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-comp.os.linux.announce+moderators Rhinocerus > Newsgroup > Newsgroup comp.lang.* 1 > Newsgroup comp.lang.fortran Root Functions: {// Variabile (/ / Variables ( ( întreg grad, // gradul polinomului Full degree / / degree polynomial real P1[], // vectorul coeficientilor polinomului Real P1 [], / / vector polynomial coefficients real P2[], // vectorul coeficie User Name Remember Me Password FAQ Community Calendar
Thread Tools Display Modes #1 (permalink) 10-10-2009, 11:39 AM Geordie La Forge @ http://MeAmI.org Guest
Posts: n/a Root Functions: {// Variabile (/ / Variables ( ( întreg grad, // gradul polinomului Full degree / / degree polynomial real P1[], // vectorul coeficientilor polinomului Real P1 [], / / vector polynomial coefficients real P2[], // vectorul coeficie Lucrarea 2 Paper 2 REZOLVAREA NUMERICA A ECUATIILOR ALGEBRICE Numerical solving of equations ALGEBRA 1. 1. SCOPUL LUCRARII Scope of Work Prezentarea unor metode de rezolvare a ecuatiilor algebrice, si implementarea acestora în limbaje de nivel înalt (în particular, C). Presentation of methods for solving algebraic equations, and their implementation in high level languages (especially C). 2. 2. PREZENTARE TEORETICA Theoretical presentation Calculul radacinilor unei ecuatii se face în doua etape: Calculation of roots of equations is done in two stages: a) Separarea radacinilor. a) Separation of roots. b) Calculul lor cu o eroare impusa. b) their calculation with an imposed error. 2.1. 2.1. SEPARAREA RADACINILOR SEPARATION ROOTS Consideram functia R, si ecuatia algebrica We consider the function R, and algebraic equation (2.1)Separarea radacinilor unei ecuatii consta în determinarea unor intervale în domeniul de definitie al functiei, în care sa existe o singura radacina reala. (2.1) Separation of roots of equations is to determine intervals in the definition of function, in which there is only one real root. Pentru separarea radacinilor reale exista mai multe metode dintre care amintim: metoda sirului lui Rolle, metoda sirului lui Sturm si metoda lui Budan- Fourier. To separate the real roots there are several ways such as: Rolle's string method, string method of Sturm and Budan-Fourier's method. 2.1.1. 2.1.1. METODA SIRULUI LUI STURM Syrian METHOD OF Sturm Consideram functia definita pe R, care îndeplineste conditiile de continuitate si derivabilitate pentru . We consider the function defined on R, satisfying the conditions of continuity and derivabilitate for. Definitia 2.1 DEFINITION 2.1 Sirul de functii f0,f1, f2......fm continue pe care satisfac conditiile: Series of functions F0, F1, F2 ...... continuous FM satisfy conditions: a) f0(x) =f(x); a) f0 (x) = f (x); b) fm(x) `0 pentru ; b) fm (x) `0 for; c) daca fi(x) =0, 1didm-1 si , atunci fi-1(x)*fi+1(x)<0; c) if fi (x) = 0, 1didm-1 and then be-1 (x) * be 1 (x) <0; d) daca f0(x)=0 pentru , atunci f'0(x)*f1 (x)>0 d) if f0 (x) = 0 for, then f'0 (x) * f1 (x)> 0 se numeste sirul lui Sturm asociat functiei f(x). Sturm's series is called associated function f (x). Numarul radacinilor ecuatiei f(x) în intervalul este dat de urmatoarea teorema: Number roots equation f (x) the range is given by the following theorem: Teorema 2.1 Theorem 2.1 Fie sirul lui Sturm f0,f1, f2......fm , atasat functiei f(x) cu conditiile f(a) `0 si f(b) `0, atunci numarul de radacini ale ecuatiei f(x)=0 în intervalul este dat de diferenta numarului de variatii de semn ale sirurilor: Let's string Sturm F0, F1, F2 ...... fm, the attachment function f (x) the terms f (a) `0 and f (b)` 0, then the number of roots of equation f (x) = 0 in the range is the difference in the number of variations of sign of strings: f0(a),f1(a), f2(a)......fm(a) f0 (a), f1 (a), f2 (a )...... fm (a) f0(b),f1(b), f2(b)......fm(b) f0 (b), f1 (b), f2 (b )...... fm (b) În cazul functiei polinom P(x) care este definita pe R, Teorema 2.1 devine: If polynomial function P (x) which is defined on R, Theorem 2.1 becomes: Teorema 2.2 Theorem 2.2 Fie P0, P1, P2,......Pm un sir de polinoame construit astfel P0=P, P1=P', iar Pi+1 este restul împartirii lui Pi-1 la Pi luat cu semn schimbat, pentru 2didm. Let P0, P1, P2 Pm ,...... built as a series of polynomials P0 = P, P1 = P 'and Pi +1 is the rest of the division of the Pi Pi-1 taken with the sign changed to 2didm. Atunci numarul de radacini ale ecuatiei P(x)=0 este egal cu diferenta dintre numarul de schimbari de semn ale sirurilor: When the number of roots of equation P (x) = 0 is equal to the difference between the number of rows changes the sign: P0(- ), P1(- ), P2(- ),......Pm(- ) P0 (-), P1 (-), P2 (- ),...... Pm (-) P0( ), P1( ), P2( ),......Pm( ) P0 (), P1 (), P2 (),...... PM () 2.1.1.1. 2.1.1.1. Algoritmul 2.1. Algorithm 2.1. Sirul lui Sturm Sturm's series {// Variabile (/ / Variables ( ( întreg grad, // gradul polinomului Full degree / / degree polynomial real P1 [], // vectorul coeficientilor polinomului Real P1 [], / / vector polynomial coefficients real P2[], // vectorul coeficientilor derivatei Real-P2 [], / / vector derivative coefficients // polinomului; / / polynomial; real Rez[], // vectorul coeficientilor restului Real Rez [], / / vector coefficients rest real C1, // pentru calculul coeficientilor restului real C1, / / to calculate the remaining coefficients real C2 Real C2 // coeficientii pentru calculul coeficientilor restului / / Coefficients to calculate the remaining coefficients ) ) { ( pentru i=grad-1 pana la 0 calculeaza P2[i]=(i +1)P1 [i+1]; for i = 0 deg-1 to calculate P2 [i] = (i +1) P1 [i +1]; daca grad=0 Rez(0)=P1(0)-P1(1) ; // Restul are gradul grad-2 if degree = 0 Rez (0) = P1 (0)-P1 (1); / / The rest is grade level-2 altfel otherwise
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