SUBROUTINE CTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
$ IWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, JOB
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
REAL DIF( * ), S( * )
COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CTGSNA estimates reciprocal condition numbers for specified
* eigenvalues and/or eigenvectors of a matrix pair (A, B).
*
* (A, B) must be in generalized Schur canonical form, that is, A and
* B are both upper triangular.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies whether condition numbers are required for
* eigenvalues (S) or eigenvectors (DIF):
* = 'E': for eigenvalues only (S);
* = 'V': for eigenvectors only (DIF);
* = 'B': for both eigenvalues and eigenvectors (S and DIF).
*
* HOWMNY (input) CHARACTER*1
* = 'A': compute condition numbers for all eigenpairs;
* = 'S': compute condition numbers for selected eigenpairs
* specified by the array SELECT.
*
* SELECT (input) LOGICAL array, dimension (N)
* If HOWMNY = 'S', SELECT specifies the eigenpairs for which
* condition numbers are required. To select condition numbers
* for the corresponding j-th eigenvalue and/or eigenvector,
* SELECT(j) must be set to .TRUE..
* If HOWMNY = 'A', SELECT is not referenced.
*
* N (input) INTEGER
* The order of the square matrix pair (A, B). N >= 0.
*
* A (input) COMPLEX array, dimension (LDA,N)
* The upper triangular matrix A in the pair (A,B).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input) COMPLEX array, dimension (LDB,N)
* The upper triangular matrix B in the pair (A, B).
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* VL (input) COMPLEX array, dimension (LDVL,M)
* IF JOB = 'E' or 'B', VL must contain left eigenvectors of
* (A, B), corresponding to the eigenpairs specified by HOWMNY
* and SELECT. The eigenvectors must be stored in consecutive
* columns of VL, as returned by CTGEVC.
* If JOB = 'V', VL is not referenced.
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1; and
* If JOB = 'E' or 'B', LDVL >= N.
*
* VR (input) COMPLEX array, dimension (LDVR,M)
* IF JOB = 'E' or 'B', VR must contain right eigenvectors of
* (A, B), corresponding to the eigenpairs specified by HOWMNY
* and SELECT. The eigenvectors must be stored in consecutive
* columns of VR, as returned by CTGEVC.
* If JOB = 'V', VR is not referenced.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1;
* If JOB = 'E' or 'B', LDVR >= N.
*
* S (output) REAL array, dimension (MM)
* If JOB = 'E' or 'B', the reciprocal condition numbers of the
* selected eigenvalues, stored in consecutive elements of the
* array.
* If JOB = 'V', S is not referenced.
*
* DIF (output) REAL array, dimension (MM)
* If JOB = 'V' or 'B', the estimated reciprocal condition
* numbers of the selected eigenvectors, stored in consecutive
* elements of the array.
* If the eigenvalues cannot be reordered to compute DIF(j),
* DIF(j) is set to 0; this can only occur when the true value
* would be very small anyway.
* For each eigenvalue/vector specified by SELECT, DIF stores
* a Frobenius norm-based estimate of Difl.
* If JOB = 'E', DIF is not referenced.
*
* MM (input) INTEGER
* The number of elements in the arrays S and DIF. MM >= M.
*
* M (output) INTEGER
* The number of elements of the arrays S and DIF used to store
* the specified condition numbers; for each selected eigenvalue
* one element is used. If HOWMNY = 'A', M is set to N.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
*
* IWORK (workspace) INTEGER array, dimension (N+2)
* If JOB = 'E', IWORK is not referenced.
*
* INFO (output) INTEGER
* = 0: Successful exit
* < 0: If INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The reciprocal of the condition number of the i-th generalized
* eigenvalue w = (a, b) is defined as
*
* S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
*
* where u and v are the right and left eigenvectors of (A, B)
* corresponding to w; |z| denotes the absolute value of the complex
* number, and norm(u) denotes the 2-norm of the vector u. The pair
* (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the
* matrix pair (A, B). If both a and b equal zero, then (A,B) is
* singular and S(I) = -1 is returned.
*
* An approximate error bound on the chordal distance between the i-th
* computed generalized eigenvalue w and the corresponding exact
* eigenvalue lambda is
*
* chord(w, lambda) <= EPS * norm(A, B) / S(I),
*
* where EPS is the machine precision.
*
* The reciprocal of the condition number of the right eigenvector u
* and left eigenvector v corresponding to the generalized eigenvalue w
* is defined as follows. Suppose
*
* (A, B) = ( a * ) ( b * ) 1
* ( 0 A22 ),( 0 B22 ) n-1
* 1 n-1 1 n-1
*
* Then the reciprocal condition number DIF(I) is
*
* Difl[(a, b), (A22, B22)] = sigma-min( Zl )
*
* where sigma-min(Zl) denotes the smallest singular value of
*
* Zl = [ kron(a, In-1) -kron(1, A22) ]
* [ kron(b, In-1) -kron(1, B22) ].
*
* Here In-1 is the identity matrix of size n-1 and X' is the conjugate
* transpose of X. kron(X, Y) is the Kronecker product between the
* matrices X and Y.
*
* We approximate the smallest singular value of Zl with an upper
* bound. This is done by CLATDF.
*
* An approximate error bound for a computed eigenvector VL(i) or
* VR(i) is given by
*
* EPS * norm(A, B) / DIF(i).
*
* See ref. [2-3] for more details and further references.
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* References
* ==========
*
* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
* M.S. Moonen et al (eds), Linear Algebra for Large Scale and
* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*
* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
* Eigenvalues of a Regular Matrix Pair (A, B) and Condition
* Estimation: Theory, Algorithms and Software, Report
* UMINF - 94.04, Department of Computing Science, Umea University,
* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
* To appear in Numerical Algorithms, 1996.
*
* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
* for Solving the Generalized Sylvester Equation and Estimating the
* Separation between Regular Matrix Pairs, Report UMINF - 93.23,
* Department of Computing Science, Umea University, S-901 87 Umea,
* Sweden, December 1993, Revised April 1994, Also as LAPACK Working
* Note 75.
* To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
INTEGER IDIFJB
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, IDIFJB = 3 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
REAL BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
COMPLEX YHAX, YHBX
* ..
* .. Local Arrays ..
COMPLEX DUMMY( 1 ), DUMMY1( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SCNRM2, SLAMCH, SLAPY2
COMPLEX CDOTC
EXTERNAL LSAME, SCNRM2, SLAMCH, SLAPY2, CDOTC
* ..
* .. External Subroutines ..
EXTERNAL CGEMV, CLACPY, CTGEXC, CTGSYL, SLABAD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CMPLX, MAX
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
*
SOMCON = LSAME( HOWMNY, 'S' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
INFO = -10
ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
INFO = -12
ELSE
*
* Set M to the number of eigenpairs for which condition numbers
* are required, and test MM.
*
IF( SOMCON ) THEN
M = 0
DO 10 K = 1, N
IF( SELECT( K ) )
$ M = M + 1
10 CONTINUE
ELSE
M = N
END IF
*
IF( N.EQ.0 ) THEN
LWMIN = 1
ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
LWMIN = 2*N*N
ELSE
LWMIN = N
END IF
WORK( 1 ) = LWMIN
*
IF( MM.LT.M ) THEN
INFO = -15
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTGSNA', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
KS = 0
DO 20 K = 1, N
*
* Determine whether condition numbers are required for the k-th
* eigenpair.
*
IF( SOMCON ) THEN
IF( .NOT.SELECT( K ) )
$ GO TO 20
END IF
*
KS = KS + 1
*
IF( WANTS ) THEN
*
* Compute the reciprocal condition number of the k-th
* eigenvalue.
*
RNRM = SCNRM2( N, VR( 1, KS ), 1 )
LNRM = SCNRM2( N, VL( 1, KS ), 1 )
CALL CGEMV( 'N', N, N, CMPLX( ONE, ZERO ), A, LDA,
$ VR( 1, KS ), 1, CMPLX( ZERO, ZERO ), WORK, 1 )
YHAX = CDOTC( N, WORK, 1, VL( 1, KS ), 1 )
CALL CGEMV( 'N', N, N, CMPLX( ONE, ZERO ), B, LDB,
$ VR( 1, KS ), 1, CMPLX( ZERO, ZERO ), WORK, 1 )
YHBX = CDOTC( N, WORK, 1, VL( 1, KS ), 1 )
COND = SLAPY2( ABS( YHAX ), ABS( YHBX ) )
IF( COND.EQ.ZERO ) THEN
S( KS ) = -ONE
ELSE
S( KS ) = COND / ( RNRM*LNRM )
END IF
END IF
*
IF( WANTDF ) THEN
IF( N.EQ.1 ) THEN
DIF( KS ) = SLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) )
ELSE
*
* Estimate the reciprocal condition number of the k-th
* eigenvectors.
*
* Copy the matrix (A, B) to the array WORK and move the
* (k,k)th pair to the (1,1) position.
*
CALL CLACPY( 'Full', N, N, A, LDA, WORK, N )
CALL CLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
IFST = K
ILST = 1
*
CALL CTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
$ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR )
*
IF( IERR.GT.0 ) THEN
*
* Ill-conditioned problem - swap rejected.
*
DIF( KS ) = ZERO
ELSE
*
* Reordering successful, solve generalized Sylvester
* equation for R and L,
* A22 * R - L * A11 = A12
* B22 * R - L * B11 = B12,
* and compute estimate of Difl[(A11,B11), (A22, B22)].
*
N1 = 1
N2 = N - N1
I = N*N + 1
CALL CTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ),
$ N, WORK, N, WORK( N1+1 ), N,
$ WORK( N*N1+N1+I ), N, WORK( I ), N,
$ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY,
$ 1, IWORK, IERR )
END IF
END IF
END IF
*
20 CONTINUE
WORK( 1 ) = LWMIN
RETURN
*
* End of CTGSNA
*
END