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
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
|
*> \brief \b DGBMV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DGBMV(TRANS,M,N,KL,KU,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA,BETA
* INTEGER INCX,INCY,KL,KU,LDA,M,N
* CHARACTER TRANS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBMV performs one of the matrix-vector operations
*>
*> y := alpha*A*x + beta*y, or y := alpha*A**T*x + beta*y,
*>
*> where alpha and beta are scalars, x and y are vectors and A is an
*> m by n band matrix, with kl sub-diagonals and ku super-diagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> On entry, TRANS specifies the operation to be performed as
*> follows:
*>
*> TRANS = 'N' or 'n' y := alpha*A*x + beta*y.
*>
*> TRANS = 'T' or 't' y := alpha*A**T*x + beta*y.
*>
*> TRANS = 'C' or 'c' y := alpha*A**T*x + beta*y.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> On entry, M specifies the number of rows of the matrix A.
*> M must be at least zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the number of columns of the matrix A.
*> N must be at least zero.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> On entry, KL specifies the number of sub-diagonals of the
*> matrix A. KL must satisfy 0 .le. KL.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> On entry, KU specifies the number of super-diagonals of the
*> matrix A. KU must satisfy 0 .le. KU.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION.
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array of DIMENSION ( LDA, n ).
*> Before entry, the leading ( kl + ku + 1 ) by n part of the
*> array A must contain the matrix of coefficients, supplied
*> column by column, with the leading diagonal of the matrix in
*> row ( ku + 1 ) of the array, the first super-diagonal
*> starting at position 2 in row ku, the first sub-diagonal
*> starting at position 1 in row ( ku + 2 ), and so on.
*> Elements in the array A that do not correspond to elements
*> in the band matrix (such as the top left ku by ku triangle)
*> are not referenced.
*> The following program segment will transfer a band matrix
*> from conventional full matrix storage to band storage:
*>
*> DO 20, J = 1, N
*> K = KU + 1 - J
*> DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL )
*> A( K + I, J ) = matrix( I, J )
*> 10 CONTINUE
*> 20 CONTINUE
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. LDA must be at least
*> ( kl + ku + 1 ).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array of DIMENSION at least
*> ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
*> and at least
*> ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
*> Before entry, the incremented array X must contain the
*> vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> On entry, INCX specifies the increment for the elements of
*> X. INCX must not be zero.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION.
*> On entry, BETA specifies the scalar beta. When BETA is
*> supplied as zero then Y need not be set on input.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array of DIMENSION at least
*> ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
*> and at least
*> ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
*> Before entry, the incremented array Y must contain the
*> vector y. On exit, Y is overwritten by the updated vector y.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> On entry, INCY specifies the increment for the elements of
*> Y. INCY must not be zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup double_blas_level2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Level 2 Blas routine.
*> The vector and matrix arguments are not referenced when N = 0, or M = 0
*>
*> -- Written on 22-October-1986.
*> Jack Dongarra, Argonne National Lab.
*> Jeremy Du Croz, Nag Central Office.
*> Sven Hammarling, Nag Central Office.
*> Richard Hanson, Sandia National Labs.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGBMV(TRANS,M,N,KL,KU,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
*
* -- Reference BLAS level2 routine (version 3.7.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER INCX,INCY,KL,KU,LDA,M,N
CHARACTER TRANS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,IX,IY,J,JX,JY,K,KUP1,KX,KY,LENX,LENY
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX,MIN
* ..
*
* Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND.
+ .NOT.LSAME(TRANS,'C')) THEN
INFO = 1
ELSE IF (M.LT.0) THEN
INFO = 2
ELSE IF (N.LT.0) THEN
INFO = 3
ELSE IF (KL.LT.0) THEN
INFO = 4
ELSE IF (KU.LT.0) THEN
INFO = 5
ELSE IF (LDA.LT. (KL+KU+1)) THEN
INFO = 8
ELSE IF (INCX.EQ.0) THEN
INFO = 10
ELSE IF (INCY.EQ.0) THEN
INFO = 13
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DGBMV ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((M.EQ.0) .OR. (N.EQ.0) .OR.
+ ((ALPHA.EQ.ZERO).AND. (BETA.EQ.ONE))) RETURN
*
* Set LENX and LENY, the lengths of the vectors x and y, and set
* up the start points in X and Y.
*
IF (LSAME(TRANS,'N')) THEN
LENX = N
LENY = M
ELSE
LENX = M
LENY = N
END IF
IF (INCX.GT.0) THEN
KX = 1
ELSE
KX = 1 - (LENX-1)*INCX
END IF
IF (INCY.GT.0) THEN
KY = 1
ELSE
KY = 1 - (LENY-1)*INCY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through the band part of A.
*
* First form y := beta*y.
*
IF (BETA.NE.ONE) THEN
IF (INCY.EQ.1) THEN
IF (BETA.EQ.ZERO) THEN
DO 10 I = 1,LENY
Y(I) = ZERO
10 CONTINUE
ELSE
DO 20 I = 1,LENY
Y(I) = BETA*Y(I)
20 CONTINUE
END IF
ELSE
IY = KY
IF (BETA.EQ.ZERO) THEN
DO 30 I = 1,LENY
Y(IY) = ZERO
IY = IY + INCY
30 CONTINUE
ELSE
DO 40 I = 1,LENY
Y(IY) = BETA*Y(IY)
IY = IY + INCY
40 CONTINUE
END IF
END IF
END IF
IF (ALPHA.EQ.ZERO) RETURN
KUP1 = KU + 1
IF (LSAME(TRANS,'N')) THEN
*
* Form y := alpha*A*x + y.
*
JX = KX
IF (INCY.EQ.1) THEN
DO 60 J = 1,N
TEMP = ALPHA*X(JX)
K = KUP1 - J
DO 50 I = MAX(1,J-KU),MIN(M,J+KL)
Y(I) = Y(I) + TEMP*A(K+I,J)
50 CONTINUE
JX = JX + INCX
60 CONTINUE
ELSE
DO 80 J = 1,N
TEMP = ALPHA*X(JX)
IY = KY
K = KUP1 - J
DO 70 I = MAX(1,J-KU),MIN(M,J+KL)
Y(IY) = Y(IY) + TEMP*A(K+I,J)
IY = IY + INCY
70 CONTINUE
JX = JX + INCX
IF (J.GT.KU) KY = KY + INCY
80 CONTINUE
END IF
ELSE
*
* Form y := alpha*A**T*x + y.
*
JY = KY
IF (INCX.EQ.1) THEN
DO 100 J = 1,N
TEMP = ZERO
K = KUP1 - J
DO 90 I = MAX(1,J-KU),MIN(M,J+KL)
TEMP = TEMP + A(K+I,J)*X(I)
90 CONTINUE
Y(JY) = Y(JY) + ALPHA*TEMP
JY = JY + INCY
100 CONTINUE
ELSE
DO 120 J = 1,N
TEMP = ZERO
IX = KX
K = KUP1 - J
DO 110 I = MAX(1,J-KU),MIN(M,J+KL)
TEMP = TEMP + A(K+I,J)*X(IX)
IX = IX + INCX
110 CONTINUE
Y(JY) = Y(JY) + ALPHA*TEMP
JY = JY + INCY
IF (J.GT.KU) KX = KX + INCX
120 CONTINUE
END IF
END IF
*
RETURN
*
* End of DGBMV .
*
END
|
