DCHKGG
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DCHKGG checks the nonsymmetric generalized eigenvalue problem
routines.
T T T
DGGHRD factors A and B as U H V and U T V , where means
transpose, H is hessenberg, T is triangular and U and V are
orthogonal.
T T
DHGEQZ factors H and T as Q S Z and Q P Z , where P 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(1),beta(1)),...,(alpha(n),beta(n)),
where alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus,
w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue
problem
det( A - w(j) B ) = 0
and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent
problem
det( m(j) A - B ) = 0
DTGEVC computes the matrix L of left eigenvectors and the matrix R
of right eigenvectors for the matrix pair ( S, P ). In the
description below, l and r are left and right eigenvectors
corresponding to the generalized eigenvalues (alpha,beta).
When DCHKGG 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, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 15
tests will be performed. The first twelve "test ratios" should be
small -- O(1). They will be compared with the threshhold THRESH:
T
(1) | A - U H V | / ( |A| n ulp )
T
(2) | B - U T V | / ( |B| n ulp )
T
(3) | I - UU | / ( n ulp )
T
(4) | I - VV | / ( n ulp )
T
(5) | H - Q S Z | / ( |H| n ulp )
T
(6) | T - Q P Z | / ( |T| n ulp )
T
(7) | I - QQ | / ( n ulp )
T
(8) | I - ZZ | / ( n ulp )
(9) max over all left eigenvalue/-vector pairs (beta/alpha,l) of
| l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )
(10) max over all left eigenvalue/-vector pairs (beta/alpha,l') of
T
| l'**H * (beta H - alpha T) | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors l' are the result of passing Q to
DTGEVC and back transforming (HOWMNY='B').
(11) max over all right eigenvalue/-vector pairs (beta/alpha,r) of
| (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )
(12) max over all right eigenvalue/-vector pairs (beta/alpha,r') of
| (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors r' are the result of passing Z to
DTGEVC and back transforming (HOWMNY='B').
The last three test ratios will usually be small, but there is no
mathematical requirement that they be so. They are therefore
compared with THRESH only if TSTDIF is .TRUE.
(13) | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )
(14) | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )
(15) max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
|beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp
In addition, the normalization of L and R are checked, and compared
with the threshhold THRSHN.
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) U ( J , J ) V where U and V are random orthogonal matrices.
(17) U ( T1, T2 ) V 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) U ( T1, T2 ) V diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) U ( T1, T2 ) V 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) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) U ( T1, T2 ) V 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) U ( big*T1, small*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) U ( small*T1, big*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) U ( small*T1, small*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) U ( big*T1, big*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) U ( T1, T2 ) V where T1 and T2 are random upper-triangular
matrices.
routines.
T T T
DGGHRD factors A and B as U H V and U T V , where means
transpose, H is hessenberg, T is triangular and U and V are
orthogonal.
T T
DHGEQZ factors H and T as Q S Z and Q P Z , where P 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(1),beta(1)),...,(alpha(n),beta(n)),
where alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus,
w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue
problem
det( A - w(j) B ) = 0
and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent
problem
det( m(j) A - B ) = 0
DTGEVC computes the matrix L of left eigenvectors and the matrix R
of right eigenvectors for the matrix pair ( S, P ). In the
description below, l and r are left and right eigenvectors
corresponding to the generalized eigenvalues (alpha,beta).
When DCHKGG 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, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 15
tests will be performed. The first twelve "test ratios" should be
small -- O(1). They will be compared with the threshhold THRESH:
T
(1) | A - U H V | / ( |A| n ulp )
T
(2) | B - U T V | / ( |B| n ulp )
T
(3) | I - UU | / ( n ulp )
T
(4) | I - VV | / ( n ulp )
T
(5) | H - Q S Z | / ( |H| n ulp )
T
(6) | T - Q P Z | / ( |T| n ulp )
T
(7) | I - QQ | / ( n ulp )
T
(8) | I - ZZ | / ( n ulp )
(9) max over all left eigenvalue/-vector pairs (beta/alpha,l) of
| l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )
(10) max over all left eigenvalue/-vector pairs (beta/alpha,l') of
T
| l'**H * (beta H - alpha T) | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors l' are the result of passing Q to
DTGEVC and back transforming (HOWMNY='B').
(11) max over all right eigenvalue/-vector pairs (beta/alpha,r) of
| (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )
(12) max over all right eigenvalue/-vector pairs (beta/alpha,r') of
| (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors r' are the result of passing Z to
DTGEVC and back transforming (HOWMNY='B').
The last three test ratios will usually be small, but there is no
mathematical requirement that they be so. They are therefore
compared with THRESH only if TSTDIF is .TRUE.
(13) | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )
(14) | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )
(15) max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
|beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp
In addition, the normalization of L and R are checked, and compared
with the threshhold THRSHN.
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) U ( J , J ) V where U and V are random orthogonal matrices.
(17) U ( T1, T2 ) V 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) U ( T1, T2 ) V diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) U ( T1, T2 ) V 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) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) U ( T1, T2 ) V 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) U ( big*T1, small*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) U ( small*T1, big*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) U ( small*T1, small*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) U ( big*T1, big*T2 ) V diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) U ( T1, T2 ) V 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,
DCHKGG does nothing. It must be at least zero. |
| NN |
(input) INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least zero. |
| NTYPES |
(input) INTEGER
The number of elements in DOTYPE. If it is zero, DCHKGG
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. 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 DCHKGG 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. It must be at least zero. |
| TSTDIF |
(input) LOGICAL
Specifies whether test ratios 13-15 will be computed and
compared with THRESH. = .FALSE.: Only test ratios 1-12 will be computed and tested. Ratios 13-15 will be set to zero. = .TRUE.: All the test ratios 1-15 will be computed and tested. |
| THRSHN |
(input) DOUBLE PRECISION
Threshhold for reporting eigenvector normalization error.
If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 5--10.) |
| 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, H, T, S1, P1, S2, and P2.
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. |
| H |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The upper Hessenberg matrix computed from A by DGGHRD.
|
| T |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by DGGHRD.
|
| S1 |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The Schur (block upper triangular) matrix computed from H by
DHGEQZ when Q and Z are also computed. |
| S2 |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The Schur (block upper triangular) matrix computed from H by
DHGEQZ when Q and Z are not computed. |
| P1 |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by DHGEQZ
when Q and Z are also computed. |
| P2 |
(workspace) DOUBLE PRECISION array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by DHGEQZ
when Q and Z are not computed. |
| U |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
The (left) orthogonal matrix computed by DGGHRD.
|
| LDU |
(input) INTEGER
The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR. It
must be at least 1 and at least max( NN ). |
| V |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
The (right) orthogonal matrix computed by DGGHRD.
|
| Q |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
The (left) orthogonal matrix computed by DHGEQZ.
|
| Z |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
The (left) orthogonal matrix computed by DHGEQZ.
|
| ALPHR1 |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| ALPHI1 |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| BETA1 |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by DHGEQZ when Q, Z, and the full Schur matrices are computed. On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th generalized eigenvalue of the matrices in A and B. |
| ALPHR3 |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| ALPHI3 |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| BETA3 |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
|
| EVECTL |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
The (block lower triangular) left eigenvector matrix for
the matrices in S1 and P1. (See DTGEVC for the format.) |
| EVEZTR |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
The (block upper triangular) right eigenvector matrix for
the matrices in S1 and P1. (See DTGEVC for the format.) |
| WORK |
(workspace) DOUBLE PRECISION array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The number of entries in WORK. This must be at least
max( 2 * N**2, 6*N, 1 ), for all N=NN(j). |
| LLWORK |
(workspace) LOGICAL array, dimension (max(NN))
|
| 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. |
| 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. |