À u
00
k þ 1
½
¼ u k
½ ; u k þ 1
½
ð0Þ ¼ u
0
k þ 1
½
ð1Þ ¼ 0;
k k þ 1
½
¼
R 1
0 u k þ 1
½
u k
½ dx
R 1
0 u 2
k þ 1
½
dx
; k ¼ 0; 1; . . .:
ð2:72Þ
Starting with u [0] (x) = 1 and k [0] = 1 we obtain, by integration of –u′′ [1] = u [0]
and insertion of boundary conditions, that u [1] (x) = –x
2 /2 + x. This in turn implies
that k 1
½ ¼
R 1
0 u 1
½ u 0
½ dx=
R 1
0 u
2
½1 dx ¼ 5=2. Compared to the exact solution k 1 = p
2 /4
the error is only 1.3%. Results for the first three iterations are given in Table 2.1,
indicating rapid convergence towards the exact eigenvalue and eigenfunction.
Already the first (normalized) function-iterate ~ u 1
½ is close to the exact eigenfunction, with an integrated RMS-error of only 3.6%. The next iterate ~ u 2
½ ðxÞ (with
RMS-error 0.4%) is virtually indistinguishable from the exact eigenfunction, when
plotted.
The eigenvalue iterates k [k+1] are upper bound estimates, k [k+1] ! k 1 . On certain
conditions one may establish two-sided bounds in terms of the so-called Schwartz’s
quotients (Flügge 1962). The method of eigenfunction iterations can be extended to
yield also higher eigenvalues and eigenfunctions. However, the Rayleigh–Ritz and
finite difference methods are more convenient for this purpose.
2.8.3 The Rayleigh–Ritz Method
The Rayleigh–Ritz Method transforms a differential EVP into a corresponding
approximate algebraic EVP. Exploiting the extremum property of the Rayleigh
quotient, the eigenvalues of the algebraic EVP become upper-bound approximations to the lowest eigenvalues of the differential EVP.
Table 2.1 Eigenfunction iterations for Example 2.18
Iteration
number k
Normalized
eigenfunction
~ u k
½ x
ð Þ ¼ u k
½ x
ð Þ=u ½k 1
ð Þ
Eigenfunction
RMS-error
R 1
0 ðu ½k Àu Exact Þ
2 dx
1=2
Eigenvalue
k [k]
Eigenvalue
error
|k [k] − k 1 |/k 1
0
1
4.8 Â 10
−1
1.0
5.9 Â 10
−1
1
−x
2 + 2x
3.6 Â 10
−2
2.5
1.3 Â 10
−2
2
x(x
3 − 4x
2 + 8)/5
4.0 Â 10
−3
2.4677
1.2 Â 10
−4
3
x(−x
5 + 6x
4
–
40x
2 + 96)/61
4.5 Â 10
−4
2.467405
1.6 Â 10
−6
Exact
sin(p/2)
0
2.467401…
(=p
2
/4)
0
76
2 Eigenvalue Problems of Vibrations and Stability
00
k þ 1
½
¼ u k
½ ; u k þ 1
½
ð0Þ ¼ u
0
k þ 1
½
ð1Þ ¼ 0;
k k þ 1
½
¼
R 1
0 u k þ 1
½
u k
½ dx
R 1
0 u 2
k þ 1
½
dx
; k ¼ 0; 1; . . .:
ð2:72Þ
Starting with u [0] (x) = 1 and k [0] = 1 we obtain, by integration of –u′′ [1] = u [0]
and insertion of boundary conditions, that u [1] (x) = –x
2 /2 + x. This in turn implies
that k 1
½ ¼
R 1
0 u 1
½ u 0
½ dx=
R 1
0 u
2
½1 dx ¼ 5=2. Compared to the exact solution k 1 = p
2 /4
the error is only 1.3%. Results for the first three iterations are given in Table 2.1,
indicating rapid convergence towards the exact eigenvalue and eigenfunction.
Already the first (normalized) function-iterate ~ u 1
½ is close to the exact eigenfunction, with an integrated RMS-error of only 3.6%. The next iterate ~ u 2
½ ðxÞ (with
RMS-error 0.4%) is virtually indistinguishable from the exact eigenfunction, when
plotted.
The eigenvalue iterates k [k+1] are upper bound estimates, k [k+1] ! k 1 . On certain
conditions one may establish two-sided bounds in terms of the so-called Schwartz’s
quotients (Flügge 1962). The method of eigenfunction iterations can be extended to
yield also higher eigenvalues and eigenfunctions. However, the Rayleigh–Ritz and
finite difference methods are more convenient for this purpose.
2.8.3 The Rayleigh–Ritz Method
The Rayleigh–Ritz Method transforms a differential EVP into a corresponding
approximate algebraic EVP. Exploiting the extremum property of the Rayleigh
quotient, the eigenvalues of the algebraic EVP become upper-bound approximations to the lowest eigenvalues of the differential EVP.
Table 2.1 Eigenfunction iterations for Example 2.18
Iteration
number k
Normalized
eigenfunction
~ u k
½ x
ð Þ ¼ u k
½ x
ð Þ=u ½k 1
ð Þ
Eigenfunction
RMS-error
R 1
0 ðu ½k Àu Exact Þ
2 dx
1=2
Eigenvalue
k [k]
Eigenvalue
error
|k [k] − k 1 |/k 1
0
1
4.8 Â 10
−1
1.0
5.9 Â 10
−1
1
−x
2 + 2x
3.6 Â 10
−2
2.5
1.3 Â 10
−2
2
x(x
3 − 4x
2 + 8)/5
4.0 Â 10
−3
2.4677
1.2 Â 10
−4
3
x(−x
5 + 6x
4
–
40x
2 + 96)/61
4.5 Â 10
−4
2.467405
1.6 Â 10
−6
Exact
sin(p/2)
0
2.467401…
(=p
2
/4)
0
76
2 Eigenvalue Problems of Vibrations and Stability