s non- perodc. In the practcal structures that wll be encountered, all these condtons hold. We shall now tae a closer loo at the dynamcs of a Marov chan, To that end, ( t) let w Pr( Z = ) represent the probablty of beng n state at tme t. Clearly, t N = The probablty of beng n state nstant t-: w ( t) = at tme t may be expressed n the state probabltes at The prevous equaton suggests the use of matrces. If we ntroduce the state dstrbuton vector

3 then the prevous equaton can succnctly be expressed n an elegant matrx/vector notaton, thus By teraton we obtan In other words, the state dstrbuton vector at tme t s the product of the state dstrbuton vector at tme t =, and the (t -)th power of the transton matrx. It s easy to see that Q t- s also a stochastc matrx. The prevous formula (3.0) s equvalent to the asserton that the n-step transton matrx s the nth power of the sngle step transton matrx Q. We note also that Q 0 = I s the ordnary dentty matrx. We shall concentrate now on the lmtng behavor of the state dstrbuton vector as t. In many cases of practcal nterest there s only one such lmtng dstrbuton vector, denoted by π = ( π,, π N ). In the long run the state dstrbuton vector converges to ths equlbrum dstrbuton vector from any vald ntal state probablty vector w () so The numberπ, s called the steady, or statonary state probablty of state. The equlbrum dstrbuton vector can be obtaned by solvng the system of lnear equatons n the N unnowns π,,π : N π Q = π Only N- of these N equatons are ndependent, so we solve the top N- along wth the normalzng condton N = π = The proof s elementary: we note that f π Q = π, then Decomposton of the ntal state vector w () n terms of the egenvectors of Q can be convenent to demonstrate the process of convergence. The matrx Q has N egenvalues { λ,,λ N }, that can be found by solvng the characterstc equaton where I s the dentty matrx, and N (left) egenvectors {u,..., u N } each of whch s a soluton of the system Provded that λ, =,., N, are dstnct, there are N ndependent egenvectors, and the egenvectors λ, =,, N, consttute a bass. The ntal state vector may be wrtten as

4 We fnd the state dstrbuton vector w (t) at nstant t: If t s assumed that the egenvalues are dstnct, the {? } can be ordered, such that λ > λ >, etc. Combnaton of prevous formulae reveals that p s an egenvector 2 λ3 wth unty egenvalue, thus λ =, We then have and convergence to p s assured snce λ <,. 2. Entropy of Marov Informaton Sources We are now n the poston to descrbe a Marov nformaton source. Gven a fnte Marov chan {Z t } and a functon? whose doman s the set of states of the chan and whose range s a fnte set G, the source alphabet, then the sequence {X t } where X t = ζ ( Z t ), s sad to be the output of a Marov nformaton source correspondng to the chan { Z t } and the functon?. In general, the number of states can be larger than the cardnalty of the source alphabet, whch means that one output symbol may correspond to more than one state. The essental feature of the Marov nformaton source s that t provdes for dependence between successve symbols, whch ntroduces redundancy n the message sequence. Each symbol conveys less nformaton than t s capable of conveyng snce t s to some extent predctable from the precedng symbol. In the foregong descrpton of an nformaton source we assumed that the symbol emtted s solely a functon of the state that s entered. Ths type of descrpton s usually called a Moore-type Marov source, In a dfferent descrpton, called the Mealy-type Marov source, the symbols emtted are a functon of the Marov chan X ˆ t = ζ ( Zt, Zt+ ). In other words, a Mealy-type Marov source s obtaned by labellng the edges of the drected graph that represents the Marov chan. Mealy- and Moore-type descrptons are equvalent. Let a Mealy-type machne be gven. By defnng a Marov nformaton source wth state set composed of trples { ( ), ζ ˆ, } and label ζˆ ( ) on the state { ( ), ζ ˆ, }, we obtan a Moore-type Marov source. The Moore-type model s referred to as the edge graph of the Mealy-type model. An example of a Mealy-type nformaton source and ts Moore-type equvalent are shown n the followng fgure.

5 (a) Example of a Mealy-type two-state Marov nformaton source, and (b) ts four-state Moore-type counterpart The dea of a Marov source has enabled us to represent certan types of structure n streams of data. We next examne the nformaton content, or entropy, of a sequence emtted by a Marov source, The entropy of a Marov nformaton source s hard to compute n most cases. For a certan class of Marov nformaton sources, termed unflar Marov nformaton source, the computaton may be greatly smplfed. The word unflar refers to the followng property. Let a Marov nformaton source wth a set of states Σ = { } alphabet G, and assocated output functon ( Z t ),, 2, N, output ζ be gven. For each state Σ, let n be the states that can be reached n one step from such that q > 0. We say s a successor of, that s, the states, f q >0. The source s sad to be unflar f for each state the symbols ζ ( ),, ζ ( n ) are dstnct. In other words, each successor of must be assocated wth a dstnct symbol. Provded ths condton s met and the ntal state of the Marov nformaton source s nown, the sequence of emtted symbols determnes the sequence of states followed by the chan, and a smple formula s avalable for the entropy of the emtted X-process. Gven a unflar Marov source, as above, let,, be the successors of n, then t s qute natural to defne the uncertanty of state as H H ( p,, pn ) =, wth H (,, ) H( p,, p ) = p log p M 2 = M p p n defned as Shannon defned the entropy of the unflar Marov source as the average of these l weghed n accordance wth the steady-state probablty of beng n a state n queston, that s, by the expresson Note that we use the notaton H{X }to express the fact that we are consderng the entropy of sequences {X} and not the functon H(.). The next numercal example may serve to llustrate the theory. Example: Consder the three-state unflar Marov chan depcted n the fgure above. From the dagram we may read the transton probablty matrx

6 What s the average probablty of beng n one of the three states? We fnd the followng system of lnear equatons that govern the steady-state probabltes: 4 7 from whch we obtan π 3 = π 2 and π 3 = 6π 2. Snce π + π2 + π3 = we have 3 The entropy of the nformaton source s found to be equal In the next secton we consder a problem whch s central to the feld of nputconstraned channels. We focus on methods to compute the maxmum amount of nformaton that can be sent over an nput-constraned channel per unt of tme. 3. Capacty of Dscrete Noseless Channels Shannon defned the capacty C of a dscrete noseless channel by where N(T) s the number of admssble sgnals of duraton T. The problem of calculaton of the capacty for constraned channels s n essence a combnatoral problem, that s, fndng the number of allowed sequences N(T). Ths fundamental defnton wll be wored out n a moment for some specfc channel models. We start, snce vrtually all channel constrants can be modeled as such, wth the computaton of the capacty of Marov nformaton sources. 3.. Capacty of Marov nformaton sources In the prevous sectons we developed a measure of nformaton content of an nformaton source that can be represented by a fnte Marov model. As dscussed, the measure of nformaton content, entropy, can be expressed n terms of the lmtng statetranston probabltes and the condtonal entropy of the states. In ths secton we address a problem that provdes the ey to answer many questons that wll emerge n the chapters to fo llow. Gven a unflar N-state Marov source wth states {, N } and

7 transton probabltes follows. Let d d qˆ, we defne the connecton matrx D = { } = f = 0 f qˆ qˆ > 0 = 0,, =,, N. d of the source as The actual values of the transton probabltes are rrelevant; the connecton (or adacency) matrx contans bnary-valued elements, and t s formed by replacng the postve elements of the transton matrx by l s. For an N-state source, the connecton matrx D s defned by d = f a transton from state to state s allowable and d = 0 otherwse. For a gven connecton matrx, we wsh to choose the transton probabltes n such a way that the entropy H N { X } = π = H s maxmzed. Such a source s called maxentropc, and sequences generated by a maxentropc unflar source are called maxentropc sequences. The maxmum entropy of a unflar Marov nformaton source, gven ts connecton matrx, s gven by { } = log, C = max H X λ 2 max where? max s the largest egenvalue of the connecton matrx D. The exstence of a postve egenvalue and correspondng egenvector wth postve elements s guaranteed by the Perron-Frobenus theorems. Essentally, there are two approaches to prove the precedng equaton. One approach, provded by Shannon, s a straghtforward routne, usng Lagrange multplers, of fndng the extreme value of a functon of several ndependent varables. The second proof of the above formula to be followed here, s establshed by enumeratng the number of dstnct sequences that a Marov source can generate. The number of dstnct sequences of length m +, m > 0, emanatng from state s denoted by N (m + ), equals the total of the numbers of sequences of unty length that emerge from s, and termnate n s multpled (snce the source s unflar) by the number of sequences of length m that start n s. Thus we fnd the followng recurrence equaton Ths s a system of N lnear homogeneous dfference equatons wth constant coeffcents, and therefore the soluton s a lnear combnaton of exponentals lambda m. To fnd the partcular {? }, we assume a soluton of the form N (m) = y *lambda m to obtan or, lettng y T = (y,,y N ), where the superscrpt T stands for transposton, we have lambda*y =Dy

8 Thus the allowable {lambda } are the egenvalues of the matrx D. For large sequence length n we may approxmate N (m) by where a s a constant ndependent of m and lambda max s the largest real egenvalue of the matrx D, or n other words, lambda max s the largest real root of the determnant equaton Prevoous equaton states that for large enough m the number of dstnct sequences grows exponentally wth the sequence length m; the growth factor s lambda max. Ths s not to say that N (m) s accurately determned by the exponental term when m s small. We have The maxmum entropy of the noseless channel may be evaluated by nvong the defnton of the capacty, or The transton probabltes q assocated wth the maxmum entropy of the source can be found wth the followng reasonng. Let p = (p, P N ) T denote the egenvector assocated wth the egenvalue lambda max or The state-transton probabltes that maxmze the entropy are To prove the above formula s a matter of substtuton. Accordng to the Perron- Frobenus theorems [5], the components of the egenvector p are postve, and thus q = 0, =, = N. Snce p = (p,.., p N ) T s an egenvector for? max we conclude and hence the matrx Q s ndeed stochastc. The entropy of a Marov nformaton source s, accordng to defnton where H s the uncertanty of state s and (p,...,p N ) s the steady-state dstrbuton. Thus,

9 Snce and we obtan Ths demonstrates that the transton probabltes gven by are ndeed maxmzng the entropy. Example (Contnued): We revert to the three-state unflar Marov chan wth transton probablty matrx The adacency matrx D s The characterstc equaton s from whch we conclude that the largest root? max = 2, and the capacty s C = log 2? max =. The egenvector assocated wth the largest egenvalue s p = (, 3, 2) T. The transton probabltes that maxmze the entropy of the Marov nformaton source are found wth

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

Convergence of random processes

Convergence of random processes DS-GA 12 Lecture notes 6 Fall 216 Convergence of random processes 1 Introducton In these notes we study convergence of dscrete random processes. Ths allows to characterze phenomena such as the law of large

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

6. Stochastic processes (2)

6. Stochastic processes (2) 6. Stochastc processes () Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brth-death processes 6. Stochastc processes () Markov process

More information

6. Stochastic processes (2)

6. Stochastic processes (2) Contents Markov processes Brth-death processes Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space

More information

Lecture 3: Shannon s Theorem

Lecture 3: Shannon s Theorem CSE 533: Error-Correctng Codes (Autumn 006 Lecture 3: Shannon s Theorem October 9, 006 Lecturer: Venkatesan Guruswam Scrbe: Wdad Machmouch 1 Communcaton Model The communcaton model we are usng conssts

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

Continuous Time Markov Chains

Continuous Time Markov Chains Contnuous Tme Markov Chans Brth and Death Processes,Transton Probablty Functon, Kolmogorov Equatons, Lmtng Probabltes, Unformzaton Chapter 6 1 Markovan Processes State Space Parameter Space (Tme) Dscrete

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Chapter 7 Channel Capacity and Coding

Chapter 7 Channel Capacity and Coding Chapter 7 Channel Capacty and Codng Contents 7. Channel models and channel capacty 7.. Channel models Bnary symmetrc channel Dscrete memoryless channels Dscrete-nput, contnuous-output channel Waveform

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Density matrix. c α (t)φ α (q)

Density matrix. c α (t)φ α (q) Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n

More information

Google PageRank with Stochastic Matrix

Google PageRank with Stochastic Matrix Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d

More information

Chapter 7 Channel Capacity and Coding

Chapter 7 Channel Capacity and Coding Wreless Informaton Transmsson System Lab. Chapter 7 Channel Capacty and Codng Insttute of Communcatons Engneerng atonal Sun Yat-sen Unversty Contents 7. Channel models and channel capacty 7.. Channel models

More information

Random Walks on Digraphs

Random Walks on Digraphs Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

More information

= z 20 z n. (k 20) + 4 z k = 4

= z 20 z n. (k 20) + 4 z k = 4 Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Lecture Space-Bounded Derandomization

Lecture Space-Bounded Derandomization Notes on Complexty Theory Last updated: October, 2008 Jonathan Katz Lecture Space-Bounded Derandomzaton 1 Space-Bounded Derandomzaton We now dscuss derandomzaton of space-bounded algorthms. Here non-trval

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

1 GSW Iterative Techniques for y = Ax

1 GSW Iterative Techniques for y = Ax 1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

Digital Signal Processing

Digital Signal Processing Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over

More information

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton

More information

Linear Regression Analysis: Terminology and Notation

Linear Regression Analysis: Terminology and Notation ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented

More information

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

More information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence

More information

EEE 241: Linear Systems

EEE 241: Linear Systems EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several

More information

Composite Hypotheses testing

Composite Hypotheses testing Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter

More information

ECE 534: Elements of Information Theory. Solutions to Midterm Exam (Spring 2006)

ECE 534: Elements of Information Theory. Solutions to Midterm Exam (Spring 2006) ECE 534: Elements of Informaton Theory Solutons to Mdterm Eam (Sprng 6) Problem [ pts.] A dscrete memoryless source has an alphabet of three letters,, =,, 3, wth probabltes.4,.4, and., respectvely. (a)

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

LECTURE 9 CANONICAL CORRELATION ANALYSIS

LECTURE 9 CANONICAL CORRELATION ANALYSIS LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Lecture 7: Boltzmann distribution & Thermodynamics of mixing

Lecture 7: Boltzmann distribution & Thermodynamics of mixing Prof. Tbbtt Lecture 7 etworks & Gels Lecture 7: Boltzmann dstrbuton & Thermodynamcs of mxng 1 Suggested readng Prof. Mark W. Tbbtt ETH Zürch 13 März 018 Molecular Drvng Forces Dll and Bromberg: Chapters

More information

Dynamic Systems on Graphs

Dynamic Systems on Graphs Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus A network s a set of nodes that collaborates to acheve what each cannot acheve alone. A network,

More information

EGR 544 Communication Theory

EGR 544 Communication Theory EGR 544 Communcaton Theory. Informaton Sources Z. Alyazcoglu Electrcal and Computer Engneerng Department Cal Poly Pomona Introducton Informaton Source x n Informaton sources Analog sources Dscrete sources

More information

Dynamical Systems and Information Theory

Dynamical Systems and Information Theory Dynamcal Systems and Informaton Theory Informaton Theory Lecture 4 Let s consder systems that evolve wth tme x F ( x, x, x,... That s, systems that can be descrbed as the evoluton of a set of state varables

More information

Hidden Markov Models & The Multivariate Gaussian (10/26/04)

Hidden Markov Models & The Multivariate Gaussian (10/26/04) CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models

More information

Channel Encoder. Channel. Figure 7.1: Communication system

Channel Encoder. Channel. Figure 7.1: Communication system Chapter 7 Processes The model of a communcaton system that we have been developng s shown n Fgure 7.. Ths model s also useful for some computaton systems. The source s assumed to emt a stream of symbols.

More information

Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty

Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,

More information

Lecture 5 Decoding Binary BCH Codes

Lecture 5 Decoding Binary BCH Codes Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

FINITE-STATE MARKOV CHAINS

FINITE-STATE MARKOV CHAINS Chapter 4 FINITE-STATE MARKOV CHAINS 4.1 Introducton The countng processes {N(t), t 0} of Chapterss 2 and 3 have the property that N(t) changes at dscrete nstants of tme, but s defned for all real t 0.

More information

Maximizing the number of nonnegative subsets

Maximizing the number of nonnegative subsets Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum

More information

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results. Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

Using T.O.M to Estimate Parameter of distributions that have not Single Exponential Family

Using T.O.M to Estimate Parameter of distributions that have not Single Exponential Family IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran

More information

The optimal delay of the second test is therefore approximately 210 hours earlier than =2.

The optimal delay of the second test is therefore approximately 210 hours earlier than =2. THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple

More information

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Classes of States and Stationary Distributions

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Classes of States and Stationary Distributions Steven R. Dunbar Department of Mathematcs 203 Avery Hall Unversty of Nebraska-Lncoln Lncoln, NE 68588-0130 http://www.math.unl.edu Voce: 402-472-3731 Fax: 402-472-8466 Topcs n Probablty Theory and Stochastc

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

CHAPTER III Neural Networks as Associative Memory

CHAPTER III Neural Networks as Associative Memory CHAPTER III Neural Networs as Assocatve Memory Introducton One of the prmary functons of the bran s assocatve memory. We assocate the faces wth names, letters wth sounds, or we can recognze the people

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Laboratory 3: Method of Least Squares

Laboratory 3: Method of Least Squares Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers. David Miller Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons

More information

PARTICIPATION FACTOR IN MODAL ANALYSIS OF POWER SYSTEMS STABILITY

PARTICIPATION FACTOR IN MODAL ANALYSIS OF POWER SYSTEMS STABILITY POZNAN UNIVE RSITY OF TE CHNOLOGY ACADE MIC JOURNALS No 86 Electrcal Engneerng 6 Volodymyr KONOVAL* Roman PRYTULA** PARTICIPATION FACTOR IN MODAL ANALYSIS OF POWER SYSTEMS STABILITY Ths paper provdes a

More information

ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013

ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013 ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run

More information

Laboratory 1c: Method of Least Squares

Laboratory 1c: Method of Least Squares Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system. Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009 College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:

More information

CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 14 GENERAL PERTURBATION THEORY CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

Continuous Time Markov Chain

Continuous Time Markov Chain Contnuous Tme Markov Chan Hu Jn Department of Electroncs and Communcaton Engneerng Hanyang Unversty ERICA Campus Contents Contnuous tme Markov Chan (CTMC) Propertes of sojourn tme Relatons Transton probablty

More information

1 Generating functions, continued

1 Generating functions, continued Generatng functons, contnued. Generatng functons and parttons We can make use of generatng functons to answer some questons a bt more restrctve than we ve done so far: Queston : Fnd a generatng functon

More information

Winter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan

Winter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments

More information

Finding Dense Subgraphs in G(n, 1/2)

Finding Dense Subgraphs in G(n, 1/2) Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016 U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

Lecture 14 (03/27/18). Channels. Decoding. Preview of the Capacity Theorem.

Lecture 14 (03/27/18). Channels. Decoding. Preview of the Capacity Theorem. Lecture 14 (03/27/18). Channels. Decodng. Prevew of the Capacty Theorem. A. Barg The concept of a communcaton channel n nformaton theory s an abstracton for transmttng dgtal (and analog) nformaton from

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

Subset Topological Spaces and Kakutani s Theorem

Subset Topological Spaces and Kakutani s Theorem MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS) Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998

More information

Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Georgia Tech PHYS 6124 Mathematical Methods of Physics I Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends

More information

7. Products and matrix elements

7. Products and matrix elements 7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ

More information

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0 Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of

More information

Solution Thermodynamics

Solution Thermodynamics Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs

More information

Markov chains. Definition of a CTMC: [2, page 381] is a continuous time, discrete value random process such that for an infinitesimal

Markov chains. Definition of a CTMC: [2, page 381] is a continuous time, discrete value random process such that for an infinitesimal Markov chans M. Veeraraghavan; March 17, 2004 [Tp: Study the MC, QT, and Lttle s law lectures together: CTMC (MC lecture), M/M/1 queue (QT lecture), Lttle s law lecture (when dervng the mean response tme

More information

THE ARIMOTO-BLAHUT ALGORITHM FOR COMPUTATION OF CHANNEL CAPACITY. William A. Pearlman. References: S. Arimoto - IEEE Trans. Inform. Thy., Jan.

THE ARIMOTO-BLAHUT ALGORITHM FOR COMPUTATION OF CHANNEL CAPACITY. William A. Pearlman. References: S. Arimoto - IEEE Trans. Inform. Thy., Jan. THE ARIMOTO-BLAHUT ALGORITHM FOR COMPUTATION OF CHANNEL CAPACITY Wllam A. Pearlman 2002 References: S. Armoto - IEEE Trans. Inform. Thy., Jan. 1972 R. Blahut - IEEE Trans. Inform. Thy., July 1972 Recall

More information

8.592J: Solutions for Assignment 7 Spring 2005

8.592J: Solutions for Assignment 7 Spring 2005 8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1

More information

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 ) Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often

More information

MMA and GCMMA two methods for nonlinear optimization

MMA and GCMMA two methods for nonlinear optimization MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information