Chandrasekhar Type Algorithms. for the Riccati Equation of Lainiotis Filter
|
|
- Ralph Blake
- 5 years ago
- Views:
Transcription
1 Cotemporary Egieerig Scieces, Vol. 3, 00, o. 4, 9-00 Chadrasekhar ype Algorithms for the Riccati Equatio of Laiiotis Filter Nicholas Assimakis Departmet of Electroics echological Educatioal Istitute of Lamia, Greece Abstract Chadrasekhar type algorithms for solvig the discrete time Riccati equatio ad Lyapuov equatio emaatig from Laiiotis filter are preseted. he Chadrasekhar type algorithms are compared to the classical per step algorithm cosistig of direct implemetatio of the recursio of the Riccati equatio or the Lyapuov equatio. It is show that Chadrasekhar type algorithms may be faster tha the classical oes. Keywords: Riccati equatio, Laiiotis filter, Chadrasekhar algorithm Itroductio he discrete time Riccati equatio arises i liear estimatio ad is associated with time ivariat systems described by the followig state space equatios: x( k + ) = Fx( k) + w( k) () zk ( ) = Hxk ( ) + vk ( ) () for k 0, where x( k ) is the dimesioal state vector at time k, zk ( ) is the m dimesioal measuremet vector, F is the system trasitio matrix, H is the output matrix, { wk ( )} ad { vk ( )} are idepedet Gaussia zero-mea white ad ucorrelated radom processes,q ad R are the plat ad measuremet oise covariace matrices respectively, ad x (0) is a Gaussia radom process with mea x 0 ad covariace P 0. he filterig/estimatio problem is to produce a estimate at time L of the state vector usig measuremets till time L, i.e. the aim is to use the measuremets set
2 9 N. Assimakis {(), z K, z( L)} i order to calculate a estimate value x( L/ L) of the state vector x( L ). he discrete time Laiiotis filter [4] is a well kow algorithm that solve the filterig problem, by computig the estimatio x( k / k ) as well as the estimatio error covariace matrix Pk ( / k ) for every k. he Laiiotis filter equatios provide a recursio for the dimetioal estimatio error covariace matrix Pk ( / k ), which is assumed to be o-egative defiite Pk ( / k) 0, the Riccati equatio emaatig from Laiiotis filter: Pk+ k+ = P+ F I+ Pk ko Pk kf (3) ( / ) [ ( / ) ] ( / ) with iitial coditio P(0 / 0) = P0, where P = Q QH AHQ F F QH AHF O F H AHF = (5) = () ad A = [ HQH + R] (7) For time ivariat systems, it is well kow [] that if the sigal process model is asymptotically stable (i.e. all eigevalues of F lie iside the uit circle), the there exists a steady state value P of the estimatio error covariace matrix. It is kow that Lyapuov equatio is derived from Riccati equatio whe R. I this case, A = 0 ad P = Q, F = F, O = 0 ad the Riccati equatio (3) becomes the Lyapuov equatio: Pk ( + / k+ ) = P+ FPk ( / kf ) (8) he discrete time Riccati equatio emaatig from the Laiiotis filter equatios has attracted eormous attetio. I view of the importace of the Riccati equatio, there exists cosiderable literature o its recursive solutios [3], [5], cocerig per step or doublig algorithms. I this paper Chadrasekhar type algorithms for solvig the discrete time Riccati equatio emaatig from Laiiotis filter are preseted ad compared to the classical per step algorithm, i.e. to the direct implemetatio of the recursio of the Riccati equatio. he paper is orgaized as follows: I sectio the classical recursive per step algorithm is preseted. I sectio 3 recursive Chadrasekhar type algorithms are preseted. I sectio 4 the computatioal requiremets of all algorithms are established ad comparisos are carried out. It is poited out that the Chadrasekhar type algorithms may be faster tha the classical per step algorithm. I additio, a rule is established i order to decide if the Chadrasekhar (4)
3 Chadrasekhar type algorithms 93 type algorithms are faster tha the classical oe. Per Step Algorithm he classical per step algorithm cosists of the direct implemetatio of the recursio of the Riccati equatio emaatig from the Laiiotis filter equatios. Also, the correspodig per step algorithm for the solutio of the Lyapuov equatio is preseted. Per Step Algorithm Riccati equatio (PSARE) he steady state solutio P is calculated by recursively implemetig the Riccati equatio (3) for k = 0,,..., with iitial coditio P(0 / 0) = P0, util the followig covergece criterio is satisfied: Pk ( + / k+ ) Pk ( / k) ε, where deotes the matrix orm ad ε is a small positive real umber pre-specified to give the steady state solutio to the accuracy desired. he steady state or limitig solutio P = lim P( k / k) of the Riccati equatio is idepedet of the k iitial coditio []. I the sequel we assume zero iitial coditio P (0 / 0) = 0, i.e. P 0 = 0. he we are able to use P(/) = P as iitial coditio. Note that the existace of [ I + P( k / k) O ] is guarateed due to the presece of the idetity matrix I. Also, the existece of A = [ HQH + R] is guarateed if R is positive defiite ( R > 0 ), which meas that o measuremet is exact. his is reasoable i physical problems. hus, the osigular measuremet oise covariaces matrix case is assumed i the sequel. Per Step Algorithm Lyapuov equatio (PSALE) Lyapuov equatio is derived from Riccati equatio whe R. he, the per step algorithm for the Lyapuov equatio cosists of the recursive implemetatio of the Lyapuov equatio (8) with iitial coditio P (/)=P. able summarizes the classical per step algorithms for solvig the Riccati ad the Lyapyov equatios emaatig from Laiiotis filter. able. Per Step Algorithms Riccati equatio Lyapuov equatio PSARE P k k P F I P k k O P k k F ( + / + ) = + [ + ( / ) ] ( / ) PSALE Pk ( + / k+ ) = P+ FPk ( / kf )
4 94 N. Assimakis 3 Chadrasekhar ype Algorithms he Chadrasekhar type algorithms use the idea of defiig the differece: δ Pk ( ) = Pk ( + / k+ ) Pk ( / k) (9) ad its factorizatio: δ Pk ( ) = YkSkY ( ) ( ) ( k) (0) with Yk ( ) of dimesio r ad Sk ( ) of dimesio r r, where 0 r = rak( δ P0 ) = rak( P ) = rak( Q) () he usig the quatity Ok ( ) = Pk ( / k) + O () the followig recursio is obvious: Ok ( + ) = Ok ( ) + YkSkY ( ) ( ) ( k) (3) Note that the o sigularity of O is guarateed if R is positive defiite ad if F is osigular. he Chadrasekhar type algorithms cosist of the recursio: Pk ( + / k+ ) = Pk ( / k) + YkSkY ( ) ( ) ( k) (4) usig recursios for the quatities Yk ( ) ad Sk ( ). wo versios of the Chadrasekhar type algorithms for the solutio of the Riccati equatio are preseted. Also, the correspodig Chadrasekhar type algorithm for the solutio of the Lyapuov equatio is preseted. Chadrasekhar ype Algorithm Riccati equatio versio (CARE) Settig Yk ( + ) = [ FO ] O ( kyk ) ( ) (5) after some algebra the followig recursio is derived [5]: Sk ( + ) = Sk ( ) SkY ( ) ( ko ) ( k+ ) YkSk ( ) ( ) () Note that the o sigularity of Ok ( ) is guarateed if O is osigular, which meas that R is positive defiite ( R > 0 ). Assumig zero iitial coditio P (0 / 0) = 0, we use the followig iitial coditios: O(0) = O Y(0) S(0) Y (0) = P Remarks.. If Q has full rak ( r = ), the we are able touse the iitial coditios Y(0) = I ad S(0) = P.. If Q = 0 ( r = 0 ), the A= R ad P = 0, F = F, O = FH R HF. So the estimatio error covariace is Pk ( / k) = P0 =0 ad the limitig value P of the
5 Chadrasekhar type algorithms 95 estimatio error covariace is P = 0. Chadrasekhar ype Algorithm Riccati equatio versio (CARE) Settig Yk ( + ) = [ FO ] O ( k+ ) Yk ( ) (7) after some algebra, workig as i [5], the followig recursio is derived: Sk ( + ) = Sk ( ) + SkY ( ) ( ko ) ( kyksk ) ( ) ( ) (8) with the same iitial coditios used i CARE. Chadrasekhar ype Algorithm Lyapuov equatio (CALE) Lyapuov equatio is derived from Riccati equatio whe R. he, the Chadrasekhar type algorithm for the Lyapuov equatio becomes: Yk ( + ) = FYk ( ) (9) Pk ( + / k+ ) = Pk ( / k) + YkY ( ) ( k) (0) with iitial coditios P (0 / 0)=0 Y(0) Y (0) = P able summarizes the Chadrasekhar type algorithms for solvig the Riccati ad the Lyapyov equatios emaatig from Laiiotis filter.. able. Chadrasekhar ype Algorithms Ok ( + ) = Ok ( ) + YkSkY ( ) ( ) ( k) Riccati equatio Lyapuov equatio CARE CARE CALE Yk FO O kyk ( + ) = [ ] ( ) ( ) ( + ) = ( ) ( ) ( ) ( + ) ( ) ( ) Sk Sk SkY ko k YkSk Pk ( + / k+ ) = Pk ( / k) + YkSkY ( ) ( ) ( k) Ok ( + ) = Ok ( ) + YkSkY ( ) ( ) ( k) Yk FO O k Yk ( ) [ + = ] ( + ) ( ) ( + ) = ( ) + ( ) ( ) ( ) ( ) ( ) Sk Sk SkY ko kyksk Pk ( + / k+ ) = Pk ( / k) + YkSkY ( ) ( ) ( k) Yk ( + ) = FYk ( ) 4 Computatioal Compariso of Algorithms Pk ( + / k+ ) = Pk ( / k) + YkY ( ) ( k) Both the per step ad the Chadrasekhar type algorithms are recursive oes. hus, the total computatioal time required for the implemetatio of each algorithm is: ta ( lg) = CBa ( lg) Sa ( lg) top () where CB( a lg) is the per recursio calculatio burde required for the o-lie calculatios of each algorithm, Sa ( lg) is the umber of recursios (steps) that
6 9 N. Assimakis each algorithm executes ad top is the time required to perform a scalar operatio. he per step ad the Chadrasekhar type algorithms preseted above are equivalet with respect to their behavior: they calculate theoretically the same steady state estimatio error variace. he, it is reasoable to assume that both algorithms compute the limitig solutio of the Riccati equatio (or the correspodig Lyapuov equatio) executig the same umber of recursios, depedig o the desired accuracy. hus, i order to compare the algorithms with respect to their computatioal time, we have to compare their per recursio calculatio burde required for the o-lie calculatios; the calculatio burde of the off-lie calculatios (iitializatio process) is ot take ito accout. he computatioal aalysis is based o the aalysis i []: scalar operatios are ivolved i matrix maipulatio operatios, which are eeded for the implemetatio of the filterig algorithms. able 3 summarizes the calculatio burde of eeded matrix operatios. able 3. Calculatio Burde of Matrix Operatios Matrix Operatio Calculatio Burde A( m) + B( m) = C( m) m A( ) + B( ) = S( ) S : I( ) + A( ) = B( ) I : idetity A( m) B( m k) = C( k) mk k A( m) B( m ) = S( ) S : m m ( ) [ A( )] 3 = B( ) he recursive computatioal requiremets of all per step ad Chadrasekhar type algorithms for solvig the Riccati equatio ad the Lyapuov equatio are summarized i able 4. he details are give i the Appedix. able 4. Per Recursio Calculatio Burde of Algorithms 3 Riccati PSARE (5 + ) equatio 3 CARE (3 3 + ) + 3r r+ 7 r Lyapuov equatio PSALE 3 3 CALE 3 r From able 4, we derive the followig coclusios:. he per recursio calculatio burdes of the classical per step algorithms deped oly o the state dimesio. he per recursio calculatio burdes of the Chadrasekhar type algorithms
7 Chadrasekhar type algorithms 97 deped o the state dimesio ad o dimesio r.. he two versios of Chadrasekhar type algorithms are equivalet with respect to their computatioal burdes. 3. Cocerig the Riccati equatio solutio algorithms: - if r =, the the classical per step algorithm is faster tha the Chadrasekhar type algorithms - if r <, the the Chadrasekhar type algorithms may be faster tha the classical per step algorithm; i fact Chadrasekhar type algorithms are faster tha the classical per step algorithm if the followig relatio holds: 3 CB( PSARE) CB( CARE) = (0 3 + ) (3r r + 7 r) > 0 () Figure depicts the relatio betwee the dimesios ad r that may hold i order to decide which algorithm is faster. 45 Chadrasekhar type algorithm vs Per step algorithm PSA dimesio r CA dimesio Figure. Chadrasekhar type algorithm may be faster tha per step algorithm he, we are able to establish the followig Rule of humb: Chadrasekhar type algorithms are faster tha the classical per step algorithm if the followig relatio holds: r < 0.4 (3) 4. Cocerig the Lyapuov equatio solutio algorithms: - if r =, the the classical per step algorithm is as fast as the Chadrasekhar type algorithm - if r <, the the Chadrasekhar type algorithm is faster tha the classical per step algorithm hus, Chadrasekhar type algorithms possess the advatage that there is a reductio i computatioal burde i compariso to the classical per step algorithm, especially whe r is eough less tha.
8 98 N. Assimakis Refereces [] B. D. O. Aderso, J. B. Moore, Optimal Filterig, Pretice Hall ic., 979. [] N. Assimakis ad M. Adam, Discrete time Kalma ad Laiiotis filters compariso, It. Joural of Mathematical Aalysis (IJMA), (007), [3] N. D. Assimakis, D. G. Laiiotis, S. K. Katsikas, F. L. Saida, A survey of recursive algorithms for the solutio of the discrete time Riccati equatio, Noliear Aalysis, heory, Methods & Applicatios, 30 (997), [4] Laiiotis D. G., Partitioed liear estimatio algorithms: Discrete case, IEEE ras. o AC, vol. AC-0, pp , 975. [5] Laiiotis D. G., Assimakis N. D., Katsikas S. K., A ew computatioally effective algorithm for solvig the discrete Riccati equatio, Joural of Mathematical Aalysis ad Applicatios, vol. 8, o. 3, pp , 994. Appedix. Calculatio burdes of algorithms A. Per Step Algorithms Per Step Algorithm Riccati equatio (PSARE) Matrix Operatio Matrix Dimesios Calculatio Burde P( k / k) O ( ) ( ) 3 ( ) + ( ) [ I + P( k / k) O ] I idetity [ I + P( k / k) O ] ( ) 3 ( ) ( ) 3 [ I + P( k / k) O ] P( k / k) F[ I + P( k / k) O] P( k / k) ( ) ( ) 3 ( ) ( ) 3 F[ I + P( k / k) O] P( k / k) F Pk ( + / k+ ) = P ( ) + ( ) + F[ I + P( k / k) O] P( k / k) F PSARE otal 3 (5 + ) Per Step Algorithm Lyapuov equatio (PSALE) Matrix Operatio Matrix Dimesios Calculatio Burde F Pk ( / k ) ( ) ( ) 3 ( ) ( ) 3 FPk ( / kf ) P( k + / k + ) = P ( ) + ( ) + FPk ( / k) F 3 PSALE otal 3
9 Chadrasekhar type algorithms 99 B. Chadrasekhar ype Algorithms Chadrasekhar ype Algorithm Riccati equatio versio / (CARE/) Matrix Calculatio Matrix Operatio Matrix Operatio Dimesios Burde YkSk ( ) ( ) YkSk ( ) ( ) ( r) ( r r) r r ( r) ( r ) YkSkY ( ) ( ) ( k ) YkSkY ( ) ( ) ( k ) r + r ( + ) Ok ( + ) = Ok ( ) + YkSkY ( ) ( ) ( k) Ok ( + ) = Ok ( ) + YkSkY ( ) ( ) ( k) O ( k) O ( k + ) ( ) + ( ) ( ) ( + ) ( 3 3 ) O ( k) Y( k) O ( k + ) Y( k) ( ) ( r) r Yk ( + ) = [ FO ] Yk ( + ) = [ FO ] ( ) ( r) r O ( k) Y( k) O ( k + ) Y( k) O ( k + ) O ( k) ( ) r r ( 3 3 ) O ( k + ) Y( k) S( k) O ( k) Y( k) S( k) ( ) ( r) r r SkY ( ) ( k) SkY ( ) ( k) ( r ) ( r) r O ( k + ) Y( k) S( k) O ( k) Y( k) S( k) + r ( r + r ) Sk ( + ) = Sk ( ) Sk ( + ) = Sk ( ) ( r r) + ( r r) SkY ( ) ( k) + SkY ( ) ( k) ( r + r) O ( k + ) Y( k) S( k) O ( k) Y( k) S( k) Pk ( + / k+ ) = Pk ( / k) Pk ( + / k+ ) = Pk ( / k) ( ) + ( ) + YkSkY ( ) ( ) ( k) + YkSkY ( ) ( ) ( k) ( + ) 3 (3 3 + ) CARE CARE otal + 3r r + 7 r
10 00 N. Assimakis Chadrasekhar ype Algorithm Lyapuov equatio (CALE) Matrix Operatio Matrix Dimesios Calculatio Burde Yk ( + ) = FYk ( ) ( ) ( r) r r YkY ( ) ( k ) ( r) ( r ) r r ( ) Pk ( + / k+ ) = ( ) + ( ) P( k / k) + Y( k) Y ( k) CALE otal 3 r Received: April, 00
Lainiotis filter implementation. via Chandrasekhar type algorithm
Joural of Computatios & Modellig, vol.1, o.1, 2011, 115-130 ISSN: 1792-7625 prit, 1792-8850 olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais
More informationTHE KALMAN FILTER RAUL ROJAS
THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider
More informationRandom Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices
Radom Matrices with Blocks of Itermediate Scale Strogly Correlated Bad Matrices Jiayi Tog Advisor: Dr. Todd Kemp May 30, 07 Departmet of Mathematics Uiversity of Califoria, Sa Diego Cotets Itroductio Notatio
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationA Simplified Derivation of Scalar Kalman Filter using Bayesian Probability Theory
6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede A Simplified Derivatio of Scalar Kalma Filter usig Bayesia Proaility Theory Gregory Kyriazis Imetro - Electrical Metrology
More informationAN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS
http://www.paper.edu.c Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () 119 15 c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC
More information6. Kalman filter implementation for linear algebraic equations. Karhunen-Loeve decomposition
6. Kalma filter implemetatio for liear algebraic equatios. Karhue-Loeve decompositio 6.1. Solvable liear algebraic systems. Probabilistic iterpretatio. Let A be a quadratic matrix (ot obligatory osigular.
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
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 informationA NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS
Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet
More informationOPTIMAL PIECEWISE UNIFORM VECTOR QUANTIZATION OF THE MEMORYLESS LAPLACIAN SOURCE
Joural of ELECTRICAL EGIEERIG, VOL. 56, O. 7-8, 2005, 200 204 OPTIMAL PIECEWISE UIFORM VECTOR QUATIZATIO OF THE MEMORYLESS LAPLACIA SOURCE Zora H. Perić Veljo Lj. Staović Alesadra Z. Jovaović Srdja M.
More informationEstimation of Backward Perturbation Bounds For Linear Least Squares Problem
dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces,
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More information5. Fast NLMS-OCF Algorithm
5. Fast LMS-OCF Algorithm The LMS-OCF algorithm preseted i Chapter, which relies o Gram-Schmidt orthogoalizatio, has a compleity O ( M ). The square-law depedece o computatioal requiremets o the umber
More informationStatistical 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 informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationCMSE 820: Math. Foundations of Data Sci.
Lecture 17 8.4 Weighted path graphs Take from [10, Lecture 3] As alluded to at the ed of the previous sectio, we ow aalyze weighted path graphs. To that ed, we prove the followig: Theorem 6 (Fiedler).
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationThe Basic Space Model
The Basic Space Model Let x i be the ith idividual s (i=,, ) reported positio o the th issue ( =,, m) ad let X 0 be the by m matrix of observed data here the 0 subscript idicates that elemets are missig
More informationVariable selection in principal components analysis of qualitative data using the accelerated ALS algorithm
Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm Masahiro Kuroda Yuichi Mori Masaya Iizuka Michio Sakakihara (Okayama Uiversity of Sciece) (Okayama Uiversity
More informationThe random version of Dvoretzky s theorem in l n
The radom versio of Dvoretzky s theorem i l Gideo Schechtma Abstract We show that with high probability a sectio of the l ball of dimesio k cε log c > 0 a uiversal costat) is ε close to a multiple of the
More informationVector Quantization: a Limiting Case of EM
. Itroductio & defiitios Assume that you are give a data set X = { x j }, j { 2,,, }, of d -dimesioal vectors. The vector quatizatio (VQ) problem requires that we fid a set of prototype vectors Z = { z
More informationA Genetic Algorithm for Solving General System of Equations
A Geetic Algorithm for Solvig Geeral System of Equatios Győző Molárka, Edit Miletics Departmet of Mathematics, Szécheyi Istvá Uiversity, Győr, Hugary molarka@sze.hu, miletics@sze.hu Abstract: For solvig
More informationBounds for the Extreme Eigenvalues Using the Trace and Determinant
ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,
More informationStatistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons
Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA Outlies. Itroductio.
More informationPC5215 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 information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationIterative method for computing a Schur form of symplectic matrix
Aals of the Uiversity of Craiova, Mathematics ad Computer Sciece Series Volume 421, 2015, Pages 158 166 ISSN: 1223-6934 Iterative method for computig a Schur form of symplectic matrix A Mesbahi, AH Betbib,
More information3/8/2016. Contents in latter part PATTERN RECOGNITION AND MACHINE LEARNING. Dynamical Systems. Dynamical Systems. Linear Dynamical Systems
Cotets i latter part PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Liear Dyamical Systems What is differet from HMM? Kalma filter Its stregth ad limitatio Particle Filter Its simple
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationDiscrete-Time Signals and Systems. Discrete-Time Signals and Systems. Signal Symmetry. Elementary Discrete-Time Signals.
Discrete-ime Sigals ad Systems Discrete-ime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. -.5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms,
More informationA Note on Effi cient Conditional Simulation of Gaussian Distributions. April 2010
A Note o Effi ciet Coditioal Simulatio of Gaussia Distributios A D D C S S, U B C, V, BC, C April 2010 A Cosider a multivariate Gaussia radom vector which ca be partitioed ito observed ad uobserved compoetswe
More informationMixtures of Gaussians and the EM Algorithm
Mixtures of Gaussias ad the EM Algorithm CSE 6363 Machie Learig Vassilis Athitsos Computer Sciece ad Egieerig Departmet Uiversity of Texas at Arligto 1 Gaussias A popular way to estimate probability desity
More informationNumerical Method for Blasius Equation on an infinite Interval
Numerical Method for Blasius Equatio o a ifiite Iterval Alexader I. Zadori Omsk departmet of Sobolev Mathematics Istitute of Siberia Brach of Russia Academy of Scieces, Russia zadori@iitam.omsk.et.ru 1
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More informationCS537. 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 informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationStudy on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm
Joural of ad Eergy Egieerig, 05, 3, 43-437 Published Olie April 05 i SciRes. http://www.scirp.org/joural/jpee http://dx.doi.org/0.436/jpee.05.34058 Study o Coal Cosumptio Curve Fittig of the Thermal Based
More informationHigher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
More informationChapter 1 Simple Linear Regression (part 6: matrix version)
Chapter Simple Liear Regressio (part 6: matrix versio) Overview Simple liear regressio model: respose variable Y, a sigle idepedet variable X Y β 0 + β X + ε Multiple liear regressio model: respose Y,
More informationA note on the modified Hermitian and skew-hermitian splitting methods for non-hermitian positive definite linear systems
A ote o the modified Hermitia ad skew-hermitia splittig methods for o-hermitia positive defiite liear systems Shi-Liag Wu Tig-Zhu Huag School of Applied Mathematics Uiversity of Electroic Sciece ad Techology
More informationki, X(n) lj (n) = (ϱ (n) ij ) 1 i,j d.
APPLICATIONES MATHEMATICAE 22,2 (1994), pp. 193 200 M. WIŚNIEWSKI (Kielce) EXTREME ORDER STATISTICS IN AN EQUALLY CORRELATED GAUSSIAN ARRAY Abstract. This paper cotais the results cocerig the wea covergece
More informationProblem Set 2 Solutions
CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S
More informationPOWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS
Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationCO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS
CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS C.PRAX ad H.SADAT Laboratoire d'etudes Thermiques,URA CNRS 403 40, Aveue du Recteur Pieau 86022 Poitiers Cedex,
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
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 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 informationECE 308 Discrete-Time Signals and Systems
ECE 38-5 ECE 38 Discrete-Time Sigals ad Systems Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa ECE 38-5 1 Additio, Multiplicatio, ad Scalig of Sequeces Amplitude Scalig: (A Costat
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationGenerating Functions for Laguerre Type Polynomials. Group Theoretic method
It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationMathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical
More informationThe Perturbation Bound for the Perron Vector of a Transition Probability Tensor
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Liear Algebra Appl. ; : 6 Published olie i Wiley IterSciece www.itersciece.wiley.com. DOI:./la The Perturbatio Boud for the Perro Vector of a Trasitio
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 informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationLinear 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 informationLast time: Moments of the Poisson distribution from its generating function. Example: Using telescope to measure intensity of an object
6.3 Stochastic Estimatio ad Cotrol, Fall 004 Lecture 7 Last time: Momets of the Poisso distributio from its geeratig fuctio. Gs () e dg µ e ds dg µ ( s) µ ( s) µ ( s) µ e ds dg X µ ds X s dg dg + ds ds
More informationChapter 12 EM algorithms The Expectation-Maximization (EM) algorithm is a maximum likelihood method for models that have hidden variables eg. Gaussian
Chapter 2 EM algorithms The Expectatio-Maximizatio (EM) algorithm is a maximum likelihood method for models that have hidde variables eg. Gaussia Mixture Models (GMMs), Liear Dyamic Systems (LDSs) ad Hidde
More informationPolynomial Multiplication and Fast Fourier Transform
Polyomial Multiplicatio ad Fast Fourier Trasform Com S 477/577 Notes Ya-Bi Jia Sep 19, 2017 I this lecture we will describe the famous algorithm of fast Fourier trasform FFT, which has revolutioized digital
More information1. Linearization of a nonlinear system given in the form of a system of ordinary differential equations
. Liearizatio of a oliear system give i the form of a system of ordiary differetial equatios We ow show how to determie a liear model which approximates the behavior of a time-ivariat oliear system i a
More informationMonte Carlo Optimization to Solve a Two-Dimensional Inverse Heat Conduction Problem
Australia Joural of Basic Applied Scieces, 5(): 097-05, 0 ISSN 99-878 Mote Carlo Optimizatio to Solve a Two-Dimesioal Iverse Heat Coductio Problem M Ebrahimi Departmet of Mathematics, Karaj Brach, Islamic
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 informationCS321. 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 informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationOn the convergence rates of Gladyshev s Hurst index estimator
Noliear Aalysis: Modellig ad Cotrol, 2010, Vol 15, No 4, 445 450 O the covergece rates of Gladyshev s Hurst idex estimator K Kubilius 1, D Melichov 2 1 Istitute of Mathematics ad Iformatics, Vilius Uiversity
More informationExact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method
Exact Solutios for a Class of Noliear Sigular Two-Poit Boudary Value Problems: The Decompositio Method Abd Elhalim Ebaid Departmet of Mathematics, Faculty of Sciece, Tabuk Uiversity, P O Box 741, Tabuki
More informationSolving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods
Applied ad Computatioal Mathematics 07; 6(6): 38-4 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 38-5605 (Prit); ISSN: 38-563 (Olie) Solvig a Noliear Equatio Usig a New Two-Step
More informationA RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a
More informationITERATIVE SOLUTION OF TWO MATRIX EQUATIONS
ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS CHUN-HUA GUO AND PETER LANCASTER Abstract We study iterative methods for fidig the maximal Hermitia positive defiite solutios of the matrix equatios X + A X 1
More informationApplied Mathematics Letters
Applied Mathematics Letters 5 (01) 03 030 Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joural homepage: www.elsevier.com/locate/aml O ew computatioal local orders of covergece
More informationA CHOLESKY LR ALGORITHM FOR THE POSITIVE DEFINITE SYMMETRIC DIAGONAL-PLUS-SEMISEPARABLE EIGENPROBLEM
A CHOLESKY LR ALGORITHM FOR THE POSITIVE DEFINITE SYMMETRIC DIAGONAL-PLUS-SEMISEPARABLE EIGENPROBLEM BOR PLESTENJAK, ELLEN VAN CAMP, AND MARC VAN BAREL Abstract. We preset a Cholesky LR algorithm with
More informationAn Alternative Scaling Factor In Broyden s Class Methods for Unconstrained Optimization
Joural of Mathematics ad Statistics 6 (): 63-67, 00 ISSN 549-3644 00 Sciece Publicatios A Alterative Scalig Factor I Broyde s Class Methods for Ucostraied Optimizatio Muhammad Fauzi bi Embog, Mustafa bi
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 informationNumerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie
More informationSlide Set 13 Linear Model with Endogenous Regressors and the GMM estimator
Slide Set 13 Liear Model with Edogeous Regressors ad the GMM estimator Pietro Coretto pcoretto@uisa.it Ecoometrics Master i Ecoomics ad Fiace (MEF) Uiversità degli Studi di Napoli Federico II Versio: Friday
More informationANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY
ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,
More informationTHE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES
COMMUN. STATIST.-STOCHASTIC MODELS, 0(3), 525-532 (994) THE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES Jack W. Silverstei Departmet of Mathematics, Box 8205 North Carolia
More informationSUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHYS309 O the reality of the quatum state Matthew F. Pusey, 1, Joatha Barrett, ad Terry Rudolph 1 1 Departmet of Physics, Imperial College Lodo, Price Cosort Road, Lodo SW7 AZ, Uited Kigdom
More informationMath 25 Solutions to practice problems
Math 5: Advaced Calculus UC Davis, Sprig 0 Math 5 Solutios to practice problems Questio For = 0,,, 3,... ad 0 k defie umbers C k C k =! k!( k)! (for k = 0 ad k = we defie C 0 = C = ). by = ( )... ( k +
More informationIN many scientific and engineering applications, one often
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several
More informationDerivative of a Determinant with Respect to an Eigenvalue in the LDU Decomposition of a Non-Symmetric Matrix
Applied Mathematics, 203, 4, 464-468 http://dx.doi.org/0.4236/am.203.43069 Published Olie March 203 (http://www.scirp.org/joural/am) Derivative of a Determiat with Respect to a Eigevalue i the LDU Decompositio
More informationUsing An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically
ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad Mid-Poit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab
More informationInfinite 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 informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES 2012 Pearso Educatio, Ic. Theorem 8: Let A be a square matrix. The the followig statemets are equivalet. That is, for a give A, the statemets
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationComplex 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 informationChapter 10 Advanced Topics in Random Processes
ery Stark ad Joh W. Woods, Probability, Statistics, ad Radom Variables for Egieers, 4th ed., Pearso Educatio Ic.,. ISBN 978--3-33-6 Chapter Advaced opics i Radom Processes Sectios. Mea-Square (m.s.) Calculus
More informationHOMEWORK #10 SOLUTIONS
Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More information