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Watt's curve, which arose in the context of early work on the steam engine, is a sextic in two variables.. One method of solving the cubic equation involves transforming variables to obtain a sextic equation having terms only of degrees 6, 3, and 0, which can be solved as a quadratic equation in the cube of the variable.
Since we study the computation of formal polynomials, we know that polynomials of very large degree require large circuits, for example, a polynomial of degree require a circuit of size roughly . So, the main goal is to prove lower bound for polynomials of small degree, say, polynomial in n . {\displaystyle n.}
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial. However, where the "degree of a polynomial" refers to the largest degree of a non-zero term of the polynomial, more typically "order" refers to the lowest degree of a non-zero term of a power series.
Many circuit complexity classes are defined in terms of class hierarchies. For each non-negative integer i, there is a class NC i, consisting of polynomial-size circuits of depth ( ()), using bounded fan-in AND, OR, and NOT gates. The union NC of all of these classes is a subject to discussion.
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
In general, the closer the approximation is required to be, the higher the degree of the polynomial and the more elements will be required in the network. [5] There are many polynomials and functions used in network synthesis for this purpose. The choice depends on which parameters of the prescribed function the designer wishes to optimise. [6]
A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: = + +. This will exactly fit a simple curve to three points. If the order of the equation is increased to a ...
Consequently, the coefficients can also be computed as the -th order derivative of a fully determined Savitzky–Golay filter with polynomial degree and a window size of +. For this, open source implementations are also available. [3]