*> \brief \b DHSEIN * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DHSEIN + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, * VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, * IFAILR, INFO ) * * .. Scalar Arguments .. * CHARACTER EIGSRC, INITV, SIDE * INTEGER INFO, LDH, LDVL, LDVR, M, MM, N * .. * .. Array Arguments .. * LOGICAL SELECT( * ) * INTEGER IFAILL( * ), IFAILR( * ) * DOUBLE PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), * \$ WI( * ), WORK( * ), WR( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DHSEIN uses inverse iteration to find specified right and/or left *> eigenvectors of a real upper Hessenberg matrix H. *> *> The right eigenvector x and the left eigenvector y of the matrix H *> corresponding to an eigenvalue w are defined by: *> *> H * x = w * x, y**h * H = w * y**h *> *> where y**h denotes the conjugate transpose of the vector y. *> \endverbatim * * Arguments: * ========== * *> \param[in] SIDE *> \verbatim *> SIDE is CHARACTER*1 *> = 'R': compute right eigenvectors only; *> = 'L': compute left eigenvectors only; *> = 'B': compute both right and left eigenvectors. *> \endverbatim *> *> \param[in] EIGSRC *> \verbatim *> EIGSRC is CHARACTER*1 *> Specifies the source of eigenvalues supplied in (WR,WI): *> = 'Q': the eigenvalues were found using DHSEQR; thus, if *> H has zero subdiagonal elements, and so is *> block-triangular, then the j-th eigenvalue can be *> assumed to be an eigenvalue of the block containing *> the j-th row/column. This property allows DHSEIN to *> perform inverse iteration on just one diagonal block. *> = 'N': no assumptions are made on the correspondence *> between eigenvalues and diagonal blocks. In this *> case, DHSEIN must always perform inverse iteration *> using the whole matrix H. *> \endverbatim *> *> \param[in] INITV *> \verbatim *> INITV is CHARACTER*1 *> = 'N': no initial vectors are supplied; *> = 'U': user-supplied initial vectors are stored in the arrays *> VL and/or VR. *> \endverbatim *> *> \param[in,out] SELECT *> \verbatim *> SELECT is LOGICAL array, dimension (N) *> Specifies the eigenvectors to be computed. To select the *> real eigenvector corresponding to a real eigenvalue WR(j), *> SELECT(j) must be set to .TRUE.. To select the complex *> eigenvector corresponding to a complex eigenvalue *> (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)), *> either SELECT(j) or SELECT(j+1) or both must be set to *> .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is *> .FALSE.. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix H. N >= 0. *> \endverbatim *> *> \param[in] H *> \verbatim *> H is DOUBLE PRECISION array, dimension (LDH,N) *> The upper Hessenberg matrix H. *> If a NaN is detected in H, the routine will return with INFO=-6. *> \endverbatim *> *> \param[in] LDH *> \verbatim *> LDH is INTEGER *> The leading dimension of the array H. LDH >= max(1,N). *> \endverbatim *> *> \param[in,out] WR *> \verbatim *> WR is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[in] WI *> \verbatim *> WI is DOUBLE PRECISION array, dimension (N) *> *> On entry, the real and imaginary parts of the eigenvalues of *> H; a complex conjugate pair of eigenvalues must be stored in *> consecutive elements of WR and WI. *> On exit, WR may have been altered since close eigenvalues *> are perturbed slightly in searching for independent *> eigenvectors. *> \endverbatim *> *> \param[in,out] VL *> \verbatim *> VL is DOUBLE PRECISION array, dimension (LDVL,MM) *> On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must *> contain starting vectors for the inverse iteration for the *> left eigenvectors; the starting vector for each eigenvector *> must be in the same column(s) in which the eigenvector will *> be stored. *> On exit, if SIDE = 'L' or 'B', the left eigenvectors *> specified by SELECT will be stored consecutively in the *> columns of VL, in the same order as their eigenvalues. A *> complex eigenvector corresponding to a complex eigenvalue is *> stored in two consecutive columns, the first holding the real *> part and the second the imaginary part. *> If SIDE = 'R', VL is not referenced. *> \endverbatim *> *> \param[in] LDVL *> \verbatim *> LDVL is INTEGER *> The leading dimension of the array VL. *> LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise. *> \endverbatim *> *> \param[in,out] VR *> \verbatim *> VR is DOUBLE PRECISION array, dimension (LDVR,MM) *> On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must *> contain starting vectors for the inverse iteration for the *> right eigenvectors; the starting vector for each eigenvector *> must be in the same column(s) in which the eigenvector will *> be stored. *> On exit, if SIDE = 'R' or 'B', the right eigenvectors *> specified by SELECT will be stored consecutively in the *> columns of VR, in the same order as their eigenvalues. A *> complex eigenvector corresponding to a complex eigenvalue is *> stored in two consecutive columns, the first holding the real *> part and the second the imaginary part. *> If SIDE = 'L', VR is not referenced. *> \endverbatim *> *> \param[in] LDVR *> \verbatim *> LDVR is INTEGER *> The leading dimension of the array VR. *> LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise. *> \endverbatim *> *> \param[in] MM *> \verbatim *> MM is INTEGER *> The number of columns in the arrays VL and/or VR. MM >= M. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The number of columns in the arrays VL and/or VR required to *> store the eigenvectors; each selected real eigenvector *> occupies one column and each selected complex eigenvector *> occupies two columns. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension ((N+2)*N) *> \endverbatim *> *> \param[out] IFAILL *> \verbatim *> IFAILL is INTEGER array, dimension (MM) *> If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left *> eigenvector in the i-th column of VL (corresponding to the *> eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the *> eigenvector converged satisfactorily. If the i-th and (i+1)th *> columns of VL hold a complex eigenvector, then IFAILL(i) and *> IFAILL(i+1) are set to the same value. *> If SIDE = 'R', IFAILL is not referenced. *> \endverbatim *> *> \param[out] IFAILR *> \verbatim *> IFAILR is INTEGER array, dimension (MM) *> If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right *> eigenvector in the i-th column of VR (corresponding to the *> eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the *> eigenvector converged satisfactorily. If the i-th and (i+1)th *> columns of VR hold a complex eigenvector, then IFAILR(i) and *> IFAILR(i+1) are set to the same value. *> If SIDE = 'L', IFAILR is not referenced. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: if INFO = i, i is the number of eigenvectors which *> failed to converge; see IFAILL and IFAILR for further *> details. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup doubleOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> Each eigenvector is normalized so that the element of largest *> magnitude has magnitude 1; here the magnitude of a complex number *> (x,y) is taken to be |x|+|y|. *> \endverbatim *> * ===================================================================== SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, \$ VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, \$ IFAILR, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER EIGSRC, INITV, SIDE INTEGER INFO, LDH, LDVL, LDVR, M, MM, N * .. * .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IFAILL( * ), IFAILR( * ) DOUBLE PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), \$ WI( * ), WORK( * ), WR( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, PAIR, RIGHTV INTEGER I, IINFO, K, KL, KLN, KR, KSI, KSR, LDWORK DOUBLE PRECISION BIGNUM, EPS3, HNORM, SMLNUM, ULP, UNFL, WKI, \$ WKR * .. * .. External Functions .. LOGICAL LSAME, DISNAN DOUBLE PRECISION DLAMCH, DLANHS EXTERNAL LSAME, DLAMCH, DLANHS, DISNAN * .. * .. External Subroutines .. EXTERNAL DLAEIN, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. Executable Statements .. * * Decode and test the input parameters. * BOTHV = LSAME( SIDE, 'B' ) RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV * FROMQR = LSAME( EIGSRC, 'Q' ) * NOINIT = LSAME( INITV, 'N' ) * * Set M to the number of columns required to store the selected * eigenvectors, and standardize the array SELECT. * M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. SELECT( K ) = .FALSE. ELSE IF( WI( K ).EQ.ZERO ) THEN IF( SELECT( K ) ) \$ M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) THEN SELECT( K ) = .TRUE. M = M + 2 END IF END IF END IF 10 CONTINUE * INFO = 0 IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -1 ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN INFO = -2 ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN INFO = -11 ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN INFO = -13 ELSE IF( MM.LT.M ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DHSEIN', -INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) \$ RETURN * * Set machine-dependent constants. * UNFL = DLAMCH( 'Safe minimum' ) ULP = DLAMCH( 'Precision' ) SMLNUM = UNFL*( N / ULP ) BIGNUM = ( ONE-ULP ) / SMLNUM * LDWORK = N + 1 * KL = 1 KLN = 0 IF( FROMQR ) THEN KR = 0 ELSE KR = N END IF KSR = 1 * DO 120 K = 1, N IF( SELECT( K ) ) THEN * * Compute eigenvector(s) corresponding to W(K). * IF( FROMQR ) THEN * * If affiliation of eigenvalues is known, check whether * the matrix splits. * * Determine KL and KR such that 1 <= KL <= K <= KR <= N * and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or * KR = N). * * Then inverse iteration can be performed with the * submatrix H(KL:N,KL:N) for a left eigenvector, and with * the submatrix H(1:KR,1:KR) for a right eigenvector. * DO 20 I = K, KL + 1, -1 IF( H( I, I-1 ).EQ.ZERO ) \$ GO TO 30 20 CONTINUE 30 CONTINUE KL = I IF( K.GT.KR ) THEN DO 40 I = K, N - 1 IF( H( I+1, I ).EQ.ZERO ) \$ GO TO 50 40 CONTINUE 50 CONTINUE KR = I END IF END IF * IF( KL.NE.KLN ) THEN KLN = KL * * Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it * has not ben computed before. * HNORM = DLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, WORK ) IF( DISNAN( HNORM ) ) THEN INFO = -6 RETURN ELSE IF( HNORM.GT.ZERO ) THEN EPS3 = HNORM*ULP ELSE EPS3 = SMLNUM END IF END IF * * Perturb eigenvalue if it is close to any previous * selected eigenvalues affiliated to the submatrix * H(KL:KR,KL:KR). Close roots are modified by EPS3. * WKR = WR( K ) WKI = WI( K ) 60 CONTINUE DO 70 I = K - 1, KL, -1 IF( SELECT( I ) .AND. ABS( WR( I )-WKR )+ \$ ABS( WI( I )-WKI ).LT.EPS3 ) THEN WKR = WKR + EPS3 GO TO 60 END IF 70 CONTINUE WR( K ) = WKR * PAIR = WKI.NE.ZERO IF( PAIR ) THEN KSI = KSR + 1 ELSE KSI = KSR END IF IF( LEFTV ) THEN * * Compute left eigenvector. * CALL DLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ), LDH, \$ WKR, WKI, VL( KL, KSR ), VL( KL, KSI ), \$ WORK, LDWORK, WORK( N*N+N+1 ), EPS3, SMLNUM, \$ BIGNUM, IINFO ) IF( IINFO.GT.0 ) THEN IF( PAIR ) THEN INFO = INFO + 2 ELSE INFO = INFO + 1 END IF IFAILL( KSR ) = K IFAILL( KSI ) = K ELSE IFAILL( KSR ) = 0 IFAILL( KSI ) = 0 END IF DO 80 I = 1, KL - 1 VL( I, KSR ) = ZERO 80 CONTINUE IF( PAIR ) THEN DO 90 I = 1, KL - 1 VL( I, KSI ) = ZERO 90 CONTINUE END IF END IF IF( RIGHTV ) THEN * * Compute right eigenvector. * CALL DLAEIN( .TRUE., NOINIT, KR, H, LDH, WKR, WKI, \$ VR( 1, KSR ), VR( 1, KSI ), WORK, LDWORK, \$ WORK( N*N+N+1 ), EPS3, SMLNUM, BIGNUM, \$ IINFO ) IF( IINFO.GT.0 ) THEN IF( PAIR ) THEN INFO = INFO + 2 ELSE INFO = INFO + 1 END IF IFAILR( KSR ) = K IFAILR( KSI ) = K ELSE IFAILR( KSR ) = 0 IFAILR( KSI ) = 0 END IF DO 100 I = KR + 1, N VR( I, KSR ) = ZERO 100 CONTINUE IF( PAIR ) THEN DO 110 I = KR + 1, N VR( I, KSI ) = ZERO 110 CONTINUE END IF END IF * IF( PAIR ) THEN KSR = KSR + 2 ELSE KSR = KSR + 1 END IF END IF 120 CONTINUE * RETURN * * End of DHSEIN * END