1: *> \brief \b DSYTRS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYTRS + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, LDB, N, NRHS
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * DOUBLE PRECISION A( LDA, * ), B( LDB, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYTRS solves a system of linear equations A*X = B with a real
39: *> symmetric matrix A using the factorization A = U*D*U**T or
40: *> A = L*D*L**T computed by DSYTRF.
41: *> \endverbatim
42: *
43: * Arguments:
44: * ==========
45: *
46: *> \param[in] UPLO
47: *> \verbatim
48: *> UPLO is CHARACTER*1
49: *> Specifies whether the details of the factorization are stored
50: *> as an upper or lower triangular matrix.
51: *> = 'U': Upper triangular, form is A = U*D*U**T;
52: *> = 'L': Lower triangular, form is A = L*D*L**T.
53: *> \endverbatim
54: *>
55: *> \param[in] N
56: *> \verbatim
57: *> N is INTEGER
58: *> The order of the matrix A. N >= 0.
59: *> \endverbatim
60: *>
61: *> \param[in] NRHS
62: *> \verbatim
63: *> NRHS is INTEGER
64: *> The number of right hand sides, i.e., the number of columns
65: *> of the matrix B. NRHS >= 0.
66: *> \endverbatim
67: *>
68: *> \param[in] A
69: *> \verbatim
70: *> A is DOUBLE PRECISION array, dimension (LDA,N)
71: *> The block diagonal matrix D and the multipliers used to
72: *> obtain the factor U or L as computed by DSYTRF.
73: *> \endverbatim
74: *>
75: *> \param[in] LDA
76: *> \verbatim
77: *> LDA is INTEGER
78: *> The leading dimension of the array A. LDA >= max(1,N).
79: *> \endverbatim
80: *>
81: *> \param[in] IPIV
82: *> \verbatim
83: *> IPIV is INTEGER array, dimension (N)
84: *> Details of the interchanges and the block structure of D
85: *> as determined by DSYTRF.
86: *> \endverbatim
87: *>
88: *> \param[in,out] B
89: *> \verbatim
90: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
91: *> On entry, the right hand side matrix B.
92: *> On exit, the solution matrix X.
93: *> \endverbatim
94: *>
95: *> \param[in] LDB
96: *> \verbatim
97: *> LDB is INTEGER
98: *> The leading dimension of the array B. LDB >= max(1,N).
99: *> \endverbatim
100: *>
101: *> \param[out] INFO
102: *> \verbatim
103: *> INFO is INTEGER
104: *> = 0: successful exit
105: *> < 0: if INFO = -i, the i-th argument had an illegal value
106: *> \endverbatim
107: *
108: * Authors:
109: * ========
110: *
111: *> \author Univ. of Tennessee
112: *> \author Univ. of California Berkeley
113: *> \author Univ. of Colorado Denver
114: *> \author NAG Ltd.
115: *
116: *> \date December 2016
117: *
118: *> \ingroup doubleSYcomputational
119: *
120: * =====================================================================
121: SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
122: *
123: * -- LAPACK computational routine (version 3.7.0) --
124: * -- LAPACK is a software package provided by Univ. of Tennessee, --
125: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
126: * December 2016
127: *
128: * .. Scalar Arguments ..
129: CHARACTER UPLO
130: INTEGER INFO, LDA, LDB, N, NRHS
131: * ..
132: * .. Array Arguments ..
133: INTEGER IPIV( * )
134: DOUBLE PRECISION A( LDA, * ), B( LDB, * )
135: * ..
136: *
137: * =====================================================================
138: *
139: * .. Parameters ..
140: DOUBLE PRECISION ONE
141: PARAMETER ( ONE = 1.0D+0 )
142: * ..
143: * .. Local Scalars ..
144: LOGICAL UPPER
145: INTEGER J, K, KP
146: DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM
147: * ..
148: * .. External Functions ..
149: LOGICAL LSAME
150: EXTERNAL LSAME
151: * ..
152: * .. External Subroutines ..
153: EXTERNAL DGEMV, DGER, DSCAL, DSWAP, XERBLA
154: * ..
155: * .. Intrinsic Functions ..
156: INTRINSIC MAX
157: * ..
158: * .. Executable Statements ..
159: *
160: INFO = 0
161: UPPER = LSAME( UPLO, 'U' )
162: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
163: INFO = -1
164: ELSE IF( N.LT.0 ) THEN
165: INFO = -2
166: ELSE IF( NRHS.LT.0 ) THEN
167: INFO = -3
168: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
169: INFO = -5
170: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
171: INFO = -8
172: END IF
173: IF( INFO.NE.0 ) THEN
174: CALL XERBLA( 'DSYTRS', -INFO )
175: RETURN
176: END IF
177: *
178: * Quick return if possible
179: *
180: IF( N.EQ.0 .OR. NRHS.EQ.0 )
181: $ RETURN
182: *
183: IF( UPPER ) THEN
184: *
185: * Solve A*X = B, where A = U*D*U**T.
186: *
187: * First solve U*D*X = B, overwriting B with X.
188: *
189: * K is the main loop index, decreasing from N to 1 in steps of
190: * 1 or 2, depending on the size of the diagonal blocks.
191: *
192: K = N
193: 10 CONTINUE
194: *
195: * If K < 1, exit from loop.
196: *
197: IF( K.LT.1 )
198: $ GO TO 30
199: *
200: IF( IPIV( K ).GT.0 ) THEN
201: *
202: * 1 x 1 diagonal block
203: *
204: * Interchange rows K and IPIV(K).
205: *
206: KP = IPIV( K )
207: IF( KP.NE.K )
208: $ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
209: *
210: * Multiply by inv(U(K)), where U(K) is the transformation
211: * stored in column K of A.
212: *
213: CALL DGER( K-1, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
214: $ B( 1, 1 ), LDB )
215: *
216: * Multiply by the inverse of the diagonal block.
217: *
218: CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
219: K = K - 1
220: ELSE
221: *
222: * 2 x 2 diagonal block
223: *
224: * Interchange rows K-1 and -IPIV(K).
225: *
226: KP = -IPIV( K )
227: IF( KP.NE.K-1 )
228: $ CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
229: *
230: * Multiply by inv(U(K)), where U(K) is the transformation
231: * stored in columns K-1 and K of A.
232: *
233: CALL DGER( K-2, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
234: $ B( 1, 1 ), LDB )
235: CALL DGER( K-2, NRHS, -ONE, A( 1, K-1 ), 1, B( K-1, 1 ),
236: $ LDB, B( 1, 1 ), LDB )
237: *
238: * Multiply by the inverse of the diagonal block.
239: *
240: AKM1K = A( K-1, K )
241: AKM1 = A( K-1, K-1 ) / AKM1K
242: AK = A( K, K ) / AKM1K
243: DENOM = AKM1*AK - ONE
244: DO 20 J = 1, NRHS
245: BKM1 = B( K-1, J ) / AKM1K
246: BK = B( K, J ) / AKM1K
247: B( K-1, J ) = ( AK*BKM1-BK ) / DENOM
248: B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM
249: 20 CONTINUE
250: K = K - 2
251: END IF
252: *
253: GO TO 10
254: 30 CONTINUE
255: *
256: * Next solve U**T *X = B, overwriting B with X.
257: *
258: * K is the main loop index, increasing from 1 to N in steps of
259: * 1 or 2, depending on the size of the diagonal blocks.
260: *
261: K = 1
262: 40 CONTINUE
263: *
264: * If K > N, exit from loop.
265: *
266: IF( K.GT.N )
267: $ GO TO 50
268: *
269: IF( IPIV( K ).GT.0 ) THEN
270: *
271: * 1 x 1 diagonal block
272: *
273: * Multiply by inv(U**T(K)), where U(K) is the transformation
274: * stored in column K of A.
275: *
276: CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ),
277: $ 1, ONE, B( K, 1 ), LDB )
278: *
279: * Interchange rows K and IPIV(K).
280: *
281: KP = IPIV( K )
282: IF( KP.NE.K )
283: $ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
284: K = K + 1
285: ELSE
286: *
287: * 2 x 2 diagonal block
288: *
289: * Multiply by inv(U**T(K+1)), where U(K+1) is the transformation
290: * stored in columns K and K+1 of A.
291: *
292: CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ),
293: $ 1, ONE, B( K, 1 ), LDB )
294: CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
295: $ A( 1, K+1 ), 1, ONE, B( K+1, 1 ), LDB )
296: *
297: * Interchange rows K and -IPIV(K).
298: *
299: KP = -IPIV( K )
300: IF( KP.NE.K )
301: $ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
302: K = K + 2
303: END IF
304: *
305: GO TO 40
306: 50 CONTINUE
307: *
308: ELSE
309: *
310: * Solve A*X = B, where A = L*D*L**T.
311: *
312: * First solve L*D*X = B, overwriting B with X.
313: *
314: * K is the main loop index, increasing from 1 to N in steps of
315: * 1 or 2, depending on the size of the diagonal blocks.
316: *
317: K = 1
318: 60 CONTINUE
319: *
320: * If K > N, exit from loop.
321: *
322: IF( K.GT.N )
323: $ GO TO 80
324: *
325: IF( IPIV( K ).GT.0 ) THEN
326: *
327: * 1 x 1 diagonal block
328: *
329: * Interchange rows K and IPIV(K).
330: *
331: KP = IPIV( K )
332: IF( KP.NE.K )
333: $ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
334: *
335: * Multiply by inv(L(K)), where L(K) is the transformation
336: * stored in column K of A.
337: *
338: IF( K.LT.N )
339: $ CALL DGER( N-K, NRHS, -ONE, A( K+1, K ), 1, B( K, 1 ),
340: $ LDB, B( K+1, 1 ), LDB )
341: *
342: * Multiply by the inverse of the diagonal block.
343: *
344: CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
345: K = K + 1
346: ELSE
347: *
348: * 2 x 2 diagonal block
349: *
350: * Interchange rows K+1 and -IPIV(K).
351: *
352: KP = -IPIV( K )
353: IF( KP.NE.K+1 )
354: $ CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
355: *
356: * Multiply by inv(L(K)), where L(K) is the transformation
357: * stored in columns K and K+1 of A.
358: *
359: IF( K.LT.N-1 ) THEN
360: CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K ), 1, B( K, 1 ),
361: $ LDB, B( K+2, 1 ), LDB )
362: CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K+1 ), 1,
363: $ B( K+1, 1 ), LDB, B( K+2, 1 ), LDB )
364: END IF
365: *
366: * Multiply by the inverse of the diagonal block.
367: *
368: AKM1K = A( K+1, K )
369: AKM1 = A( K, K ) / AKM1K
370: AK = A( K+1, K+1 ) / AKM1K
371: DENOM = AKM1*AK - ONE
372: DO 70 J = 1, NRHS
373: BKM1 = B( K, J ) / AKM1K
374: BK = B( K+1, J ) / AKM1K
375: B( K, J ) = ( AK*BKM1-BK ) / DENOM
376: B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
377: 70 CONTINUE
378: K = K + 2
379: END IF
380: *
381: GO TO 60
382: 80 CONTINUE
383: *
384: * Next solve L**T *X = B, overwriting B with X.
385: *
386: * K is the main loop index, decreasing from N to 1 in steps of
387: * 1 or 2, depending on the size of the diagonal blocks.
388: *
389: K = N
390: 90 CONTINUE
391: *
392: * If K < 1, exit from loop.
393: *
394: IF( K.LT.1 )
395: $ GO TO 100
396: *
397: IF( IPIV( K ).GT.0 ) THEN
398: *
399: * 1 x 1 diagonal block
400: *
401: * Multiply by inv(L**T(K)), where L(K) is the transformation
402: * stored in column K of A.
403: *
404: IF( K.LT.N )
405: $ CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
406: $ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
407: *
408: * Interchange rows K and IPIV(K).
409: *
410: KP = IPIV( K )
411: IF( KP.NE.K )
412: $ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
413: K = K - 1
414: ELSE
415: *
416: * 2 x 2 diagonal block
417: *
418: * Multiply by inv(L**T(K-1)), where L(K-1) is the transformation
419: * stored in columns K-1 and K of A.
420: *
421: IF( K.LT.N ) THEN
422: CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
423: $ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
424: CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
425: $ LDB, A( K+1, K-1 ), 1, ONE, B( K-1, 1 ),
426: $ LDB )
427: END IF
428: *
429: * Interchange rows K and -IPIV(K).
430: *
431: KP = -IPIV( K )
432: IF( KP.NE.K )
433: $ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
434: K = K - 2
435: END IF
436: *
437: GO TO 90
438: 100 CONTINUE
439: END IF
440: *
441: RETURN
442: *
443: * End of DSYTRS
444: *
445: END
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