SGET24
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
SGET24 checks the nonsymmetric eigenvalue (Schur form) problem
expert driver SGEESX.
If COMP = .FALSE., the first 13 of the following tests will be
be performed on the input matrix A, and also tests 14 and 15
if LWORK is sufficiently large.
If COMP = .TRUE., all 17 test will be performed.
(1) 0 if T is in Schur form, 1/ulp otherwise
(no sorting of eigenvalues)
(2) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (no sorting of eigenvalues).
(3) | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).
(4) 0 if WR+sqrt(-1)*WI are eigenvalues of T
1/ulp otherwise
(no sorting of eigenvalues)
(5) 0 if T(with VS) = T(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(6) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(7) 0 if T is in Schur form, 1/ulp otherwise
(with sorting of eigenvalues)
(8) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (with sorting of eigenvalues).
(9) | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).
(10) 0 if WR+sqrt(-1)*WI are eigenvalues of T
1/ulp otherwise
If workspace sufficient, also compare WR, WI with and
without reciprocal condition numbers
(with sorting of eigenvalues)
(11) 0 if T(with VS) = T(without VS),
1/ulp otherwise
If workspace sufficient, also compare T with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(12) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
If workspace sufficient, also compare VS with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(13) if sorting worked and SDIM is the number of
eigenvalues which were SELECTed
If workspace sufficient, also compare SDIM with and
without reciprocal condition numbers
(14) if RCONDE the same no matter if VS and/or RCONDV computed
(15) if RCONDV the same no matter if VS and/or RCONDE computed
(16) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal average eigenvalue condition number
computed by SGEESX and RCDEIN (the precomputed true value)
is supplied as input. cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.
(17) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right invariant subspace condition
number computed by SGEESX and RCDVIN (the precomputed true
value) is supplied as input. cond(RCONDV) is the condition
number of RCONDV, and takes errors in computing RCONDV into
account, so that the resulting quantity should be O(ULP).
cond(RCONDV) is essentially given by norm(A)/RCONDE.
expert driver SGEESX.
If COMP = .FALSE., the first 13 of the following tests will be
be performed on the input matrix A, and also tests 14 and 15
if LWORK is sufficiently large.
If COMP = .TRUE., all 17 test will be performed.
(1) 0 if T is in Schur form, 1/ulp otherwise
(no sorting of eigenvalues)
(2) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (no sorting of eigenvalues).
(3) | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).
(4) 0 if WR+sqrt(-1)*WI are eigenvalues of T
1/ulp otherwise
(no sorting of eigenvalues)
(5) 0 if T(with VS) = T(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(6) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(7) 0 if T is in Schur form, 1/ulp otherwise
(with sorting of eigenvalues)
(8) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (with sorting of eigenvalues).
(9) | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).
(10) 0 if WR+sqrt(-1)*WI are eigenvalues of T
1/ulp otherwise
If workspace sufficient, also compare WR, WI with and
without reciprocal condition numbers
(with sorting of eigenvalues)
(11) 0 if T(with VS) = T(without VS),
1/ulp otherwise
If workspace sufficient, also compare T with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(12) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
If workspace sufficient, also compare VS with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(13) if sorting worked and SDIM is the number of
eigenvalues which were SELECTed
If workspace sufficient, also compare SDIM with and
without reciprocal condition numbers
(14) if RCONDE the same no matter if VS and/or RCONDV computed
(15) if RCONDV the same no matter if VS and/or RCONDE computed
(16) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal average eigenvalue condition number
computed by SGEESX and RCDEIN (the precomputed true value)
is supplied as input. cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.
(17) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right invariant subspace condition
number computed by SGEESX and RCDVIN (the precomputed true
value) is supplied as input. cond(RCONDV) is the condition
number of RCONDV, and takes errors in computing RCONDV into
account, so that the resulting quantity should be O(ULP).
cond(RCONDV) is essentially given by norm(A)/RCONDE.
Arguments
| COMP |
(input) LOGICAL
COMP describes which input tests to perform:
= .FALSE. if the computed condition numbers are not to be tested against RCDVIN and RCDEIN = .TRUE. if they are to be compared |
| JTYPE |
(input) INTEGER
Type of input matrix. Used to label output if error occurs.
|
| ISEED |
(input) INTEGER array, dimension (4)
If COMP = .FALSE., the random number generator seed
used to produce matrix. If COMP = .TRUE., ISEED(1) = the number of the example. Used to label output if error occurs. |
| THRESH |
(input) REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. |
| NOUNIT |
(input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.) |
| N |
(input) INTEGER
The dimension of A. N must be at least 0.
|
| A |
(input/output) REAL array, dimension (LDA, N)
Used to hold the matrix whose eigenvalues are to be
computed. |
| LDA |
(input) INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least N. |
| H |
(workspace) REAL array, dimension (LDA, N)
Another copy of the test matrix A, modified by SGEESX.
|
| HT |
(workspace) REAL array, dimension (LDA, N)
Yet another copy of the test matrix A, modified by SGEESX.
|
| WR |
(workspace) REAL array, dimension (N)
|
| WI |
(workspace) REAL array, dimension (N)
The real and imaginary parts of the eigenvalues of A.
On exit, WR + WI*i are the eigenvalues of the matrix in A. |
| WRT |
(workspace) REAL array, dimension (N)
|
| WIT |
(workspace) REAL array, dimension (N)
Like WR, WI, these arrays contain the eigenvalues of A,
but those computed when SGEESX only computes a partial eigendecomposition, i.e. not Schur vectors |
| WRTMP |
(workspace) REAL array, dimension (N)
|
| WITMP |
(workspace) REAL array, dimension (N)
Like WR, WI, these arrays contain the eigenvalues of A,
but sorted by increasing real part. |
| VS |
(workspace) REAL array, dimension (LDVS, N)
VS holds the computed Schur vectors.
|
| LDVS |
(input) INTEGER
Leading dimension of VS. Must be at least max(1, N).
|
| VS1 |
(workspace) REAL array, dimension (LDVS, N)
VS1 holds another copy of the computed Schur vectors.
|
| RCDEIN |
(input) REAL
When COMP = .TRUE. RCDEIN holds the precomputed reciprocal
condition number for the average of selected eigenvalues. |
| RCDVIN |
(input) REAL
When COMP = .TRUE. RCDVIN holds the precomputed reciprocal
condition number for the selected right invariant subspace. |
| NSLCT |
(input) INTEGER
When COMP = .TRUE. the number of selected eigenvalues
corresponding to the precomputed values RCDEIN and RCDVIN. |
| ISLCT |
(input) INTEGER array, dimension (NSLCT)
When COMP = .TRUE. ISLCT selects the eigenvalues of the
input matrix corresponding to the precomputed values RCDEIN and RCDVIN. For I=1, ... ,NSLCT, if ISLCT(I) = J, then the eigenvalue with the J-th largest real part is selected. Not referenced if COMP = .FALSE. |
| RESULT |
(output) REAL array, dimension (17)
The values computed by the 17 tests described above.
The values are currently limited to 1/ulp, to avoid overflow. |
| WORK |
(workspace) REAL array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The number of entries in WORK to be passed to SGEESX. This
must be at least 3*N, and N+N**2 if tests 14--16 are to be performed. |
| IWORK |
(workspace) INTEGER array, dimension (N*N)
|
| BWORK |
(workspace) LOGICAL array, dimension (N)
|
| INFO |
(output) INTEGER
If 0, successful exit.
If <0, input parameter -INFO had an incorrect value. If >0, SGEESX returned an error code, the absolute value of which is returned. |