LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zsytrs_3()

 subroutine zsytrs_3 ( character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO )

ZSYTRS_3

Purpose:
``` ZSYTRS_3 solves a system of linear equations A * X = B with a complex
symmetric matrix A using the factorization computed
by ZSYTRF_RK or ZSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This algorithm is using Level 3 BLAS.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.``` [in] A ``` A is COMPLEX*16 array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] E ``` E is COMPLEX*16 array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_RK or ZSYTRF_BK.``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Contributors:
```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 163 of file zsytrs_3.f.

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 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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
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