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 good performance, LWORK should generally be larger. 101*> 102*> If LWORK = -1, then a workspace query is assumed; the routine 103*> only calculates the optimal size of the WORK array, returns 104*> this value as the first entry of the WORK array, and no error 105*> message related to LWORK is issued by XERBLA. 106*> \endverbatim 107*> 108*> \param[out] INFO 109*> \verbatim 110*> INFO is INTEGER 111*> = 0: successful exit 112*> < 0: if INFO = -i, the i-th argument had an illegal value. 113*> \endverbatim 114* 115* Authors: 116* ======== 117* 118*> \author Univ. of Tennessee 119*> \author Univ. of California Berkeley 120*> \author Univ. of Colorado Denver 121*> \author NAG Ltd. 122* 123*> \ingroup complex16GEcomputational 124* 125*> \par Further Details: 126* ===================== 127*> 128*> \verbatim 129*> 130*> The matrix Q is represented as a product of (ihi-ilo) elementary 131*> reflectors 132*> 133*> Q = H(ilo) H(ilo+1) . . . H(ihi-1). 134*> 135*> Each H(i) has the form 136*> 137*> H(i) = I - tau * v * v**H 138*> 139*> where tau is a complex scalar, and v is a complex vector with 140*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on 141*> exit in A(i+2:ihi,i), and tau in TAU(i). 142*> 143*> The contents of A are illustrated by the following example, with 144*> n = 7, ilo = 2 and ihi = 6: 145*> 146*> on entry, on exit, 147*> 148*> ( a a a a a a a ) ( a a h h h h a ) 149*> ( a a a a a a ) ( a h h h h a ) 150*> ( a a a a a a ) ( h h h h h h ) 151*> ( a a a a a a ) ( v2 h h h h h ) 152*> ( a a a a a a ) ( v2 v3 h h h h ) 153*> ( a a a a a a ) ( v2 v3 v4 h h h ) 154*> ( a ) ( a ) 155*> 156*> where a denotes an element of the original matrix A, h denotes a 157*> modified element of the upper Hessenberg matrix H, and vi denotes an 158*> element of the vector defining H(i). 159*> 160*> This file is a slight modification of LAPACK-3.0's ZGEHRD 161*> subroutine incorporating improvements proposed by Quintana-Orti and 162*> Van de Geijn (2006). (See ZLAHR2.) 163*> \endverbatim 164*> 165* ===================================================================== 166 SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) 167* 168* -- LAPACK computational routine -- 169* -- LAPACK is a software package provided by Univ. of Tennessee, -- 170* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 171* 172* .. Scalar Arguments .. 173 INTEGER IHI, ILO, INFO, LDA, LWORK, N 174* .. 175* .. Array Arguments .. 176 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) 177* .. 178* 179* ===================================================================== 180* 181* .. Parameters .. 182 INTEGER NBMAX, LDT, TSIZE 183 PARAMETER ( NBMAX = 64, LDT = NBMAX+1, 184 $ TSIZE = LDT*NBMAX ) 185 COMPLEX*16 ZERO, ONE 186 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), 187 $ ONE = ( 1.0D+0, 0.0D+0 ) ) 188* .. 189* .. Local Scalars .. 190 LOGICAL LQUERY 191 INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB, 192 $ NBMIN, NH, NX 193 COMPLEX*16 EI 194* .. 195* .. External Subroutines .. 196 EXTERNAL ZAXPY, ZGEHD2, ZGEMM, ZLAHR2, ZLARFB, ZTRMM, 197 $ XERBLA 198* .. 199* .. Intrinsic Functions .. 200 INTRINSIC MAX, MIN 201* .. 202* .. External Functions .. 203 INTEGER ILAENV 204 EXTERNAL ILAENV 205* .. 206* .. Executable Statements .. 207* 208* Test the input parameters 209* 210 INFO = 0 211 LQUERY = ( LWORK.EQ.-1 ) 212 IF( N.LT.0 ) THEN 213 INFO = -1 214 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN 215 INFO = -2 216 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN 217 INFO = -3 218 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 219 INFO = -5 220 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN 221 INFO = -8 222 END IF 223* 224 IF( INFO.EQ.0 ) THEN 225* 226* Compute the workspace requirements 227* 228 NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) 229 LWKOPT = N*NB + TSIZE 230 WORK( 1 ) = LWKOPT 231 ENDIF 232* 233 IF( INFO.NE.0 ) THEN 234 CALL XERBLA( 'ZGEHRD', -INFO ) 235 RETURN 236 ELSE IF( LQUERY ) THEN 237 RETURN 238 END IF 239* 240* Set elements 1:ILO-1 and IHI:N-1 of TAU to zero 241* 242 DO 10 I = 1, ILO - 1 243 TAU( I ) = ZERO 244 10 CONTINUE 245 DO 20 I = MAX( 1, IHI ), N - 1 246 TAU( I ) = ZERO 247 20 CONTINUE 248* 249* Quick return if possible 250* 251 NH = IHI - ILO + 1 252 IF( NH.LE.1 ) THEN 253 WORK( 1 ) = 1 254 RETURN 255 END IF 256* 257* Determine the block size 258* 259 NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) 260 NBMIN = 2 261 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 IF( LWORK.LT.N*NB+TSIZE ) THEN 272* 273* Not enough workspace to use optimal NB: determine the 274* minimum value of NB, and reduce NB or force use of 275* unblocked code 276* 277 NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI, 278 $ -1 ) ) 279 IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN 280 NB = (LWORK-TSIZE) / N 281 ELSE 282 NB = 1 283 END IF 284 END IF 285 END IF 286 END IF 287 LDWORK = N 288* 289 IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN 290* 291* Use unblocked code below 292* 293 I = ILO 294* 295 ELSE 296* 297* Use blocked code 298* 299 IWT = 1 + N*NB 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 ), 308 $ WORK( IWT ), LDT, 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, 339 $ WORK( IWT ), LDT, A( I+1, I+IB ), LDA, 340 $ WORK, LDWORK ) 341 40 CONTINUE 342 END IF 343* 344* Use unblocked code to reduce the rest of the matrix 345* 346 CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) 347 WORK( 1 ) = LWKOPT 348* 349 RETURN 350* 351* End of ZGEHRD 352* 353 END 354