DDRGES
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DDRGES checks the nonsymmetric generalized eigenvalue (Schur form)
problem driver DGGES.
DGGES factors A and B as Q S Z' and Q T Z' , where ' means
transpose, T is upper triangular, S is in generalized Schur form
(block upper triangular, with 1x1 and 2x2 blocks on the diagonal,
the 2x2 blocks corresponding to complex conjugate pairs of
generalized eigenvalues), and Q and Z are orthogonal. 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 reorder the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms.
When DDRGES is called, a number of matrix "sizes" ("N's") and a
number of matrix "TYPES" are specified. For each size ("N")
and each TYPE of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following 13 tests
will be performed and compared with the threshhold THRESH except
the tests (5), (11) and (13).
(1) | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
(2) | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
(3) | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
(4) | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
(5) if A is in Schur form (i.e. quasi-triangular form)
(no sorting of eigenvalues)
(6) if eigenvalues = diagonal blocks of the Schur form (S, T),
i.e., test the maximum over j of D(j) where:
if alpha(j) is real:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
if alpha(j) is complex:
| det( s S - w T ) |
D(j) = ---------------------------------------------------
ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
and S and T are here the 2 x 2 diagonal blocks of S and T
corresponding to the j-th and j+1-th eigenvalues.
(no sorting of eigenvalues)
(7) | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp )
(with sorting of eigenvalues).
(8) | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
(9) | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
(10) if A is in Schur form (i.e. quasi-triangular form)
(with sorting of eigenvalues).
(11) if eigenvalues = diagonal blocks of the Schur form (S, T),
i.e. test the maximum over j of D(j) where:
if alpha(j) is real:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
if alpha(j) is complex:
| det( s S - w T ) |
D(j) = ---------------------------------------------------
ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
and S and T are here the 2 x 2 diagonal blocks of S and T
corresponding to the j-th and j+1-th eigenvalues.
(with sorting of eigenvalues).
(12) if sorting worked and SDIM is the number of eigenvalues
which were SELECTed.
problem driver DGGES.
DGGES factors A and B as Q S Z' and Q T Z' , where ' means
transpose, T is upper triangular, S is in generalized Schur form
(block upper triangular, with 1x1 and 2x2 blocks on the diagonal,
the 2x2 blocks corresponding to complex conjugate pairs of
generalized eigenvalues), and Q and Z are orthogonal. 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 reorder the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms.
When DDRGES is called, a number of matrix "sizes" ("N's") and a
number of matrix "TYPES" are specified. For each size ("N")
and each TYPE of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following 13 tests
will be performed and compared with the threshhold THRESH except
the tests (5), (11) and (13).
(1) | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
(2) | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
(3) | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
(4) | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
(5) if A is in Schur form (i.e. quasi-triangular form)
(no sorting of eigenvalues)
(6) if eigenvalues = diagonal blocks of the Schur form (S, T),
i.e., test the maximum over j of D(j) where:
if alpha(j) is real:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
if alpha(j) is complex:
| det( s S - w T ) |
D(j) = ---------------------------------------------------
ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
and S and T are here the 2 x 2 diagonal blocks of S and T
corresponding to the j-th and j+1-th eigenvalues.
(no sorting of eigenvalues)
(7) | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp )
(with sorting of eigenvalues).
(8) | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
(9) | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
(10) if A is in Schur form (i.e. quasi-triangular form)
(with sorting of eigenvalues).
(11) if eigenvalues = diagonal blocks of the Schur form (S, T),
i.e. test the maximum over j of D(j) where:
if alpha(j) is real:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
if alpha(j) is complex:
| det( s S - w T ) |
D(j) = ---------------------------------------------------
ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
and S and T are here the 2 x 2 diagonal blocks of S and T
corresponding to the j-th and j+1-th eigenvalues.
(with sorting of eigenvalues).
(12) if sorting worked and SDIM is the number of eigenvalues
which were SELECTed.
Test Matrices
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
Arguments
| NSIZES |
(input) INTEGER
The number of sizes of matrices to use. If it is zero,
DDRGES does nothing. NSIZES >= 0. |
| NN |
(input) INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. NN >= 0. |
| NTYPES |
(input) INTEGER
The number of elements in DOTYPE. If it is zero, DDRGES
does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A on input. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . |
| DOTYPE |
(input) LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. |
| ISEED |
(input/output) INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRGES to continue the same random number sequence. |
| THRESH |
(input) DOUBLE PRECISION
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. |
| NOUNIT |
(input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.) |
| A |
(input/workspace) DOUBLE PRECISION array,
dimension(LDA, max(NN))
Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. |
| LDA |
(input) INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ). |
| B |
(input/workspace) DOUBLE PRECISION array,
dimension(LDA, max(NN))
Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. |
| S |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The Schur form matrix computed from A by DGGES. On exit, S
contains the Schur form matrix corresponding to the matrix in A. |
| T |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by DGGES.
|
| Q |
(workspace) DOUBLE PRECISION array, dimension (LDQ, max(NN))
The (left) orthogonal matrix computed by DGGES.
|
| LDQ |
(input) INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ). |
| Z |
(workspace) DOUBLE PRECISION array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by DGGES.
|
| ALPHAR |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| ALPHAI |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| BETA |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by DGGES.
( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. |
| WORK |
(workspace) DOUBLE PRECISION array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The dimension of the array WORK.
LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest matrix dimension. |
| RESULT |
(output) DOUBLE PRECISION array, dimension (15)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow. |
| BWORK |
(workspace) LOGICAL array, dimension (N)
|
| 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. INFO is the absolute value of the INFO value returned. |