SUBROUTINE CTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
$ LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK,
$ INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH.
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, JOB
INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
REAL RWORK( * ), S( * ), SEP( * )
COMPLEX T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( LDWORK, * )
* ..
*
* Purpose
* =======
*
* CTRSNA estimates reciprocal condition numbers for specified
* eigenvalues and/or right eigenvectors of a complex upper triangular
* matrix T (or of any matrix Q*T*Q**H with Q unitary).
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies whether condition numbers are required for
* eigenvalues (S) or eigenvectors (SEP):
* = 'E': for eigenvalues only (S);
* = 'V': for eigenvectors only (SEP);
* = 'B': for both eigenvalues and eigenvectors (S and SEP).
*
* 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 j-th eigenpair, SELECT(j) must be set to .TRUE..
* If HOWMNY = 'A', SELECT is not referenced.
*
* N (input) INTEGER
* The order of the matrix T. N >= 0.
*
* T (input) COMPLEX array, dimension (LDT,N)
* The upper triangular matrix T.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= max(1,N).
*
* VL (input) COMPLEX array, dimension (LDVL,M)
* If JOB = 'E' or 'B', VL must contain left eigenvectors of T
* (or of any Q*T*Q**H with Q unitary), corresponding to the
* eigenpairs specified by HOWMNY and SELECT. The eigenvectors
* must be stored in consecutive columns of VL, as returned by
* CHSEIN or CTREVC.
* 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 T
* (or of any Q*T*Q**H with Q unitary), corresponding to the
* eigenpairs specified by HOWMNY and SELECT. The eigenvectors
* must be stored in consecutive columns of VR, as returned by
* CHSEIN or CTREVC.
* If JOB = 'V', VR is not referenced.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR.
* LDVR >= 1; and 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. Thus S(j), SEP(j), and the j-th columns of VL and VR
* all correspond to the same eigenpair (but not in general the
* j-th eigenpair, unless all eigenpairs are selected).
* If JOB = 'V', S is not referenced.
*
* SEP (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 JOB = 'E', SEP is not referenced.
*
* MM (input) INTEGER
* The number of elements in the arrays S (if JOB = 'E' or 'B')
* and/or SEP (if JOB = 'V' or 'B'). MM >= M.
*
* M (output) INTEGER
* The number of elements of the arrays S and/or SEP actually
* used to store the estimated condition numbers.
* If HOWMNY = 'A', M is set to N.
*
* WORK (workspace) COMPLEX array, dimension (LDWORK,N+6)
* If JOB = 'E', WORK is not referenced.
*
* LDWORK (input) INTEGER
* The leading dimension of the array WORK.
* LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
*
* RWORK (workspace) REAL array, dimension (N)
* If JOB = 'E', RWORK 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 an eigenvalue lambda is
* defined as
*
* S(lambda) = |v'*u| / (norm(u)*norm(v))
*
* where u and v are the right and left eigenvectors of T corresponding
* to lambda; v' denotes the conjugate transpose of v, and norm(u)
* denotes the Euclidean norm. These reciprocal condition numbers always
* lie between zero (very badly conditioned) and one (very well
* conditioned). If n = 1, S(lambda) is defined to be 1.
*
* An approximate error bound for a computed eigenvalue W(i) is given by
*
* EPS * norm(T) / S(i)
*
* where EPS is the machine precision.
*
* The reciprocal of the condition number of the right eigenvector u
* corresponding to lambda is defined as follows. Suppose
*
* T = ( lambda c )
* ( 0 T22 )
*
* Then the reciprocal condition number is
*
* SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
*
* where sigma-min denotes the smallest singular value. We approximate
* the smallest singular value by the reciprocal of an estimate of the
* one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
* defined to be abs(T(1,1)).
*
* An approximate error bound for a computed right eigenvector VR(i)
* is given by
*
* EPS * norm(T) / SEP(i)
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0+0 )
* ..
* .. Local Scalars ..
LOGICAL SOMCON, WANTBH, WANTS, WANTSP
CHARACTER NORMIN
INTEGER I, IERR, IX, J, K, KASE, KS
REAL BIGNUM, EPS, EST, LNRM, RNRM, SCALE, SMLNUM,
$ XNORM
COMPLEX CDUM, PROD
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
COMPLEX DUMMY( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ICAMAX
REAL SCNRM2, SLAMCH
COMPLEX CDOTC
EXTERNAL LSAME, ICAMAX, SCNRM2, SLAMCH, CDOTC
* ..
* .. External Subroutines ..
EXTERNAL CLACN2, CLACPY, CLATRS, CSRSCL, CTREXC, SLABAD,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MAX, REAL
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
*
SOMCON = LSAME( HOWMNY, 'S' )
*
* Set M to the number of eigenpairs for which condition numbers are
* to be computed.
*
IF( SOMCON ) THEN
M = 0
DO 10 J = 1, N
IF( SELECT( J ) )
$ M = M + 1
10 CONTINUE
ELSE
M = N
END IF
*
INFO = 0
IF( .NOT.WANTS .AND. .NOT.WANTSP ) 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( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDVL.LT.1 .OR. ( WANTS .AND. LDVL.LT.N ) ) THEN
INFO = -8
ELSE IF( LDVR.LT.1 .OR. ( WANTS .AND. LDVR.LT.N ) ) THEN
INFO = -10
ELSE IF( MM.LT.M ) THEN
INFO = -13
ELSE IF( LDWORK.LT.1 .OR. ( WANTSP .AND. LDWORK.LT.N ) ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTRSNA', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( SOMCON ) THEN
IF( .NOT.SELECT( 1 ) )
$ RETURN
END IF
IF( WANTS )
$ S( 1 ) = ONE
IF( WANTSP )
$ SEP( 1 ) = ABS( T( 1, 1 ) )
RETURN
END IF
*
* Get machine constants
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
*
KS = 1
DO 50 K = 1, N
*
IF( SOMCON ) THEN
IF( .NOT.SELECT( K ) )
$ GO TO 50
END IF
*
IF( WANTS ) THEN
*
* Compute the reciprocal condition number of the k-th
* eigenvalue.
*
PROD = CDOTC( N, VR( 1, KS ), 1, VL( 1, KS ), 1 )
RNRM = SCNRM2( N, VR( 1, KS ), 1 )
LNRM = SCNRM2( N, VL( 1, KS ), 1 )
S( KS ) = ABS( PROD ) / ( RNRM*LNRM )
*
END IF
*
IF( WANTSP ) THEN
*
* Estimate the reciprocal condition number of the k-th
* eigenvector.
*
* Copy the matrix T to the array WORK and swap the k-th
* diagonal element to the (1,1) position.
*
CALL CLACPY( 'Full', N, N, T, LDT, WORK, LDWORK )
CALL CTREXC( 'No Q', N, WORK, LDWORK, DUMMY, 1, K, 1, IERR )
*
* Form C = T22 - lambda*I in WORK(2:N,2:N).
*
DO 20 I = 2, N
WORK( I, I ) = WORK( I, I ) - WORK( 1, 1 )
20 CONTINUE
*
* Estimate a lower bound for the 1-norm of inv(C'). The 1st
* and (N+1)th columns of WORK are used to store work vectors.
*
SEP( KS ) = ZERO
EST = ZERO
KASE = 0
NORMIN = 'N'
30 CONTINUE
CALL CLACN2( N-1, WORK( 1, N+1 ), WORK, EST, KASE, ISAVE )
*
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve C'*x = scale*b
*
CALL CLATRS( 'Upper', 'Conjugate transpose',
$ 'Nonunit', NORMIN, N-1, WORK( 2, 2 ),
$ LDWORK, WORK, SCALE, RWORK, IERR )
ELSE
*
* Solve C*x = scale*b
*
CALL CLATRS( 'Upper', 'No transpose', 'Nonunit',
$ NORMIN, N-1, WORK( 2, 2 ), LDWORK, WORK,
$ SCALE, RWORK, IERR )
END IF
NORMIN = 'Y'
IF( SCALE.NE.ONE ) THEN
*
* Multiply by 1/SCALE if doing so will not cause
* overflow.
*
IX = ICAMAX( N-1, WORK, 1 )
XNORM = CABS1( WORK( IX, 1 ) )
IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 40
CALL CSRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 30
END IF
*
SEP( KS ) = ONE / MAX( EST, SMLNUM )
END IF
*
40 CONTINUE
KS = KS + 1
50 CONTINUE
RETURN
*
* End of CTRSNA
*
END