1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
|
SUBROUTINE PZQRT16( TRANS, M, N, NRHS, A, IA, JA, DESCA, X, IX,
$ JX, DESCX, B, IB, JB, DESCB, RWORK, RESID )
*
* -- ScaLAPACK routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IA, IB, IX, JA, JB, JX, M, N, NRHS
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
INTEGER DESCA( * ), DESCB( * ), DESCX( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( * ), B( * ), X( * )
* ..
*
* Purpose
* =======
*
* PZQRT16 computes the residual for a solution of a system of linear
* equations sub( A )*sub( X ) = B or sub( A' )*sub( X ) = B:
* RESID = norm(B - sub( A )*sub( X ) ) /
* ( max(m,n) * norm(sub( A ) ) * norm(sub( X ) ) * EPS ),
* where EPS is the machine epsilon, sub( A ) denotes
* A(IA:IA+N-1,JA,JA+N-1), and sub( X ) denotes
* X(IX:IX+N-1, JX:JX+NRHS-1).
*
* Notes
* =====
*
* Each global data object is described by an associated description
* vector. This vector stores the information required to establish
* the mapping between an object element and its corresponding process
* and memory location.
*
* Let A be a generic term for any 2D block cyclicly distributed array.
* Such a global array has an associated description vector DESCA.
* In the following comments, the character _ should be read as
* "of the global array".
*
* NOTATION STORED IN EXPLANATION
* --------------- -------------- --------------------------------------
* DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
* DTYPE_A = 1.
* CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
* the BLACS process grid A is distribu-
* ted over. The context itself is glo-
* bal, but the handle (the integer
* value) may vary.
* M_A (global) DESCA( M_ ) The number of rows in the global
* array A.
* N_A (global) DESCA( N_ ) The number of columns in the global
* array A.
* MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
* the rows of the array.
* NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
* the columns of the array.
* RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
* row of the array A is distributed.
* CSRC_A (global) DESCA( CSRC_ ) The process column over which the
* first column of the array A is
* distributed.
* LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
* array. LLD_A >= MAX(1,LOCr(M_A)).
*
* Let K be the number of rows or columns of a distributed matrix,
* and assume that its process grid has dimension p x q.
* LOCr( K ) denotes the number of elements of K that a process
* would receive if K were distributed over the p processes of its
* process column.
* Similarly, LOCc( K ) denotes the number of elements of K that a
* process would receive if K were distributed over the q processes of
* its process row.
* The values of LOCr() and LOCc() may be determined via a call to the
* ScaLAPACK tool function, NUMROC:
* LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
* LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
* An upper bound for these quantities may be computed by:
* LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
* LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
*
* Arguments
* =========
*
* TRANS (global input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': sub( A )*sub( X ) = sub( B )
* = 'T': sub( A' )*sub( X )= sub( B ), where A' is the
* transpose of sub( A ).
* = 'C': sub( A' )*sub( X )= B, where A' is the conjugate
* transpose of sub( A ).
*
* M (global input) INTEGER
* The number of rows to be operated on, i.e. the number of rows
* of the distributed submatrix sub( A ). M >= 0.
*
* N (global input) INTEGER
* The number of columns to be operated on, i.e. the number of
* columns of the distributed submatrix sub( A ). N >= 0.
*
* NRHS (global input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the distributed submatrix sub( B ). NRHS >= 0.
*
* A (local input) COMPLEX*16 pointer into the local
* memory to an array of dimension (LLD_A,LOCc(JA+N-1)).
* The original M x N matrix A.
*
* IA (global input) INTEGER
* The row index in the global array A indicating the first
* row of sub( A ).
*
* JA (global input) INTEGER
* The column index in the global array A indicating the
* first column of sub( A ).
*
* DESCA (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix A.
*
* X (local input) COMPLEX*16 pointer into the local
* memory to an array of dimension (LLD_X,LOCc(JX+NRHS-1)). This
* array contains the local pieces of the computed solution
* distributed vectors for the system of linear equations.
*
* IX (global input) INTEGER
* The row index in the global array X indicating the first
* row of sub( X ).
*
* JX (global input) INTEGER
* The column index in the global array X indicating the
* first column of sub( X ).
*
* DESCX (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix X.
*
* B (local input/local output) COMPLEX*16 pointer into
* the local memory to an array of dimension
* (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the
* local pieces of the distributes right hand side vectors for
* the system of linear equations. On exit, sub( B ) is over-
* written with the difference sub( B ) - sub( A )*sub( X ) or
* sub( B ) - sub( A )'*sub( X ).
*
* IB (global input) INTEGER
* The row index in the global array B indicating the first
* row of sub( B ).
*
* JB (global input) INTEGER
* The column index in the global array B indicating the
* first column of sub( B ).
*
* DESCB (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix B.
*
* RWORK (local workspace) DOUBLE PRECISION array, dimension (LRWORK)
* LWORK >= Nq0 if TRANS = 'N', and LRWORK >= Mp0 otherwise.
*
* where
*
* IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ),
* IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ),
* IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
* Mp0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ),
* Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ),
*
* INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW,
* MYCOL, NPROW and NPCOL can be determined by calling the
* subroutine BLACS_GRIDINFO.
*
* RESID (global output) DOUBLE PRECISION
* The maximum over the number of right hand sides of
* norm( sub( B )- sub( A )*sub( X ) ) /
* ( max(m,n) * norm( sub( A ) ) * norm( sub( X ) ) * EPS ).
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
$ LLD_, MB_, M_, NB_, N_, RSRC_
PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
$ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
$ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER ICTXT, IDUMM, J, MYCOL, MYROW, N1, N2, NPCOL,
$ NPROW
DOUBLE PRECISION ANORM, BNORM, EPS, XNORM
* ..
* .. Local Arrays ..
DOUBLE PRECISION TEMP( 2 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION PDLAMCH, PZLANGE
EXTERNAL LSAME, PDLAMCH, PDLANGE
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, DGAMX2D, PDZASUM,
$ PZGEMM
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Get grid parameters
*
ICTXT = DESCA( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
* Quick exit if M = 0 or N = 0 or NRHS = 0
*
IF( M.LE.0 .OR. N.LE.0 .OR. NRHS.EQ.0 ) THEN
RESID = ZERO
RETURN
END IF
*
IF( LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' ) ) THEN
ANORM = PZLANGE( 'I', M, N, A, IA, JA, DESCA, RWORK )
N1 = N
N2 = M
ELSE
ANORM = PZLANGE( '1', M, N, A, IA, JA, DESCA, RWORK )
N1 = M
N2 = N
END IF
*
EPS = PDLAMCH( ICTXT, 'Epsilon' )
*
* Compute B - sub( A )*sub( X ) (or B - sub( A' )*sub( X ) ) and
* store in B.
*
CALL PZGEMM( TRANS, 'No transpose', N1, NRHS, N2, -CONE, A, IA,
$ JA, DESCA, X, IX, JX, DESCX, CONE, B, IB, JB, DESCB )
*
* Compute the maximum over the number of right hand sides of
* norm( sub( B ) - sub( A )*sub( X ) ) /
* ( max(m,n) * norm( sub( A ) ) * norm( sub( X ) ) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
*
CALL PDZASUM( N1, BNORM, B, IB, JB+J-1, DESCB, 1 )
CALL PDZASUM( N2, XNORM, X, IX, JX+J-1, DESCX, 1 )
*
* Only the process columns owning the vector operands will have
* the correct result, the other will have zero.
*
TEMP( 1 ) = BNORM
TEMP( 2 ) = XNORM
CALL DGAMX2D( ICTXT, 'All', ' ', 2, 1, TEMP, 2, IDUMM, IDUMM,
$ -1, -1, IDUMM )
BNORM = TEMP( 1 )
XNORM = TEMP( 2 )
*
* Every processes have ANORM, BNORM and XNORM now.
*
IF( ANORM.EQ.ZERO .AND. BNORM.EQ.ZERO ) THEN
RESID = ZERO
ELSE IF( ANORM.LE.ZERO .OR. XNORM.LE.ZERO ) THEN
RESID = ONE / EPS
ELSE
RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) /
$ ( MAX( M, N )*EPS ) )
END IF
*
10 CONTINUE
*
RETURN
*
* End of PZQRT16
*
END
|
