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
371
372
373
374
375
376
377
378
379
380
|
*> \brief \b CHBMV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CHBMV(UPLO,N,K,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
*
* .. Scalar Arguments ..
* COMPLEX ALPHA,BETA
* INTEGER INCX,INCY,K,LDA,N
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* COMPLEX A(LDA,*),X(*),Y(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CHBMV performs the matrix-vector operation
*>
*> y := alpha*A*x + beta*y,
*>
*> where alpha and beta are scalars, x and y are n element vectors and
*> A is an n by n hermitian band matrix, with k super-diagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the band matrix A is being supplied as
*> follows:
*>
*> UPLO = 'U' or 'u' The upper triangular part of A is
*> being supplied.
*>
*> UPLO = 'L' or 'l' The lower triangular part of A is
*> being supplied.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the matrix A.
*> N must be at least zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> On entry, K specifies the number of super-diagonals of the
*> matrix A. K must satisfy 0 .le. K.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is COMPLEX
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX array, dimension ( LDA, N )
*> Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
*> by n part of the array A must contain the upper triangular
*> band part of the hermitian matrix, supplied column by
*> column, with the leading diagonal of the matrix in row
*> ( k + 1 ) of the array, the first super-diagonal starting at
*> position 2 in row k, and so on. The top left k by k triangle
*> of the array A is not referenced.
*> The following program segment will transfer the upper
*> triangular part of a hermitian band matrix from conventional
*> full matrix storage to band storage:
*>
*> DO 20, J = 1, N
*> M = K + 1 - J
*> DO 10, I = MAX( 1, J - K ), J
*> A( M + I, J ) = matrix( I, J )
*> 10 CONTINUE
*> 20 CONTINUE
*>
*> Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
*> by n part of the array A must contain the lower triangular
*> band part of the hermitian matrix, supplied column by
*> column, with the leading diagonal of the matrix in row 1 of
*> the array, the first sub-diagonal starting at position 1 in
*> row 2, and so on. The bottom right k by k triangle of the
*> array A is not referenced.
*> The following program segment will transfer the lower
*> triangular part of a hermitian band matrix from conventional
*> full matrix storage to band storage:
*>
*> DO 20, J = 1, N
*> M = 1 - J
*> DO 10, I = J, MIN( N, J + K )
*> A( M + I, J ) = matrix( I, J )
*> 10 CONTINUE
*> 20 CONTINUE
*>
*> Note that the imaginary parts of the diagonal elements need
*> not be set and are assumed to be zero.
*> \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
*> ( k + 1 ).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is COMPLEX array, dimension at least
*> ( 1 + ( n - 1 )*abs( INCX ) ).
*> 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 COMPLEX
*> On entry, BETA specifies the scalar beta.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is COMPLEX array, dimension at least
*> ( 1 + ( n - 1 )*abs( INCY ) ).
*> 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 complex_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 CHBMV(UPLO,N,K,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 ..
COMPLEX ALPHA,BETA
INTEGER INCX,INCY,K,LDA,N
CHARACTER UPLO
* ..
* .. Array Arguments ..
COMPLEX A(LDA,*),X(*),Y(*)
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ONE
PARAMETER (ONE= (1.0E+0,0.0E+0))
COMPLEX ZERO
PARAMETER (ZERO= (0.0E+0,0.0E+0))
* ..
* .. Local Scalars ..
COMPLEX TEMP1,TEMP2
INTEGER I,INFO,IX,IY,J,JX,JY,KPLUS1,KX,KY,L
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC CONJG,MAX,MIN,REAL
* ..
*
* Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
INFO = 1
ELSE IF (N.LT.0) THEN
INFO = 2
ELSE IF (K.LT.0) THEN
INFO = 3
ELSE IF (LDA.LT. (K+1)) THEN
INFO = 6
ELSE IF (INCX.EQ.0) THEN
INFO = 8
ELSE IF (INCY.EQ.0) THEN
INFO = 11
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('CHBMV ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((N.EQ.0) .OR. ((ALPHA.EQ.ZERO).AND. (BETA.EQ.ONE))) RETURN
*
* Set up the start points in X and Y.
*
IF (INCX.GT.0) THEN
KX = 1
ELSE
KX = 1 - (N-1)*INCX
END IF
IF (INCY.GT.0) THEN
KY = 1
ELSE
KY = 1 - (N-1)*INCY
END IF
*
* Start the operations. In this version the elements of the array A
* are accessed sequentially with one pass through 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,N
Y(I) = ZERO
10 CONTINUE
ELSE
DO 20 I = 1,N
Y(I) = BETA*Y(I)
20 CONTINUE
END IF
ELSE
IY = KY
IF (BETA.EQ.ZERO) THEN
DO 30 I = 1,N
Y(IY) = ZERO
IY = IY + INCY
30 CONTINUE
ELSE
DO 40 I = 1,N
Y(IY) = BETA*Y(IY)
IY = IY + INCY
40 CONTINUE
END IF
END IF
END IF
IF (ALPHA.EQ.ZERO) RETURN
IF (LSAME(UPLO,'U')) THEN
*
* Form y when upper triangle of A is stored.
*
KPLUS1 = K + 1
IF ((INCX.EQ.1) .AND. (INCY.EQ.1)) THEN
DO 60 J = 1,N
TEMP1 = ALPHA*X(J)
TEMP2 = ZERO
L = KPLUS1 - J
DO 50 I = MAX(1,J-K),J - 1
Y(I) = Y(I) + TEMP1*A(L+I,J)
TEMP2 = TEMP2 + CONJG(A(L+I,J))*X(I)
50 CONTINUE
Y(J) = Y(J) + TEMP1*REAL(A(KPLUS1,J)) + ALPHA*TEMP2
60 CONTINUE
ELSE
JX = KX
JY = KY
DO 80 J = 1,N
TEMP1 = ALPHA*X(JX)
TEMP2 = ZERO
IX = KX
IY = KY
L = KPLUS1 - J
DO 70 I = MAX(1,J-K),J - 1
Y(IY) = Y(IY) + TEMP1*A(L+I,J)
TEMP2 = TEMP2 + CONJG(A(L+I,J))*X(IX)
IX = IX + INCX
IY = IY + INCY
70 CONTINUE
Y(JY) = Y(JY) + TEMP1*REAL(A(KPLUS1,J)) + ALPHA*TEMP2
JX = JX + INCX
JY = JY + INCY
IF (J.GT.K) THEN
KX = KX + INCX
KY = KY + INCY
END IF
80 CONTINUE
END IF
ELSE
*
* Form y when lower triangle of A is stored.
*
IF ((INCX.EQ.1) .AND. (INCY.EQ.1)) THEN
DO 100 J = 1,N
TEMP1 = ALPHA*X(J)
TEMP2 = ZERO
Y(J) = Y(J) + TEMP1*REAL(A(1,J))
L = 1 - J
DO 90 I = J + 1,MIN(N,J+K)
Y(I) = Y(I) + TEMP1*A(L+I,J)
TEMP2 = TEMP2 + CONJG(A(L+I,J))*X(I)
90 CONTINUE
Y(J) = Y(J) + ALPHA*TEMP2
100 CONTINUE
ELSE
JX = KX
JY = KY
DO 120 J = 1,N
TEMP1 = ALPHA*X(JX)
TEMP2 = ZERO
Y(JY) = Y(JY) + TEMP1*REAL(A(1,J))
L = 1 - J
IX = JX
IY = JY
DO 110 I = J + 1,MIN(N,J+K)
IX = IX + INCX
IY = IY + INCY
Y(IY) = Y(IY) + TEMP1*A(L+I,J)
TEMP2 = TEMP2 + CONJG(A(L+I,J))*X(IX)
110 CONTINUE
Y(JY) = Y(JY) + ALPHA*TEMP2
JX = JX + INCX
JY = JY + INCY
120 CONTINUE
END IF
END IF
*
RETURN
*
* End of CHBMV .
*
END
|
