 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dsysv_rk()

 subroutine dsysv_rk ( character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

Purpose:
``` DSYSV_RK computes the solution to a real system of linear
equations A * X = B, where A is an N-by-N symmetric matrix
and X and B are N-by-NRHS matrices.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
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.

DSYTRF_RK is called to compute the factorization of a real
symmetric matrix.  The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine DSYTRS_3.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The number of linear equations, i.e., 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,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK: 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 are stored on exit 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. For more info see the description of DSYTRF_RK routine.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] E ``` E is DOUBLE PRECISION array, dimension (N) On exit, contains the output computed by the factorization routine DSYTRF_RK, i.e. 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) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases. For more info see the description of DSYTRF_RK routine.``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D, as determined by DSYTRF_RK. For more info see the description of DSYTRF_RK routine.``` [in,out] B ``` B is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for DSYTRF_RK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.```
Contributors:
```  December 2016,  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 226 of file dsysv_rk.f.

228 *
229 * -- LAPACK driver routine --
230 * -- LAPACK is a software package provided by Univ. of Tennessee, --
231 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232 *
233 * .. Scalar Arguments ..
234  CHARACTER UPLO
235  INTEGER INFO, LDA, LDB, LWORK, N, NRHS
236 * ..
237 * .. Array Arguments ..
238  INTEGER IPIV( * )
239  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
240 * ..
241 *
242 * =====================================================================
243 *
244 * .. Local Scalars ..
245  LOGICAL LQUERY
246  INTEGER LWKOPT
247 * ..
248 * .. External Functions ..
249  LOGICAL LSAME
250  EXTERNAL lsame
251 * ..
252 * .. External Subroutines ..
253  EXTERNAL xerbla, dsytrf_rk, dsytrs_3
254 * ..
255 * .. Intrinsic Functions ..
256  INTRINSIC max
257 * ..
258 * .. Executable Statements ..
259 *
260 * Test the input parameters.
261 *
262  info = 0
263  lquery = ( lwork.EQ.-1 )
264  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
265  info = -1
266  ELSE IF( n.LT.0 ) THEN
267  info = -2
268  ELSE IF( nrhs.LT.0 ) THEN
269  info = -3
270  ELSE IF( lda.LT.max( 1, n ) ) THEN
271  info = -5
272  ELSE IF( ldb.LT.max( 1, n ) ) THEN
273  info = -9
274  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
275  info = -11
276  END IF
277 *
278  IF( info.EQ.0 ) THEN
279  IF( n.EQ.0 ) THEN
280  lwkopt = 1
281  ELSE
282  CALL dsytrf_rk( uplo, n, a, lda, e, ipiv, work, -1, info )
283  lwkopt = work(1)
284  END IF
285  work( 1 ) = lwkopt
286  END IF
287 *
288  IF( info.NE.0 ) THEN
289  CALL xerbla( 'DSYSV_RK ', -info )
290  RETURN
291  ELSE IF( lquery ) THEN
292  RETURN
293  END IF
294 *
295 * Compute the factorization A = P*U*D*(U**T)*(P**T) or
296 * A = P*U*D*(U**T)*(P**T).
297 *
298  CALL dsytrf_rk( uplo, n, a, lda, e, ipiv, work, lwork, info )
299 *
300  IF( info.EQ.0 ) THEN
301 *
302 * Solve the system A*X = B with BLAS3 solver, overwriting B with X.
303 *
304  CALL dsytrs_3( uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info )
305 *
306  END IF
307 *
308  work( 1 ) = lwkopt
309 *
310  RETURN
311 *
312 * End of DSYSV_RK
313 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dsytrf_rk(UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Ka...
Definition: dsytrf_rk.f:259
subroutine dsytrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
DSYTRS_3
Definition: dsytrs_3.f:165
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