LAPACK  3.8.0 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.```
Date
December 2016
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 230 of file dsysv_rk.f.

230 *
231 * -- LAPACK driver routine (version 3.7.0) --
232 * -- LAPACK is a software package provided by Univ. of Tennessee, --
233 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
234 * December 2016
235 *
236 * .. Scalar Arguments ..
237  CHARACTER uplo
238  INTEGER info, lda, ldb, lwork, n, nrhs
239 * ..
240 * .. Array Arguments ..
241  INTEGER ipiv( * )
242  DOUBLE PRECISION a( lda, * ), b( ldb, * ), e( * ), work( * )
243 * ..
244 *
245 * =====================================================================
246 *
247 * .. Local Scalars ..
248  LOGICAL lquery
249  INTEGER lwkopt
250 * ..
251 * .. External Functions ..
252  LOGICAL lsame
253  EXTERNAL lsame
254 * ..
255 * .. External Subroutines ..
256  EXTERNAL xerbla, dsytrf_rk, dsytrs_3
257 * ..
258 * .. Intrinsic Functions ..
259  INTRINSIC max
260 * ..
261 * .. Executable Statements ..
262 *
263 * Test the input parameters.
264 *
265  info = 0
266  lquery = ( lwork.EQ.-1 )
267  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
268  info = -1
269  ELSE IF( n.LT.0 ) THEN
270  info = -2
271  ELSE IF( nrhs.LT.0 ) THEN
272  info = -3
273  ELSE IF( lda.LT.max( 1, n ) ) THEN
274  info = -5
275  ELSE IF( ldb.LT.max( 1, n ) ) THEN
276  info = -9
277  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
278  info = -11
279  END IF
280 *
281  IF( info.EQ.0 ) THEN
282  IF( n.EQ.0 ) THEN
283  lwkopt = 1
284  ELSE
285  CALL dsytrf_rk( uplo, n, a, lda, e, ipiv, work, -1, info )
286  lwkopt = work(1)
287  END IF
288  work( 1 ) = lwkopt
289  END IF
290 *
291  IF( info.NE.0 ) THEN
292  CALL xerbla( 'DSYSV_RK ', -info )
293  RETURN
294  ELSE IF( lquery ) THEN
295  RETURN
296  END IF
297 *
298 * Compute the factorization A = P*U*D*(U**T)*(P**T) or
299 * A = P*U*D*(U**T)*(P**T).
300 *
301  CALL dsytrf_rk( uplo, n, a, lda, e, ipiv, work, lwork, info )
302 *
303  IF( info.EQ.0 ) THEN
304 *
305 * Solve the system A*X = B with BLAS3 solver, overwriting B with X.
306 *
307  CALL dsytrs_3( uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info )
308 *
309  END IF
310 *
311  work( 1 ) = lwkopt
312 *
313  RETURN
314 *
315 * End of DSYSV_RK
316 *
subroutine dsytrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
DSYTRS_3
Definition: dsytrs_3.f:167
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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:261
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