*> \brief \b SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAHQR + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, * ILOZ, IHIZ, Z, LDZ, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N * LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. * REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAHQR is an auxiliary routine called by SHSEQR to update the *> eigenvalues and Schur decomposition already computed by SHSEQR, by *> dealing with the Hessenberg submatrix in rows and columns ILO to *> IHI. *> \endverbatim * * Arguments: * ========== * *> \param[in] WANTT *> \verbatim *> WANTT is LOGICAL *> = .TRUE. : the full Schur form T is required; *> = .FALSE.: only eigenvalues are required. *> \endverbatim *> *> \param[in] WANTZ *> \verbatim *> WANTZ is LOGICAL *> = .TRUE. : the matrix of Schur vectors Z is required; *> = .FALSE.: Schur vectors are not required. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix H. N >= 0. *> \endverbatim *> *> \param[in] ILO *> \verbatim *> ILO is INTEGER *> \endverbatim *> *> \param[in] IHI *> \verbatim *> IHI is INTEGER *> It is assumed that H is already upper quasi-triangular in *> rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless *> ILO = 1). SLAHQR works primarily with the Hessenberg *> submatrix in rows and columns ILO to IHI, but applies *> transformations to all of H if WANTT is .TRUE.. *> 1 <= ILO <= max(1,IHI); IHI <= N. *> \endverbatim *> *> \param[in,out] H *> \verbatim *> H is REAL array, dimension (LDH,N) *> On entry, the upper Hessenberg matrix H. *> On exit, if INFO is zero and if WANTT is .TRUE., H is upper *> quasi-triangular in rows and columns ILO:IHI, with any *> 2-by-2 diagonal blocks in standard form. If INFO is zero *> and WANTT is .FALSE., the contents of H are unspecified on *> exit. The output state of H if INFO is nonzero is given *> below under the description of INFO. *> \endverbatim *> *> \param[in] LDH *> \verbatim *> LDH is INTEGER *> The leading dimension of the array H. LDH >= max(1,N). *> \endverbatim *> *> \param[out] WR *> \verbatim *> WR is REAL array, dimension (N) *> \endverbatim *> *> \param[out] WI *> \verbatim *> WI is REAL array, dimension (N) *> The real and imaginary parts, respectively, of the computed *> eigenvalues ILO to IHI are stored in the corresponding *> elements of WR and WI. If two eigenvalues are computed as a *> complex conjugate pair, they are stored in consecutive *> elements of WR and WI, say the i-th and (i+1)th, with *> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the *> eigenvalues are stored in the same order as on the diagonal *> of the Schur form returned in H, with WR(i) = H(i,i), and, if *> H(i:i+1,i:i+1) is a 2-by-2 diagonal block, *> WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). *> \endverbatim *> *> \param[in] ILOZ *> \verbatim *> ILOZ is INTEGER *> \endverbatim *> *> \param[in] IHIZ *> \verbatim *> IHIZ is INTEGER *> Specify the rows of Z to which transformations must be *> applied if WANTZ is .TRUE.. *> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is REAL array, dimension (LDZ,N) *> If WANTZ is .TRUE., on entry Z must contain the current *> matrix Z of transformations accumulated by SHSEQR, and on *> exit Z has been updated; transformations are applied only to *> the submatrix Z(ILOZ:IHIZ,ILO:IHI). *> If WANTZ is .FALSE., Z is not referenced. *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. LDZ >= max(1,N). *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> .GT. 0: If INFO = i, SLAHQR failed to compute all the *> eigenvalues ILO to IHI in a total of 30 iterations *> per eigenvalue; elements i+1:ihi of WR and WI *> contain those eigenvalues which have been *> successfully computed. *> *> If INFO .GT. 0 and WANTT is .FALSE., then on exit, *> the remaining unconverged eigenvalues are the *> eigenvalues of the upper Hessenberg matrix rows *> and columns ILO thorugh INFO of the final, output *> value of H. *> *> If INFO .GT. 0 and WANTT is .TRUE., then on exit *> (*) (initial value of H)*U = U*(final value of H) *> where U is an orthognal matrix. The final *> value of H is upper Hessenberg and triangular in *> rows and columns INFO+1 through IHI. *> *> If INFO .GT. 0 and WANTZ is .TRUE., then on exit *> (final value of Z) = (initial value of Z)*U *> where U is the orthogonal matrix in (*) *> (regardless of the value of WANTT.) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2015 * *> \ingroup realOTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> 02-96 Based on modifications by *> David Day, Sandia National Laboratory, USA *> *> 12-04 Further modifications by *> Ralph Byers, University of Kansas, USA *> This is a modified version of SLAHQR from LAPACK version 3.0. *> It is (1) more robust against overflow and underflow and *> (2) adopts the more conservative Ahues & Tisseur stopping *> criterion (LAWN 122, 1997). *> \endverbatim *> * ===================================================================== SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, $ ILOZ, IHIZ, Z, LDZ, INFO ) * * -- LAPACK auxiliary routine (version 3.6.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2015 * * .. Scalar Arguments .. INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * ) * .. * * ========================================================= * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0, TWO = 2.0e0 ) REAL DAT1, DAT2 PARAMETER ( DAT1 = 3.0e0 / 4.0e0, DAT2 = -0.4375e0 ) * .. * .. Local Scalars .. REAL AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S, $ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX, $ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST, $ ULP, V2, V3 INTEGER I, I1, I2, ITS, ITMAX, J, K, L, M, NH, NR, NZ * .. * .. Local Arrays .. REAL V( 3 ) * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. External Subroutines .. EXTERNAL SCOPY, SLABAD, SLANV2, SLARFG, SROT * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, REAL, SQRT * .. * .. Executable Statements .. * INFO = 0 * * Quick return if possible * IF( N.EQ.0 ) $ RETURN IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF * * ==== clear out the trash ==== DO 10 J = ILO, IHI - 3 H( J+2, J ) = ZERO H( J+3, J ) = ZERO 10 CONTINUE IF( ILO.LE.IHI-2 ) $ H( IHI, IHI-2 ) = ZERO * NH = IHI - ILO + 1 NZ = IHIZ - ILOZ + 1 * * Set machine-dependent constants for the stopping criterion. * SAFMIN = SLAMCH( 'SAFE MINIMUM' ) SAFMAX = ONE / SAFMIN CALL SLABAD( SAFMIN, SAFMAX ) ULP = SLAMCH( 'PRECISION' ) SMLNUM = SAFMIN*( REAL( NH ) / ULP ) * * I1 and I2 are the indices of the first row and last column of H * to which transformations must be applied. If eigenvalues only are * being computed, I1 and I2 are set inside the main loop. * IF( WANTT ) THEN I1 = 1 I2 = N END IF * * ITMAX is the total number of QR iterations allowed. * ITMAX = 30 * MAX( 10, NH ) * * The main loop begins here. I is the loop index and decreases from * IHI to ILO in steps of 1 or 2. Each iteration of the loop works * with the active submatrix in rows and columns L to I. * Eigenvalues I+1 to IHI have already converged. Either L = ILO or * H(L,L-1) is negligible so that the matrix splits. * I = IHI 20 CONTINUE L = ILO IF( I.LT.ILO ) $ GO TO 160 * * Perform QR iterations on rows and columns ILO to I until a * submatrix of order 1 or 2 splits off at the bottom because a * subdiagonal element has become negligible. * DO 140 ITS = 0, ITMAX * * Look for a single small subdiagonal element. * DO 30 K = I, L + 1, -1 IF( ABS( H( K, K-1 ) ).LE.SMLNUM ) $ GO TO 40 TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST.EQ.ZERO ) THEN IF( K-2.GE.ILO ) $ TST = TST + ABS( H( K-1, K-2 ) ) IF( K+1.LE.IHI ) $ TST = TST + ABS( H( K+1, K ) ) END IF * ==== The following is a conservative small subdiagonal * . deflation criterion due to Ahues & Tisseur (LAWN 122, * . 1997). It has better mathematical foundation and * . improves accuracy in some cases. ==== IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) ) BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) ) AA = MAX( ABS( H( K, K ) ), $ ABS( H( K-1, K-1 )-H( K, K ) ) ) BB = MIN( ABS( H( K, K ) ), $ ABS( H( K-1, K-1 )-H( K, K ) ) ) S = AA + AB IF( BA*( AB / S ).LE.MAX( SMLNUM, $ ULP*( BB*( AA / S ) ) ) )GO TO 40 END IF 30 CONTINUE 40 CONTINUE L = K IF( L.GT.ILO ) THEN * * H(L,L-1) is negligible * H( L, L-1 ) = ZERO END IF * * Exit from loop if a submatrix of order 1 or 2 has split off. * IF( L.GE.I-1 ) $ GO TO 150 * * Now the active submatrix is in rows and columns L to I. If * eigenvalues only are being computed, only the active submatrix * need be transformed. * IF( .NOT.WANTT ) THEN I1 = L I2 = I END IF * IF( ITS.EQ.10 ) THEN * * Exceptional shift. * S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) ) H11 = DAT1*S + H( L, L ) H12 = DAT2*S H21 = S H22 = H11 ELSE IF( ITS.EQ.20 ) THEN * * Exceptional shift. * S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) H11 = DAT1*S + H( I, I ) H12 = DAT2*S H21 = S H22 = H11 ELSE * * Prepare to use Francis' double shift * (i.e. 2nd degree generalized Rayleigh quotient) * H11 = H( I-1, I-1 ) H21 = H( I, I-1 ) H12 = H( I-1, I ) H22 = H( I, I ) END IF S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 ) IF( S.EQ.ZERO ) THEN RT1R = ZERO RT1I = ZERO RT2R = ZERO RT2I = ZERO ELSE H11 = H11 / S H21 = H21 / S H12 = H12 / S H22 = H22 / S TR = ( H11+H22 ) / TWO DET = ( H11-TR )*( H22-TR ) - H12*H21 RTDISC = SQRT( ABS( DET ) ) IF( DET.GE.ZERO ) THEN * * ==== complex conjugate shifts ==== * RT1R = TR*S RT2R = RT1R RT1I = RTDISC*S RT2I = -RT1I ELSE * * ==== real shifts (use only one of them) ==== * RT1R = TR + RTDISC RT2R = TR - RTDISC IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN RT1R = RT1R*S RT2R = RT1R ELSE RT2R = RT2R*S RT1R = RT2R END IF RT1I = ZERO RT2I = ZERO END IF END IF * * Look for two consecutive small subdiagonal elements. * DO 50 M = I - 2, L, -1 * Determine the effect of starting the double-shift QR * iteration at row M, and see if this would make H(M,M-1) * negligible. (The following uses scaling to avoid * overflows and most underflows.) * H21S = H( M+1, M ) S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S ) H21S = H( M+1, M ) / S V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )* $ ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S ) V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R ) V( 3 ) = H21S*H( M+2, M+1 ) S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) ) V( 1 ) = V( 1 ) / S V( 2 ) = V( 2 ) / S V( 3 ) = V( 3 ) / S IF( M.EQ.L ) $ GO TO 60 IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE. $ ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M, $ M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60 50 CONTINUE 60 CONTINUE * * Double-shift QR step * DO 130 K = M, I - 1 * * The first iteration of this loop determines a reflection G * from the vector V and applies it from left and right to H, * thus creating a nonzero bulge below the subdiagonal. * * Each subsequent iteration determines a reflection G to * restore the Hessenberg form in the (K-1)th column, and thus * chases the bulge one step toward the bottom of the active * submatrix. NR is the order of G. * NR = MIN( 3, I-K+1 ) IF( K.GT.M ) $ CALL SCOPY( NR, H( K, K-1 ), 1, V, 1 ) CALL SLARFG( NR, V( 1 ), V( 2 ), 1, T1 ) IF( K.GT.M ) THEN H( K, K-1 ) = V( 1 ) H( K+1, K-1 ) = ZERO IF( K.LT.I-1 ) $ H( K+2, K-1 ) = ZERO ELSE IF( M.GT.L ) THEN * ==== Use the following instead of * . H( K, K-1 ) = -H( K, K-1 ) to * . avoid a bug when v(2) and v(3) * . underflow. ==== H( K, K-1 ) = H( K, K-1 )*( ONE-T1 ) END IF V2 = V( 2 ) T2 = T1*V2 IF( NR.EQ.3 ) THEN V3 = V( 3 ) T3 = T1*V3 * * Apply G from the left to transform the rows of the matrix * in columns K to I2. * DO 70 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 H( K+2, J ) = H( K+2, J ) - SUM*T3 70 CONTINUE * * Apply G from the right to transform the columns of the * matrix in rows I1 to min(K+3,I). * DO 80 J = I1, MIN( K+3, I ) SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 H( J, K+2 ) = H( J, K+2 ) - SUM*T3 80 CONTINUE * IF( WANTZ ) THEN * * Accumulate transformations in the matrix Z * DO 90 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3 90 CONTINUE END IF ELSE IF( NR.EQ.2 ) THEN * * Apply G from the left to transform the rows of the matrix * in columns K to I2. * DO 100 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 100 CONTINUE * * Apply G from the right to transform the columns of the * matrix in rows I1 to min(K+3,I). * DO 110 J = I1, I SUM = H( J, K ) + V2*H( J, K+1 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 110 CONTINUE * IF( WANTZ ) THEN * * Accumulate transformations in the matrix Z * DO 120 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 120 CONTINUE END IF END IF 130 CONTINUE * 140 CONTINUE * * Failure to converge in remaining number of iterations * INFO = I RETURN * 150 CONTINUE * IF( L.EQ.I ) THEN * * H(I,I-1) is negligible: one eigenvalue has converged. * WR( I ) = H( I, I ) WI( I ) = ZERO ELSE IF( L.EQ.I-1 ) THEN * * H(I-1,I-2) is negligible: a pair of eigenvalues have converged. * * Transform the 2-by-2 submatrix to standard Schur form, * and compute and store the eigenvalues. * CALL SLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ), $ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ), $ CS, SN ) * IF( WANTT ) THEN * * Apply the transformation to the rest of H. * IF( I2.GT.I ) $ CALL SROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH, $ CS, SN ) CALL SROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN ) END IF IF( WANTZ ) THEN * * Apply the transformation to Z. * CALL SROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN ) END IF END IF * * return to start of the main loop with new value of I. * I = L - 1 GO TO 20 * 160 CONTINUE RETURN * * End of SLAHQR * END