CGET52
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
CGET52 does an eigenvector check for the generalized eigenvalue
problem.
The basic test for right eigenvectors is:
| b(i) A E(i) - a(i) B E(i) |
RESULT(1) = max -------------------------------
i n ulp max( |b(i) A|, |a(i) B| )
using the 1-norm. Here, a(i)/b(i) = w is the i-th generalized
eigenvalue of A - w B, or, equivalently, b(i)/a(i) = m is the i-th
generalized eigenvalue of m A - B.
H H _ _
For left eigenvectors, A , B , a, and b are used.
CGET52 also tests the normalization of E. Each eigenvector is
supposed to be normalized so that the maximum "absolute value"
of its elements is 1, where in this case, "absolute value"
of a complex value x is |Re(x)| + |Im(x)| ; let us call this
maximum "absolute value" norm of a vector v M(v).
if a(i)=b(i)=0, then the eigenvector is set to be the jth coordinate
vector. The normalization test is:
RESULT(2) = max | M(v(i)) - 1 | / ( n ulp )
eigenvectors v(i)
problem.
The basic test for right eigenvectors is:
| b(i) A E(i) - a(i) B E(i) |
RESULT(1) = max -------------------------------
i n ulp max( |b(i) A|, |a(i) B| )
using the 1-norm. Here, a(i)/b(i) = w is the i-th generalized
eigenvalue of A - w B, or, equivalently, b(i)/a(i) = m is the i-th
generalized eigenvalue of m A - B.
H H _ _
For left eigenvectors, A , B , a, and b are used.
CGET52 also tests the normalization of E. Each eigenvector is
supposed to be normalized so that the maximum "absolute value"
of its elements is 1, where in this case, "absolute value"
of a complex value x is |Re(x)| + |Im(x)| ; let us call this
maximum "absolute value" norm of a vector v M(v).
if a(i)=b(i)=0, then the eigenvector is set to be the jth coordinate
vector. The normalization test is:
RESULT(2) = max | M(v(i)) - 1 | / ( n ulp )
eigenvectors v(i)
Arguments
| LEFT |
(input) LOGICAL
=.TRUE.: The eigenvectors in the columns of E are assumed
to be *left* eigenvectors. =.FALSE.: The eigenvectors in the columns of E are assumed to be *right* eigenvectors. |
| N |
(input) INTEGER
The size of the matrices. If it is zero, CGET52 does
nothing. It must be at least zero. |
| A |
(input) COMPLEX array, dimension (LDA, N)
The matrix A.
|
| LDA |
(input) INTEGER
The leading dimension of A. It must be at least 1
and at least N. |
| B |
(input) COMPLEX array, dimension (LDB, N)
The matrix B.
|
| LDB |
(input) INTEGER
The leading dimension of B. It must be at least 1
and at least N. |
| E |
(input) COMPLEX array, dimension (LDE, N)
The matrix of eigenvectors. It must be O( 1 ).
|
| LDE |
(input) INTEGER
The leading dimension of E. It must be at least 1 and at
least N. |
| ALPHA |
(input) COMPLEX array, dimension (N)
The values a(i) as described above, which, along with b(i),
define the generalized eigenvalues. |
| BETA |
(input) COMPLEX array, dimension (N)
The values b(i) as described above, which, along with a(i),
define the generalized eigenvalues. |
| WORK |
(workspace) COMPLEX array, dimension (N**2)
|
| RWORK |
(workspace) REAL array, dimension (N)
|
| RESULT |
(output) REAL array, dimension (2)
The values computed by the test described above. If A E or
B E is likely to overflow, then RESULT(1:2) is set to 10 / ulp. |