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LAPACK
3.6.1
LAPACK: Linear Algebra PACKage
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LAPACK
3.6.1
LAPACK: Linear Algebra PACKage
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| subroutine cchkgg | ( | integer | NSIZES, |
| integer, dimension( * ) | NN, | ||
| integer | NTYPES, | ||
| logical, dimension( * ) | DOTYPE, | ||
| integer, dimension( 4 ) | ISEED, | ||
| real | THRESH, | ||
| logical | TSTDIF, | ||
| real | THRSHN, | ||
| integer | NOUNIT, | ||
| complex, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| complex, dimension( lda, * ) | B, | ||
| complex, dimension( lda, * ) | H, | ||
| complex, dimension( lda, * ) | T, | ||
| complex, dimension( lda, * ) | S1, | ||
| complex, dimension( lda, * ) | S2, | ||
| complex, dimension( lda, * ) | P1, | ||
| complex, dimension( lda, * ) | P2, | ||
| complex, dimension( ldu, * ) | U, | ||
| integer | LDU, | ||
| complex, dimension( ldu, * ) | V, | ||
| complex, dimension( ldu, * ) | Q, | ||
| complex, dimension( ldu, * ) | Z, | ||
| complex, dimension( * ) | ALPHA1, | ||
| complex, dimension( * ) | BETA1, | ||
| complex, dimension( * ) | ALPHA3, | ||
| complex, dimension( * ) | BETA3, | ||
| complex, dimension( ldu, * ) | EVECTL, | ||
| complex, dimension( ldu, * ) | EVECTR, | ||
| complex, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| real, dimension( * ) | RWORK, | ||
| logical, dimension( * ) | LLWORK, | ||
| real, dimension( 15 ) | RESULT, | ||
| integer | INFO | ||
| ) |
CCHKGG
CCHKGG checks the nonsymmetric generalized eigenvalue problem
routines.
H H H
CGGHRD factors A and B as U H V and U T V , where means conjugate
transpose, H is hessenberg, T is triangular and U and V are unitary.
H H
CHGEQZ factors H and T as Q S Z and Q P Z , where P and S are upper
triangular and Q and Z are unitary. 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
CTGEVC 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 CCHKGG 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, 13
tests will be performed. The first twelve "test ratios" should be
small -- O(1). They will be compared with the threshold THRESH:
H
(1) | A - U H V | / ( |A| n ulp )
H
(2) | B - U T V | / ( |B| n ulp )
H
(3) | I - UU | / ( n ulp )
H
(4) | I - VV | / ( n ulp )
H
(5) | H - Q S Z | / ( |H| n ulp )
H
(6) | T - Q P Z | / ( |T| n ulp )
H
(7) | I - QQ | / ( n ulp )
H
(8) | I - ZZ | / ( n ulp )
(9) max over all left eigenvalue/-vector pairs (beta/alpha,l) of
H
| (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) )
(10) max over all left eigenvalue/-vector pairs (beta/alpha,l') of
H
| (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors l' are the result of passing Q to
STGEVC and back transforming (JOB='B').
(11) max over all right eigenvalue/-vector pairs (beta/alpha,r) of
| (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )
(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
STGEVC and back transforming (JOB='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 threshold 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 P*D1, P is a random unitary diagonal
matrix (i.e., with random magnitude 1 entries
on the diagonal), and D1=diag( 0, 1,..., N-1 )
(i.e., a diagonal matrix with D1(1,1)=0,
D1(2,2)=1, ..., D1(N,N)=N-1.)
(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=P*diag( 0, 0, 1, ..., N-3, 0 ) and
D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and
P and Q are random unitary diagonal matrices.
t t
(16) U ( J , J ) V where U and V are random unitary 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) =
P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
Q*( 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) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) U ( small*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) U ( big*T1, big*T2 ) V diag(T1) = P*( 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. | [in] | NSIZES | NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CCHKGG does nothing. It must be at least zero. |
| [in] | NN | NN is 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. |
| [in] | NTYPES | NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CCHKGG
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. . |
| [in] | DOTYPE | DOTYPE is 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. |
| [in,out] | ISEED | ISEED is 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 CCHKGG to continue the same random number
sequence. |
| [in] | THRESH | THRESH is 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. |
| [in] | TSTDIF | TSTDIF is 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. |
| [in] | THRSHN | THRSHN is REAL
Threshold 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.) |
| [in] | NOUNIT | NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.) |
| [in,out] | A | A is COMPLEX 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. |
| [in] | LDA | LDA is 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 ). |
| [in,out] | B | B is COMPLEX 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. |
| [out] | H | H is COMPLEX array, dimension (LDA, max(NN))
The upper Hessenberg matrix computed from A by CGGHRD. |
| [out] | T | T is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGHRD. |
| [out] | S1 | S1 is COMPLEX array, dimension (LDA, max(NN))
The Schur (upper triangular) matrix computed from H by CHGEQZ
when Q and Z are also computed. |
| [out] | S2 | S2 is COMPLEX array, dimension (LDA, max(NN))
The Schur (upper triangular) matrix computed from H by CHGEQZ
when Q and Z are not computed. |
| [out] | P1 | P1 is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by CHGEQZ
when Q and Z are also computed. |
| [out] | P2 | P2 is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by CHGEQZ
when Q and Z are not computed. |
| [out] | U | U is COMPLEX array, dimension (LDU, max(NN))
The (left) unitary matrix computed by CGGHRD. |
| [in] | LDU | LDU is INTEGER
The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It
must be at least 1 and at least max( NN ). |
| [out] | V | V is COMPLEX array, dimension (LDU, max(NN))
The (right) unitary matrix computed by CGGHRD. |
| [out] | Q | Q is COMPLEX array, dimension (LDU, max(NN))
The (left) unitary matrix computed by CHGEQZ. |
| [out] | Z | Z is COMPLEX array, dimension (LDU, max(NN))
The (left) unitary matrix computed by CHGEQZ. |
| [out] | ALPHA1 | ALPHA1 is COMPLEX array, dimension (max(NN)) |
| [out] | BETA1 | BETA1 is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CHGEQZ
when Q, Z, and the full Schur matrices are computed. |
| [out] | ALPHA3 | ALPHA3 is COMPLEX array, dimension (max(NN)) |
| [out] | BETA3 | BETA3 is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CHGEQZ
when neither Q, Z, nor the Schur matrices are computed. |
| [out] | EVECTL | EVECTL is COMPLEX array, dimension (LDU, max(NN))
The (lower triangular) left eigenvector matrix for the
matrices in S1 and P1. |
| [out] | EVECTR | EVECTR is COMPLEX array, dimension (LDU, max(NN))
The (upper triangular) right eigenvector matrix for the
matrices in S1 and P1. |
| [out] | WORK | WORK is COMPLEX array, dimension (LWORK) |
| [in] | LWORK | LWORK is INTEGER
The number of entries in WORK. This must be at least
max( 4*N, 2 * N**2, 1 ), for all N=NN(j). |
| [out] | RWORK | RWORK is REAL array, dimension (2*max(NN)) |
| [out] | LLWORK | LLWORK is LOGICAL array, dimension (max(NN)) |
| [out] | RESULT | RESULT is REAL array, dimension (15)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow. |
| [out] | INFO | INFO is 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. |
Definition at line 505 of file cchkgg.f.
1.8.10