SGET22
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
SGET22 does an eigenvector check.
The basic test is:
RESULT(1) = | A E - E W | / ( |A| |E| ulp )
using the 1-norm. It also tests the normalization of E:
RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
j
where E(j) is the j-th eigenvector, and m-norm is the max-norm of a
vector. If an eigenvector is complex, as determined from WI(j)
nonzero, then the max-norm of the vector ( er + i*ei ) is the maximum
of
|er(1)| + |ei(1)|, ... , |er(n)| + |ei(n)|
W is a block diagonal matrix, with a 1 by 1 block for each real
eigenvalue and a 2 by 2 block for each complex conjugate pair.
If eigenvalues j and j+1 are a complex conjugate pair, so that
WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 by 2
block corresponding to the pair will be:
( wr wi )
( -wi wr )
Such a block multiplying an n by 2 matrix ( ur ui ) on the right
will be the same as multiplying ur + i*ui by wr + i*wi.
To handle various schemes for storage of left eigenvectors, there are
options to use A-transpose instead of A, E-transpose instead of E,
and/or W-transpose instead of W.
The basic test is:
RESULT(1) = | A E - E W | / ( |A| |E| ulp )
using the 1-norm. It also tests the normalization of E:
RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
j
where E(j) is the j-th eigenvector, and m-norm is the max-norm of a
vector. If an eigenvector is complex, as determined from WI(j)
nonzero, then the max-norm of the vector ( er + i*ei ) is the maximum
of
|er(1)| + |ei(1)|, ... , |er(n)| + |ei(n)|
W is a block diagonal matrix, with a 1 by 1 block for each real
eigenvalue and a 2 by 2 block for each complex conjugate pair.
If eigenvalues j and j+1 are a complex conjugate pair, so that
WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 by 2
block corresponding to the pair will be:
( wr wi )
( -wi wr )
Such a block multiplying an n by 2 matrix ( ur ui ) on the right
will be the same as multiplying ur + i*ui by wr + i*wi.
To handle various schemes for storage of left eigenvectors, there are
options to use A-transpose instead of A, E-transpose instead of E,
and/or W-transpose instead of W.
Arguments
| TRANSA |
(input) CHARACTER*1
Specifies whether or not A is transposed.
= 'N': No transpose = 'T': Transpose = 'C': Conjugate transpose (= Transpose) |
| TRANSE |
(input) CHARACTER*1
Specifies whether or not E is transposed.
= 'N': No transpose, eigenvectors are in columns of E = 'T': Transpose, eigenvectors are in rows of E = 'C': Conjugate transpose (= Transpose) |
| TRANSW |
(input) CHARACTER*1
Specifies whether or not W is transposed.
= 'N': No transpose = 'T': Transpose, use -WI(j) instead of WI(j) = 'C': Conjugate transpose, use -WI(j) instead of WI(j) |
| N |
(input) INTEGER
The order of the matrix A. N >= 0.
|
| A |
(input) REAL array, dimension (LDA,N)
The matrix whose eigenvectors are in E.
|
| LDA |
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
|
| E |
(input) REAL array, dimension (LDE,N)
The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors
are stored in the columns of E, if TRANSE = 'T' or 'C', the eigenvectors are stored in the rows of E. |
| LDE |
(input) INTEGER
The leading dimension of the array E. LDE >= max(1,N).
|
| WR |
(input) REAL array, dimension (N)
|
| WI |
(input) REAL array, dimension (N)
The real and imaginary parts of the eigenvalues of A.
Purely real eigenvalues are indicated by WI(j) = 0. Complex conjugate pairs are indicated by WR(j)=WR(j+1) and WI(j) = - WI(j+1) non-zero; the real part is assumed to be stored in the j-th row/column and the imaginary part in the (j+1)-th row/column. |
| WORK |
(workspace) REAL array, dimension (N*(N+1))
|
| RESULT |
(output) REAL array, dimension (2)
RESULT(1) = | A E - E W | / ( |A| |E| ulp )
RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp ) |