1*> \brief \b ZGEHRD 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZGEHRD + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgehrd.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgehrd.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgehrd.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) 22* 23* .. Scalar Arguments .. 24* INTEGER IHI, ILO, INFO, LDA, LWORK, N 25* .. 26* .. Array Arguments .. 27* COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) 28* .. 29* 30* 31*> \par Purpose: 32* ============= 33*> 34*> \verbatim 35*> 36*> ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by 37*> an unitary similarity transformation: Q**H * A * Q = H . 38*> \endverbatim 39* 40* Arguments: 41* ========== 42* 43*> \param[in] N 44*> \verbatim 45*> N is INTEGER 46*> The order of the matrix A. N >= 0. 47*> \endverbatim 48*> 49*> \param[in] ILO 50*> \verbatim 51*> ILO is INTEGER 52*> \endverbatim 53*> 54*> \param[in] IHI 55*> \verbatim 56*> IHI is INTEGER 57*> 58*> It is assumed that A is already upper triangular in rows 59*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally 60*> set by a previous call to ZGEBAL; otherwise they should be 61*> set to 1 and N respectively. See Further Details. 62*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. 63*> \endverbatim 64*> 65*> \param[in,out] A 66*> \verbatim 67*> A is COMPLEX*16 array, dimension (LDA,N) 68*> On entry, the N-by-N general matrix to be reduced. 69*> On exit, the upper triangle and the first subdiagonal of A 70*> are overwritten with the upper Hessenberg matrix H, and the 71*> elements below the first subdiagonal, with the array TAU, 72*> represent the unitary matrix Q as a product of elementary 73*> reflectors. See Further Details. 74*> \endverbatim 75*> 76*> \param[in] LDA 77*> \verbatim 78*> LDA is INTEGER 79*> The leading dimension of the array A. LDA >= max(1,N). 80*> \endverbatim 81*> 82*> \param[out] TAU 83*> \verbatim 84*> TAU is COMPLEX*16 array, dimension (N-1) 85*> The scalar factors of the elementary reflectors (see Further 86*> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to 87*> zero. 88*> \endverbatim 89*> 90*> \param[out] WORK 91*> \verbatim 92*> WORK is COMPLEX*16 array, dimension (LWORK) 93*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 94*> \endverbatim 95*> 96*> \param[in] LWORK 97*> \verbatim 98*> LWORK is INTEGER 99*> The length of the array WORK. LWORK >= max(1,N). 100*> For optimum performance LWORK >= N*NB, where NB is the 101*> optimal blocksize. 102*> 103*> If LWORK = -1, then a workspace query is assumed; the routine 104*> only calculates the optimal size of the WORK array, returns 105*> this value as the first entry of the WORK array, and no error 106*> message related to LWORK is issued by XERBLA. 107*> \endverbatim 108*> 109*> \param[out] INFO 110*> \verbatim 111*> INFO is INTEGER 112*> = 0: successful exit 113*> < 0: if INFO = -i, the i-th argument had an illegal value. 114*> \endverbatim 115* 116* Authors: 117* ======== 118* 119*> \author Univ. of Tennessee 120*> \author Univ. of California Berkeley 121*> \author Univ. of Colorado Denver 122*> \author NAG Ltd. 123* 124*> \date November 2011 125* 126*> \ingroup complex16GEcomputational 127* 128*> \par Further Details: 129* ===================== 130*> 131*> \verbatim 132*> 133*> The matrix Q is represented as a product of (ihi-ilo) elementary 134*> reflectors 135*> 136*> Q = H(ilo) H(ilo+1) . . . H(ihi-1). 137*> 138*> Each H(i) has the form 139*> 140*> H(i) = I - tau * v * v**H 141*> 142*> where tau is a complex scalar, and v is a complex vector with 143*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on 144*> exit in A(i+2:ihi,i), and tau in TAU(i). 145*> 146*> The contents of A are illustrated by the following example, with 147*> n = 7, ilo = 2 and ihi = 6: 148*> 149*> on entry, on exit, 150*> 151*> ( a a a a a a a ) ( a a h h h h a ) 152*> ( a a a a a a ) ( a h h h h a ) 153*> ( a a a a a a ) ( h h h h h h ) 154*> ( a a a a a a ) ( v2 h h h h h ) 155*> ( a a a a a a ) ( v2 v3 h h h h ) 156*> ( a a a a a a ) ( v2 v3 v4 h h h ) 157*> ( a ) ( a ) 158*> 159*> where a denotes an element of the original matrix A, h denotes a 160*> modified element of the upper Hessenberg matrix H, and vi denotes an 161*> element of the vector defining H(i). 162*> 163*> This file is a slight modification of LAPACK-3.0's DGEHRD 164*> subroutine incorporating improvements proposed by Quintana-Orti and 165*> Van de Geijn (2006). (See DLAHR2.) 166*> \endverbatim 167*> 168* ===================================================================== 169 SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) 170* 171* -- LAPACK computational routine (version 3.4.0) -- 172* -- LAPACK is a software package provided by Univ. of Tennessee, -- 173* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 174* November 2011 175* 176* .. Scalar Arguments .. 177 INTEGER IHI, ILO, INFO, LDA, LWORK, N 178* .. 179* .. Array Arguments .. 180 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) 181* .. 182* 183* ===================================================================== 184* 185* .. Parameters .. 186 INTEGER NBMAX, LDT 187 PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) 188 COMPLEX*16 ZERO, ONE 189 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), 190 $ ONE = ( 1.0D+0, 0.0D+0 ) ) 191* .. 192* .. Local Scalars .. 193 LOGICAL LQUERY 194 INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB, 195 $ NBMIN, NH, NX 196 COMPLEX*16 EI 197* .. 198* .. Local Arrays .. 199 COMPLEX*16 T( LDT, NBMAX ) 200* .. 201* .. External Subroutines .. 202 EXTERNAL ZAXPY, ZGEHD2, ZGEMM, ZLAHR2, ZLARFB, ZTRMM, 203 $ XERBLA 204* .. 205* .. Intrinsic Functions .. 206 INTRINSIC MAX, MIN 207* .. 208* .. External Functions .. 209 INTEGER ILAENV 210 EXTERNAL ILAENV 211* .. 212* .. Executable Statements .. 213* 214* Test the input parameters 215* 216 INFO = 0 217 NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) 218 LWKOPT = N*NB 219 WORK( 1 ) = LWKOPT 220 LQUERY = ( LWORK.EQ.-1 ) 221 IF( N.LT.0 ) THEN 222 INFO = -1 223 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN 224 INFO = -2 225 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN 226 INFO = -3 227 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 228 INFO = -5 229 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN 230 INFO = -8 231 END IF 232 IF( INFO.NE.0 ) THEN 233 CALL XERBLA( 'ZGEHRD', -INFO ) 234 RETURN 235 ELSE IF( LQUERY ) THEN 236 RETURN 237 END IF 238* 239* Set elements 1:ILO-1 and IHI:N-1 of TAU to zero 240* 241 DO 10 I = 1, ILO - 1 242 TAU( I ) = ZERO 243 10 CONTINUE 244 DO 20 I = MAX( 1, IHI ), N - 1 245 TAU( I ) = ZERO 246 20 CONTINUE 247* 248* Quick return if possible 249* 250 NH = IHI - ILO + 1 251 IF( NH.LE.1 ) THEN 252 WORK( 1 ) = 1 253 RETURN 254 END IF 255* 256* Determine the block size 257* 258 NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) 259 NBMIN = 2 260 IWS = 1 261 IF( NB.GT.1 .AND. NB.LT.NH ) THEN 262* 263* Determine when to cross over from blocked to unblocked code 264* (last block is always handled by unblocked code) 265* 266 NX = MAX( NB, ILAENV( 3, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) 267 IF( NX.LT.NH ) THEN 268* 269* Determine if workspace is large enough for blocked code 270* 271 IWS = N*NB 272 IF( LWORK.LT.IWS ) THEN 273* 274* Not enough workspace to use optimal NB: determine the 275* minimum value of NB, and reduce NB or force use of 276* unblocked code 277* 278 NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI, 279 $ -1 ) ) 280 IF( LWORK.GE.N*NBMIN ) THEN 281 NB = LWORK / N 282 ELSE 283 NB = 1 284 END IF 285 END IF 286 END IF 287 END IF 288 LDWORK = N 289* 290 IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN 291* 292* Use unblocked code below 293* 294 I = ILO 295* 296 ELSE 297* 298* Use blocked code 299* 300 DO 40 I = ILO, IHI - 1 - NX, NB 301 IB = MIN( NB, IHI-I ) 302* 303* Reduce columns i:i+ib-1 to Hessenberg form, returning the 304* matrices V and T of the block reflector H = I - V*T*V**H 305* which performs the reduction, and also the matrix Y = A*V*T 306* 307 CALL ZLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT, 308 $ WORK, LDWORK ) 309* 310* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the 311* right, computing A := A - Y * V**H. V(i+ib,ib-1) must be set 312* to 1 313* 314 EI = A( I+IB, I+IB-1 ) 315 A( I+IB, I+IB-1 ) = ONE 316 CALL ZGEMM( 'No transpose', 'Conjugate transpose', 317 $ IHI, IHI-I-IB+1, 318 $ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE, 319 $ A( 1, I+IB ), LDA ) 320 A( I+IB, I+IB-1 ) = EI 321* 322* Apply the block reflector H to A(1:i,i+1:i+ib-1) from the 323* right 324* 325 CALL ZTRMM( 'Right', 'Lower', 'Conjugate transpose', 326 $ 'Unit', I, IB-1, 327 $ ONE, A( I+1, I ), LDA, WORK, LDWORK ) 328 DO 30 J = 0, IB-2 329 CALL ZAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1, 330 $ A( 1, I+J+1 ), 1 ) 331 30 CONTINUE 332* 333* Apply the block reflector H to A(i+1:ihi,i+ib:n) from the 334* left 335* 336 CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward', 337 $ 'Columnwise', 338 $ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT, 339 $ A( I+1, I+IB ), LDA, WORK, LDWORK ) 340 40 CONTINUE 341 END IF 342* 343* Use unblocked code to reduce the rest of the matrix 344* 345 CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) 346 WORK( 1 ) = IWS 347* 348 RETURN 349* 350* End of ZGEHRD 351* 352 END 353c $Id$ 354