DGET52
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DGET52 does an eigenvector check for the generalized eigenvalue
problem.
The basic test for right eigenvectors is:
| b(j) A E(j) - a(j) B E(j) |
RESULT(1) = max -------------------------------
j n ulp max( |b(j) A|, |a(j) B| )
using the 1-norm. Here, a(j)/b(j) = w is the j-th generalized
eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
generalized eigenvalue of m A - B.
For real eigenvalues, the test is straightforward. For complex
eigenvalues, E(j) and a(j) are complex, represented by
Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
eigenvector becomes
max( |Wr|, |Wi| )
n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
where
Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
T T _
For left eigenvectors, A , B , a, and b are used.
DGET52 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(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
vector. The normalization test is:
RESULT(2) = max | M(v(j)) - 1 | / ( n ulp )
eigenvectors v(j)
problem.
The basic test for right eigenvectors is:
| b(j) A E(j) - a(j) B E(j) |
RESULT(1) = max -------------------------------
j n ulp max( |b(j) A|, |a(j) B| )
using the 1-norm. Here, a(j)/b(j) = w is the j-th generalized
eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
generalized eigenvalue of m A - B.
For real eigenvalues, the test is straightforward. For complex
eigenvalues, E(j) and a(j) are complex, represented by
Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
eigenvector becomes
max( |Wr|, |Wi| )
n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
where
Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
T T _
For left eigenvectors, A , B , a, and b are used.
DGET52 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(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
vector. The normalization test is:
RESULT(2) = max | M(v(j)) - 1 | / ( n ulp )
eigenvectors v(j)
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, DGET52 does
nothing. It must be at least zero. |
| A |
(input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDE, N)
The matrix of eigenvectors. It must be O( 1 ). Complex
eigenvalues and eigenvectors always come in pairs, the eigenvalue and its conjugate being stored in adjacent elements of ALPHAR, ALPHAI, and BETA. Thus, if a(j)/b(j) and a(j+1)/b(j+1) are a complex conjugate pair of generalized eigenvalues, then E(,j) contains the real part of the eigenvector and E(,j+1) contains the imaginary part. Note that whether E(,j) is a real eigenvector or part of a complex one is specified by whether ALPHAI(j) is zero or not. |
| LDE |
(input) INTEGER
The leading dimension of E. It must be at least 1 and at
least N. |
| ALPHAR |
(input) DOUBLE PRECISION array, dimension (N)
The real parts of the values a(j) as described above, which,
along with b(j), define the generalized eigenvalues. Complex eigenvalues always come in complex conjugate pairs a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent elements in ALPHAR, ALPHAI, and BETA. Thus, if the j-th and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1) is assumed to be equal to ALPHAR(j)/BETA(j). |
| ALPHAI |
(input) DOUBLE PRECISION array, dimension (N)
The imaginary parts of the values a(j) as described above,
which, along with b(j), define the generalized eigenvalues. If ALPHAI(j)=0, then the eigenvalue is real, otherwise it is part of a complex conjugate pair. Complex eigenvalues always come in complex conjugate pairs a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent elements in ALPHAR, ALPHAI, and BETA. Thus, if the j-th and (j+1)-st eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to be equal to -ALPHAI(j)/BETA(j). Also, nonzero values in ALPHAI are assumed to always come in adjacent pairs. |
| BETA |
(input) DOUBLE PRECISION array, dimension (N)
The values b(j) as described above, which, along with a(j),
define the generalized eigenvalues. |
| WORK |
(workspace) DOUBLE PRECISION array, dimension (N**2+N)
|
| RESULT |
(output) DOUBLE PRECISION 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. |