Graphics Programs Reference
In-Depth Information
u
n
−
3
−
4
u
n
−
2
+
+
α
)
u
n
−
1
−
4
u
n
=
λ
−
u
n
−
2
+
2
u
n
−
1
−
(6
(
u
n
)
u
n
−
2
−
4
u
n
−
1
+
(5
+
α
)
u
n
=
λ
(
−
u
n
−
1
+
2
u
n
)
where
PL
2
E I
Write aprogram thatcomputes the lowest three buckling loads
P
and the corre-
sponding modeshapes.Runthe programwith
kL
4
kh
4
E I
=
kL
4
E I
Ph
2
E I
=
1
1
α
=
λ
=
(
n
+
1)
4
(
n
+
1)
2
/
(
E I
)
=
1000 and
n
=
25.
17.
Find smallest fiveeigenvalues of the 20
×
20 matrix
⎡
⎣
⎤
⎦
2100
···
0 1
121 0
···
00
0 12 1
···
00
.
.
.
.
.
.
.
.
.
.
.
.
.
A
=
00
···
1210
00
···
0 121
1 0
···
0012
Note
: this is adifficult matrix thathas many pairsofdouble eigenvalues.
MATLAB Functions
MATLAB'sfunction for solving eigenvalue problems is
eig
. Its usagefor the standard
eigenvalue problem
Ax
=
λ
x
is
eVals = eig(A)
returns the eigenvalues of the matrix
A
(
A
can be unsymmetric).
[X,D] = eig(A)
returns the eigenvectormatrix
X
and the diagonal matrix
D
that
contains the eigenvalues onits diagonal; that is,
eVals = diag(D)
.
For the nonstandard form
Ax
=
λ
Bx
, the calls are
eVals = eig(A,B)
[X,D] = eig(A,B)
The method of solutionis based on
Schur's factorization
:
PA P
T
T
, where
P
and
T
are unitary and triangular matrices, respectively. Schur'sfactorizationis not
coveredinthistext.
=
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