1: *> \brief \b DTBSV
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 DTBSV(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: * DOUBLE PRECISION A(LDA,*),X(*)
19: * ..
20: *
21: *
22: *> \par Purpose:
23: * =============
24: *>
25: *> \verbatim
26: *>
27: *> DTBSV 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 double_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 DTBSV(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: DOUBLE PRECISION A(LDA,*),X(*)
200: * ..
201: *
202: * =====================================================================
203: *
204: * .. Parameters ..
205: DOUBLE PRECISION ZERO
206: PARAMETER (ZERO=0.0D+0)
207: * ..
208: * .. Local Scalars ..
209: DOUBLE PRECISION 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('DTBSV ',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 DTBSV
397: *
398: END
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