1      SUBROUTINE ZTBMV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX)
2*     .. Scalar Arguments ..
3      INTEGER INCX,K,LDA,N
4      CHARACTER DIAG,TRANS,UPLO
5*     ..
6*     .. Array Arguments ..
7      DOUBLE COMPLEX A(LDA,*),X(*)
8*     ..
9*
10*  Purpose
11*  =======
12*
13*  ZTBMV  performs one of the matrix-vector operations
14*
15*     x := A*x,   or   x := A'*x,   or   x := conjg( A' )*x,
16*
17*  where x is an n element vector and  A is an n by n unit, or non-unit,
18*  upper or lower triangular band matrix, with ( k + 1 ) diagonals.
19*
20*  Arguments
21*  ==========
22*
23*  UPLO   - CHARACTER*1.
24*           On entry, UPLO specifies whether the matrix is an upper or
25*           lower triangular matrix as follows:
26*
27*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
28*
29*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
30*
31*           Unchanged on exit.
32*
33*  TRANS  - CHARACTER*1.
34*           On entry, TRANS specifies the operation to be performed as
35*           follows:
36*
37*              TRANS = 'N' or 'n'   x := A*x.
38*
39*              TRANS = 'T' or 't'   x := A'*x.
40*
41*              TRANS = 'C' or 'c'   x := conjg( A' )*x.
42*
43*           Unchanged on exit.
44*
45*  DIAG   - CHARACTER*1.
46*           On entry, DIAG specifies whether or not A is unit
47*           triangular as follows:
48*
49*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
50*
51*              DIAG = 'N' or 'n'   A is not assumed to be unit
52*                                  triangular.
53*
54*           Unchanged on exit.
55*
56*  N      - INTEGER.
57*           On entry, N specifies the order of the matrix A.
58*           N must be at least zero.
59*           Unchanged on exit.
60*
61*  K      - INTEGER.
62*           On entry with UPLO = 'U' or 'u', K specifies the number of
63*           super-diagonals of the matrix A.
64*           On entry with UPLO = 'L' or 'l', K specifies the number of
65*           sub-diagonals of the matrix A.
66*           K must satisfy  0 .le. K.
67*           Unchanged on exit.
68*
69*  A      - COMPLEX*16       array of DIMENSION ( LDA, n ).
70*           Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
71*           by n part of the array A must contain the upper triangular
72*           band part of the matrix of coefficients, supplied column by
73*           column, with the leading diagonal of the matrix in row
74*           ( k + 1 ) of the array, the first super-diagonal starting at
75*           position 2 in row k, and so on. The top left k by k triangle
76*           of the array A is not referenced.
77*           The following program segment will transfer an upper
78*           triangular band matrix from conventional full matrix storage
79*           to band storage:
80*
81*                 DO 20, J = 1, N
82*                    M = K + 1 - J
83*                    DO 10, I = MAX( 1, J - K ), J
84*                       A( M + I, J ) = matrix( I, J )
85*              10    CONTINUE
86*              20 CONTINUE
87*
88*           Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
89*           by n part of the array A must contain the lower triangular
90*           band part of the matrix of coefficients, supplied column by
91*           column, with the leading diagonal of the matrix in row 1 of
92*           the array, the first sub-diagonal starting at position 1 in
93*           row 2, and so on. The bottom right k by k triangle of the
94*           array A is not referenced.
95*           The following program segment will transfer a lower
96*           triangular band matrix from conventional full matrix storage
97*           to band storage:
98*
99*                 DO 20, J = 1, N
100*                    M = 1 - J
101*                    DO 10, I = J, MIN( N, J + K )
102*                       A( M + I, J ) = matrix( I, J )
103*              10    CONTINUE
104*              20 CONTINUE
105*
106*           Note that when DIAG = 'U' or 'u' the elements of the array A
107*           corresponding to the diagonal elements of the matrix are not
108*           referenced, but are assumed to be unity.
109*           Unchanged on exit.
110*
111*  LDA    - INTEGER.
112*           On entry, LDA specifies the first dimension of A as declared
113*           in the calling (sub) program. LDA must be at least
114*           ( k + 1 ).
115*           Unchanged on exit.
116*
117*  X      - COMPLEX*16       array of dimension at least
118*           ( 1 + ( n - 1 )*abs( INCX ) ).
119*           Before entry, the incremented array X must contain the n
120*           element vector x. On exit, X is overwritten with the
121*           tranformed vector x.
122*
123*  INCX   - INTEGER.
124*           On entry, INCX specifies the increment for the elements of
125*           X. INCX must not be zero.
126*           Unchanged on exit.
127*
128*  Further Details
129*  ===============
130*
131*  Level 2 Blas routine.
132*
133*  -- Written on 22-October-1986.
134*     Jack Dongarra, Argonne National Lab.
135*     Jeremy Du Croz, Nag Central Office.
136*     Sven Hammarling, Nag Central Office.
137*     Richard Hanson, Sandia National Labs.
138*
139*  =====================================================================
140*
141*     .. Parameters ..
142      DOUBLE COMPLEX ZERO
143      PARAMETER (ZERO= (0.0D+0,0.0D+0))
144*     ..
145*     .. Local Scalars ..
146      DOUBLE COMPLEX TEMP
147      INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
148      LOGICAL NOCONJ,NOUNIT
149*     ..
150*     .. External Functions ..
151      LOGICAL LSAME
152      EXTERNAL LSAME
153*     ..
154*     .. External Subroutines ..
155      EXTERNAL XERBLA
156*     ..
157*     .. Intrinsic Functions ..
158      INTRINSIC DCONJG,MAX,MIN
159*     ..
160*
161*     Test the input parameters.
162*
163      INFO = 0
164      IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
165          INFO = 1
166      ELSE IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND.
167     +         .NOT.LSAME(TRANS,'C')) THEN
168          INFO = 2
169      ELSE IF (.NOT.LSAME(DIAG,'U') .AND. .NOT.LSAME(DIAG,'N')) THEN
170          INFO = 3
171      ELSE IF (N.LT.0) THEN
172          INFO = 4
173      ELSE IF (K.LT.0) THEN
174          INFO = 5
175      ELSE IF (LDA.LT. (K+1)) THEN
176          INFO = 7
177      ELSE IF (INCX.EQ.0) THEN
178          INFO = 9
179      END IF
180      IF (INFO.NE.0) THEN
181          CALL XERBLA('ZTBMV ',INFO)
182          RETURN
183      END IF
184*
185*     Quick return if possible.
186*
187      IF (N.EQ.0) RETURN
188*
189      NOCONJ = LSAME(TRANS,'T')
190      NOUNIT = LSAME(DIAG,'N')
191*
192*     Set up the start point in X if the increment is not unity. This
193*     will be  ( N - 1 )*INCX   too small for descending loops.
194*
195      IF (INCX.LE.0) THEN
196          KX = 1 - (N-1)*INCX
197      ELSE IF (INCX.NE.1) THEN
198          KX = 1
199      END IF
200*
201*     Start the operations. In this version the elements of A are
202*     accessed sequentially with one pass through A.
203*
204      IF (LSAME(TRANS,'N')) THEN
205*
206*         Form  x := A*x.
207*
208          IF (LSAME(UPLO,'U')) THEN
209              KPLUS1 = K + 1
210              IF (INCX.EQ.1) THEN
211                  DO 20 J = 1,N
212                      IF (X(J).NE.ZERO) THEN
213                          TEMP = X(J)
214                          L = KPLUS1 - J
215                          DO 10 I = MAX(1,J-K),J - 1
216                              X(I) = X(I) + TEMP*A(L+I,J)
217   10                     CONTINUE
218                          IF (NOUNIT) X(J) = X(J)*A(KPLUS1,J)
219                      END IF
220   20             CONTINUE
221              ELSE
222                  JX = KX
223                  DO 40 J = 1,N
224                      IF (X(JX).NE.ZERO) THEN
225                          TEMP = X(JX)
226                          IX = KX
227                          L = KPLUS1 - J
228                          DO 30 I = MAX(1,J-K),J - 1
229                              X(IX) = X(IX) + TEMP*A(L+I,J)
230                              IX = IX + INCX
231   30                     CONTINUE
232                          IF (NOUNIT) X(JX) = X(JX)*A(KPLUS1,J)
233                      END IF
234                      JX = JX + INCX
235                      IF (J.GT.K) KX = KX + INCX
236   40             CONTINUE
237              END IF
238          ELSE
239              IF (INCX.EQ.1) THEN
240                  DO 60 J = N,1,-1
241                      IF (X(J).NE.ZERO) THEN
242                          TEMP = X(J)
243                          L = 1 - J
244                          DO 50 I = MIN(N,J+K),J + 1,-1
245                              X(I) = X(I) + TEMP*A(L+I,J)
246   50                     CONTINUE
247                          IF (NOUNIT) X(J) = X(J)*A(1,J)
248                      END IF
249   60             CONTINUE
250              ELSE
251                  KX = KX + (N-1)*INCX
252                  JX = KX
253                  DO 80 J = N,1,-1
254                      IF (X(JX).NE.ZERO) THEN
255                          TEMP = X(JX)
256                          IX = KX
257                          L = 1 - J
258                          DO 70 I = MIN(N,J+K),J + 1,-1
259                              X(IX) = X(IX) + TEMP*A(L+I,J)
260                              IX = IX - INCX
261   70                     CONTINUE
262                          IF (NOUNIT) X(JX) = X(JX)*A(1,J)
263                      END IF
264                      JX = JX - INCX
265                      IF ((N-J).GE.K) KX = KX - INCX
266   80             CONTINUE
267              END IF
268          END IF
269      ELSE
270*
271*        Form  x := A'*x  or  x := conjg( A' )*x.
272*
273          IF (LSAME(UPLO,'U')) THEN
274              KPLUS1 = K + 1
275              IF (INCX.EQ.1) THEN
276                  DO 110 J = N,1,-1
277                      TEMP = X(J)
278                      L = KPLUS1 - J
279                      IF (NOCONJ) THEN
280                          IF (NOUNIT) TEMP = TEMP*A(KPLUS1,J)
281                          DO 90 I = J - 1,MAX(1,J-K),-1
282                              TEMP = TEMP + A(L+I,J)*X(I)
283   90                     CONTINUE
284                      ELSE
285                          IF (NOUNIT) TEMP = TEMP*DCONJG(A(KPLUS1,J))
286                          DO 100 I = J - 1,MAX(1,J-K),-1
287                              TEMP = TEMP + DCONJG(A(L+I,J))*X(I)
288  100                     CONTINUE
289                      END IF
290                      X(J) = TEMP
291  110             CONTINUE
292              ELSE
293                  KX = KX + (N-1)*INCX
294                  JX = KX
295                  DO 140 J = N,1,-1
296                      TEMP = X(JX)
297                      KX = KX - INCX
298                      IX = KX
299                      L = KPLUS1 - J
300                      IF (NOCONJ) THEN
301                          IF (NOUNIT) TEMP = TEMP*A(KPLUS1,J)
302                          DO 120 I = J - 1,MAX(1,J-K),-1
303                              TEMP = TEMP + A(L+I,J)*X(IX)
304                              IX = IX - INCX
305  120                     CONTINUE
306                      ELSE
307                          IF (NOUNIT) TEMP = TEMP*DCONJG(A(KPLUS1,J))
308                          DO 130 I = J - 1,MAX(1,J-K),-1
309                              TEMP = TEMP + DCONJG(A(L+I,J))*X(IX)
310                              IX = IX - INCX
311  130                     CONTINUE
312                      END IF
313                      X(JX) = TEMP
314                      JX = JX - INCX
315  140             CONTINUE
316              END IF
317          ELSE
318              IF (INCX.EQ.1) THEN
319                  DO 170 J = 1,N
320                      TEMP = X(J)
321                      L = 1 - J
322                      IF (NOCONJ) THEN
323                          IF (NOUNIT) TEMP = TEMP*A(1,J)
324                          DO 150 I = J + 1,MIN(N,J+K)
325                              TEMP = TEMP + A(L+I,J)*X(I)
326  150                     CONTINUE
327                      ELSE
328                          IF (NOUNIT) TEMP = TEMP*DCONJG(A(1,J))
329                          DO 160 I = J + 1,MIN(N,J+K)
330                              TEMP = TEMP + DCONJG(A(L+I,J))*X(I)
331  160                     CONTINUE
332                      END IF
333                      X(J) = TEMP
334  170             CONTINUE
335              ELSE
336                  JX = KX
337                  DO 200 J = 1,N
338                      TEMP = X(JX)
339                      KX = KX + INCX
340                      IX = KX
341                      L = 1 - J
342                      IF (NOCONJ) THEN
343                          IF (NOUNIT) TEMP = TEMP*A(1,J)
344                          DO 180 I = J + 1,MIN(N,J+K)
345                              TEMP = TEMP + A(L+I,J)*X(IX)
346                              IX = IX + INCX
347  180                     CONTINUE
348                      ELSE
349                          IF (NOUNIT) TEMP = TEMP*DCONJG(A(1,J))
350                          DO 190 I = J + 1,MIN(N,J+K)
351                              TEMP = TEMP + DCONJG(A(L+I,J))*X(IX)
352                              IX = IX + INCX
353  190                     CONTINUE
354                      END IF
355                      X(JX) = TEMP
356                      JX = JX + INCX
357  200             CONTINUE
358              END IF
359          END IF
360      END IF
361*
362      RETURN
363*
364*     End of ZTBMV .
365*
366      END
367