UnEncrypted
.pdfMulti-scale models for drug resistant tuberculosis |
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Rodrigues, P.; Rebelo, C.; Gomes, M.G.M. ................................................................. |
1071 |
A new approach for adaptive linear discrimination in brain computer interfaces |
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Rodríguez-Bermúdez, G.; García-Laencina, P.J.; Roca-González, J. ........................ |
1083 |
A Bicriterion Server Allocation Problem for a Queueing Loss System |
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Sá Esteves, J. .............................................................................................................. |
1087 |
Numerical Methods for the Computation of Stability boundaries for Structured |
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population models |
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Sánchez, J.; Getto, P.; de Roos, A.M.; Lessard, J.P. .................................................. |
1094 |
Input-Output Systems in the Study of Dichotomy and Trichotomy of Discrete |
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Dynamical Systems |
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Sasu, A.L.; Sasu, B. ..................................................................................................... |
1096 |
A Comparative Study on the Dichotomy Robustness of Discrete Dynamical Systems |
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Sasu, B.; Sasu, A.L. ..................................................................................................... |
1099 |
CRYSCOR, a program for post Hartree-Fock calculations on periodic systems |
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Schütz, M. ..................................................................................................................... |
1101 |
Parallelization of the interpolation process in the Koetter-Vardy soft-decision list |
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decoding algorithm |
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Simarro-Haro, M.A.; Moreira, J.; Fernández, M.; Soriano, M.; González, A.; Martínez- |
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Zaldívar, F.J. ................................................................................................................ |
1102 |
Numerical simulation of a receptor-toxin-antibody interaction |
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Skakauskas, V.; Katauskis, P.; Skvortsov, A. .............................................................. |
1111 |
Fractional calculus and superdiffusion in epidemiology: shift of critical thresholds |
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Skwara, U.; Martins, J.; Ghaffari, P.; Aguiar, M.; Boto, J.; Stollenwerk, N. ................. 1118 |
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Completed Richardson Extrapolation for Option Pricing |
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Tangman, D.Y. ............................................................................................................. |
1130 |
Parallelization and Performance Analysis of a Brownian Dynamics Simulation |
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using OpenMP |
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Teijeiro, C.; Sutmann, G.; Taboada, G.L.; Touriño, J. ................................................. |
1143 |
Forward-Backward Differential Equations: Approximation of Small Solutions |
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Teodoro, M.F.; Lima, P.M.; Ford, N.J.; Lumb, P.M. ..................................................... |
1155 |
Measuring the Impact of Configuration Parameters in CUDA Through |
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Benchmarking |
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Torres, Y.; González-Escribano, A.; Llanos, D.R. ....................................................... 1161 |
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A Factorized Novel Bound Analysis For Multivariate Data Modelling: |
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Interval Factorized HDMR |
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Tunga, B.; Demiralp, M. ............................................................................................... |
1173 |
Probabilistic Evolution of the State Variable Expected Values in Liouville Equation Perspective, for a Many Particle System Interacting Via Elastic Forces
Tunga, B.; Demiralp, M. ............................................................................................... |
1186 |
Page 13 of 1573
Solution for a two-dimensional Lamb´s problem using GFDM
Ureña, F.; Benito, J.J.; Gavete, L.; Salete, E.; Alonso, A. ........................................... 1198
A not so common boundary problem related to the membrane equilibrium equations
Viglialoro, G.; Murcia, J. ............................................................................................... |
1206 |
Strategy for selecting the frequencies in trigonometrically-fitted Störmer/Verlet type methods
Vigo-Aguiar, J.; Ramos, H. .......................................................................................... 1212
The method of increments: an extension to the multi-refererence treatment in metals
Voloshina, E.; Paulus, B. .............................................................................................. |
1223 |
Exponential time differenting schemes for reaction-diffusion problems |
|
Wade, B. A. .................................................................................................................. |
1227 |
File fragment classification: An application of a neural network and linear programming based discriminant model
Wilgenbus, E.; Kruger, H.; du Toit, T. .......................................................................... |
1237 |
Page 14 of 1573
?
Volume IV.................................................................................................................... |
1249 |
Index............................................................................................................................. |
1251 |
Mathematical model to predict the effects of pregnancy on antibody response during viral infection
Abdulhafid, A.; Andreansky, S.; Haskell, E.C. ............................................................. |
1267 |
Models for copper(I)-binding sites in proteins |
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Ahte, P.; Eller, N.A.; Palumaa, P.; Tamm, T. ............................................................... |
1275 |
Comparison of eigensolvers efficiency in quadratic eigenvalue problems |
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Aires, S.M.; d' Almeida, F.D. ........................................................................................ |
1279 |
An efficient and reliable model to simulate elastic, 1-D transversal waves |
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Alcaraz, M.; Morales, J.L.; Alhama, I.; Alhama, F. ....................................................... |
1284 |
Density driven fluid flow and heat transport in porous: Numerical simulation by network method
Alhama, I.; Canovas, M.; Alhama, F. ........................................................................... |
1290 |
Tilted Bianchi Type IX Cosmological Model in General Relativity |
|
Bagora(Menaria), A.; Purohit R..................................................................................... |
1298 |
Catalytic reactions of free gold and palladium clusters in an ion trap |
|
Bernhardt, T.M. ............................................................................................................. |
1309 |
Prediction of Stable Low Density Materials Inspired by Nanocluster Building Block Assembly
Bromley, S.T. .............................................................................................................. 1314
Numerical Methods for the Intrinsic Analysis of Fluid Interfaces: Applications to Ionic Liquids
Cordeiro, M.N.D.S.; Jorge, M. ...................................................................................... |
1318 |
Mathematical Model for Food Gums Using Non-Integer Order Calculus |
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David S. A.; Katayama, A.H.; de Oliveira, C. ............................................................... |
1321 |
Page 15 of 1573
Improving Metadata Management in a Distributed File System |
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Díaz, A.F.; Anguita, M.; Ortega, J. ............................................................................... |
1333 |
Computational soft modeling of video images of a gas-liquid transfer |
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experiment |
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Ferreira, M.M.C.; Gurden, S.P.; de Faria, C.G ............................................................ |
1337 |
A Direct Algorithm for Finding Nash Equilibrium |
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Gao, L.S. ...................................................................................................................... |
1338 |
Pole: A Planning Tool to Maximize the Network Lifetime in Wireless Sensor |
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Networks |
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Garcia-Sanchez, A.J.; Garcia-Sanchez, F.; Rodenas-Herraiz, D.; Garcia-Haro, J. .... |
1345 |
Gallium Clusters: from superheating to superatoms |
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Gaston, N.; Schebarchov, D.; Steenbergen, K.G. ....................................................... |
1357 |
Born Oppenheimer DFT molecular dynamics and DFT-MD methods for |
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biomolecules |
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Goursot, A.; Mineva, T.; Salahub, D.R. ........................................................................ |
1361 |
The Optimum Performance of Air-conditioning, Ventilation and Heat |
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Insulation Systems of Crew and Passenger Cabins of Airplanes |
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Gusev, S.A.; Nikolaev, V.N........................................................................................... |
1366 |
Atomistic Simulations of Functional Gold Nanoparticles in Biological |
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Environment |
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Heikkilä, E.; Gurtovenko, A.A.; Martinez-Seara, H.; Vattulainen, I.; Häkkinen, H.; |
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Akola, J. ....................................................................................................................... 1376 |
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A general purpose non-linear optimization framework based on Particle Swarm Optimization
Izquierdo, J.; Montalvo, I.; Herrera, M.; Pérez-García, R. ........................................... 1385
Quantum-chemical studies of organic molecular crystals - structure and |
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spectroscopy |
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Jacob, C.R.; Tonner, R. ............................................................................................... |
1397 |
Computational study of solids irradiated by intense x-ray free-electron lasers |
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Kitamura, H. ................................................................................................................. |
1402 |
Computational Methods for Problems of Viscoelastic Solid Deformation with |
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Application to the Diagnosis of Coronary Heart Disease |
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Kruse, C.; Maischak, M.; Shaw, S.; Whiteman, J.; Greenwald, S.; Brewin, M.; |
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Birch, M.; Banks, H.T.; Kenz, Z.; Hu, S. ....................................................................... |
1412 |
Modeling Earthen Dikes: Sensitivity Analysis and Calibration of Soil |
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Properties Based on Sensor Data |
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Krzhizhanovskaya, V.V.; Melnikova, N.B...................................................................... |
1414 |
Modeling of the Charge Density for Long and Short Channel Double Gate MOSFET Transistor
Latreche, S.; Smali B. .................................................................................................. 1425 |
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Effective rate constants for nanostructured heterogeneous catalysts |
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Lund, N.; Zhang, X.Y.; Gaston, N.; Hendy, S.C............................................................ |
1436 |
Page 16 of 1573
A Fast Recursive Blocked Algorithm for Dense Matrix Inversion |
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Mahfoudhi, R.; Mahjoub, Z............................................................................................ |
1440 |
Data Mining with Enhanced Neural Networks |
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Martínez, A.; Castellanos, A.; Sotto, A.; Mingo, L.F. ................................................... 1450 |
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Numerical methods for unsteady blood flow interaction with nonlinear |
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viscoelastic arterial vessel wall |
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Mihai, F.; Youn, I.; Seshaiyer, P. ................................................................................. 1462 |
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A mathematical model for the Container Stowage and Ship Routing Problem |
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Moura, A.; Oliveira, J.; Pimentel, C. ............................................................................. |
1473 |
Dimensional control of tunnels using topographic profiles: a functional |
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approach |
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Ordóñez, C.; Argüelles, R.; Martínez, J.; García-Cortés, S. ........................................ |
1485 |
Dynamic Analysis of Orthotropic Plates and Bridges Structure to Moving |
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Load |
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Rachid, L.; Meriem, O. ................................................................................................. |
1492 |
Optimal control strategies of Aedesaegypti mosquito population using the |
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sterile insect technique and insecticide |
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Rafikova, E.; Rafikov, M.; Mo Yang, H. ........................................................................ |
1504 |
Determining the thermal properties of drill cuttings using the point source |
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method: Thermal model and experiment procedure |
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Rey-Ronco, M.A.; Alonso-Sánchez, T.; Coppen-Rodríguez, J.; Castro-Gª, M.P. ....... |
1509 |
Electron Transfer and Other Reactions in Proteins – Towards an |
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Understanding of the Effects of Quantum Decoherence |
|
Salahub, D.R................................................................................................................. |
1521 |
Travelling wave solutions for ring topology neural fields |
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Salomon, F.; Haskell, E.C. ........................................................................................... |
1523 |
High-Pressure Simulations – Squeezing the Hell out of Atoms |
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Schwerdtfeger, P.; Biering, S.; Hasanbulli, M.; Hermann, A.; Wiebke, J.; Wormit, |
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M.; Pahl, E..................................................................................................................... |
1532 |
GA algorithm for generating geometric random variables of order k |
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Shmerling, E. ................................................................................................................ |
1534 |
Data analysis of photometric observations by HDAC onboard Cassini: 3D mapping and in-flight calibrations
Skorov, Y.;Reulke, R.; Keller, H.U.; Glassmeier, K.H. ................................................. |
1538 |
The transport properties of the near-surface porous layers of a commentary nucleus: Transition-probability and effective thermal conductivity
Skorov, Y.; Schmidt, H.; Blum, J.; Keller, H.U. ............................................................ |
1543 |
Dynamics of Conformational Modes in Biopolymers |
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Stepanova M; Potapov A. ............................................................................................ |
1547 |
Page 17 of 1573
Classification of Workers according to their Risk of Musculoskeletal Discomfort using the K-Nearest Neighbour Technique
Suárez Sánchez, A.; de Cos Juez, F.J.; Iglesias Rodríguez, F.J.; Sánchez |
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Lasheras, F.; García Nieto, P.J. ................................................................................... |
1548 |
On the Problem of Efficient Search of the Entire Set of Suboptimal Routes in a Transportation Network
Valuev, A. ..................................................................................................................... |
1560 |
Reactions of Aun+ (n = 1-4) with SiH4 and Finite Temperature Simulations of Aun (n = 24-40)
Vey, J.; Hamilton, I.P. ................................................................................................... |
1564 |
Computer vision algorithmization and intelligent traffic monitoring
Vinogradov, A. .............................................................................................................. |
1567 |
|
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Addendum:
A viability analysis for a stock/price model |
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Jerry C. and Raissi N..................................................................................................................... |
1574 |
Computing alphalimit sets: a note on their definition. |
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García-Guirao J.L. and Lampart M............................................................................................... |
1585 |
. |
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A Software Infrastructure for Test-Driven Learningin Introductory |
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Programming Courses |
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Guil F., Barón J, Corral A., Martínez M., Martín I........................................................................ |
1593 |
Page 18 of 1573
Proceedings of the 12th International Conference on Computational and Mathematical Methods
in Science and Engineering, CMMSE 2012 July, 2-5, 2012.
Principal Logarithm of matrix by recursive methods
J. Abderram´an Marrero1, R. Ben Taher2 and M. Rachidi2
1 Department of Mathematics Applied to Information Technologies, Telecommunication Engineering School, U.P.M. Technical University of Madrid, Spain
2 Group of DEFA - Department of Mathematics and Informatics, Faculty of Sciences,
University My Ismail, Meknes, Morocco
emails: jc.abderraman@upm.es, bentaher89@hotmail.fr, mu.rachidi@hotmail.fr
Abstract
We propose new methods for producing explicit representations of the principal matrix logarithm, without to use the Jordan canonical form of the original matrix. They are based on the H¨orner decomposition of the matrix and the Binet formula for the general solution of linear recurrence relations.
Key words: Binet formula, Recurrence relations, Matrix powers, Logarithm of a matrix.
MSC 2000: Primary 15A99, 40A05 Secondary 40A25, 45M05, 15A18.
1Decomposition of the principal logarithm of matrix
The logarithm of matrix occurs in various fields of mathematics, applied sciences and engineering; e.g. see [8]. Particularly, computing logarithms of real matrices turns out to be crucial in some class of problems of medical imaging and system of identification [1]. Various approaches, methods and algorithms are expanded in producing representations of the matrix logarithm (see [4, 6, 8]), and its computation still an exciting area. On the other hand, the numerical aspect of the matrix logarithm is also an interesting research subject (see [8]).
Originally, for a matrix B in Md(C), the algebra of square matrices, the problem consists in finding X Md(K), satisfying the equation eX = T . Any solution of this equation, denoted X = log(B), is called logarithm of B. It was shown in [6] that a matrix B has a logarithm (not necessary real) if and only if B is invertible. Moreover, the matrix equation exp(X) = B may have infinitely many solutions. Meanwhile, if B GL(d, C) have no
c CMMSE |
Page 19 of 1573 |
ISBN:978-84-615-5392-1 |
Principal Logarithm of matrix by recursive methods
eigenvalues on the closed negative real axis, there exists a unique logarithm X of B, called the principal logarithm of B and denoted by X = Log(B), e.g. see [4, 6, 8]. This unique matrix logarithm has all its eigenvalues into the horizontal strip determined by the condition {λi(X) C : |Im(λi(X))| < π}. In addition, if B is a real matrix then its principal logarithm is real.
The simplest way to define Log(B) is the Taylor series Log(B) = n≥0(−1)n (B−Id)n+1 ,
n+1
which makes sense when A = Id − B (Id is the identity matrix), satisfies A < 1, for any matrix norm . , or the spectral radius verifies ρ(A) = max{|λ|; λ σ(A)} < 1, σ(A) is the spectrum of A. More precisely, the computation of Log(B) was based in various
papers (see [4, 6]) on the powers series Log(Id − tA) = − |
n≥1 |
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An, when it converges. |
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For A = Id − B and t = 1 we recover the series of Log(B). Let A be in Md(C) such that |
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matrix). The decomposition of An (n ≥ r) in the H¨orner basis A0 = Id, A1 = A − a0Id,
· · · , Ar−1 = Ar−1 − a0Ar−2 − · · · − ar−2Id, is
An = unA0 + un−1A1 + ... + un−r+1Ar−1, for n ≥ r |
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(see [2, 3]) where u0 = 1, u−1 = · · · = u−r+1 = 0, and for n ≥ 1 we show that the term un verifies the linear recursive relation of order r,
un+1 = a0un + · · · + ar−1un−r+1, |
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where a0, a1, · · · , ar−1 are specified as the coe cients of {un}n≥−r+1 (see [5]). Using (1) we derive that, for t R with σ(Id − tA) ∩ (R− {0}) = and |t| < 1/ A , we have
Theorem 1 [H¨orner decomposition]. Under the preceding data, we obtain
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(−ϕs(t)) As, where ϕs(t) = |
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for n r. Therefore, the polynomial decomposition of Log(I |
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2Binet formula for the principal logarithm of matrix
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The solutions of sequence (2) can be written using the Binet formula un = |
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for all n N, where the λi (1 ≤ i ≤ l) are the roots of the polynomial R(z), of multi-
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ci,j nj λin = un(= δ0,−n), n = −r + 1, ..., −1, 0 (see [5]). Set 1 = {i (1 ≤ i ≤ |
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c CMMSE |
Page 20 of 1573 |
ISBN:978-84-615-5392-1 |
J. Abderraman´ Marrero, R. Ben Taher and M. Rachidi
l) : mi = 1} and |
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nk λn. For reason of generality we suppose that |
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Combining (3) of Theorem 1 with the Binet formula of (2), we derive the following main results on the new explicit representation for the principal logarithm of matrix.
Theorem 2 Let A be in Md(C) such that A < 1 and P (A) = Θd, where P (z) =
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(ar |
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= 0). Then, for t |
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tA) |
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we have Log(Id − tA) = |
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(Qs(t) + Φs(t) + Ψs(t))As, such that Qs(t) is a polynomial of |
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degree ≤ r − 1, given by |
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Qs(t) = |
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ciλin−s + |
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The function Φs(t) is |
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Φs(t) = |
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and finally the rational function Ψs(t) is |
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Ψs(t) = |
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(4)
(5)
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where D = t |
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satisfying Pn+1(t) = t(1 + t) |
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Example 1 Let consider B = |
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is a (real) companion matrix. |
Binet formula |
yields un = 4 |
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For- |
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mulas of Theorems 1 and 2 provide us the principal logarithm of matrix B; |
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basis of the matrix A. A direct computation shows that, |
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c CMMSE |
Page 21 of 1573 |
ISBN:978-84-615-5392-1 |
Principal Logarithm of matrix by recursive methods
3Concluding remarks and perspective
In the best of our knowledge expressions (4), (5) and (6) are not current in the literature. Generally in various studies the Jordan canonical form of the matrix B plays an important role for determining Log(Id − tA) (or Log(B)) (see [4, 8]). Meanwhile, the Jordan canonical form is not necessary for exhibiting the H¨orner (or polynomial) decomposition of Log(Id − tA), and also for the usage of Binet formula in Theorem 2 and Example 1.
Our methods have some interesting perspective. We have already obtained some results. Particularly, the problem of computing the principal logarithm of matrices of order 2 and 3 are detailed. Moreover, for σ(A) D = {z C : |z − 1| < 1}, our methods work by considering some matrix transformations.
References
[1]V. Arsigny, X. Pennec and N. Ayache, Polyrigid and polya ne transformations: a novel geometrical tool to deal with non-rigid deformations application to the registration of histological slices, Medical Image Analysis, 9 (2005), 507–523.
[2]R. Ben Taher and M. Rachidi, On the matrix powers and exponential by r- generalized Fibonacci sequences methods: the companion matrix case, Linear Algebra and Its Applications, 370 (2003) 341–353.
[3]R. Ben Taher, M. Mouline and M. Rachidi, Fibonacci-Horner decomposition of the matrix exponential and the fundamental system of solutions, Electron. J. Linear Algebra, 15 (2006) 178–190.
[4]W. J. Culver, On the existence and uniqueness of the real logarithm of matrix, Proc. of the Amer. Math. Soc., 17, No. 5 (1966) 1146–1151.
[5]F. Dubeau, W. Motta, M. Rachidi and O. Saeki, On weighted r-generalized Fibonacci sequences, Fibonacci Quarterly, 35 (1997) 102–110.
[6]F. R. Gantmacher, Theory of Matrices, Vol. I, Chelsea, New York, 1960.
[7]G. Golub and C. Van Loan, Matrix Computations, 3rd Ed., Johns Hopkins Press, Baltimore, 1996.
[8]N. Higham, Functions of Matrices: Theory and Computation. SIAM Philadelphia, 2008.
c CMMSE |
Page 22 of 1573 |
ISBN:978-84-615-5392-1 |