On the calculation of Jacobi Matrices

Computation of connection coefficients and measure modifications for orthogonal polynomials

“…This result is essentially a rearrangement of the results in [19]; the authors there indirectly state this result and obtain it without the use of the Christoffel modifica- …”

“…For N degrees of freedom, the sparse connection result for polynomial modifications can be used to effect an O(N)-cost transform between expansion coefficients for π and those for p. This result can be extended to show that if J is the Jacobi matrix for π, then a leading principal submatrix of q(J) is positive and the Cholesky decomposition of this matrix gives exactly the sought connection coefficients. This is a variant of the classical Kautsky/Golub Jacobi matrix characterization of q(J) [19] -we extend this to cases when q is a non-polynomial.…”

The structure of iterative methods for symmetric linear discrete ill-posed problems

“…Let the entries of the symmetric (infinite) tridiagonal matrices T andT be the recurrence coefficients of the polynomials p j andp j , respectively. If β is an even positive integer, thenT can be determined from T by application of β /2 steps of the QR algorithm to T ; see Kautsky and Golub [12] and Buhmann and Iserles [3] for discussions. When T is positive definite and β > 0 is an odd integer, then one step of the symmetric LR method is required to constructT from T (in addition to (β − 1)/2 steps with the QR algorithm); see [12].…”

“…If β is an even positive integer, thenT can be determined from T by application of β /2 steps of the QR algorithm to T ; see Kautsky and Golub [12] and Buhmann and Iserles [3] for discussions. When T is positive definite and β > 0 is an odd integer, then one step of the symmetric LR method is required to constructT from T (in addition to (β − 1)/2 steps with the QR algorithm); see [12]. A recent discussion on how symmetric tridiagonal matrices whose entries are recursion coefficients for orthonormal polynomials are changed when the measure is modified by a polynomial or rational function is provided by Gautschi [7].…”

“…The matrix Q (1) k+1 can be expressed as a product of k Givens rotations, 12) where G j ∈ R (k+1)×(k+1) is a rotation in the planes j and j +1. Thus, G j is the identity matrix except for a 2 × 2 block in the rows and columns j and j + 1.…”