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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
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10-10-2009, 11:39 AM
Geordie La Forge @ http://MeAmI.org
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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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