LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sgbmv()

 subroutine sgbmv ( character trans, integer m, integer n, integer kl, integer ku, real alpha, real, dimension(lda,*) a, integer lda, real, dimension(*) x, integer incx, real beta, real, dimension(*) y, integer incy )

SGBMV

Purpose:
``` SGBMV  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.```
Parameters
 [in] TRANS ``` 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.``` [in] M ``` M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero.``` [in] N ``` N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero.``` [in] KL ``` KL is INTEGER On entry, KL specifies the number of sub-diagonals of the matrix A. KL must satisfy 0 .le. KL.``` [in] KU ``` KU is INTEGER On entry, KU specifies the number of super-diagonals of the matrix A. KU must satisfy 0 .le. KU.``` [in] ALPHA ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha.``` [in] A ``` A is REAL array, 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``` [in] LDA ``` 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 ).``` [in] X ``` X is REAL array, 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.``` [in] INCX ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.``` [in] BETA ``` BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input.``` [in,out] Y ``` Y is REAL array, 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. If either m or n is zero, then Y not referenced and the function performs a quick return.``` [in] INCY ``` INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.```
Further Details:
```  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.```

Definition at line 186 of file sgbmv.f.

188*
189* -- Reference BLAS level2 routine --
190* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
191* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
192*
193* .. Scalar Arguments ..
194 REAL ALPHA,BETA
195 INTEGER INCX,INCY,KL,KU,LDA,M,N
196 CHARACTER TRANS
197* ..
198* .. Array Arguments ..
199 REAL A(LDA,*),X(*),Y(*)
200* ..
201*
202* =====================================================================
203*
204* .. Parameters ..
205 REAL ONE,ZERO
206 parameter(one=1.0e+0,zero=0.0e+0)
207* ..
208* .. Local Scalars ..
209 REAL TEMP
210 INTEGER I,INFO,IX,IY,J,JX,JY,K,KUP1,KX,KY,LENX,LENY
211* ..
212* .. External Functions ..
213 LOGICAL LSAME
214 EXTERNAL lsame
215* ..
216* .. External Subroutines ..
217 EXTERNAL xerbla
218* ..
219* .. Intrinsic Functions ..
220 INTRINSIC max,min
221* ..
222*
223* Test the input parameters.
224*
225 info = 0
226 IF (.NOT.lsame(trans,'N') .AND. .NOT.lsame(trans,'T') .AND.
227 + .NOT.lsame(trans,'C')) THEN
228 info = 1
229 ELSE IF (m.LT.0) THEN
230 info = 2
231 ELSE IF (n.LT.0) THEN
232 info = 3
233 ELSE IF (kl.LT.0) THEN
234 info = 4
235 ELSE IF (ku.LT.0) THEN
236 info = 5
237 ELSE IF (lda.LT. (kl+ku+1)) THEN
238 info = 8
239 ELSE IF (incx.EQ.0) THEN
240 info = 10
241 ELSE IF (incy.EQ.0) THEN
242 info = 13
243 END IF
244 IF (info.NE.0) THEN
245 CALL xerbla('SGBMV ',info)
246 RETURN
247 END IF
248*
249* Quick return if possible.
250*
251 IF ((m.EQ.0) .OR. (n.EQ.0) .OR.
252 + ((alpha.EQ.zero).AND. (beta.EQ.one))) RETURN
253*
254* Set LENX and LENY, the lengths of the vectors x and y, and set
255* up the start points in X and Y.
256*
257 IF (lsame(trans,'N')) THEN
258 lenx = n
259 leny = m
260 ELSE
261 lenx = m
262 leny = n
263 END IF
264 IF (incx.GT.0) THEN
265 kx = 1
266 ELSE
267 kx = 1 - (lenx-1)*incx
268 END IF
269 IF (incy.GT.0) THEN
270 ky = 1
271 ELSE
272 ky = 1 - (leny-1)*incy
273 END IF
274*
275* Start the operations. In this version the elements of A are
276* accessed sequentially with one pass through the band part of A.
277*
278* First form y := beta*y.
279*
280 IF (beta.NE.one) THEN
281 IF (incy.EQ.1) THEN
282 IF (beta.EQ.zero) THEN
283 DO 10 i = 1,leny
284 y(i) = zero
285 10 CONTINUE
286 ELSE
287 DO 20 i = 1,leny
288 y(i) = beta*y(i)
289 20 CONTINUE
290 END IF
291 ELSE
292 iy = ky
293 IF (beta.EQ.zero) THEN
294 DO 30 i = 1,leny
295 y(iy) = zero
296 iy = iy + incy
297 30 CONTINUE
298 ELSE
299 DO 40 i = 1,leny
300 y(iy) = beta*y(iy)
301 iy = iy + incy
302 40 CONTINUE
303 END IF
304 END IF
305 END IF
306 IF (alpha.EQ.zero) RETURN
307 kup1 = ku + 1
308 IF (lsame(trans,'N')) THEN
309*
310* Form y := alpha*A*x + y.
311*
312 jx = kx
313 IF (incy.EQ.1) THEN
314 DO 60 j = 1,n
315 temp = alpha*x(jx)
316 k = kup1 - j
317 DO 50 i = max(1,j-ku),min(m,j+kl)
318 y(i) = y(i) + temp*a(k+i,j)
319 50 CONTINUE
320 jx = jx + incx
321 60 CONTINUE
322 ELSE
323 DO 80 j = 1,n
324 temp = alpha*x(jx)
325 iy = ky
326 k = kup1 - j
327 DO 70 i = max(1,j-ku),min(m,j+kl)
328 y(iy) = y(iy) + temp*a(k+i,j)
329 iy = iy + incy
330 70 CONTINUE
331 jx = jx + incx
332 IF (j.GT.ku) ky = ky + incy
333 80 CONTINUE
334 END IF
335 ELSE
336*
337* Form y := alpha*A**T*x + y.
338*
339 jy = ky
340 IF (incx.EQ.1) THEN
341 DO 100 j = 1,n
342 temp = zero
343 k = kup1 - j
344 DO 90 i = max(1,j-ku),min(m,j+kl)
345 temp = temp + a(k+i,j)*x(i)
346 90 CONTINUE
347 y(jy) = y(jy) + alpha*temp
348 jy = jy + incy
349 100 CONTINUE
350 ELSE
351 DO 120 j = 1,n
352 temp = zero
353 ix = kx
354 k = kup1 - j
355 DO 110 i = max(1,j-ku),min(m,j+kl)
356 temp = temp + a(k+i,j)*x(ix)
357 ix = ix + incx
358 110 CONTINUE
359 y(jy) = y(jy) + alpha*temp
360 jy = jy + incy
361 IF (j.GT.ku) kx = kx + incx
362 120 CONTINUE
363 END IF
364 END IF
365*
366 RETURN
367*
368* End of SGBMV
369*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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