Page 323 - Matrix Analysis & Applied Linear Algebra

5.5 Gram–Schmidt Procedure 319

5.5.10. Depending on how the inner products r ij are defined, verify that the fol-
lowing code implements both the classical and modified Gram–Schmidt
algorithms applied to a set of vectors {x 1 , x 2 ,..., x n } .
For j =1 to n
u j ←− x j
For i =1 to j − 1

u i x j (classical Gram–Schmidt)
r ij ←−
u i u j (modified Gram–Schmidt)
u j ←− u j − r ij u i
End
r jj ←− u j
If r jj =0
quit (because x j ∈ span {x 1 , x 2 ,..., x j−1 } )
Else u j ←− u j /r jj
End
If exact arithmetic is used, will the inner products r ij be the same for
both implementations?

5.5.11. Let V be the inner-product space of real-valued continuous functions
defined on the interval [−1, 1], where the inner product is defined by

1
f g = f(x)g(x)dx,
−1
and let S be the subspace of V that is spanned by the three linearly
2
independent polynomials q 0 =1, q 1 = x, q 2 = x .
(a) Use the Gram–Schmidt process to determine an orthonormal set
of polynomials {p 0 ,p 1 ,p 2 } that spans S. These polynomials
45
are the first three normalized Legendre polynomials.
(b) Verify that p n satisfies Legendre’s differential equation
2
(1 − x )y − 2xy + n(n +1)y =0

for n =0, 1, 2. This equation and its solutions are of consider-
able importance in applied mathematics.
45
Adrien–Marie Legendre (1752–1833) was one of the most eminent French mathematicians of
the eighteenth century. His primary work in higher mathematics concerned number theory
and the study of elliptic functions. But he was also instrumental in the development of the
theory of least squares, and some people believe that Legendre should receive the credit that
is often afforded to Gauss for the introduction of the method of least squares. Like Gauss and
many other successful mathematicians, Legendre spent substantial time engaged in diligent
and painstaking computation. It is reported that in 1824 Legendre refused to vote for the
government’s candidate for Institut National, so his pension was stopped, and he died in
poverty.