1 SUBROUTINE PZGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, 2 $ LWORK, INFO ) 3* 4* -- ScaLAPACK routine (version 1.7) -- 5* University of Tennessee, Knoxville, Oak Ridge National Laboratory, 6* and University of California, Berkeley. 7* May 25, 2001 8* 9* .. Scalar Arguments .. 10 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N 11* .. 12* .. Array Arguments .. 13 INTEGER DESCA( * ) 14 COMPLEX*16 A( * ), TAU( * ), WORK( * ) 15* .. 16* 17* Purpose 18* ======= 19* 20* PZGEHRD reduces a complex general distributed matrix sub( A ) 21* to upper Hessenberg form H by an unitary similarity transformation: 22* Q' * sub( A ) * Q = H, where 23* sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). 24* 25* Notes 26* ===== 27* 28* Each global data object is described by an associated description 29* vector. This vector stores the information required to establish 30* the mapping between an object element and its corresponding process 31* and memory location. 32* 33* Let A be a generic term for any 2D block cyclicly distributed array. 34* Such a global array has an associated description vector DESCA. 35* In the following comments, the character _ should be read as 36* "of the global array". 37* 38* NOTATION STORED IN EXPLANATION 39* --------------- -------------- -------------------------------------- 40* DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, 41* DTYPE_A = 1. 42* CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating 43* the BLACS process grid A is distribu- 44* ted over. The context itself is glo- 45* bal, but the handle (the integer 46* value) may vary. 47* M_A (global) DESCA( M_ ) The number of rows in the global 48* array A. 49* N_A (global) DESCA( N_ ) The number of columns in the global 50* array A. 51* MB_A (global) DESCA( MB_ ) The blocking factor used to distribute 52* the rows of the array. 53* NB_A (global) DESCA( NB_ ) The blocking factor used to distribute 54* the columns of the array. 55* RSRC_A (global) DESCA( RSRC_ ) The process row over which the first 56* row of the array A is distributed. 57* CSRC_A (global) DESCA( CSRC_ ) The process column over which the 58* first column of the array A is 59* distributed. 60* LLD_A (local) DESCA( LLD_ ) The leading dimension of the local 61* array. LLD_A >= MAX(1,LOCr(M_A)). 62* 63* Let K be the number of rows or columns of a distributed matrix, 64* and assume that its process grid has dimension p x q. 65* LOCr( K ) denotes the number of elements of K that a process 66* would receive if K were distributed over the p processes of its 67* process column. 68* Similarly, LOCc( K ) denotes the number of elements of K that a 69* process would receive if K were distributed over the q processes of 70* its process row. 71* The values of LOCr() and LOCc() may be determined via a call to the 72* ScaLAPACK tool function, NUMROC: 73* LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), 74* LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). 75* An upper bound for these quantities may be computed by: 76* LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A 77* LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A 78* 79* Arguments 80* ========= 81* 82* N (global input) INTEGER 83* The number of rows and columns to be operated on, i.e. the 84* order of the distributed submatrix sub( A ). N >= 0. 85* 86* ILO (global input) INTEGER 87* IHI (global input) INTEGER 88* It is assumed that sub( A ) is already upper triangular in 89* rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 90* and JA+IHI:JA+N-1. See Further Details. If N > 0, 91* 1 <= ILO <= IHI <= N; otherwise set ILO = 1, IHI = N. 92* 93* A (local input/local output) COMPLEX*16 pointer into the 94* local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). 95* On entry, this array contains the local pieces of the N-by-N 96* general distributed matrix sub( A ) to be reduced. On exit, 97* the upper triangle and the first subdiagonal of sub( A ) are 98* overwritten with the upper Hessenberg matrix H, and the ele- 99* ments below the first subdiagonal, with the array TAU, repre- 100* sent the unitary matrix Q as a product of elementary 101* reflectors. See Further Details. 102* 103* IA (global input) INTEGER 104* The row index in the global array A indicating the first 105* row of sub( A ). 106* 107* JA (global input) INTEGER 108* The column index in the global array A indicating the 109* first column of sub( A ). 110* 111* DESCA (global and local input) INTEGER array of dimension DLEN_. 112* The array descriptor for the distributed matrix A. 113* 114* TAU (local output) COMPLEX*16 array, dimension LOCc(JA+N-2) 115* The scalar factors of the elementary reflectors (see Further 116* Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are 117* set to zero. TAU is tied to the distributed matrix A. 118* 119* WORK (local workspace/local output) COMPLEX*16 array, 120* dimension (LWORK) 121* On exit, WORK( 1 ) returns the minimal and optimal LWORK. 122* 123* LWORK (local or global input) INTEGER 124* The dimension of the array WORK. 125* LWORK is local input and must be at least 126* LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ ) 127* 128* where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), 129* ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ), 130* IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), 131* IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), 132* ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), 133* IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ), 134* ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ), 135* INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ), 136* 137* INDXG2P and NUMROC are ScaLAPACK tool functions; 138* MYROW, MYCOL, NPROW and NPCOL can be determined by calling 139* the subroutine BLACS_GRIDINFO. 140* 141* If LWORK = -1, then LWORK is global input and a workspace 142* query is assumed; the routine only calculates the minimum 143* and optimal size for all work arrays. Each of these 144* values is returned in the first entry of the corresponding 145* work array, and no error message is issued by PXERBLA. 146* 147* INFO (global output) INTEGER 148* = 0: successful exit 149* < 0: If the i-th argument is an array and the j-entry had 150* an illegal value, then INFO = -(i*100+j), if the i-th 151* argument is a scalar and had an illegal value, then 152* INFO = -i. 153* 154* Further Details 155* =============== 156* 157* The matrix Q is represented as a product of (ihi-ilo) elementary 158* reflectors 159* 160* Q = H(ilo) H(ilo+1) . . . H(ihi-1). 161* 162* Each H(i) has the form 163* 164* H(i) = I - tau * v * v' 165* 166* where tau is a complex scalar, and v is a complex vector with 167* v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on 168* exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2). 169* 170* The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- 171* ing example, with N = 7, ILO = 2 and IHI = 6: 172* 173* on entry on exit 174* 175* ( a a a a a a a ) ( a a h h h h a ) 176* ( a a a a a a ) ( a h h h h a ) 177* ( a a a a a a ) ( h h h h h h ) 178* ( a a a a a a ) ( v2 h h h h h ) 179* ( a a a a a a ) ( v2 v3 h h h h ) 180* ( a a a a a a ) ( v2 v3 v4 h h h ) 181* ( a ) ( a ) 182* 183* where a denotes an element of the original matrix sub( A ), H denotes 184* a modified element of the upper Hessenberg matrix H, and vi denotes 185* an element of the vector defining H(JA+ILO+I-2). 186* 187* Alignment requirements 188* ====================== 189* 190* The distributed submatrix sub( A ) must verify some alignment proper- 191* ties, namely the following expression should be true: 192* ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) 193* 194* ===================================================================== 195* 196* .. Parameters .. 197 INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_, 198 $ LLD_, MB_, M_, NB_, N_, RSRC_ 199 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, 200 $ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, 201 $ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) 202 COMPLEX*16 ONE, ZERO 203 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ), 204 $ ZERO = ( 0.0D+0, 0.0D+0 ) ) 205* .. 206* .. Local Scalars .. 207 LOGICAL LQUERY 208 CHARACTER COLCTOP, ROWCTOP 209 INTEGER I, IACOL, IAROW, IB, ICOFFA, ICTXT, IHIP, 210 $ IHLP, IIA, IINFO, ILCOL, ILROW, IMCOL, INLQ, 211 $ IOFF, IPT, IPW, IPY, IROFFA, J, JJ, JJA, JY, 212 $ K, L, LWMIN, MYCOL, MYROW, NB, NPCOL, NPROW, 213 $ NQ 214 COMPLEX*16 EI 215* .. 216* .. Local Arrays .. 217 INTEGER DESCY( DLEN_ ), IDUM1( 3 ), IDUM2( 3 ) 218* .. 219* .. External Subroutines .. 220 EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, INFOG1L, 221 $ INFOG2L, PCHK1MAT, PB_TOPGET, PB_TOPSET, 222 $ PXERBLA, PZGEMM, PZGEHD2, PZLAHRD, PZLARFB 223* .. 224* .. External Functions .. 225 INTEGER INDXG2P, NUMROC 226 EXTERNAL INDXG2P, NUMROC 227* .. 228* .. Intrinsic Functions .. 229 INTRINSIC DBLE, DCMPLX, MAX, MIN, MOD 230* .. 231* .. Executable Statements .. 232* 233* Get grid parameters 234* 235 ICTXT = DESCA( CTXT_ ) 236 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) 237* 238* Test the input parameters 239* 240 INFO = 0 241 IF( NPROW.EQ.-1 ) THEN 242 INFO = -(700+CTXT_) 243 ELSE 244 CALL CHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, INFO ) 245 IF( INFO.EQ.0 ) THEN 246 NB = DESCA( NB_ ) 247 IROFFA = MOD( IA-1, NB ) 248 ICOFFA = MOD( JA-1, NB ) 249 CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, 250 $ IIA, JJA, IAROW, IACOL ) 251 IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ) 252 IOFF = MOD( IA+ILO-2, NB ) 253 ILROW = INDXG2P( IA+ILO-1, NB, MYROW, DESCA( RSRC_ ), 254 $ NPROW ) 255 IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ) 256 ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, DESCA( CSRC_ ), 257 $ NPCOL ) 258 INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ) 259 LWMIN = NB*( NB + MAX( IHIP+1, IHLP+INLQ ) ) 260* 261 WORK( 1 ) = DCMPLX( DBLE( LWMIN ) ) 262 LQUERY = ( LWORK.EQ.-1 ) 263 IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN 264 INFO = -2 265 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN 266 INFO = -3 267 ELSE IF( IROFFA.NE.ICOFFA .OR. IROFFA.NE.0 ) THEN 268 INFO = -6 269 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN 270 INFO = -(700+NB_) 271 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 272 INFO = -10 273 END IF 274 END IF 275 IDUM1( 1 ) = ILO 276 IDUM2( 1 ) = 2 277 IDUM1( 2 ) = IHI 278 IDUM2( 2 ) = 3 279 IF( LWORK.EQ.-1 ) THEN 280 IDUM1( 3 ) = -1 281 ELSE 282 IDUM1( 3 ) = 1 283 END IF 284 IDUM2( 3 ) = 10 285 CALL PCHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, 3, IDUM1, IDUM2, 286 $ INFO ) 287 END IF 288* 289 IF( INFO.NE.0 ) THEN 290 CALL PXERBLA( ICTXT, 'PZGEHRD', -INFO ) 291 RETURN 292 ELSE IF( LQUERY ) THEN 293 RETURN 294 END IF 295* 296* Set elements JA:JA+ILO-2 and JA+JHI-1:JA+N-2 of TAU to zero. 297* 298 NQ = NUMROC( JA+N-2, NB, MYCOL, DESCA( CSRC_ ), NPCOL ) 299 CALL INFOG1L( JA+ILO-2, NB, NPCOL, MYCOL, DESCA( CSRC_ ), JJ, 300 $ IMCOL ) 301 DO 10 J = JJA, MIN( JJ, NQ ) 302 TAU( J ) = ZERO 303 10 CONTINUE 304* 305 CALL INFOG1L( JA+IHI-1, NB, NPCOL, MYCOL, DESCA( CSRC_ ), JJ, 306 $ IMCOL ) 307 DO 20 J = JJ, NQ 308 TAU( J ) = ZERO 309 20 CONTINUE 310* 311* Quick return if possible 312* 313 IF( IHI-ILO.LE.0 ) 314 $ RETURN 315* 316 CALL PB_TOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) 317 CALL PB_TOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) 318 CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' ) 319 CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' ) 320* 321 IPT = 1 322 IPY = IPT + NB * NB 323 IPW = IPY + IHIP * NB 324 CALL DESCSET( DESCY, IHI+IROFFA, NB, NB, NB, IAROW, ILCOL, ICTXT, 325 $ MAX( 1, IHIP ) ) 326* 327 K = ILO 328 IB = NB - IOFF 329 JY = IOFF + 1 330* 331* Loop over remaining block of columns 332* 333 DO 30 L = 1, IHI-ILO+IOFF-NB, NB 334 I = IA + K - 1 335 J = JA + K - 1 336* 337* Reduce columns j:j+ib-1 to Hessenberg form, returning the 338* matrices V and T of the block reflector H = I - V*T*V' 339* which performs the reduction, and also the matrix Y = A*V*T 340* 341 CALL PZLAHRD( IHI, K, IB, A, IA, J, DESCA, TAU, WORK( IPT ), 342 $ WORK( IPY ), 1, JY, DESCY, WORK( IPW ) ) 343* 344* Apply the block reflector H to A(ia:ia+ihi-1,j+ib:ja+ihi-1) 345* from the right, computing A := A - Y * V'. 346* V(i+ib,ib-1) must be set to 1. 347* 348 CALL PZELSET2( EI, A, I+IB, J+IB-1, DESCA, ONE ) 349 CALL PZGEMM( 'No transpose', 'Conjugate transpose', IHI, 350 $ IHI-K-IB+1, IB, -ONE, WORK( IPY ), 1, JY, DESCY, 351 $ A, I+IB, J, DESCA, ONE, A, IA, J+IB, DESCA ) 352 CALL PZELSET( A, I+IB, J+IB-1, DESCA, EI ) 353* 354* Apply the block reflector H to A(i+1:ia+ihi-1,j+ib:ja+n-1) from 355* the left 356* 357 CALL PZLARFB( 'Left', 'Conjugate transpose', 'Forward', 358 $ 'Columnwise', IHI-K, N-K-IB+1, IB, A, I+1, J, 359 $ DESCA, WORK( IPT ), A, I+1, J+IB, DESCA, 360 $ WORK( IPY ) ) 361* 362 K = K + IB 363 IB = NB 364 JY = 1 365 DESCY( CSRC_ ) = MOD( DESCY( CSRC_ ) + 1, NPCOL ) 366* 367 30 CONTINUE 368* 369* Use unblocked code to reduce the rest of the matrix 370* 371 CALL PZGEHD2( N, K, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, 372 $ IINFO ) 373* 374 CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) 375 CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) 376* 377 WORK( 1 ) = DCMPLX( DBLE( LWMIN ) ) 378* 379 RETURN 380* 381* End of PZGEHRD 382* 383 END 384