LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ ssysv_rk()

 subroutine ssysv_rk ( character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, integer lwork, integer info )

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

Purpose:
``` SSYSV_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.

SSYTRF_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 SSYTRS_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 REAL 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 SSYTRF_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 REAL 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 SSYTRF_RK. For more info see the description of DSYTRF_RK routine.``` [in,out] B ``` B is REAL 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 REAL 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 ssysv_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 REAL 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 REAL SROUNDUP_LWORK
251 EXTERNAL lsame, sroundup_lwork
252* ..
253* .. External Subroutines ..
254 EXTERNAL xerbla, ssytrf_rk, ssytrs_3
255* ..
256* .. Intrinsic Functions ..
257 INTRINSIC max
258* ..
259* .. Executable Statements ..
260*
261* Test the input parameters.
262*
263 info = 0
264 lquery = ( lwork.EQ.-1 )
265 IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
266 info = -1
267 ELSE IF( n.LT.0 ) THEN
268 info = -2
269 ELSE IF( nrhs.LT.0 ) THEN
270 info = -3
271 ELSE IF( lda.LT.max( 1, n ) ) THEN
272 info = -5
273 ELSE IF( ldb.LT.max( 1, n ) ) THEN
274 info = -9
275 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
276 info = -11
277 END IF
278*
279 IF( info.EQ.0 ) THEN
280 IF( n.EQ.0 ) THEN
281 lwkopt = 1
282 ELSE
283 CALL ssytrf_rk( uplo, n, a, lda, e, ipiv, work, -1, info )
284 lwkopt = int( work( 1 ) )
285 END IF
286 work( 1 ) = sroundup_lwork(lwkopt)
287 END IF
288*
289 IF( info.NE.0 ) THEN
290 CALL xerbla( 'SSYSV_RK ', -info )
291 RETURN
292 ELSE IF( lquery ) THEN
293 RETURN
294 END IF
295*
296* Compute the factorization A = P*U*D*(U**T)*(P**T) or
297* A = P*U*D*(U**T)*(P**T).
298*
299 CALL ssytrf_rk( uplo, n, a, lda, e, ipiv, work, lwork, info )
300*
301 IF( info.EQ.0 ) THEN
302*
303* Solve the system A*X = B with BLAS3 solver, overwriting B with X.
304*
305 CALL ssytrs_3( uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info )
306*
307 END IF
308*
309 work( 1 ) = sroundup_lwork(lwkopt)
310*
311 RETURN
312*
313* End of SSYSV_RK
314*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine ssytrf_rk(uplo, n, a, lda, e, ipiv, work, lwork, info)
SSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Ka...
Definition ssytrf_rk.f:259
subroutine ssytrs_3(uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info)
SSYTRS_3
Definition ssytrs_3.f:165
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
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