monic.polynomial.recurrences {orthopolynom} | R Documentation |
This function returns a data frame with parameters required to construct monic orthogonal polynomials based on the standard recurrence relation for the non-monic polynomials. The recurrence relation for monic orthogonal polynomials is as follows.
q_{k + 1} ( x ) = ( {x - a_k } );q_k ( x ) - b_k ;q_{k - 1} ( x )
We require that q_{-1} ( x ) = 0 and q_0 ( x ) = 1. The recurrence for non-monic orthogonal polynomials is given by
c_k ;p_{k + 1} ( x ) = ( {d_k + e_k ;x} );p_k ( x ) - f_k ;p_{k - 1} ( x )
We require that p_{-1} ( x ) = 0 and p_0 ( x ) = 1. The monic polynomial recurrence parameters, a and b, are related to the non-monic polynomial parameter vectors c, d, e and f in the following manner.
a_k = - frac{{d_k }}{{e_k }}
b_k = frac{{c_{k - 1} ;f_k }}{{e_{k - 1} ;e_k }}
with b_0 = 0.
monic.polynomial.recurrences(recurrences)
recurrences |
the data frame of recurrence parameter vectors c, d, e and f |
A data frame with $n$+1 rows and two named columns, a and b.
Frederick Novomestky fnovomes@poly.edu
Press, et al (1992)
r <- chebyshev.t.recurrences( 10, normalized=TRUE ) m.r <- monic.polynomial.recurrences( r )