1*> \brief \b STBSV
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*  Definition:
9*  ===========
10*
11*       SUBROUTINE STBSV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX)
12*
13*       .. Scalar Arguments ..
14*       INTEGER INCX,K,LDA,N
15*       CHARACTER DIAG,TRANS,UPLO
16*       ..
17*       .. Array Arguments ..
18*       REAL A(LDA,*),X(*)
19*       ..
20*
21*
22*> \par Purpose:
23*  =============
24*>
25*> \verbatim
26*>
27*> STBSV  solves one of the systems of equations
28*>
29*>    A*x = b,   or   A**T*x = b,
30*>
31*> where b and x are n element vectors and A is an n by n unit, or
32*> non-unit, upper or lower triangular band matrix, with ( k + 1 )
33*> diagonals.
34*>
35*> No test for singularity or near-singularity is included in this
36*> routine. Such tests must be performed before calling this routine.
37*> \endverbatim
38*
39*  Arguments:
40*  ==========
41*
42*> \param[in] UPLO
43*> \verbatim
44*>          UPLO is CHARACTER*1
45*>           On entry, UPLO specifies whether the matrix is an upper or
46*>           lower triangular matrix as follows:
47*>
48*>              UPLO = 'U' or 'u'   A is an upper triangular matrix.
49*>
50*>              UPLO = 'L' or 'l'   A is a lower triangular matrix.
51*> \endverbatim
52*>
53*> \param[in] TRANS
54*> \verbatim
55*>          TRANS is CHARACTER*1
56*>           On entry, TRANS specifies the equations to be solved as
57*>           follows:
58*>
59*>              TRANS = 'N' or 'n'   A*x = b.
60*>
61*>              TRANS = 'T' or 't'   A**T*x = b.
62*>
63*>              TRANS = 'C' or 'c'   A**T*x = b.
64*> \endverbatim
65*>
66*> \param[in] DIAG
67*> \verbatim
68*>          DIAG is CHARACTER*1
69*>           On entry, DIAG specifies whether or not A is unit
70*>           triangular as follows:
71*>
72*>              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
73*>
74*>              DIAG = 'N' or 'n'   A is not assumed to be unit
75*>                                  triangular.
76*> \endverbatim
77*>
78*> \param[in] N
79*> \verbatim
80*>          N is INTEGER
81*>           On entry, N specifies the order of the matrix A.
82*>           N must be at least zero.
83*> \endverbatim
84*>
85*> \param[in] K
86*> \verbatim
87*>          K is INTEGER
88*>           On entry with UPLO = 'U' or 'u', K specifies the number of
89*>           super-diagonals of the matrix A.
90*>           On entry with UPLO = 'L' or 'l', K specifies the number of
91*>           sub-diagonals of the matrix A.
92*>           K must satisfy  0 .le. K.
93*> \endverbatim
94*>
95*> \param[in] A
96*> \verbatim
97*>          A is REAL array, dimension ( LDA, N )
98*>           Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
99*>           by n part of the array A must contain the upper triangular
100*>           band part of the matrix of coefficients, supplied column by
101*>           column, with the leading diagonal of the matrix in row
102*>           ( k + 1 ) of the array, the first super-diagonal starting at
103*>           position 2 in row k, and so on. The top left k by k triangle
104*>           of the array A is not referenced.
105*>           The following program segment will transfer an upper
106*>           triangular band matrix from conventional full matrix storage
107*>           to band storage:
108*>
109*>                 DO 20, J = 1, N
110*>                    M = K + 1 - J
111*>                    DO 10, I = MAX( 1, J - K ), J
112*>                       A( M + I, J ) = matrix( I, J )
113*>              10    CONTINUE
114*>              20 CONTINUE
115*>
116*>           Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
117*>           by n part of the array A must contain the lower triangular
118*>           band part of the matrix of coefficients, supplied column by
119*>           column, with the leading diagonal of the matrix in row 1 of
120*>           the array, the first sub-diagonal starting at position 1 in
121*>           row 2, and so on. The bottom right k by k triangle of the
122*>           array A is not referenced.
123*>           The following program segment will transfer a lower
124*>           triangular band matrix from conventional full matrix storage
125*>           to band storage:
126*>
127*>                 DO 20, J = 1, N
128*>                    M = 1 - J
129*>                    DO 10, I = J, MIN( N, J + K )
130*>                       A( M + I, J ) = matrix( I, J )
131*>              10    CONTINUE
132*>              20 CONTINUE
133*>
134*>           Note that when DIAG = 'U' or 'u' the elements of the array A
135*>           corresponding to the diagonal elements of the matrix are not
136*>           referenced, but are assumed to be unity.
137*> \endverbatim
138*>
139*> \param[in] LDA
140*> \verbatim
141*>          LDA is INTEGER
142*>           On entry, LDA specifies the first dimension of A as declared
143*>           in the calling (sub) program. LDA must be at least
144*>           ( k + 1 ).
145*> \endverbatim
146*>
147*> \param[in,out] X
148*> \verbatim
149*>          X is REAL array, dimension at least
150*>           ( 1 + ( n - 1 )*abs( INCX ) ).
151*>           Before entry, the incremented array X must contain the n
152*>           element right-hand side vector b. On exit, X is overwritten
153*>           with the solution vector x.
154*> \endverbatim
155*>
156*> \param[in] INCX
157*> \verbatim
158*>          INCX is INTEGER
159*>           On entry, INCX specifies the increment for the elements of
160*>           X. INCX must not be zero.
161*> \endverbatim
162*
163*  Authors:
164*  ========
165*
166*> \author Univ. of Tennessee
167*> \author Univ. of California Berkeley
168*> \author Univ. of Colorado Denver
169*> \author NAG Ltd.
170*
171*> \ingroup single_blas_level2
172*
173*> \par Further Details:
174*  =====================
175*>
176*> \verbatim
177*>
178*>  Level 2 Blas routine.
179*>
180*>  -- Written on 22-October-1986.
181*>     Jack Dongarra, Argonne National Lab.
182*>     Jeremy Du Croz, Nag Central Office.
183*>     Sven Hammarling, Nag Central Office.
184*>     Richard Hanson, Sandia National Labs.
185*> \endverbatim
186*>
187*  =====================================================================
188      SUBROUTINE STBSV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX)
189*
190*  -- Reference BLAS level2 routine --
191*  -- Reference BLAS is a software package provided by Univ. of Tennessee,    --
192*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193*
194*     .. Scalar Arguments ..
195      INTEGER INCX,K,LDA,N
196      CHARACTER DIAG,TRANS,UPLO
197*     ..
198*     .. Array Arguments ..
199      REAL A(LDA,*),X(*)
200*     ..
201*
202*  =====================================================================
203*
204*     .. Parameters ..
205      REAL ZERO
206      PARAMETER (ZERO=0.0E+0)
207*     ..
208*     .. Local Scalars ..
209      REAL TEMP
210      INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
211      LOGICAL NOUNIT
212*     ..
213*     .. External Functions ..
214      LOGICAL LSAME
215      EXTERNAL LSAME
216*     ..
217*     .. External Subroutines ..
218      EXTERNAL XERBLA
219*     ..
220*     .. Intrinsic Functions ..
221      INTRINSIC MAX,MIN
222*     ..
223*
224*     Test the input parameters.
225*
226      INFO = 0
227      IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
228          INFO = 1
229      ELSE IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND.
230     +         .NOT.LSAME(TRANS,'C')) THEN
231          INFO = 2
232      ELSE IF (.NOT.LSAME(DIAG,'U') .AND. .NOT.LSAME(DIAG,'N')) THEN
233          INFO = 3
234      ELSE IF (N.LT.0) THEN
235          INFO = 4
236      ELSE IF (K.LT.0) THEN
237          INFO = 5
238      ELSE IF (LDA.LT. (K+1)) THEN
239          INFO = 7
240      ELSE IF (INCX.EQ.0) THEN
241          INFO = 9
242      END IF
243      IF (INFO.NE.0) THEN
244          CALL XERBLA('STBSV ',INFO)
245          RETURN
246      END IF
247*
248*     Quick return if possible.
249*
250      IF (N.EQ.0) RETURN
251*
252      NOUNIT = LSAME(DIAG,'N')
253*
254*     Set up the start point in X if the increment is not unity. This
255*     will be  ( N - 1 )*INCX  too small for descending loops.
256*
257      IF (INCX.LE.0) THEN
258          KX = 1 - (N-1)*INCX
259      ELSE IF (INCX.NE.1) THEN
260          KX = 1
261      END IF
262*
263*     Start the operations. In this version the elements of A are
264*     accessed by sequentially with one pass through A.
265*
266      IF (LSAME(TRANS,'N')) THEN
267*
268*        Form  x := inv( A )*x.
269*
270          IF (LSAME(UPLO,'U')) THEN
271              KPLUS1 = K + 1
272              IF (INCX.EQ.1) THEN
273                  DO 20 J = N,1,-1
274                      IF (X(J).NE.ZERO) THEN
275                          L = KPLUS1 - J
276                          IF (NOUNIT) X(J) = X(J)/A(KPLUS1,J)
277                          TEMP = X(J)
278                          DO 10 I = J - 1,MAX(1,J-K),-1
279                              X(I) = X(I) - TEMP*A(L+I,J)
280   10                     CONTINUE
281                      END IF
282   20             CONTINUE
283              ELSE
284                  KX = KX + (N-1)*INCX
285                  JX = KX
286                  DO 40 J = N,1,-1
287                      KX = KX - INCX
288                      IF (X(JX).NE.ZERO) THEN
289                          IX = KX
290                          L = KPLUS1 - J
291                          IF (NOUNIT) X(JX) = X(JX)/A(KPLUS1,J)
292                          TEMP = X(JX)
293                          DO 30 I = J - 1,MAX(1,J-K),-1
294                              X(IX) = X(IX) - TEMP*A(L+I,J)
295                              IX = IX - INCX
296   30                     CONTINUE
297                      END IF
298                      JX = JX - INCX
299   40             CONTINUE
300              END IF
301          ELSE
302              IF (INCX.EQ.1) THEN
303                  DO 60 J = 1,N
304                      IF (X(J).NE.ZERO) THEN
305                          L = 1 - J
306                          IF (NOUNIT) X(J) = X(J)/A(1,J)
307                          TEMP = X(J)
308                          DO 50 I = J + 1,MIN(N,J+K)
309                              X(I) = X(I) - TEMP*A(L+I,J)
310   50                     CONTINUE
311                      END IF
312   60             CONTINUE
313              ELSE
314                  JX = KX
315                  DO 80 J = 1,N
316                      KX = KX + INCX
317                      IF (X(JX).NE.ZERO) THEN
318                          IX = KX
319                          L = 1 - J
320                          IF (NOUNIT) X(JX) = X(JX)/A(1,J)
321                          TEMP = X(JX)
322                          DO 70 I = J + 1,MIN(N,J+K)
323                              X(IX) = X(IX) - TEMP*A(L+I,J)
324                              IX = IX + INCX
325   70                     CONTINUE
326                      END IF
327                      JX = JX + INCX
328   80             CONTINUE
329              END IF
330          END IF
331      ELSE
332*
333*        Form  x := inv( A**T)*x.
334*
335          IF (LSAME(UPLO,'U')) THEN
336              KPLUS1 = K + 1
337              IF (INCX.EQ.1) THEN
338                  DO 100 J = 1,N
339                      TEMP = X(J)
340                      L = KPLUS1 - J
341                      DO 90 I = MAX(1,J-K),J - 1
342                          TEMP = TEMP - A(L+I,J)*X(I)
343   90                 CONTINUE
344                      IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
345                      X(J) = TEMP
346  100             CONTINUE
347              ELSE
348                  JX = KX
349                  DO 120 J = 1,N
350                      TEMP = X(JX)
351                      IX = KX
352                      L = KPLUS1 - J
353                      DO 110 I = MAX(1,J-K),J - 1
354                          TEMP = TEMP - A(L+I,J)*X(IX)
355                          IX = IX + INCX
356  110                 CONTINUE
357                      IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
358                      X(JX) = TEMP
359                      JX = JX + INCX
360                      IF (J.GT.K) KX = KX + INCX
361  120             CONTINUE
362              END IF
363          ELSE
364              IF (INCX.EQ.1) THEN
365                  DO 140 J = N,1,-1
366                      TEMP = X(J)
367                      L = 1 - J
368                      DO 130 I = MIN(N,J+K),J + 1,-1
369                          TEMP = TEMP - A(L+I,J)*X(I)
370  130                 CONTINUE
371                      IF (NOUNIT) TEMP = TEMP/A(1,J)
372                      X(J) = TEMP
373  140             CONTINUE
374              ELSE
375                  KX = KX + (N-1)*INCX
376                  JX = KX
377                  DO 160 J = N,1,-1
378                      TEMP = X(JX)
379                      IX = KX
380                      L = 1 - J
381                      DO 150 I = MIN(N,J+K),J + 1,-1
382                          TEMP = TEMP - A(L+I,J)*X(IX)
383                          IX = IX - INCX
384  150                 CONTINUE
385                      IF (NOUNIT) TEMP = TEMP/A(1,J)
386                      X(JX) = TEMP
387                      JX = JX - INCX
388                      IF ((N-J).GE.K) KX = KX - INCX
389  160             CONTINUE
390              END IF
391          END IF
392      END IF
393*
394      RETURN
395*
396*     End of STBSV
397*
398      END
399