SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, \$ VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, \$ IFAILR, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. 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( * ) * .. * * Purpose * ======= * * 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. * * Arguments * ========= * * SIDE (input) CHARACTER*1 * = 'R': compute right eigenvectors only; * = 'L': compute left eigenvectors only; * = 'B': compute both right and left eigenvectors. * * EIGSRC (input) 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. * * INITV (input) CHARACTER*1 * = 'N': no initial vectors are supplied; * = 'U': user-supplied initial vectors are stored in the arrays * VL and/or VR. * * SELECT (input/output) 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.. * * N (input) INTEGER * The order of the matrix H. N >= 0. * * H (input) DOUBLE PRECISION array, dimension (LDH,N) * The upper Hessenberg matrix H. * * LDH (input) INTEGER * The leading dimension of the array H. LDH >= max(1,N). * * WR (input/output) DOUBLE PRECISION array, dimension (N) * WI (input) 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. * * VL (input/output) 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. * * LDVL (input) INTEGER * The leading dimension of the array VL. * LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise. * * VR (input/output) 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. * * LDVR (input) INTEGER * The leading dimension of the array VR. * LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise. * * MM (input) INTEGER * The number of columns in the arrays VL and/or VR. MM >= M. * * M (output) 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. * * WORK (workspace) DOUBLE PRECISION array, dimension ((N+2)*N) * * IFAILL (output) 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. * * IFAILR (output) 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. * * INFO (output) 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. * * Further Details * =============== * * 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|. * * ===================================================================== * * .. 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 DOUBLE PRECISION DLAMCH, DLANHS EXTERNAL LSAME, DLAMCH, DLANHS * .. * .. 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( 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