304 lines
11 KiB
FortranFixed
304 lines
11 KiB
FortranFixed
SUBROUTINE DGET52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR,
|
|
$ ALPHAI, BETA, WORK, RESULT )
|
|
*
|
|
* -- LAPACK test routine (version 3.1) --
|
|
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
|
|
* November 2006
|
|
*
|
|
* .. Scalar Arguments ..
|
|
LOGICAL LEFT
|
|
INTEGER LDA, LDB, LDE, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
|
$ B( LDB, * ), BETA( * ), E( LDE, * ),
|
|
$ RESULT( 2 ), WORK( * )
|
|
* ..
|
|
*
|
|
* Purpose
|
|
* =======
|
|
*
|
|
* DGET52 does an eigenvector check for the generalized eigenvalue
|
|
* problem.
|
|
*
|
|
* The basic test for right eigenvectors is:
|
|
*
|
|
* | b(j) A E(j) - a(j) B E(j) |
|
|
* RESULT(1) = max -------------------------------
|
|
* j n ulp max( |b(j) A|, |a(j) B| )
|
|
*
|
|
* using the 1-norm. Here, a(j)/b(j) = w is the j-th generalized
|
|
* eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
|
|
* generalized eigenvalue of m A - B.
|
|
*
|
|
* For real eigenvalues, the test is straightforward. For complex
|
|
* eigenvalues, E(j) and a(j) are complex, represented by
|
|
* Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
|
|
* eigenvector becomes
|
|
*
|
|
* max( |Wr|, |Wi| )
|
|
* --------------------------------------------
|
|
* n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
|
|
*
|
|
* where
|
|
*
|
|
* Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
|
|
*
|
|
* Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
|
|
*
|
|
* T T _
|
|
* For left eigenvectors, A , B , a, and b are used.
|
|
*
|
|
* DGET52 also tests the normalization of E. Each eigenvector is
|
|
* supposed to be normalized so that the maximum "absolute value"
|
|
* of its elements is 1, where in this case, "absolute value"
|
|
* of a complex value x is |Re(x)| + |Im(x)| ; let us call this
|
|
* maximum "absolute value" norm of a vector v M(v).
|
|
* if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
|
|
* vector. The normalization test is:
|
|
*
|
|
* RESULT(2) = max | M(v(j)) - 1 | / ( n ulp )
|
|
* eigenvectors v(j)
|
|
*
|
|
* Arguments
|
|
* =========
|
|
*
|
|
* LEFT (input) LOGICAL
|
|
* =.TRUE.: The eigenvectors in the columns of E are assumed
|
|
* to be *left* eigenvectors.
|
|
* =.FALSE.: The eigenvectors in the columns of E are assumed
|
|
* to be *right* eigenvectors.
|
|
*
|
|
* N (input) INTEGER
|
|
* The size of the matrices. If it is zero, DGET52 does
|
|
* nothing. It must be at least zero.
|
|
*
|
|
* A (input) DOUBLE PRECISION array, dimension (LDA, N)
|
|
* The matrix A.
|
|
*
|
|
* LDA (input) INTEGER
|
|
* The leading dimension of A. It must be at least 1
|
|
* and at least N.
|
|
*
|
|
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
|
|
* The matrix B.
|
|
*
|
|
* LDB (input) INTEGER
|
|
* The leading dimension of B. It must be at least 1
|
|
* and at least N.
|
|
*
|
|
* E (input) DOUBLE PRECISION array, dimension (LDE, N)
|
|
* The matrix of eigenvectors. It must be O( 1 ). Complex
|
|
* eigenvalues and eigenvectors always come in pairs, the
|
|
* eigenvalue and its conjugate being stored in adjacent
|
|
* elements of ALPHAR, ALPHAI, and BETA. Thus, if a(j)/b(j)
|
|
* and a(j+1)/b(j+1) are a complex conjugate pair of
|
|
* generalized eigenvalues, then E(,j) contains the real part
|
|
* of the eigenvector and E(,j+1) contains the imaginary part.
|
|
* Note that whether E(,j) is a real eigenvector or part of a
|
|
* complex one is specified by whether ALPHAI(j) is zero or not.
|
|
*
|
|
* LDE (input) INTEGER
|
|
* The leading dimension of E. It must be at least 1 and at
|
|
* least N.
|
|
*
|
|
* ALPHAR (input) DOUBLE PRECISION array, dimension (N)
|
|
* The real parts of the values a(j) as described above, which,
|
|
* along with b(j), define the generalized eigenvalues.
|
|
* Complex eigenvalues always come in complex conjugate pairs
|
|
* a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent
|
|
* elements in ALPHAR, ALPHAI, and BETA. Thus, if the j-th
|
|
* and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)
|
|
* is assumed to be equal to ALPHAR(j)/BETA(j).
|
|
*
|
|
* ALPHAI (input) DOUBLE PRECISION array, dimension (N)
|
|
* The imaginary parts of the values a(j) as described above,
|
|
* which, along with b(j), define the generalized eigenvalues.
|
|
* If ALPHAI(j)=0, then the eigenvalue is real, otherwise it
|
|
* is part of a complex conjugate pair. Complex eigenvalues
|
|
* always come in complex conjugate pairs a(j)/b(j) and
|
|
* a(j+1)/b(j+1), which are stored in adjacent elements in
|
|
* ALPHAR, ALPHAI, and BETA. Thus, if the j-th and (j+1)-st
|
|
* eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to
|
|
* be equal to -ALPHAI(j)/BETA(j). Also, nonzero values in
|
|
* ALPHAI are assumed to always come in adjacent pairs.
|
|
*
|
|
* BETA (input) DOUBLE PRECISION array, dimension (N)
|
|
* The values b(j) as described above, which, along with a(j),
|
|
* define the generalized eigenvalues.
|
|
*
|
|
* WORK (workspace) DOUBLE PRECISION array, dimension (N**2+N)
|
|
*
|
|
* RESULT (output) DOUBLE PRECISION array, dimension (2)
|
|
* The values computed by the test described above. If A E or
|
|
* B E is likely to overflow, then RESULT(1:2) is set to
|
|
* 10 / ulp.
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE, TEN
|
|
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL ILCPLX
|
|
CHARACTER NORMAB, TRANS
|
|
INTEGER J, JVEC
|
|
DOUBLE PRECISION ABMAX, ACOEF, ALFMAX, ANORM, BCOEFI, BCOEFR,
|
|
$ BETMAX, BNORM, ENORM, ENRMER, ERRNRM, SAFMAX,
|
|
$ SAFMIN, SALFI, SALFR, SBETA, SCALE, TEMP1, ULP
|
|
* ..
|
|
* .. External Functions ..
|
|
DOUBLE PRECISION DLAMCH, DLANGE
|
|
EXTERNAL DLAMCH, DLANGE
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DGEMV
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC ABS, DBLE, MAX
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
RESULT( 1 ) = ZERO
|
|
RESULT( 2 ) = ZERO
|
|
IF( N.LE.0 )
|
|
$ RETURN
|
|
*
|
|
SAFMIN = DLAMCH( 'Safe minimum' )
|
|
SAFMAX = ONE / SAFMIN
|
|
ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
|
|
*
|
|
IF( LEFT ) THEN
|
|
TRANS = 'T'
|
|
NORMAB = 'I'
|
|
ELSE
|
|
TRANS = 'N'
|
|
NORMAB = 'O'
|
|
END IF
|
|
*
|
|
* Norm of A, B, and E:
|
|
*
|
|
ANORM = MAX( DLANGE( NORMAB, N, N, A, LDA, WORK ), SAFMIN )
|
|
BNORM = MAX( DLANGE( NORMAB, N, N, B, LDB, WORK ), SAFMIN )
|
|
ENORM = MAX( DLANGE( 'O', N, N, E, LDE, WORK ), ULP )
|
|
ALFMAX = SAFMAX / MAX( ONE, BNORM )
|
|
BETMAX = SAFMAX / MAX( ONE, ANORM )
|
|
*
|
|
* Compute error matrix.
|
|
* Column i = ( b(i) A - a(i) B ) E(i) / max( |a(i) B| |b(i) A| )
|
|
*
|
|
ILCPLX = .FALSE.
|
|
DO 10 JVEC = 1, N
|
|
IF( ILCPLX ) THEN
|
|
*
|
|
* 2nd Eigenvalue/-vector of pair -- do nothing
|
|
*
|
|
ILCPLX = .FALSE.
|
|
ELSE
|
|
SALFR = ALPHAR( JVEC )
|
|
SALFI = ALPHAI( JVEC )
|
|
SBETA = BETA( JVEC )
|
|
IF( SALFI.EQ.ZERO ) THEN
|
|
*
|
|
* Real eigenvalue and -vector
|
|
*
|
|
ABMAX = MAX( ABS( SALFR ), ABS( SBETA ) )
|
|
IF( ABS( SALFR ).GT.ALFMAX .OR. ABS( SBETA ).GT.
|
|
$ BETMAX .OR. ABMAX.LT.ONE ) THEN
|
|
SCALE = ONE / MAX( ABMAX, SAFMIN )
|
|
SALFR = SCALE*SALFR
|
|
SBETA = SCALE*SBETA
|
|
END IF
|
|
SCALE = ONE / MAX( ABS( SALFR )*BNORM,
|
|
$ ABS( SBETA )*ANORM, SAFMIN )
|
|
ACOEF = SCALE*SBETA
|
|
BCOEFR = SCALE*SALFR
|
|
CALL DGEMV( TRANS, N, N, ACOEF, A, LDA, E( 1, JVEC ), 1,
|
|
$ ZERO, WORK( N*( JVEC-1 )+1 ), 1 )
|
|
CALL DGEMV( TRANS, N, N, -BCOEFR, B, LDA, E( 1, JVEC ),
|
|
$ 1, ONE, WORK( N*( JVEC-1 )+1 ), 1 )
|
|
ELSE
|
|
*
|
|
* Complex conjugate pair
|
|
*
|
|
ILCPLX = .TRUE.
|
|
IF( JVEC.EQ.N ) THEN
|
|
RESULT( 1 ) = TEN / ULP
|
|
RETURN
|
|
END IF
|
|
ABMAX = MAX( ABS( SALFR )+ABS( SALFI ), ABS( SBETA ) )
|
|
IF( ABS( SALFR )+ABS( SALFI ).GT.ALFMAX .OR.
|
|
$ ABS( SBETA ).GT.BETMAX .OR. ABMAX.LT.ONE ) THEN
|
|
SCALE = ONE / MAX( ABMAX, SAFMIN )
|
|
SALFR = SCALE*SALFR
|
|
SALFI = SCALE*SALFI
|
|
SBETA = SCALE*SBETA
|
|
END IF
|
|
SCALE = ONE / MAX( ( ABS( SALFR )+ABS( SALFI ) )*BNORM,
|
|
$ ABS( SBETA )*ANORM, SAFMIN )
|
|
ACOEF = SCALE*SBETA
|
|
BCOEFR = SCALE*SALFR
|
|
BCOEFI = SCALE*SALFI
|
|
IF( LEFT ) THEN
|
|
BCOEFI = -BCOEFI
|
|
END IF
|
|
*
|
|
CALL DGEMV( TRANS, N, N, ACOEF, A, LDA, E( 1, JVEC ), 1,
|
|
$ ZERO, WORK( N*( JVEC-1 )+1 ), 1 )
|
|
CALL DGEMV( TRANS, N, N, -BCOEFR, B, LDA, E( 1, JVEC ),
|
|
$ 1, ONE, WORK( N*( JVEC-1 )+1 ), 1 )
|
|
CALL DGEMV( TRANS, N, N, BCOEFI, B, LDA, E( 1, JVEC+1 ),
|
|
$ 1, ONE, WORK( N*( JVEC-1 )+1 ), 1 )
|
|
*
|
|
CALL DGEMV( TRANS, N, N, ACOEF, A, LDA, E( 1, JVEC+1 ),
|
|
$ 1, ZERO, WORK( N*JVEC+1 ), 1 )
|
|
CALL DGEMV( TRANS, N, N, -BCOEFI, B, LDA, E( 1, JVEC ),
|
|
$ 1, ONE, WORK( N*JVEC+1 ), 1 )
|
|
CALL DGEMV( TRANS, N, N, -BCOEFR, B, LDA, E( 1, JVEC+1 ),
|
|
$ 1, ONE, WORK( N*JVEC+1 ), 1 )
|
|
END IF
|
|
END IF
|
|
10 CONTINUE
|
|
*
|
|
ERRNRM = DLANGE( 'One', N, N, WORK, N, WORK( N**2+1 ) ) / ENORM
|
|
*
|
|
* Compute RESULT(1)
|
|
*
|
|
RESULT( 1 ) = ERRNRM / ULP
|
|
*
|
|
* Normalization of E:
|
|
*
|
|
ENRMER = ZERO
|
|
ILCPLX = .FALSE.
|
|
DO 40 JVEC = 1, N
|
|
IF( ILCPLX ) THEN
|
|
ILCPLX = .FALSE.
|
|
ELSE
|
|
TEMP1 = ZERO
|
|
IF( ALPHAI( JVEC ).EQ.ZERO ) THEN
|
|
DO 20 J = 1, N
|
|
TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) ) )
|
|
20 CONTINUE
|
|
ENRMER = MAX( ENRMER, TEMP1-ONE )
|
|
ELSE
|
|
ILCPLX = .TRUE.
|
|
DO 30 J = 1, N
|
|
TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) )+
|
|
$ ABS( E( J, JVEC+1 ) ) )
|
|
30 CONTINUE
|
|
ENRMER = MAX( ENRMER, TEMP1-ONE )
|
|
END IF
|
|
END IF
|
|
40 CONTINUE
|
|
*
|
|
* Compute RESULT(2) : the normalization error in E.
|
|
*
|
|
RESULT( 2 ) = ENRMER / ( DBLE( N )*ULP )
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DGET52
|
|
*
|
|
END
|