روش‌هاي مقياس‌بندي و مربع‌سازي براي محاسبه‌‌ي تابع‌نمايي‌ ماتريسي

چكيده انگليسي :

Scaling and Squaring Methods for Computing the Matrix Exponential ZEINAB MEMAR September 16 2019 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mehdi tatari mtatari@cc iut ac irAdvisor Dr Amir Hashemi amir hashemi@cc iut ac ir2000 MSC 65F60 15A60 65F30Keywords matrix exponential scaling and squaring method Pad e approximation backward error analysis over scaling Abstract Modelling most of the problems caused by physical economic and biological processes leads to differential equa tions where the exponential function of a matrix eA is used For example the matrix exponential appeared in solvingthe linear first order ordinary differential equation Therefore it is important to present a proper numerical methodfor approximating the solutions In 2005 Higham presented an algorithm based on scaling and squaring technique 11 The scaling and squaringtechnique is widely used to computing the matrix exponential so that MATLAB s expm function is based on this smethod The scaling and squaring method exploit eA e2 A 2 rm 2 s A 2 where rm x is the m m s sPad e approximation to ex where the approximation is formed by repeating squarings s times Non negative integerm and s are selected optimally The approximation of eA by a Pad e approximation for small A can be computed accurately near the origin When A is large the value of s selected a large integer by the scaling and squaring method which has harmful effect onaccuracy and it also has high computational costs In 2009 Higham and Al Mohy reform this algorithm by using two technique 1 In this thesis we study their im provement The rst method is for triangular matrices In this idea errors in diagonal elements are removed and errorin non diagonal elements are reduced by computing Di expm 2 i diag T at each steps instead of computingDs rm 2 s diag T and then repeatedly squaring are performed In the second method an idea is presented to overcome overscaling It uses practical bound for norm of matrix powerseries and it tries to improve the backward error This algorithm employs more re ned truncation error bounds based 1 which can be smaller than the bounds based on A that was used in Ak kon the members of the sequence 1 Ak followed by the choice of k The value of k kscaling and squaring method It is necessary to employmust be chosen in such a way as to strike the balance between overscaling and creating an appropriate error bound 1 1 A2p A2p 2As a result by focusing on the backward error bounds we find that max can be used 2p 2p 2 instead of A where 2m 2p p 1 For non normal matrices this expression can be much smaller As an im 1portant point it is not necessary to compute powers of A and A to get values of Ak k insted MATLAB snormest and normAm functions can be implemented By IEEE double precision arithmetic the solutions providedlead to algorithm that is either high computationally accurate or low cost or both occur Numerical experiments are performed to compare scaling and squaring algorithms with presented improvement Thealgorithms are implemented on a set of matrices and the computational cost relative error and values of s are com pared These numerical experiments show that by applying the new algorithm in most cases the values of s decreasesand thus the relative errors reduced

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روش‌هاي مقياس‌بندي و مربع‌سازي براي محاسبه‌‌ي تابع‌نمايي‌ ماتريسي

مقياس‌بندي و مربع‌سازي , تقريب پاده , تابع نمايي ماتريس , آناليز خطاي پسرو