*> \brief \b ZLAHQR 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 ZLAHQR + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, * IHIZ, Z, LDZ, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N * LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. * COMPLEX*16 H( LDH, * ), W( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLAHQR is an auxiliary routine called by CHSEQR to update the *> eigenvalues and Schur decomposition already computed by CHSEQR, 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 triangular in rows and *> columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). *> ZLAHQR 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 COMPLEX*16 array, dimension (LDH,N) *> On entry, the upper Hessenberg matrix H. *> On exit, if INFO is zero and if WANTT is .TRUE., then H *> is upper triangular in rows and columns ILO:IHI. If INFO *> is zero and if WANTT is .FALSE., then the contents of H *> are unspecified on exit. The output state of H in case *> INF is positive is 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] W *> \verbatim *> W is COMPLEX*16 array, dimension (N) *> The computed eigenvalues ILO to IHI are stored in the *> corresponding elements of W. If WANTT is .TRUE., the *> eigenvalues are stored in the same order as on the diagonal *> of the Schur form returned in H, with W(i) = H(i,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 COMPLEX*16 array, dimension (LDZ,N) *> If WANTZ is .TRUE., on entry Z must contain the current *> matrix Z of transformations accumulated by CHSEQR, 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, ZLAHQR failed to compute all the *> eigenvalues ILO to IHI in a total of 30 iterations *> per eigenvalue; elements i+1:ihi of W 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 December 2016 * *> \ingroup complex16OTHERauxiliary * *> \par Contributors: * ================== *> *> \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 ZLAHQR 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 ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, $ IHIZ, Z, LDZ, INFO ) * * -- LAPACK auxiliary routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. COMPLEX*16 H( LDH, * ), W( * ), Z( LDZ, * ) * .. * * ========================================================= * * .. Parameters .. COMPLEX*16 ZERO, ONE PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), $ ONE = ( 1.0d0, 0.0d0 ) ) DOUBLE PRECISION RZERO, RONE, HALF PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0, HALF = 0.5d0 ) DOUBLE PRECISION DAT1 PARAMETER ( DAT1 = 3.0d0 / 4.0d0 ) * .. * .. Local Scalars .. COMPLEX*16 CDUM, H11, H11S, H22, SC, SUM, T, T1, TEMP, U, $ V2, X, Y DOUBLE PRECISION AA, AB, BA, BB, H10, H21, RTEMP, S, SAFMAX, $ SAFMIN, SMLNUM, SX, T2, TST, ULP INTEGER I, I1, I2, ITS, ITMAX, J, JHI, JLO, K, L, M, $ NH, NZ * .. * .. Local Arrays .. COMPLEX*16 V( 2 ) * .. * .. External Functions .. COMPLEX*16 ZLADIV DOUBLE PRECISION DLAMCH EXTERNAL ZLADIV, DLAMCH * .. * .. External Subroutines .. EXTERNAL DLABAD, ZCOPY, ZLARFG, ZSCAL * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT * .. * .. Statement Function definitions .. CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) ) * .. * .. Executable Statements .. * INFO = 0 * * Quick return if possible * IF( N.EQ.0 ) $ RETURN IF( ILO.EQ.IHI ) THEN W( ILO ) = H( ILO, ILO ) 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 * ==== ensure that subdiagonal entries are real ==== IF( WANTT ) THEN JLO = 1 JHI = N ELSE JLO = ILO JHI = IHI END IF DO 20 I = ILO + 1, IHI IF( DIMAG( H( I, I-1 ) ).NE.RZERO ) THEN * ==== The following redundant normalization * . avoids problems with both gradual and * . sudden underflow in ABS(H(I,I-1)) ==== SC = H( I, I-1 ) / CABS1( H( I, I-1 ) ) SC = DCONJG( SC ) / ABS( SC ) H( I, I-1 ) = ABS( H( I, I-1 ) ) CALL ZSCAL( JHI-I+1, SC, H( I, I ), LDH ) CALL ZSCAL( MIN( JHI, I+1 )-JLO+1, DCONJG( SC ), $ H( JLO, I ), 1 ) IF( WANTZ ) $ CALL ZSCAL( IHIZ-ILOZ+1, DCONJG( SC ), Z( ILOZ, I ), 1 ) END IF 20 CONTINUE * NH = IHI - ILO + 1 NZ = IHIZ - ILOZ + 1 * * Set machine-dependent constants for the stopping criterion. * SAFMIN = DLAMCH( 'SAFE MINIMUM' ) SAFMAX = RONE / SAFMIN CALL DLABAD( SAFMIN, SAFMAX ) ULP = DLAMCH( 'PRECISION' ) SMLNUM = SAFMIN*( DBLE( 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. 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 30 CONTINUE IF( I.LT.ILO ) $ GO TO 150 * * Perform QR iterations on rows and columns ILO to I until a * submatrix of order 1 splits off at the bottom because a * subdiagonal element has become negligible. * L = ILO DO 130 ITS = 0, ITMAX * * Look for a single small subdiagonal element. * DO 40 K = I, L + 1, -1 IF( CABS1( H( K, K-1 ) ).LE.SMLNUM ) $ GO TO 50 TST = CABS1( H( K-1, K-1 ) ) + CABS1( H( K, K ) ) IF( TST.EQ.ZERO ) THEN IF( K-2.GE.ILO ) $ TST = TST + ABS( DBLE( H( K-1, K-2 ) ) ) IF( K+1.LE.IHI ) $ TST = TST + ABS( DBLE( 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 examples. ==== IF( ABS( DBLE( H( K, K-1 ) ) ).LE.ULP*TST ) THEN AB = MAX( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) ) BA = MIN( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) ) AA = MAX( CABS1( H( K, K ) ), $ CABS1( H( K-1, K-1 )-H( K, K ) ) ) BB = MIN( CABS1( H( K, K ) ), $ CABS1( H( K-1, K-1 )-H( K, K ) ) ) S = AA + AB IF( BA*( AB / S ).LE.MAX( SMLNUM, $ ULP*( BB*( AA / S ) ) ) )GO TO 50 END IF 40 CONTINUE 50 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 has split off. * IF( L.GE.I ) $ GO TO 140 * * 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 = DAT1*ABS( DBLE( H( L+1, L ) ) ) T = S + H( L, L ) ELSE IF( ITS.EQ.20 ) THEN * * Exceptional shift. * S = DAT1*ABS( DBLE( H( I, I-1 ) ) ) T = S + H( I, I ) ELSE * * Wilkinson's shift. * T = H( I, I ) U = SQRT( H( I-1, I ) )*SQRT( H( I, I-1 ) ) S = CABS1( U ) IF( S.NE.RZERO ) THEN X = HALF*( H( I-1, I-1 )-T ) SX = CABS1( X ) S = MAX( S, CABS1( X ) ) Y = S*SQRT( ( X / S )**2+( U / S )**2 ) IF( SX.GT.RZERO ) THEN IF( DBLE( X / SX )*DBLE( Y )+DIMAG( X / SX )* $ DIMAG( Y ).LT.RZERO )Y = -Y END IF T = T - U*ZLADIV( U, ( X+Y ) ) END IF END IF * * Look for two consecutive small subdiagonal elements. * DO 60 M = I - 1, L + 1, -1 * * Determine the effect of starting the single-shift QR * iteration at row M, and see if this would make H(M,M-1) * negligible. * H11 = H( M, M ) H22 = H( M+1, M+1 ) H11S = H11 - T H21 = DBLE( H( M+1, M ) ) S = CABS1( H11S ) + ABS( H21 ) H11S = H11S / S H21 = H21 / S V( 1 ) = H11S V( 2 ) = H21 H10 = DBLE( H( M, M-1 ) ) IF( ABS( H10 )*ABS( H21 ).LE.ULP* $ ( CABS1( H11S )*( CABS1( H11 )+CABS1( H22 ) ) ) ) $ GO TO 70 60 CONTINUE H11 = H( L, L ) H22 = H( L+1, L+1 ) H11S = H11 - T H21 = DBLE( H( L+1, L ) ) S = CABS1( H11S ) + ABS( H21 ) H11S = H11S / S H21 = H21 / S V( 1 ) = H11S V( 2 ) = H21 70 CONTINUE * * Single-shift QR step * DO 120 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. * * V(2) is always real before the call to ZLARFG, and hence * after the call T2 ( = T1*V(2) ) is also real. * IF( K.GT.M ) $ CALL ZCOPY( 2, H( K, K-1 ), 1, V, 1 ) CALL ZLARFG( 2, V( 1 ), V( 2 ), 1, T1 ) IF( K.GT.M ) THEN H( K, K-1 ) = V( 1 ) H( K+1, K-1 ) = ZERO END IF V2 = V( 2 ) T2 = DBLE( T1*V2 ) * * Apply G from the left to transform the rows of the matrix * in columns K to I2. * DO 80 J = K, I2 SUM = DCONJG( T1 )*H( K, J ) + T2*H( K+1, J ) H( K, J ) = H( K, J ) - SUM H( K+1, J ) = H( K+1, J ) - SUM*V2 80 CONTINUE * * Apply G from the right to transform the columns of the * matrix in rows I1 to min(K+2,I). * DO 90 J = I1, MIN( K+2, I ) SUM = T1*H( J, K ) + T2*H( J, K+1 ) H( J, K ) = H( J, K ) - SUM H( J, K+1 ) = H( J, K+1 ) - SUM*DCONJG( V2 ) 90 CONTINUE * IF( WANTZ ) THEN * * Accumulate transformations in the matrix Z * DO 100 J = ILOZ, IHIZ SUM = T1*Z( J, K ) + T2*Z( J, K+1 ) Z( J, K ) = Z( J, K ) - SUM Z( J, K+1 ) = Z( J, K+1 ) - SUM*DCONJG( V2 ) 100 CONTINUE END IF * IF( K.EQ.M .AND. M.GT.L ) THEN * * If the QR step was started at row M > L because two * consecutive small subdiagonals were found, then extra * scaling must be performed to ensure that H(M,M-1) remains * real. * TEMP = ONE - T1 TEMP = TEMP / ABS( TEMP ) H( M+1, M ) = H( M+1, M )*DCONJG( TEMP ) IF( M+2.LE.I ) $ H( M+2, M+1 ) = H( M+2, M+1 )*TEMP DO 110 J = M, I IF( J.NE.M+1 ) THEN IF( I2.GT.J ) $ CALL ZSCAL( I2-J, TEMP, H( J, J+1 ), LDH ) CALL ZSCAL( J-I1, DCONJG( TEMP ), H( I1, J ), 1 ) IF( WANTZ ) THEN CALL ZSCAL( NZ, DCONJG( TEMP ), Z( ILOZ, J ), $ 1 ) END IF END IF 110 CONTINUE END IF 120 CONTINUE * * Ensure that H(I,I-1) is real. * TEMP = H( I, I-1 ) IF( DIMAG( TEMP ).NE.RZERO ) THEN RTEMP = ABS( TEMP ) H( I, I-1 ) = RTEMP TEMP = TEMP / RTEMP IF( I2.GT.I ) $ CALL ZSCAL( I2-I, DCONJG( TEMP ), H( I, I+1 ), LDH ) CALL ZSCAL( I-I1, TEMP, H( I1, I ), 1 ) IF( WANTZ ) THEN CALL ZSCAL( NZ, TEMP, Z( ILOZ, I ), 1 ) END IF END IF * 130 CONTINUE * * Failure to converge in remaining number of iterations * INFO = I RETURN * 140 CONTINUE * * H(I,I-1) is negligible: one eigenvalue has converged. * W( I ) = H( I, I ) * * return to start of the main loop with new value of I. * I = L - 1 GO TO 30 * 150 CONTINUE RETURN * * End of ZLAHQR * END