CDRGSX
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
Purpose
CDRGSX checks the nonsymmetric generalized eigenvalue (Schur form)
problem expert driver CGGESX.
CGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate
transpose, S and T are upper triangular (i.e., in generalized Schur
form), and Q and Z are unitary. It also computes the generalized
eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus,
w(j) = alpha(j)/beta(j) is a root of the characteristic equation
det( A - w(j) B ) = 0
Optionally it also reorders the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms; computes a reciprocal condition number for the average
of the selected eigenvalues; and computes a reciprocal condition
number for the right and left deflating subspaces corresponding to
the selected eigenvalues.
When CDRGSX is called with NSIZE > 0, five (5) types of built-in
matrix pairs are used to test the routine CGGESX.
When CDRGSX is called with NSIZE = 0, it reads in test matrix data
to test CGGESX.
(need more details on what kind of read-in data are needed).
For each matrix pair, the following tests will be performed and
compared with the threshhold THRESH except for the tests (7) and (9):
(1) | A - Q S Z' | / ( |A| n ulp )
(2) | B - Q T Z' | / ( |B| n ulp )
(3) | I - QQ' | / ( n ulp )
(4) | I - ZZ' | / ( n ulp )
(5) if A is in Schur form (i.e. triangular form)
(6) maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(7) if sorting worked and SDIM is the number of eigenvalues
which were selected.
(8) the estimated value DIF does not differ from the true values of
Difu and Difl more than a factor 10*THRESH. If the estimate DIF
equals zero the corresponding true values of Difu and Difl
should be less than EPS*norm(A, B). If the true value of Difu
and Difl equal zero, the estimate DIF should be less than
EPS*norm(A, B).
(9) If INFO = N+3 is returned by CGGESX, the reordering "failed"
and we check that DIF = PL = PR = 0 and that the true value of
Difu and Difl is < EPS*norm(A, B). We count the events when
INFO=N+3.
For read-in test matrices, the same tests are run except that the
exact value for DIF (and PL) is input data. Additionally, there is
one more test run for read-in test matrices:
(10) the estimated value PL does not differ from the true value of
PLTRU more than a factor THRESH. If the estimate PL equals
zero the corresponding true value of PLTRU should be less than
EPS*norm(A, B). If the true value of PLTRU equal zero, the
estimate PL should be less than EPS*norm(A, B).
Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1)
matrix pairs are generated and tested. NSIZE should be kept small.
SVD (routine CGESVD) is used for computing the true value of DIF_u
and DIF_l when testing the built-in test problems.
problem expert driver CGGESX.
CGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate
transpose, S and T are upper triangular (i.e., in generalized Schur
form), and Q and Z are unitary. It also computes the generalized
eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus,
w(j) = alpha(j)/beta(j) is a root of the characteristic equation
det( A - w(j) B ) = 0
Optionally it also reorders the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms; computes a reciprocal condition number for the average
of the selected eigenvalues; and computes a reciprocal condition
number for the right and left deflating subspaces corresponding to
the selected eigenvalues.
When CDRGSX is called with NSIZE > 0, five (5) types of built-in
matrix pairs are used to test the routine CGGESX.
When CDRGSX is called with NSIZE = 0, it reads in test matrix data
to test CGGESX.
(need more details on what kind of read-in data are needed).
For each matrix pair, the following tests will be performed and
compared with the threshhold THRESH except for the tests (7) and (9):
(1) | A - Q S Z' | / ( |A| n ulp )
(2) | B - Q T Z' | / ( |B| n ulp )
(3) | I - QQ' | / ( n ulp )
(4) | I - ZZ' | / ( n ulp )
(5) if A is in Schur form (i.e. triangular form)
(6) maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(7) if sorting worked and SDIM is the number of eigenvalues
which were selected.
(8) the estimated value DIF does not differ from the true values of
Difu and Difl more than a factor 10*THRESH. If the estimate DIF
equals zero the corresponding true values of Difu and Difl
should be less than EPS*norm(A, B). If the true value of Difu
and Difl equal zero, the estimate DIF should be less than
EPS*norm(A, B).
(9) If INFO = N+3 is returned by CGGESX, the reordering "failed"
and we check that DIF = PL = PR = 0 and that the true value of
Difu and Difl is < EPS*norm(A, B). We count the events when
INFO=N+3.
For read-in test matrices, the same tests are run except that the
exact value for DIF (and PL) is input data. Additionally, there is
one more test run for read-in test matrices:
(10) the estimated value PL does not differ from the true value of
PLTRU more than a factor THRESH. If the estimate PL equals
zero the corresponding true value of PLTRU should be less than
EPS*norm(A, B). If the true value of PLTRU equal zero, the
estimate PL should be less than EPS*norm(A, B).
Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1)
matrix pairs are generated and tested. NSIZE should be kept small.
SVD (routine CGESVD) is used for computing the true value of DIF_u
and DIF_l when testing the built-in test problems.
Built-in Test Matrices
All built-in test matrices are the 2 by 2 block of triangular
matrices
A = [ A11 A12 ] and B = [ B11 B12 ]
[ A22 ] [ B22 ]
where for different type of A11 and A22 are given as the following.
A12 and B12 are chosen so that the generalized Sylvester equation
A11*R - L*A22 = -A12
B11*R - L*B22 = -B12
have prescribed solution R and L.
Type 1: A11 = J_m(1,-1) and A_22 = J_k(1-a,1).
B11 = I_m, B22 = I_k
where J_k(a,b) is the k-by-k Jordan block with ``a'' on
diagonal and ``b'' on superdiagonal.
Type 2: A11 = (a_ij) = ( 2(.5-sin(i)) ) and
B11 = (b_ij) = ( 2(.5-sin(ij)) ) for i=1,...,m, j=i,...,m
A22 = (a_ij) = ( 2(.5-sin(i+j)) ) and
B22 = (b_ij) = ( 2(.5-sin(ij)) ) for i=m+1,...,k, j=i,...,k
Type 3: A11, A22 and B11, B22 are chosen as for Type 2, but each
second diagonal block in A_11 and each third diagonal block
in A_22 are made as 2 by 2 blocks.
Type 4: A11 = ( 20(.5 - sin(ij)) ) and B22 = ( 2(.5 - sin(i+j)) )
for i=1,...,m, j=1,...,m and
A22 = ( 20(.5 - sin(i+j)) ) and B22 = ( 2(.5 - sin(ij)) )
for i=m+1,...,k, j=m+1,...,k
Type 5: (A,B) and have potentially close or common eigenvalues and
very large departure from block diagonality A_11 is chosen
as the m x m leading submatrix of A_1:
| 1 b |
| -b 1 |
| 1+d b |
| -b 1+d |
A_1 = | d 1 |
| -1 d |
| -d 1 |
| -1 -d |
| 1 |
and A_22 is chosen as the k x k leading submatrix of A_2:
| -1 b |
| -b -1 |
| 1-d b |
| -b 1-d |
A_2 = | d 1+b |
| -1-b d |
| -d 1+b |
| -1+b -d |
| 1-d |
and matrix B are chosen as identity matrices (see SLATM5).
matrices
A = [ A11 A12 ] and B = [ B11 B12 ]
[ A22 ] [ B22 ]
where for different type of A11 and A22 are given as the following.
A12 and B12 are chosen so that the generalized Sylvester equation
A11*R - L*A22 = -A12
B11*R - L*B22 = -B12
have prescribed solution R and L.
Type 1: A11 = J_m(1,-1) and A_22 = J_k(1-a,1).
B11 = I_m, B22 = I_k
where J_k(a,b) is the k-by-k Jordan block with ``a'' on
diagonal and ``b'' on superdiagonal.
Type 2: A11 = (a_ij) = ( 2(.5-sin(i)) ) and
B11 = (b_ij) = ( 2(.5-sin(ij)) ) for i=1,...,m, j=i,...,m
A22 = (a_ij) = ( 2(.5-sin(i+j)) ) and
B22 = (b_ij) = ( 2(.5-sin(ij)) ) for i=m+1,...,k, j=i,...,k
Type 3: A11, A22 and B11, B22 are chosen as for Type 2, but each
second diagonal block in A_11 and each third diagonal block
in A_22 are made as 2 by 2 blocks.
Type 4: A11 = ( 20(.5 - sin(ij)) ) and B22 = ( 2(.5 - sin(i+j)) )
for i=1,...,m, j=1,...,m and
A22 = ( 20(.5 - sin(i+j)) ) and B22 = ( 2(.5 - sin(ij)) )
for i=m+1,...,k, j=m+1,...,k
Type 5: (A,B) and have potentially close or common eigenvalues and
very large departure from block diagonality A_11 is chosen
as the m x m leading submatrix of A_1:
| 1 b |
| -b 1 |
| 1+d b |
| -b 1+d |
A_1 = | d 1 |
| -1 d |
| -d 1 |
| -1 -d |
| 1 |
and A_22 is chosen as the k x k leading submatrix of A_2:
| -1 b |
| -b -1 |
| 1-d b |
| -b 1-d |
A_2 = | d 1+b |
| -1-b d |
| -d 1+b |
| -1+b -d |
| 1-d |
and matrix B are chosen as identity matrices (see SLATM5).
Arguments
| NSIZE |
(input) INTEGER
The maximum size of the matrices to use. NSIZE >= 0.
If NSIZE = 0, no built-in tests matrices are used, but read-in test matrices are used to test SGGESX. |
| NCMAX |
(input) INTEGER
Maximum allowable NMAX for generating Kroneker matrix
in call to CLAKF2 |
| 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. THRESH >= 0. |
| NIN |
(input) INTEGER
The FORTRAN unit number for reading in the data file of
problems to solve. |
| NOUT |
(input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.) |
| A |
(workspace) COMPLEX array, dimension (LDA, NSIZE)
Used to store the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used. |
| LDA |
(input) INTEGER
The leading dimension of A, B, AI, BI, Z and Q,
LDA >= max( 1, NSIZE ). For the read-in test, LDA >= max( 1, N ), N is the size of the test matrices. |
| B |
(workspace) COMPLEX array, dimension (LDA, NSIZE)
Used to store the matrix whose eigenvalues are to be
computed. On exit, B contains the last matrix actually used. |
| AI |
(workspace) COMPLEX array, dimension (LDA, NSIZE)
Copy of A, modified by CGGESX.
|
| BI |
(workspace) COMPLEX array, dimension (LDA, NSIZE)
Copy of B, modified by CGGESX.
|
| Z |
(workspace) COMPLEX array, dimension (LDA, NSIZE)
Z holds the left Schur vectors computed by CGGESX.
|
| Q |
(workspace) COMPLEX array, dimension (LDA, NSIZE)
Q holds the right Schur vectors computed by CGGESX.
|
| ALPHA |
(workspace) COMPLEX array, dimension (NSIZE)
|
| BETA |
(workspace) COMPLEX array, dimension (NSIZE)
On exit, ALPHA/BETA are the eigenvalues.
|
| C |
(workspace) COMPLEX array, dimension (LDC, LDC)
Store the matrix generated by subroutine CLAKF2, this is the
matrix formed by Kronecker products used for estimating DIF. |
| LDC |
(input) INTEGER
The leading dimension of C. LDC >= max(1, LDA*LDA/2 ).
|
| S |
(workspace) REAL array, dimension (LDC)
Singular values of C
|
| WORK |
(workspace) COMPLEX array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The dimension of the array WORK. LWORK >= 3*NSIZE*NSIZE/2
|
| RWORK |
(workspace) REAL array,
dimension (5*NSIZE*NSIZE/2 - 4)
|
| IWORK |
(workspace) INTEGER array, dimension (LIWORK)
|
| LIWORK |
(input) INTEGER
The dimension of the array IWORK. LIWORK >= NSIZE + 2.
|
| BWORK |
(workspace) LOGICAL array, dimension (NSIZE)
|
| INFO |
(output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value. > 0: A routine returned an error code. |