CGET23
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
CGET23 checks the nonsymmetric eigenvalue problem driver CGEEVX.
If COMP = .FALSE., the first 8 of the following tests will be
performed on the input matrix A, and also test 9 if LWORK is
sufficiently large.
if COMP is .TRUE. all 11 tests will be performed.
(1) | A * VR - VR * W | / ( n |A| ulp )
Here VR is the matrix of unit right eigenvectors.
W is a diagonal matrix with diagonal entries W(j).
(2) | A**H * VL - VL * W**H | / ( n |A| ulp )
Here VL is the matrix of unit left eigenvectors, A**H is the
conjugate transpose of A, and W is as above.
(3) | |VR(i)| - 1 | / ulp and largest component real
VR(i) denotes the i-th column of VR.
(4) | |VL(i)| - 1 | / ulp and largest component real
VL(i) denotes the i-th column of VL.
(5) 0 if W(full) = W(partial), 1/ulp otherwise
W(full) denotes the eigenvalues computed when VR, VL, RCONDV
and RCONDE are also computed, and W(partial) denotes the
eigenvalues computed when only some of VR, VL, RCONDV, and
RCONDE are computed.
(6) 0 if VR(full) = VR(partial), 1/ulp otherwise
VR(full) denotes the right eigenvectors computed when VL, RCONDV
and RCONDE are computed, and VR(partial) denotes the result
when only some of VL and RCONDV are computed.
(7) 0 if VL(full) = VL(partial), 1/ulp otherwise
VL(full) denotes the left eigenvectors computed when VR, RCONDV
and RCONDE are computed, and VL(partial) denotes the result
when only some of VR and RCONDV are computed.
(8) 0 if SCALE, ILO, IHI, ABNRM (full) =
SCALE, ILO, IHI, ABNRM (partial)
1/ulp otherwise
SCALE, ILO, IHI and ABNRM describe how the matrix is balanced.
(full) is when VR, VL, RCONDE and RCONDV are also computed, and
(partial) is when some are not computed.
(9) 0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise
RCONDV(full) denotes the reciprocal condition numbers of the
right eigenvectors computed when VR, VL and RCONDE are also
computed. RCONDV(partial) denotes the reciprocal condition
numbers when only some of VR, VL and RCONDE are computed.
(10) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right eigenvector condition number
computed by CGEEVX 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.
(11) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal eigenvalue condition number
computed by CGEEVX 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.
If COMP = .FALSE., the first 8 of the following tests will be
performed on the input matrix A, and also test 9 if LWORK is
sufficiently large.
if COMP is .TRUE. all 11 tests will be performed.
(1) | A * VR - VR * W | / ( n |A| ulp )
Here VR is the matrix of unit right eigenvectors.
W is a diagonal matrix with diagonal entries W(j).
(2) | A**H * VL - VL * W**H | / ( n |A| ulp )
Here VL is the matrix of unit left eigenvectors, A**H is the
conjugate transpose of A, and W is as above.
(3) | |VR(i)| - 1 | / ulp and largest component real
VR(i) denotes the i-th column of VR.
(4) | |VL(i)| - 1 | / ulp and largest component real
VL(i) denotes the i-th column of VL.
(5) 0 if W(full) = W(partial), 1/ulp otherwise
W(full) denotes the eigenvalues computed when VR, VL, RCONDV
and RCONDE are also computed, and W(partial) denotes the
eigenvalues computed when only some of VR, VL, RCONDV, and
RCONDE are computed.
(6) 0 if VR(full) = VR(partial), 1/ulp otherwise
VR(full) denotes the right eigenvectors computed when VL, RCONDV
and RCONDE are computed, and VR(partial) denotes the result
when only some of VL and RCONDV are computed.
(7) 0 if VL(full) = VL(partial), 1/ulp otherwise
VL(full) denotes the left eigenvectors computed when VR, RCONDV
and RCONDE are computed, and VL(partial) denotes the result
when only some of VR and RCONDV are computed.
(8) 0 if SCALE, ILO, IHI, ABNRM (full) =
SCALE, ILO, IHI, ABNRM (partial)
1/ulp otherwise
SCALE, ILO, IHI and ABNRM describe how the matrix is balanced.
(full) is when VR, VL, RCONDE and RCONDV are also computed, and
(partial) is when some are not computed.
(9) 0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise
RCONDV(full) denotes the reciprocal condition numbers of the
right eigenvectors computed when VR, VL and RCONDE are also
computed. RCONDV(partial) denotes the reciprocal condition
numbers when only some of VR, VL and RCONDE are computed.
(10) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right eigenvector condition number
computed by CGEEVX 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.
(11) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal eigenvalue condition number
computed by CGEEVX 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.
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 |
| ISRT |
(input) INTEGER
If COMP = .TRUE., ISRT indicates in how the eigenvalues
corresponding to values in RCDVIN and RCDEIN are ordered: = 0 means the eigenvalues are sorted by increasing real part = 1 means the eigenvalues are sorted by increasing imaginary part If COMP = .FALSE., ISRT is not referenced. |
| BALANC |
(input) CHARACTER
Describes the balancing option to be tested.
= 'N' for no permuting or diagonal scaling = 'P' for permuting but no diagonal scaling = 'S' for no permuting but diagonal scaling = 'B' for permuting and diagonal scaling |
| JTYPE |
(input) INTEGER
Type of input matrix. 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. |
| 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. |
| 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) COMPLEX 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) COMPLEX array, dimension (LDA,N)
Another copy of the test matrix A, modified by CGEEVX.
|
| W |
(workspace) COMPLEX array, dimension (N)
Contains the eigenvalues of A.
|
| W1 |
(workspace) COMPLEX array, dimension (N)
Like W, this array contains the eigenvalues of A,
but those computed when CGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. |
| VL |
(workspace) COMPLEX array, dimension (LDVL,N)
VL holds the computed left eigenvectors.
|
| LDVL |
(input) INTEGER
Leading dimension of VL. Must be at least max(1,N).
|
| VR |
(workspace) COMPLEX array, dimension (LDVR,N)
VR holds the computed right eigenvectors.
|
| LDVR |
(input) INTEGER
Leading dimension of VR. Must be at least max(1,N).
|
| LRE |
(workspace) COMPLEX array, dimension (LDLRE,N)
LRE holds the computed right or left eigenvectors.
|
| LDLRE |
(input) INTEGER
Leading dimension of LRE. Must be at least max(1,N).
|
| RCONDV |
(workspace) REAL array, dimension (N)
RCONDV holds the computed reciprocal condition numbers
for eigenvectors. |
| RCNDV1 |
(workspace) REAL array, dimension (N)
RCNDV1 holds more computed reciprocal condition numbers
for eigenvectors. |
| RCDVIN |
(input) REAL array, dimension (N)
When COMP = .TRUE. RCDVIN holds the precomputed reciprocal
condition numbers for eigenvectors to be compared with RCONDV. |
| RCONDE |
(workspace) REAL array, dimension (N)
RCONDE holds the computed reciprocal condition numbers
for eigenvalues. |
| RCNDE1 |
(workspace) REAL array, dimension (N)
RCNDE1 holds more computed reciprocal condition numbers
for eigenvalues. |
| RCDEIN |
(input) REAL array, dimension (N)
When COMP = .TRUE. RCDEIN holds the precomputed reciprocal
condition numbers for eigenvalues to be compared with RCONDE. |
| SCALE |
(workspace) REAL array, dimension (N)
Holds information describing balancing of matrix.
|
| SCALE1 |
(workspace) REAL array, dimension (N)
Holds information describing balancing of matrix.
|
| RESULT |
(output) REAL array, dimension (11)
The values computed by the 11 tests described above.
The values are currently limited to 1/ulp, to avoid overflow. |
| WORK |
(workspace) COMPLEX array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The number of entries in WORK. This must be at least
2*N, and 2*N+N**2 if tests 9, 10 or 11 are to be performed. |
| RWORK |
(workspace) REAL array, dimension (2*N)
|
| INFO |
(output) INTEGER
If 0, successful exit.
If <0, input parameter -INFO had an incorrect value. If >0, CGEEVX returned an error code, the absolute value of which is returned. |