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If x is a simple root of the polynomial (), then Laguerre's method converges cubically whenever the initial guess, (), is close enough to the root . On the other hand, when x 1 {\displaystyle x_{1}} is a multiple root convergence is merely linear, with the penalty of calculating values for the polynomial and its first and second derivatives at ...
So, except for very low degrees, root finding of polynomials consists of finding approximations of the roots. By the fundamental theorem of algebra, a polynomial of degree n has exactly n real or complex roots counting multiplicities. It follows that the problem of root finding for polynomials may be split in three different subproblems;
Wilkinson's polynomial is often used to illustrate the undesirability of naively computing eigenvalues of a matrix by first calculating the coefficients of the matrix's characteristic polynomial and then finding its roots, since using the coefficients as an intermediate step may introduce an extreme ill-conditioning even if the original problem ...
Horner's method evaluates a polynomial using repeated bracketing: + + + + + = + (+ (+ (+ + (+)))). This method reduces the number of multiplications and additions to just Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: b 0 is the number of connected components; b 1 is the number of one-dimensional or "circular" holes;
To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers.
Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree. Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform. This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.
Just like the vertex cover problem is polynomial for tree graphs but NP-hard for general graphs, the square covering problem is linear for hole-free polygons but NP-hard for general polygons. It is possible to use the linear algorithm to get a 2-approximation; i.e., a covering with at most 2 opt squares, where opt is the number of squares in a ...