X r
p¼1
K p u ¼ kLu
ð2:109Þ
and if each partial problem K p u = k
(p) Lu with B l = 0 is self-adjoint and
completely definite with smallest eigenvalue k 1
(p)
. Then for the smallest
eigenvalue k 1 of the composite EVP it holds that:
k 1 !
X r
p¼1
k
ðpÞ
1
!
ð2:110Þ
2.8.7 The Rayleigh Quotient Estimate
and Its Accuracy
The Rayleigh quotient (Sects. 2.6.8 and 2.7.4) provides estimates of the lowest
eigenvalues which are typically much more accurate than the trial functions used.
For the general differential EVP (2.2) this is difficult to prove (Temple and Bickley
1933), but for the algebraic EVP (2.1) it is rather simple. Since differential EVPs
can be approximated by algebraic EVPs (e.g. using Rayleigh-Ritz, finite difference,
or finite element methods), the calculation of the accuracy of Rayleigh quotient
estimates for standard linear algebraic EVPs is instructive.
For the algebraic EVP (2.1) the Rayleigh quotient becomes, similarly to (2.50):
RðuÞ ¼
u
T Ku
u T Lu
¼ ~ k ! k 1 ;
ð2:111Þ
where u is the guess or trial vector for the lowest eigenvector u 1 ; ~ k is the Rayleigh
quotient estimate of the true eigenvalue k 1 corresponding to u, and by (2.1)
Rðu 1 Þ ¼ k 1 . To estimate the error ~ k 1 À k 1 ; we consider the vector variable u ¼
u 1 þ e as composed of the true lowest eigenvector u 1 and an error vector e , and
calculate the corresponding change in the Rayleigh quotient by Taylor-expanding
for e
j j ¼ e ( 1:
RðuÞ ¼ Rðu 1 þ eÞ ¼ ~ k 1 ¼ Rðu 1 Þ þ rRðu 1 Þ
ð
Þ
T e þ Oðe
2
Þ;
ð2:112Þ
where O(e
2 ) denotes small terms of order of magnitude e
2 . Calculating the gradient:
rRðuÞ ¼
@R
@u
¼
@
@u
u
T Ku
u T Lu
¼
@
@u u
T Ku
ð
Þu
T Lu À u
T Ku
@
@u u
T Lu
ð
Þ
u T Lu
ð
Þ
2
¼ 2
Ku
ð Þu
T Lu À u
T Ku Lu
ð Þ
u T Lu
ð
Þ
2
¼¼ 2
Ku À
u
T Ku
u T Lu Lu
u T Lu
;
ð2:113Þ
86
2 Eigenvalue Problems of Vibrations and Stability
p¼1
K p u ¼ kLu
ð2:109Þ
and if each partial problem K p u = k
(p) Lu with B l = 0 is self-adjoint and
completely definite with smallest eigenvalue k 1
(p)
. Then for the smallest
eigenvalue k 1 of the composite EVP it holds that:
k 1 !
X r
p¼1
k
ðpÞ
1
!
ð2:110Þ
2.8.7 The Rayleigh Quotient Estimate
and Its Accuracy
The Rayleigh quotient (Sects. 2.6.8 and 2.7.4) provides estimates of the lowest
eigenvalues which are typically much more accurate than the trial functions used.
For the general differential EVP (2.2) this is difficult to prove (Temple and Bickley
1933), but for the algebraic EVP (2.1) it is rather simple. Since differential EVPs
can be approximated by algebraic EVPs (e.g. using Rayleigh-Ritz, finite difference,
or finite element methods), the calculation of the accuracy of Rayleigh quotient
estimates for standard linear algebraic EVPs is instructive.
For the algebraic EVP (2.1) the Rayleigh quotient becomes, similarly to (2.50):
RðuÞ ¼
u
T Ku
u T Lu
¼ ~ k ! k 1 ;
ð2:111Þ
where u is the guess or trial vector for the lowest eigenvector u 1 ; ~ k is the Rayleigh
quotient estimate of the true eigenvalue k 1 corresponding to u, and by (2.1)
Rðu 1 Þ ¼ k 1 . To estimate the error ~ k 1 À k 1 ; we consider the vector variable u ¼
u 1 þ e as composed of the true lowest eigenvector u 1 and an error vector e , and
calculate the corresponding change in the Rayleigh quotient by Taylor-expanding
for e
j j ¼ e ( 1:
RðuÞ ¼ Rðu 1 þ eÞ ¼ ~ k 1 ¼ Rðu 1 Þ þ rRðu 1 Þ
ð
Þ
T e þ Oðe
2
Þ;
ð2:112Þ
where O(e
2 ) denotes small terms of order of magnitude e
2 . Calculating the gradient:
rRðuÞ ¼
@R
@u
¼
@
@u
u
T Ku
u T Lu
¼
@
@u u
T Ku
ð
Þu
T Lu À u
T Ku
@
@u u
T Lu
ð
Þ
u T Lu
ð
Þ
2
¼ 2
Ku
ð Þu
T Lu À u
T Ku Lu
ð Þ
u T Lu
ð
Þ
2
¼¼ 2
Ku À
u
T Ku
u T Lu Lu
u T Lu
;
ð2:113Þ
86
2 Eigenvalue Problems of Vibrations and Stability