1*> \brief \b SSYTRS
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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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/ssytrs.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SSYTRS( 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*       REAL               A( LDA, * ), B( LDB, * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> SSYTRS 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 SSYTRF.
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 REAL 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 SSYTRF.
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 SSYTRF.
86*> \endverbatim
87*>
88*> \param[in,out] B
89*> \verbatim
90*>          B is REAL 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 November 2011
117*
118*> \ingroup realSYcomputational
119*
120*  =====================================================================
121      SUBROUTINE SSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
122*
123*  -- LAPACK computational routine (version 3.4.0) --
124*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
125*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
126*     November 2011
127*
128*     .. Scalar Arguments ..
129      CHARACTER          UPLO
130      INTEGER            INFO, LDA, LDB, N, NRHS
131*     ..
132*     .. Array Arguments ..
133      INTEGER            IPIV( * )
134      REAL               A( LDA, * ), B( LDB, * )
135*     ..
136*
137*  =====================================================================
138*
139*     .. Parameters ..
140      REAL               ONE
141      PARAMETER          ( ONE = 1.0E+0 )
142*     ..
143*     .. Local Scalars ..
144      LOGICAL            UPPER
145      INTEGER            J, K, KP
146      REAL               AK, AKM1, AKM1K, BK, BKM1, DENOM
147*     ..
148*     .. External Functions ..
149      LOGICAL            LSAME
150      EXTERNAL           LSAME
151*     ..
152*     .. External Subroutines ..
153      EXTERNAL           SGEMV, SGER, SSCAL, SSWAP, 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( 'SSYTRS', -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 SSWAP( 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 SGER( 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 SSCAL( 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 SSWAP( 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 SGER( K-2, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
234     $                 B( 1, 1 ), LDB )
235            CALL SGER( 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 SGEMV( '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 SSWAP( 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 SGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ),
293     $                  1, ONE, B( K, 1 ), LDB )
294            CALL SGEMV( '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 SSWAP( 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 SSWAP( 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 SGER( 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 SSCAL( 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 SSWAP( 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 SGER( N-K-1, NRHS, -ONE, A( K+2, K ), 1, B( K, 1 ),
361     $                    LDB, B( K+2, 1 ), LDB )
362               CALL SGER( 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 SGEMV( '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 SSWAP( 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 SGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
423     $                     LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
424               CALL SGEMV( '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 SSWAP( 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 SSYTRS
444*
445      END
446