# Tema 3J - CNA-TEMA 3 1.Se considera matricea: ---- = 3 1 2...

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Unformatted text preview: CNA-TEMA 3 1.Se considera matricea: ---- = 3 1 2 5 1 2 3 3 A . Sa se calculeze « de mana« fara a folosi functii Matlab : Comparati rezultatele obtinute cu cele date prin aplicarea functiilor Matlab corespunzatoare a) detA= -6+(-6)-[-10-18]=16 ----- = 2 2 3 5 3 1 1 3 T A iar -- = 12 6 2 4 2 2 4 6 6 * A ---- ⋅ = ⋅ = ⇒- 12 6 2 4 2 2 4 6 6 16 1 * det 1 1 A A A In Matlab >> F=A^(-1) sau F=inv(A) F = 0.3750 0.3750 -0.2500 0.1250 0.1250 0.2500 0.1250 -0.3750 0.7500 b) 2 ln A e G = ) 8 cos( ) ) ) 2 ln 1 A H c e G b A F a A π = = =- X e G A X = ⇒ ---- = ⋅ ---- = ⋅ = 2 ln 3 2 ln 2 ln 2 2 ln 5 2 ln 2 ln 2 2 ln 3 2 ln 3 2 ln 3 1 2 5 1 2 3 3 2 ln Pentru determinarea valorii lui X se foloseste un algorim(algoritmul 3.1) in care matricea X este descompusa in vectori si valorii proprii. In Matlab >> [G]=alg31('exp',A*log(2)) G = 10.0000 -18.0000 12.0000-6.0000 22.0000 -12.0000-6.0000 18.0000 -8.0000 unde algoritmul de calcul este urmatorul : function [G]=alg31(f,A) [n,n]=size(A); [V,L]=eig(A); D=zeros(n,n); for i=1:n D(i,i)=feval(f,L(i,i)); end G=V*D/V; G=real(F); c) ⋅ = A H 8 cos π Y e H A Y = ⇒ ---- = ⋅ ---- = ⋅ = 8 3 8 8 2 8 5 8 8 2 8 3 8 3 8 3 1 2 5 1 2 3 3 8 π π π π π π π π π π In Matlab: (prin aplicarea aceluiasi algoritm se obtine): >> [H]=alg31('cos',A*pi/8) H = 0.3536 1.0607 -0.7071 0.3536 -0.3536 0.7071 0.3536 -1.0607 1.4142 2.Este adevarata urmatoarea afirmatie: Daca AB=BA atunci B A B A e e e ⋅ = + Argumentati. Egalitatea este adevarata in pe baza propozitiilor: 1.f(BAB-1 )=Bf(A)B-1 2.Af(A)=f(A)A deoarece:B=ABA-1 =>B=f(A). 3.Se va scrie programul Matlab pentru implementarea algoritmului de calcul al exponentialei matriciale bazat pe aproximatia Taylor si se va testa pe exemple numerice semnificative (m,n>20). Se vor compara solutiile calculate cu acest program cu solutiile calculate de celelalte programme....
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## This note was uploaded on 11/05/2011 for the course AUTOMATIC 18 taught by Professor Corneliupopeea during the Spring '11 term at UPB Colombia.

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Tema 3J - CNA-TEMA 3 1.Se considera matricea: ---- = 3 1 2...

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