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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 schkgg | ( | integer | NSIZES, |
| integer, dimension( * ) | NN, | ||
| integer | NTYPES, | ||
| logical, dimension( * ) | DOTYPE, | ||
| integer, dimension( 4 ) | ISEED, | ||
| real | THRESH, | ||
| logical | TSTDIF, | ||
| real | THRSHN, | ||
| integer | NOUNIT, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( lda, * ) | B, | ||
| real, dimension( lda, * ) | H, | ||
| real, dimension( lda, * ) | T, | ||
| real, dimension( lda, * ) | S1, | ||
| real, dimension( lda, * ) | S2, | ||
| real, dimension( lda, * ) | P1, | ||
| real, dimension( lda, * ) | P2, | ||
| real, dimension( ldu, * ) | U, | ||
| integer | LDU, | ||
| real, dimension( ldu, * ) | V, | ||
| real, dimension( ldu, * ) | Q, | ||
| real, dimension( ldu, * ) | Z, | ||
| real, dimension( * ) | ALPHR1, | ||
| real, dimension( * ) | ALPHI1, | ||
| real, dimension( * ) | BETA1, | ||
| real, dimension( * ) | ALPHR3, | ||
| real, dimension( * ) | ALPHI3, | ||
| real, dimension( * ) | BETA3, | ||
| real, dimension( ldu, * ) | EVECTL, | ||
| real, dimension( ldu, * ) | EVECTR, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| logical, dimension( * ) | LLWORK, | ||
| real, dimension( 15 ) | RESULT, | ||
| integer | INFO | ||
| ) |
SCHKGG
SCHKGG checks the nonsymmetric generalized eigenvalue problem
routines.
T T T
SGGHRD 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
SHGEQZ 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
STGEVC 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 SCHKGG 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 threshold 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
STGEVC 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
STGEVC 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 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 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. | [in] | NSIZES | NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
SCHKGG 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, SCHKGG
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 SCHKGG 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 REAL 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 REAL 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 REAL array, dimension (LDA, max(NN))
The upper Hessenberg matrix computed from A by SGGHRD. |
| [out] | T | T is REAL array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by SGGHRD. |
| [out] | S1 | S1 is REAL array, dimension (LDA, max(NN))
The Schur (block upper triangular) matrix computed from H by
SHGEQZ when Q and Z are also computed. |
| [out] | S2 | S2 is REAL array, dimension (LDA, max(NN))
The Schur (block upper triangular) matrix computed from H by
SHGEQZ when Q and Z are not computed. |
| [out] | P1 | P1 is REAL array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by SHGEQZ
when Q and Z are also computed. |
| [out] | P2 | P2 is REAL array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by SHGEQZ
when Q and Z are not computed. |
| [out] | U | U is REAL array, dimension (LDU, max(NN))
The (left) orthogonal matrix computed by SGGHRD. |
| [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 REAL array, dimension (LDU, max(NN))
The (right) orthogonal matrix computed by SGGHRD. |
| [out] | Q | Q is REAL array, dimension (LDU, max(NN))
The (left) orthogonal matrix computed by SHGEQZ. |
| [out] | Z | Z is REAL array, dimension (LDU, max(NN))
The (left) orthogonal matrix computed by SHGEQZ. |
| [out] | ALPHR1 | ALPHR1 is REAL array, dimension (max(NN)) |
| [out] | ALPHI1 | ALPHI1 is REAL array, dimension (max(NN)) |
| [out] | BETA1 | BETA1 is REAL array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by SHGEQZ
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. |
| [out] | ALPHR3 | ALPHR3 is REAL array, dimension (max(NN)) |
| [out] | ALPHI3 | ALPHI3 is REAL array, dimension (max(NN)) |
| [out] | BETA3 | BETA3 is REAL array, dimension (max(NN)) |
| [out] | EVECTL | EVECTL is REAL array, dimension (LDU, max(NN))
The (block lower triangular) left eigenvector matrix for
the matrices in S1 and P1. (See STGEVC for the format.) |
| [out] | EVECTR | EVECTR is REAL array, dimension (LDU, max(NN))
The (block upper triangular) right eigenvector matrix for
the matrices in S1 and P1. (See STGEVC for the format.) |
| [out] | WORK | WORK is REAL array, dimension (LWORK) |
| [in] | LWORK | LWORK is INTEGER
The number of entries in WORK. This must be at least
max( 2 * N**2, 6*N, 1 ), for all N=NN(j). |
| [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 513 of file schkgg.f.
1.8.10