Algebra polynomial equation

In particular, the standard CCSD method is obtained by setting rriA = 2 in Eq. (9). In the CCSDT method, ttia is set at 3 in the CCSDTQ approach, rriA = 4, etc. The cluster operator or the cluster amplitudes define it are obtained by solving the system of nonlinear, energy-independent, algebraic (polynomial) equations, which can be given the following symbolic form ... 

This relation is equivalent to an algebraic equation of degree n in the unknown X and therefore has n roots, some of which may be repeated (degenerate). These roots are the characteristic values or eigenvalues of the matrix B, When the determinant of Eq. (69) is expanded, the result is the polynomial equation... 

For the CSTR we have S algebraic species mass-balance equations. We can eliminate S R of these to obtain R irreducible algebraic polynomials in R of the S species which we... 

Find the factors and roots of simple polynomial equations using either algebraic or graphical procedures... 

An Algebraic Method for Finding Roots of Polynomial Equations... 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

Suppose we have some family of polynomial equations over k. We can then form a most general possible solution of the equations as follows. Take a polynomial ring over k, with one indeterminate for each variable in the equations. Divide by the ideal generated by the relations which the equations express. Call the quotient algebra A. From the equation for SL2, for instance, we get A — X12, X2u X22 I[X X22 12 21 ) The... 

A number that is the root of an algebraic polynomial. For example, sqrt(2) is an algebraic number because it is a solution of the equation x2=2. alphameric... 

Polynomial equations through the fourth degree can be solved algebraically, but some equations of fifth and higher degree cannot be solved algebraically. 

Mathematica can carry out both symbolic and numerical solutions of equations, including single algebraic equations, simultaneous algebraic equations, and differential equations, which we discuss later. Mathematica contains the rules needed for the symbolic solution of polynomial equations up to the fourth degree, and can... 

Definition 1. A closed algebraic subset of kn is a set consisting of all roots of a finite collection of polynomial equations i.e.,... 

The simplest useful polynomial equation of state is one that is cubic in molar volume, for such an expression is capable of yielding the ideal gas equation in the limit as V ->> oo, and of representing both liquid-and vapor-like volumes for sufficiently low temperatures. If we require that the equation be explicit in pressure, then algebraic arguments lead us to a five-parameter expression of the form (I)... 

Classical approach to solving this problem consists in assuming analytical model of the approximative function, for example taking the form of algebraic polynomial. In general it is going to be a certain vector function X t, C), where C is a vector of parameters sought after. The C vector is set upon all discrete observations in the network by solution of equations... 

Once again, algebraic manipulations lead to a third order polynomial equation of this form - aX - hX = Q that can be written as l -X) X +aX+b = 0 where... 

The third problem of antiquity states that for a given circle construct a square with the same area as the circle using ouly a compass and straightedge. The solution of the problem requires the coustructiou of the number >/ji because the formula for the area of a circle or radius r is Because % is trauscendental number, which means that Jt is not a solution of any algebraic equation (a polynomial equation in one variable having rational coefficients), the construction of % by compass and straightedge is impossible. In 1882, F. Lindemann (1852-1939) proved the transcendence of %. 

Real numbers are often divided into two subsets. One subset, the algebraic numbers, are real numbers which solve a polynomial equation in one variable with integer coefficients. For example is an... 

The accuracy of calculations based on these curves can be significantly improved when a curve is described, with the aid of the proper computer software, as a polynomial equation. Using a computer permits, without referring to graphs, the calculation of the strength or deflection of a plastic just by using polynomial equations. A polynomial is the sum of two or more algebraic expressions or the sum of a finite number of terms that are each composed of a positive power of a variable that is multiplied by a constant. 

Note recall the discussion in Section 2.1.1 about the term linear. Mathematicians distinguish between linear equations, for which the methods of linear algebra lead to a closed-form solution, and non-linear equations that cannot be so solved. In this terminology, polynomial equations are linear, even... 

Cubic and quartic polynomial equations can be solved algebraically, but it is probably best to apply approximation techniques rather than to attempt an algebraic solution. Equations containing sines, cosines, logarithms, exponentials, and so on frequently must be solved by approximations. There are two approaches for obtaining an approximate solution to an equation. One approach is to approximate the equation by making simplifying assumptions, and the other is to seek a numerical approximation to the root. 

Substitution of the known values of (x, f x,)) in Eq. (3.112) yields a set of ( + 1) simultaneous linear algebraic equations whose unknowns are the coefficients, ..., a of the polynomial equation. The. solution of this set of linear algebraic equations may be obtained using one of the algorithms discussed in Chap. 2. However, this solution results in an ill-conditioned linear system therefore, other methods have been favored in the development of interpolating polynomials. 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

Any algebraic equation may be written as a polynomial of nth degree in x of the form... 

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