LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
zsytrs_3.f
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1 *> \brief \b ZSYTRS_3
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
22 * INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, LDA, LDB, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IPIV( * )
30 * COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *> ZSYTRS_3 solves a system of linear equations A * X = B with a complex
39 *> symmetric matrix A using the factorization computed
40 *> by ZSYTRF_RK or ZSYTRF_BK:
41 *>
42 *> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
43 *>
44 *> where U (or L) is unit upper (or lower) triangular matrix,
45 *> U**T (or L**T) is the transpose of U (or L), P is a permutation
46 *> matrix, P**T is the transpose of P, and D is symmetric and block
47 *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
48 *>
49 *> This algorithm is using Level 3 BLAS.
50 *> \endverbatim
51 *
52 * Arguments:
53 * ==========
54 *
55 *> \param[in] UPLO
56 *> \verbatim
57 *> UPLO is CHARACTER*1
58 *> Specifies whether the details of the factorization are
59 *> stored as an upper or lower triangular matrix:
60 *> = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
61 *> = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
62 *> \endverbatim
63 *>
64 *> \param[in] N
65 *> \verbatim
66 *> N is INTEGER
67 *> The order of the matrix A. N >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in] NRHS
71 *> \verbatim
72 *> NRHS is INTEGER
73 *> The number of right hand sides, i.e., the number of columns
74 *> of the matrix B. NRHS >= 0.
75 *> \endverbatim
76 *>
77 *> \param[in] A
78 *> \verbatim
79 *> A is COMPLEX*16 array, dimension (LDA,N)
80 *> Diagonal of the block diagonal matrix D and factors U or L
81 *> as computed by ZSYTRF_RK and ZSYTRF_BK:
82 *> a) ONLY diagonal elements of the symmetric block diagonal
83 *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
84 *> (superdiagonal (or subdiagonal) elements of D
85 *> should be provided on entry in array E), and
86 *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
87 *> If UPLO = 'L': factor L in the subdiagonal part of A.
88 *> \endverbatim
89 *>
90 *> \param[in] LDA
91 *> \verbatim
92 *> LDA is INTEGER
93 *> The leading dimension of the array A. LDA >= max(1,N).
94 *> \endverbatim
95 *>
96 *> \param[in] E
97 *> \verbatim
98 *> E is COMPLEX*16 array, dimension (N)
99 *> On entry, contains the superdiagonal (or subdiagonal)
100 *> elements of the symmetric block diagonal matrix D
101 *> with 1-by-1 or 2-by-2 diagonal blocks, where
102 *> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
103 *> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
104 *>
105 *> NOTE: For 1-by-1 diagonal block D(k), where
106 *> 1 <= k <= N, the element E(k) is not referenced in both
107 *> UPLO = 'U' or UPLO = 'L' cases.
108 *> \endverbatim
109 *>
110 *> \param[in] IPIV
111 *> \verbatim
112 *> IPIV is INTEGER array, dimension (N)
113 *> Details of the interchanges and the block structure of D
114 *> as determined by ZSYTRF_RK or ZSYTRF_BK.
115 *> \endverbatim
116 *>
117 *> \param[in,out] B
118 *> \verbatim
119 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
120 *> On entry, the right hand side matrix B.
121 *> On exit, the solution matrix X.
122 *> \endverbatim
123 *>
124 *> \param[in] LDB
125 *> \verbatim
126 *> LDB is INTEGER
127 *> The leading dimension of the array B. LDB >= max(1,N).
128 *> \endverbatim
129 *>
130 *> \param[out] INFO
131 *> \verbatim
132 *> INFO is INTEGER
133 *> = 0: successful exit
134 *> < 0: if INFO = -i, the i-th argument had an illegal value
135 *> \endverbatim
136 *
137 * Authors:
138 * ========
139 *
140 *> \author Univ. of Tennessee
141 *> \author Univ. of California Berkeley
142 *> \author Univ. of Colorado Denver
143 *> \author NAG Ltd.
144 *
145 *> \ingroup complex16SYcomputational
146 *
147 *> \par Contributors:
148 * ==================
149 *>
150 *> \verbatim
151 *>
152 *> June 2017, Igor Kozachenko,
153 *> Computer Science Division,
154 *> University of California, Berkeley
155 *>
156 *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
157 *> School of Mathematics,
158 *> University of Manchester
159 *>
160 *> \endverbatim
161 *
162 * =====================================================================
163  SUBROUTINE zsytrs_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
164  \$ INFO )
165 *
166 * -- LAPACK computational routine --
167 * -- LAPACK is a software package provided by Univ. of Tennessee, --
168 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169 *
170 * .. Scalar Arguments ..
171  CHARACTER UPLO
172  INTEGER INFO, LDA, LDB, N, NRHS
173 * ..
174 * .. Array Arguments ..
175  INTEGER IPIV( * )
176  COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * )
177 * ..
178 *
179 * =====================================================================
180 *
181 * .. Parameters ..
182  COMPLEX*16 ONE
183  parameter( one = ( 1.0d+0,0.0d+0 ) )
184 * ..
185 * .. Local Scalars ..
186  LOGICAL UPPER
187  INTEGER I, J, K, KP
188  COMPLEX*16 AK, AKM1, AKM1K, BK, BKM1, DENOM
189 * ..
190 * .. External Functions ..
191  LOGICAL LSAME
192  EXTERNAL lsame
193 * ..
194 * .. External Subroutines ..
195  EXTERNAL zscal, zswap, ztrsm, xerbla
196 * ..
197 * .. Intrinsic Functions ..
198  INTRINSIC abs, max
199 * ..
200 * .. Executable Statements ..
201 *
202  info = 0
203  upper = lsame( uplo, 'U' )
204  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
205  info = -1
206  ELSE IF( n.LT.0 ) THEN
207  info = -2
208  ELSE IF( nrhs.LT.0 ) THEN
209  info = -3
210  ELSE IF( lda.LT.max( 1, n ) ) THEN
211  info = -5
212  ELSE IF( ldb.LT.max( 1, n ) ) THEN
213  info = -9
214  END IF
215  IF( info.NE.0 ) THEN
216  CALL xerbla( 'ZSYTRS_3', -info )
217  RETURN
218  END IF
219 *
220 * Quick return if possible
221 *
222  IF( n.EQ.0 .OR. nrhs.EQ.0 )
223  \$ RETURN
224 *
225  IF( upper ) THEN
226 *
227 * Begin Upper
228 *
229 * Solve A*X = B, where A = U*D*U**T.
230 *
231 * P**T * B
232 *
233 * Interchange rows K and IPIV(K) of matrix B in the same order
234 * that the formation order of IPIV(I) vector for Upper case.
235 *
236 * (We can do the simple loop over IPIV with decrement -1,
237 * since the ABS value of IPIV(I) represents the row index
238 * of the interchange with row i in both 1x1 and 2x2 pivot cases)
239 *
240  DO k = n, 1, -1
241  kp = abs( ipiv( k ) )
242  IF( kp.NE.k ) THEN
243  CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
244  END IF
245  END DO
246 *
247 * Compute (U \P**T * B) -> B [ (U \P**T * B) ]
248 *
249  CALL ztrsm( 'L', 'U', 'N', 'U', n, nrhs, one, a, lda, b, ldb )
250 *
251 * Compute D \ B -> B [ D \ (U \P**T * B) ]
252 *
253  i = n
254  DO WHILE ( i.GE.1 )
255  IF( ipiv( i ).GT.0 ) THEN
256  CALL zscal( nrhs, one / a( i, i ), b( i, 1 ), ldb )
257  ELSE IF ( i.GT.1 ) THEN
258  akm1k = e( i )
259  akm1 = a( i-1, i-1 ) / akm1k
260  ak = a( i, i ) / akm1k
261  denom = akm1*ak - one
262  DO j = 1, nrhs
263  bkm1 = b( i-1, j ) / akm1k
264  bk = b( i, j ) / akm1k
265  b( i-1, j ) = ( ak*bkm1-bk ) / denom
266  b( i, j ) = ( akm1*bk-bkm1 ) / denom
267  END DO
268  i = i - 1
269  END IF
270  i = i - 1
271  END DO
272 *
273 * Compute (U**T \ B) -> B [ U**T \ (D \ (U \P**T * B) ) ]
274 *
275  CALL ztrsm( 'L', 'U', 'T', 'U', n, nrhs, one, a, lda, b, ldb )
276 *
277 * P * B [ P * (U**T \ (D \ (U \P**T * B) )) ]
278 *
279 * Interchange rows K and IPIV(K) of matrix B in reverse order
280 * from the formation order of IPIV(I) vector for Upper case.
281 *
282 * (We can do the simple loop over IPIV with increment 1,
283 * since the ABS value of IPIV(I) represents the row index
284 * of the interchange with row i in both 1x1 and 2x2 pivot cases)
285 *
286  DO k = 1, n, 1
287  kp = abs( ipiv( k ) )
288  IF( kp.NE.k ) THEN
289  CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
290  END IF
291  END DO
292 *
293  ELSE
294 *
295 * Begin Lower
296 *
297 * Solve A*X = B, where A = L*D*L**T.
298 *
299 * P**T * B
300 * Interchange rows K and IPIV(K) of matrix B in the same order
301 * that the formation order of IPIV(I) vector for Lower case.
302 *
303 * (We can do the simple loop over IPIV with increment 1,
304 * since the ABS value of IPIV(I) represents the row index
305 * of the interchange with row i in both 1x1 and 2x2 pivot cases)
306 *
307  DO k = 1, n, 1
308  kp = abs( ipiv( k ) )
309  IF( kp.NE.k ) THEN
310  CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
311  END IF
312  END DO
313 *
314 * Compute (L \P**T * B) -> B [ (L \P**T * B) ]
315 *
316  CALL ztrsm( 'L', 'L', 'N', 'U', n, nrhs, one, a, lda, b, ldb )
317 *
318 * Compute D \ B -> B [ D \ (L \P**T * B) ]
319 *
320  i = 1
321  DO WHILE ( i.LE.n )
322  IF( ipiv( i ).GT.0 ) THEN
323  CALL zscal( nrhs, one / a( i, i ), b( i, 1 ), ldb )
324  ELSE IF( i.LT.n ) THEN
325  akm1k = e( i )
326  akm1 = a( i, i ) / akm1k
327  ak = a( i+1, i+1 ) / akm1k
328  denom = akm1*ak - one
329  DO j = 1, nrhs
330  bkm1 = b( i, j ) / akm1k
331  bk = b( i+1, j ) / akm1k
332  b( i, j ) = ( ak*bkm1-bk ) / denom
333  b( i+1, j ) = ( akm1*bk-bkm1 ) / denom
334  END DO
335  i = i + 1
336  END IF
337  i = i + 1
338  END DO
339 *
340 * Compute (L**T \ B) -> B [ L**T \ (D \ (L \P**T * B) ) ]
341 *
342  CALL ztrsm('L', 'L', 'T', 'U', n, nrhs, one, a, lda, b, ldb )
343 *
344 * P * B [ P * (L**T \ (D \ (L \P**T * B) )) ]
345 *
346 * Interchange rows K and IPIV(K) of matrix B in reverse order
347 * from the formation order of IPIV(I) vector for Lower case.
348 *
349 * (We can do the simple loop over IPIV with decrement -1,
350 * since the ABS value of IPIV(I) represents the row index
351 * of the interchange with row i in both 1x1 and 2x2 pivot cases)
352 *
353  DO k = n, 1, -1
354  kp = abs( ipiv( k ) )
355  IF( kp.NE.k ) THEN
356  CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
357  END IF
358  END DO
359 *
360 * END Lower
361 *
362  END IF
363 *
364  RETURN
365 *
366 * End of ZSYTRS_3
367 *
368  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine ztrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
ZTRSM
Definition: ztrsm.f:180
subroutine zsytrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
ZSYTRS_3
Definition: zsytrs_3.f:165