1       SUBROUTINE ZTBSV(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 *  ZTBSV  solves one of the systems of equations
 14 *
 15 *     A*x = b,   or   A**T*x = b,   or   A**H*x = b,
 16 *
 17 *  where b and x are n element vectors and A is an n by n unit, or
 18 *  non-unit, upper or lower triangular band matrix, with ( k + 1 )
 19 *  diagonals.
 20 *
 21 *  No test for singularity or near-singularity is included in this
 22 *  routine. Such tests must be performed before calling this routine.
 23 *
 24 *  Arguments
 25 *  ==========
 26 *
 27 *  UPLO   - CHARACTER*1.
 28 *           On entry, UPLO specifies whether the matrix is an upper or
 29 *           lower triangular matrix as follows:
 30 *
 31 *              UPLO = 'U' or 'u'   A is an upper triangular matrix.
 32 *
 33 *              UPLO = 'L' or 'l'   A is a lower triangular matrix.
 34 *
 35 *           Unchanged on exit.
 36 *
 37 *  TRANS  - CHARACTER*1.
 38 *           On entry, TRANS specifies the equations to be solved as
 39 *           follows:
 40 *
 41 *              TRANS = 'N' or 'n'   A*x = b.
 42 *
 43 *              TRANS = 'T' or 't'   A**T*x = b.
 44 *
 45 *              TRANS = 'C' or 'c'   A**H*x = b.
 46 *
 47 *           Unchanged on exit.
 48 *
 49 *  DIAG   - CHARACTER*1.
 50 *           On entry, DIAG specifies whether or not A is unit
 51 *           triangular as follows:
 52 *
 53 *              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
 54 *
 55 *              DIAG = 'N' or 'n'   A is not assumed to be unit
 56 *                                  triangular.
 57 *
 58 *           Unchanged on exit.
 59 *
 60 *  N      - INTEGER.
 61 *           On entry, N specifies the order of the matrix A.
 62 *           N must be at least zero.
 63 *           Unchanged on exit.
 64 *
 65 *  K      - INTEGER.
 66 *           On entry with UPLO = 'U' or 'u', K specifies the number of
 67 *           super-diagonals of the matrix A.
 68 *           On entry with UPLO = 'L' or 'l', K specifies the number of
 69 *           sub-diagonals of the matrix A.
 70 *           K must satisfy  0 .le. K.
 71 *           Unchanged on exit.
 72 *
 73 *  A      - COMPLEX*16       array of DIMENSION ( LDA, n ).
 74 *           Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
 75 *           by n part of the array A must contain the upper triangular
 76 *           band part of the matrix of coefficients, supplied column by
 77 *           column, with the leading diagonal of the matrix in row
 78 *           ( k + 1 ) of the array, the first super-diagonal starting at
 79 *           position 2 in row k, and so on. The top left k by k triangle
 80 *           of the array A is not referenced.
 81 *           The following program segment will transfer an upper
 82 *           triangular band matrix from conventional full matrix storage
 83 *           to band storage:
 84 *
 85 *                 DO 20, J = 1, N
 86 *                    M = K + 1 - J
 87 *                    DO 10, I = MAX( 1, J - K ), J
 88 *                       A( M + I, J ) = matrix( I, J )
 89 *              10    CONTINUE
 90 *              20 CONTINUE
 91 *
 92 *           Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
 93 *           by n part of the array A must contain the lower triangular
 94 *           band part of the matrix of coefficients, supplied column by
 95 *           column, with the leading diagonal of the matrix in row 1 of
 96 *           the array, the first sub-diagonal starting at position 1 in
 97 *           row 2, and so on. The bottom right k by k triangle of the
 98 *           array A is not referenced.
 99 *           The following program segment will transfer a lower
100 *           triangular band matrix from conventional full matrix storage
101 *           to band storage:
102 *
103 *                 DO 20, J = 1, N
104 *                    M = 1 - J
105 *                    DO 10, I = J, MIN( N, J + K )
106 *                       A( M + I, J ) = matrix( I, J )
107 *              10    CONTINUE
108 *              20 CONTINUE
109 *
110 *           Note that when DIAG = 'U' or 'u' the elements of the array A
111 *           corresponding to the diagonal elements of the matrix are not
112 *           referenced, but are assumed to be unity.
113 *           Unchanged on exit.
114 *
115 *  LDA    - INTEGER.
116 *           On entry, LDA specifies the first dimension of A as declared
117 *           in the calling (sub) program. LDA must be at least
118 *           ( k + 1 ).
119 *           Unchanged on exit.
120 *
121 *  X      - COMPLEX*16       array of dimension at least
122 *           ( 1 + ( n - 1 )*abs( INCX ) ).
123 *           Before entry, the incremented array X must contain the n
124 *           element right-hand side vector b. On exit, X is overwritten
125 *           with the solution vector x.
126 *
127 *  INCX   - INTEGER.
128 *           On entry, INCX specifies the increment for the elements of
129 *           X. INCX must not be zero.
130 *           Unchanged on exit.
131 *
132 *  Further Details
133 *  ===============
134 *
135 *  Level 2 Blas routine.
136 *
137 *  -- Written on 22-October-1986.
138 *     Jack Dongarra, Argonne National Lab.
139 *     Jeremy Du Croz, Nag Central Office.
140 *     Sven Hammarling, Nag Central Office.
141 *     Richard Hanson, Sandia National Labs.
142 *
143 *  =====================================================================
144 *
145 *     .. Parameters ..
146       DOUBLE COMPLEX ZERO
147       PARAMETER (ZERO= (0.0D+0,0.0D+0))
148 *     ..
149 *     .. Local Scalars ..
150       DOUBLE COMPLEX TEMP
151       INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
152       LOGICAL NOCONJ,NOUNIT
153 *     ..
154 *     .. External Functions ..
155       LOGICAL LSAME
156       EXTERNAL LSAME
157 *     ..
158 *     .. External Subroutines ..
159       EXTERNAL XERBLA
160 *     ..
161 *     .. Intrinsic Functions ..
162       INTRINSIC DCONJG,MAX,MIN
163 *     ..
164 *
165 *     Test the input parameters.
166 *
167       INFO = 0
168       IF (.NOT.LSAME(UPLO,'U'.AND. .NOT.LSAME(UPLO,'L')) THEN
169           INFO = 1
170       ELSE IF (.NOT.LSAME(TRANS,'N'.AND. .NOT.LSAME(TRANS,'T'.AND.
171      +         .NOT.LSAME(TRANS,'C')) THEN
172           INFO = 2
173       ELSE IF (.NOT.LSAME(DIAG,'U'.AND. .NOT.LSAME(DIAG,'N')) THEN
174           INFO = 3
175       ELSE IF (N.LT.0THEN
176           INFO = 4
177       ELSE IF (K.LT.0THEN
178           INFO = 5
179       ELSE IF (LDA.LT. (K+1)) THEN
180           INFO = 7
181       ELSE IF (INCX.EQ.0THEN
182           INFO = 9
183       END IF
184       IF (INFO.NE.0THEN
185           CALL XERBLA('ZTBSV ',INFO)
186           RETURN
187       END IF
188 *
189 *     Quick return if possible.
190 *
191       IF (N.EQ.0RETURN
192 *
193       NOCONJ = LSAME(TRANS,'T')
194       NOUNIT = LSAME(DIAG,'N')
195 *
196 *     Set up the start point in X if the increment is not unity. This
197 *     will be  ( N - 1 )*INCX  too small for descending loops.
198 *
199       IF (INCX.LE.0THEN
200           KX = 1 - (N-1)*INCX
201       ELSE IF (INCX.NE.1THEN
202           KX = 1
203       END IF
204 *
205 *     Start the operations. In this version the elements of A are
206 *     accessed by sequentially with one pass through A.
207 *
208       IF (LSAME(TRANS,'N')) THEN
209 *
210 *        Form  x := inv( A )*x.
211 *
212           IF (LSAME(UPLO,'U')) THEN
213               KPLUS1 = K + 1
214               IF (INCX.EQ.1THEN
215                   DO 20 J = N,1,-1
216                       IF (X(J).NE.ZERO) THEN
217                           L = KPLUS1 - J
218                           IF (NOUNIT) X(J) = X(J)/A(KPLUS1,J)
219                           TEMP = X(J)
220                           DO 10 I = J - 1,MAX(1,J-K),-1
221                               X(I) = X(I) - TEMP*A(L+I,J)
222    10                     CONTINUE
223                       END IF
224    20             CONTINUE
225               ELSE
226                   KX = KX + (N-1)*INCX
227                   JX = KX
228                   DO 40 J = N,1,-1
229                       KX = KX - INCX
230                       IF (X(JX).NE.ZERO) THEN
231                           IX = KX
232                           L = KPLUS1 - J
233                           IF (NOUNIT) X(JX) = X(JX)/A(KPLUS1,J)
234                           TEMP = X(JX)
235                           DO 30 I = J - 1,MAX(1,J-K),-1
236                               X(IX) = X(IX) - TEMP*A(L+I,J)
237                               IX = IX - INCX
238    30                     CONTINUE
239                       END IF
240                       JX = JX - INCX
241    40             CONTINUE
242               END IF
243           ELSE
244               IF (INCX.EQ.1THEN
245                   DO 60 J = 1,N
246                       IF (X(J).NE.ZERO) THEN
247                           L = 1 - J
248                           IF (NOUNIT) X(J) = X(J)/A(1,J)
249                           TEMP = X(J)
250                           DO 50 I = J + 1,MIN(N,J+K)
251                               X(I) = X(I) - TEMP*A(L+I,J)
252    50                     CONTINUE
253                       END IF
254    60             CONTINUE
255               ELSE
256                   JX = KX
257                   DO 80 J = 1,N
258                       KX = KX + INCX
259                       IF (X(JX).NE.ZERO) THEN
260                           IX = KX
261                           L = 1 - J
262                           IF (NOUNIT) X(JX) = X(JX)/A(1,J)
263                           TEMP = X(JX)
264                           DO 70 I = J + 1,MIN(N,J+K)
265                               X(IX) = X(IX) - TEMP*A(L+I,J)
266                               IX = IX + INCX
267    70                     CONTINUE
268                       END IF
269                       JX = JX + INCX
270    80             CONTINUE
271               END IF
272           END IF
273       ELSE
274 *
275 *        Form  x := inv( A**T )*x  or  x := inv( A**H )*x.
276 *
277           IF (LSAME(UPLO,'U')) THEN
278               KPLUS1 = K + 1
279               IF (INCX.EQ.1THEN
280                   DO 110 J = 1,N
281                       TEMP = X(J)
282                       L = KPLUS1 - J
283                       IF (NOCONJ) THEN
284                           DO 90 I = MAX(1,J-K),J - 1
285                               TEMP = TEMP - A(L+I,J)*X(I)
286    90                     CONTINUE
287                           IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
288                       ELSE
289                           DO 100 I = MAX(1,J-K),J - 1
290                               TEMP = TEMP - DCONJG(A(L+I,J))*X(I)
291   100                     CONTINUE
292                           IF (NOUNIT) TEMP = TEMP/DCONJG(A(KPLUS1,J))
293                       END IF
294                       X(J) = TEMP
295   110             CONTINUE
296               ELSE
297                   JX = KX
298                   DO 140 J = 1,N
299                       TEMP = X(JX)
300                       IX = KX
301                       L = KPLUS1 - J
302                       IF (NOCONJ) THEN
303                           DO 120 I = MAX(1,J-K),J - 1
304                               TEMP = TEMP - A(L+I,J)*X(IX)
305                               IX = IX + INCX
306   120                     CONTINUE
307                           IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
308                       ELSE
309                           DO 130 I = MAX(1,J-K),J - 1
310                               TEMP = TEMP - DCONJG(A(L+I,J))*X(IX)
311                               IX = IX + INCX
312   130                     CONTINUE
313                           IF (NOUNIT) TEMP = TEMP/DCONJG(A(KPLUS1,J))
314                       END IF
315                       X(JX) = TEMP
316                       JX = JX + INCX
317                       IF (J.GT.K) KX = KX + INCX
318   140             CONTINUE
319               END IF
320           ELSE
321               IF (INCX.EQ.1THEN
322                   DO 170 J = N,1,-1
323                       TEMP = X(J)
324                       L = 1 - J
325                       IF (NOCONJ) THEN
326                           DO 150 I = MIN(N,J+K),J + 1,-1
327                               TEMP = TEMP - A(L+I,J)*X(I)
328   150                     CONTINUE
329                           IF (NOUNIT) TEMP = TEMP/A(1,J)
330                       ELSE
331                           DO 160 I = MIN(N,J+K),J + 1,-1
332                               TEMP = TEMP - DCONJG(A(L+I,J))*X(I)
333   160                     CONTINUE
334                           IF (NOUNIT) TEMP = TEMP/DCONJG(A(1,J))
335                       END IF
336                       X(J) = TEMP
337   170             CONTINUE
338               ELSE
339                   KX = KX + (N-1)*INCX
340                   JX = KX
341                   DO 200 J = N,1,-1
342                       TEMP = X(JX)
343                       IX = KX
344                       L = 1 - J
345                       IF (NOCONJ) THEN
346                           DO 180 I = MIN(N,J+K),J + 1,-1
347                               TEMP = TEMP - A(L+I,J)*X(IX)
348                               IX = IX - INCX
349   180                     CONTINUE
350                           IF (NOUNIT) TEMP = TEMP/A(1,J)
351                       ELSE
352                           DO 190 I = MIN(N,J+K),J + 1,-1
353                               TEMP = TEMP - DCONJG(A(L+I,J))*X(IX)
354                               IX = IX - INCX
355   190                     CONTINUE
356                           IF (NOUNIT) TEMP = TEMP/DCONJG(A(1,J))
357                       END IF
358                       X(JX) = TEMP
359                       JX = JX - INCX
360                       IF ((N-J).GE.K) KX = KX - INCX
361   200             CONTINUE
362               END IF
363           END IF
364       END IF
365 *
366       RETURN
367 *
368 *     End of ZTBSV .
369 *
370       END