The modified Chebyshev algorithm constructs the three-term recurrence relation of orthogonal polynomials
from moments expressed in an auxiliary polynomial basis. For a finite positive
measure 
on the real line, let the known monic
polynomials 
satisfy
 |
(1)
|
with 
and 
. For a positive
integer 
, the input consists of the modified moments
 |
(2)
|
for 
, 1, ..., 
,
with 
. Assume the polynomial inner product is positive
definite through degree 
.
The desired monic polynomials satisfy
 |
(3)
|
with 
and 
(Gautschi 1982).
Set 
, 
, 
, and 
. The intermediate quantities 
obey
 |
(4)
|
for 
, ..., 
and 
,
..., 
. The recurrence coefficients
are then
 |  |  |
(5)
|
 |  |  |
(6)
|
The value 
records the mass normalization and does not affect the zero term 
. Choosing 
gives 
,
reducing the method to the original power-moment construction.
The diagonal entries 
and off-diagonal entries 
, 
,
form the associated Jacobi matrix. The Golub-Welsch
algorithm then gives Gaussian quadrature
nodes and weights. A suitable auxiliary family can improve
conditioning, but this is not automatic. The modified moments must themselves be
computed accurately; changing basis after errors have entered the power moments does
not restore the lost information (Wheeler 1974, Gautschi 1982).
