1 SUBROUTINE PCGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, 2 $ LWORK, INFO ) 3* 4* -- ScaLAPACK auxiliary routine (version 1.7) -- 5* University of Tennessee, Knoxville, Oak Ridge National Laboratory, 6* and University of California, Berkeley. 7* May 1, 1997 8* 9* .. Scalar Arguments .. 10 INTEGER IA, IHI, ILO, INFO, JA, LWORK, N 11* .. 12* .. Array Arguments .. 13 INTEGER DESCA( * ) 14 COMPLEX A( * ), TAU( * ), WORK( * ) 15* .. 16* 17* Purpose 18* ======= 19* 20* PCGEHD2 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+JLO-2 90* and JA+JHI: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 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 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 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 + MAX( NpA0, NB ) 127* 128* where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), 129* IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), 130* NpA0 = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), 131* 132* INDXG2P and NUMROC are ScaLAPACK tool functions; 133* MYROW, MYCOL, NPROW and NPCOL can be determined by calling 134* the subroutine BLACS_GRIDINFO. 135* 136* If LWORK = -1, then LWORK is global input and a workspace 137* query is assumed; the routine only calculates the minimum 138* and optimal size for all work arrays. Each of these 139* values is returned in the first entry of the corresponding 140* work array, and no error message is issued by PXERBLA. 141* 142* INFO (local output) INTEGER 143* = 0: successful exit 144* < 0: If the i-th argument is an array and the j-entry had 145* an illegal value, then INFO = -(i*100+j), if the i-th 146* argument is a scalar and had an illegal value, then 147* INFO = -i. 148* 149* Further Details 150* =============== 151* 152* The matrix Q is represented as a product of (ihi-ilo) elementary 153* reflectors 154* 155* Q = H(ilo) H(ilo+1) . . . H(ihi-1). 156* 157* Each H(i) has the form 158* 159* H(i) = I - tau * v * v' 160* 161* where tau is a complex scalar, and v is a complex vector with 162* v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on 163* exit in A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in TAU(ja+ilo+i-2). 164* 165* The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follo- 166* wing example, with n = 7, ilo = 2 and ihi = 6: 167* 168* on entry on exit 169* 170* ( a a a a a a a ) ( a a h h h h a ) 171* ( a a a a a a ) ( a h h h h a ) 172* ( a a a a a a ) ( h h h h h h ) 173* ( a a a a a a ) ( v2 h h h h h ) 174* ( a a a a a a ) ( v2 v3 h h h h ) 175* ( a a a a a a ) ( v2 v3 v4 h h h ) 176* ( a ) ( a ) 177* 178* where a denotes an element of the original matrix sub( A ), h denotes 179* a modified element of the upper Hessenberg matrix H, and vi denotes 180* an element of the vector defining H(ja+ilo+i-2). 181* 182* Alignment requirements 183* ====================== 184* 185* The distributed submatrix sub( A ) must verify some alignment proper- 186* ties, namely the following expression should be true: 187* ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) 188* 189* ===================================================================== 190* 191* .. Parameters .. 192 INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_, 193 $ LLD_, MB_, M_, NB_, N_, RSRC_ 194 PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, 195 $ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, 196 $ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 ) 197 COMPLEX ONE 198 PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) ) 199* .. 200* .. Local Scalars .. 201 LOGICAL LQUERY 202 INTEGER I, IAROW, ICOFFA, ICTXT, IROFFA, J, K, LWMIN, 203 $ MYCOL, MYROW, NPA0, NPCOL, NPROW 204 COMPLEX AII 205* .. 206* .. External Subroutines .. 207 EXTERNAL BLACS_ABORT, BLACS_GRIDINFO, CHK1MAT, PCELSET, 208 $ PCLARF, PCLARFC, PCLARFG, PXERBLA 209* .. 210* .. External Functions .. 211 INTEGER INDXG2P, NUMROC 212 EXTERNAL INDXG2P, NUMROC 213* .. 214* .. Intrinsic Functions .. 215 INTRINSIC CMPLX, MAX, MIN, MOD, REAL 216* .. 217* .. Executable Statements .. 218* 219* Get grid parameters 220* 221 ICTXT = DESCA( CTXT_ ) 222 CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) 223* 224* Test the input parameters 225* 226 INFO = 0 227 IF( NPROW.EQ.-1 ) THEN 228 INFO = -(700+CTXT_) 229 ELSE 230 CALL CHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, INFO ) 231 IF( INFO.EQ.0 ) THEN 232 IROFFA = MOD( IA-1, DESCA( MB_ ) ) 233 ICOFFA = MOD( JA-1, DESCA( NB_ ) ) 234 IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), 235 $ NPROW ) 236 NPA0 = NUMROC( IHI+IROFFA, DESCA( MB_ ), MYROW, IAROW, 237 $ NPROW ) 238 LWMIN = DESCA( NB_ ) + MAX( NPA0, DESCA( NB_ ) ) 239* 240 WORK( 1 ) = CMPLX( REAL( LWMIN ) ) 241 LQUERY = ( LWORK.EQ.-1 ) 242 IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN 243 INFO = -2 244 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN 245 INFO = -3 246 ELSE IF( IROFFA.NE.ICOFFA ) THEN 247 INFO = -6 248 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN 249 INFO = -(700+NB_) 250 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 251 INFO = -10 252 END IF 253 END IF 254 END IF 255* 256 IF( INFO.NE.0 ) THEN 257 CALL PXERBLA( ICTXT, 'PCGEHD2', -INFO ) 258 CALL BLACS_ABORT( ICTXT, 1 ) 259 RETURN 260 ELSE IF( LQUERY ) THEN 261 RETURN 262 END IF 263* 264 DO 10 K = ILO, IHI-1 265 I = IA + K - 1 266 J = JA + K - 1 267* 268* Compute elementary reflector H(j) to annihilate 269* A(i+2:ihi+ia-1,j) 270* 271 CALL PCLARFG( IHI-K, AII, I+1, J, A, MIN( I+2, N+IA-1 ), J, 272 $ DESCA, 1, TAU ) 273 CALL PCELSET( A, I+1, J, DESCA, ONE ) 274* 275* Apply H(k) to A(ia:ihi+ia-1,j+1:ihi+ja-1) from the right 276* 277 CALL PCLARF( 'Right', IHI, IHI-K, A, I+1, J, DESCA, 1, TAU, A, 278 $ IA, J+1, DESCA, WORK ) 279* 280* Apply H(j) to A(i+1:ia+ihi-1,j+1:ja+n-1) from the left 281* 282 CALL PCLARFC( 'Left', IHI-K, N-K, A, I+1, J, DESCA, 1, TAU, A, 283 $ I+1, J+1, DESCA, WORK ) 284* 285 CALL PCELSET( A, I+1, J, DESCA, AII ) 286 10 CONTINUE 287* 288 WORK( 1 ) = CMPLX( REAL( LWMIN ) ) 289* 290 RETURN 291* 292* End of PCGEHD2 293* 294 END 295