OpenRadioss 2025.1.11
OpenRadioss project
Loading...
Searching...
No Matches
complex
LAPACK » Symmetric Matrix » Linear Solve

Functions

subroutine csysv (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
  CSYSV computes the solution to system of linear equations A * X = B for SY matrices
subroutine csysv_aa (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
  CSYSV_AA computes the solution to system of linear equations A * X = B for SY matrices
subroutine csysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
  CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices
subroutine csysv_rook (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
  CSYSV_ROOK computes the solution to system of linear equations A * X = B for SY matrices
subroutine csysvx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info)
  CSYSVX computes the solution to system of linear equations A * X = B for SY matrices
subroutine csysvxx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
  CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices

Detailed Description

This is the group of complex solve driver functions for SY matrices

Function Documentation

◆ csysv()

subroutine csysv ( character uplo,
integer n,
integer nrhs,
complex, dimension( lda, * ) a,
integer lda,
integer, dimension( * ) ipiv,
complex, dimension( ldb, * ) b,
integer ldb,
complex, dimension( * ) work,
integer lwork,
integer info )

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

Download CSYSV + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> CSYSV computes the solution to a complex 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 diagonal pivoting method is used to factor A as
!>    A = U * D * U**T,  if UPLO = 'U', or
!>    A = L * D * L**T,  if UPLO = 'L',
!> where U (or L) is a product of permutation and unit upper (lower)
!> triangular matrices, and D is symmetric and block diagonal with
!> 1-by-1 and 2-by-2 diagonal blocks.  The factored form of A is then
!> used to solve the system of equations A * X = B.
!>
Parameters
[in]UPLO
!>          UPLO is CHARACTER*1
!>          = '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 COMPLEX 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, the block diagonal matrix D and the
!>          multipliers used to obtain the factor U or L from the
!>          factorization A = U*D*U**T or A = L*D*L**T as computed by
!>          CSYTRF.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[out]IPIV
!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D, as
!>          determined by CSYTRF.  If IPIV(k) > 0, then rows and columns
!>          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
!>          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
!>          then rows and columns k-1 and -IPIV(k) were interchanged and
!>          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
!>          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
!>          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
!>          diagonal block.
!>
[in,out]B
!>          B is COMPLEX 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 COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= 1, and for best performance
!>          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
!>          CSYTRF.
!>          for LWORK < N, TRS will be done with Level BLAS 2
!>          for LWORK >= N, TRS will be done with Level BLAS 3
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, 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 = -i, the i-th argument had an illegal value
!>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
!>               has been completed, but the block diagonal matrix D is
!>               exactly singular, so the solution could not be computed.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 169 of file csysv.f.

171*
172* -- LAPACK driver routine --
173* -- LAPACK is a software package provided by Univ. of Tennessee, --
174* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175*
176* .. Scalar Arguments ..
177 CHARACTER UPLO
178 INTEGER INFO, LDA, LDB, LWORK, N, NRHS
179* ..
180* .. Array Arguments ..
181 INTEGER IPIV( * )
182 COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
183* ..
184*
185* =====================================================================
186*
187* .. Local Scalars ..
188 LOGICAL LQUERY
189 INTEGER LWKOPT
190* ..
191* .. External Functions ..
192 LOGICAL LSAME
193 EXTERNAL lsame
194* ..
195* .. External Subroutines ..
196 EXTERNAL xerbla, csytrf, csytrs, csytrs2
197* ..
198* .. Intrinsic Functions ..
199 INTRINSIC max
200* ..
201* .. Executable Statements ..
202*
203* Test the input parameters.
204*
205 info = 0
206 lquery = ( lwork.EQ.-1 )
207 IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
208 info = -1
209 ELSE IF( n.LT.0 ) THEN
210 info = -2
211 ELSE IF( nrhs.LT.0 ) THEN
212 info = -3
213 ELSE IF( lda.LT.max( 1, n ) ) THEN
214 info = -5
215 ELSE IF( ldb.LT.max( 1, n ) ) THEN
216 info = -8
217 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
218 info = -10
219 END IF
220*
221 IF( info.EQ.0 ) THEN
222 IF( n.EQ.0 ) THEN
223 lwkopt = 1
224 ELSE
225 CALL csytrf( uplo, n, a, lda, ipiv, work, -1, info )
226 lwkopt = real( work(1) )
227 END IF
228 work( 1 ) = lwkopt
229 END IF
230*
231 IF( info.NE.0 ) THEN
232 CALL xerbla( 'CSYSV ', -info )
233 RETURN
234 ELSE IF( lquery ) THEN
235 RETURN
236 END IF
237*
238* Compute the factorization A = U*D*U**T or A = L*D*L**T.
239*
240 CALL csytrf( uplo, n, a, lda, ipiv, work, lwork, info )
241 IF( info.EQ.0 ) THEN
242*
243* Solve the system A*X = B, overwriting B with X.
244*
245 IF ( lwork.LT.n ) THEN
246*
247* Solve with TRS ( Use Level BLAS 2)
248*
249 CALL csytrs( uplo, n, nrhs, a, lda, ipiv, b, ldb, info )
250*
251 ELSE
252*
253* Solve with TRS2 ( Use Level BLAS 3)
254*
255 CALL csytrs2( uplo,n,nrhs,a,lda,ipiv,b,ldb,work,info )
256*
257 END IF
258*
259 END IF
260*
261 work( 1 ) = lwkopt
262*
263 RETURN
264*
265* End of CSYSV
266*
xerbla
subroutine xerbla(srname, info)
XERBLA
Definition xerbla.f:60
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:53
csytrs2
subroutine csytrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info)
CSYTRS2
Definition csytrs2.f:132
csytrs
subroutine csytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
CSYTRS
Definition csytrs.f:120
csytrf
subroutine csytrf(uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRF
Definition csytrf.f:182
max
#define max(a, b)
Definition macros.h:21

◆ csysv_aa()

subroutine csysv_aa ( character uplo,
integer n,
integer nrhs,
complex, dimension( lda, * ) a,
integer lda,
integer, dimension( * ) ipiv,
complex, dimension( ldb, * ) b,
integer ldb,
complex, dimension( * ) work,
integer lwork,
integer info )

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

Download CSYSV_AA + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> CSYSV computes the solution to a complex 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.
!>
!> Aasen's algorithm is used to factor A as
!>    A = U**T * T * U,  if UPLO = 'U', or
!>    A = L * T * L**T,  if UPLO = 'L',
!> where U (or L) is a product of permutation and unit upper (lower)
!> triangular matrices, and T is symmetric tridiagonal. The factored
!> form of A is then used to solve the system of equations A * X = B.
!>
Parameters
[in]UPLO
!>          UPLO is CHARACTER*1
!>          = '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 COMPLEX 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, the tridiagonal matrix T and the
!>          multipliers used to obtain the factor U or L from the
!>          factorization A = U**T*T*U or A = L*T*L**T as computed by
!>          CSYTRF.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[out]IPIV
!>          IPIV is INTEGER array, dimension (N)
!>          On exit, it contains the details of the interchanges, i.e.,
!>          the row and column k of A were interchanged with the
!>          row and column IPIV(k).
!>
[in,out]B
!>          B is COMPLEX 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 COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= MAX(2*N, 3*N-2), and for
!>          the best performance, LWORK >= max(1,N*NB), where NB is
!>          the optimal blocksize for CSYTRF_AA.
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, 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 = -i, the i-th argument had an illegal value
!>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
!>               has been completed, but the block diagonal matrix D is
!>               exactly singular, so the solution could not be computed.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 160 of file csysv_aa.f.

162*
163* -- LAPACK driver routine --
164* -- LAPACK is a software package provided by Univ. of Tennessee, --
165* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166*
167* .. Scalar Arguments ..
168 CHARACTER UPLO
169 INTEGER INFO, LDA, LDB, LWORK, N, NRHS
170* ..
171* .. Array Arguments ..
172 INTEGER IPIV( * )
173 COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
174* ..
175*
176* =====================================================================
177*
178* .. Local Scalars ..
179 LOGICAL LQUERY
180 INTEGER LWKOPT, LWKOPT_SYTRF, LWKOPT_SYTRS
181* ..
182* .. External Functions ..
183 LOGICAL LSAME
184 INTEGER ILAENV
185 EXTERNAL ilaenv, lsame
186* ..
187* .. External Subroutines ..
188 EXTERNAL xerbla, csytrf_aa, csytrs_aa
189* ..
190* .. Intrinsic Functions ..
191 INTRINSIC max
192* ..
193* .. Executable Statements ..
194*
195* Test the input parameters.
196*
197 info = 0
198 lquery = ( lwork.EQ.-1 )
199 IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
200 info = -1
201 ELSE IF( n.LT.0 ) THEN
202 info = -2
203 ELSE IF( nrhs.LT.0 ) THEN
204 info = -3
205 ELSE IF( lda.LT.max( 1, n ) ) THEN
206 info = -5
207 ELSE IF( ldb.LT.max( 1, n ) ) THEN
208 info = -8
209 ELSE IF( lwork.LT.max(2*n, 3*n-2) .AND. .NOT.lquery ) THEN
210 info = -10
211 END IF
212*
213 IF( info.EQ.0 ) THEN
214 CALL csytrf_aa( uplo, n, a, lda, ipiv, work, -1, info )
215 lwkopt_sytrf = int( work(1) )
216 CALL csytrs_aa( uplo, n, nrhs, a, lda, ipiv, b, ldb, work,
217 $ -1, info )
218 lwkopt_sytrs = int( work(1) )
219 lwkopt = max( lwkopt_sytrf, lwkopt_sytrs )
220 work( 1 ) = lwkopt
221 END IF
222*
223 IF( info.NE.0 ) THEN
224 CALL xerbla( 'CSYSV_AA ', -info )
225 RETURN
226 ELSE IF( lquery ) THEN
227 RETURN
228 END IF
229*
230* Compute the factorization A = U**T*T*U or A = L*T*L**T.
231*
232 CALL csytrf_aa( uplo, n, a, lda, ipiv, work, lwork, info )
233 IF( info.EQ.0 ) THEN
234*
235* Solve the system A*X = B, overwriting B with X.
236*
237 CALL csytrs_aa( uplo, n, nrhs, a, lda, ipiv, b, ldb, work,
238 $ lwork, info )
239*
240 END IF
241*
242 work( 1 ) = lwkopt
243*
244 RETURN
245*
246* End of CSYSV_AA
247*
ilaenv
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
csytrf_aa
subroutine csytrf_aa(uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRF_AA
Definition csytrf_aa.f:132
csytrs_aa
subroutine csytrs_aa(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
CSYTRS_AA
Definition csytrs_aa.f:131

◆ csysv_rk()

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

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

Download CSYSV_RK + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!> CSYSV_RK computes the solution to a complex 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.
!>
!> CSYTRF_RK is called to compute the factorization of a complex
!> symmetric matrix.  The factored form of A is then used to solve
!> the system of equations A * X = B by calling BLAS3 routine CSYTRS_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 COMPLEX 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 CSYTRF_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 CSYTRF_RK routine.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[out]E
!>          E is COMPLEX array, dimension (N)
!>          On exit, contains the output computed by the factorization
!>          routine CSYTRF_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 CSYTRF_RK routine.
!>
[out]IPIV
!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D,
!>          as determined by CSYTRF_RK.
!>
!>          For more info see the description of CSYTRF_RK routine.
!>
[in,out]B
!>          B is COMPLEX 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 COMPLEX 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 CSYTRF_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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 csysv_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 COMPLEX 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, csytrf_rk, csytrs_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 csytrf_rk( uplo, n, a, lda, e, ipiv, work, -1, info )
283 lwkopt = real( work(1) )
284 END IF
285 work( 1 ) = lwkopt
286 END IF
287*
288 IF( info.NE.0 ) THEN
289 CALL xerbla( 'CSYSV_RK ', -info )
290 RETURN
291 ELSE IF( lquery ) THEN
292 RETURN
293 END IF
294*
295* Compute the factorization A = U*D*U**T or A = L*D*L**T.
296*
297 CALL csytrf_rk( uplo, n, a, lda, e, ipiv, work, lwork, info )
298*
299 IF( info.EQ.0 ) THEN
300*
301* Solve the system A*X = B with BLAS3 solver, overwriting B with X.
302*
303 CALL csytrs_3( uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info )
304*
305 END IF
306*
307 work( 1 ) = lwkopt
308*
309 RETURN
310*
311* End of CSYSV_RK
312*
csytrf_rk
subroutine csytrf_rk(uplo, n, a, lda, e, ipiv, work, lwork, info)
CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch...
Definition csytrf_rk.f:259
csytrs_3
subroutine csytrs_3(uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info)
CSYTRS_3
Definition csytrs_3.f:165

◆ csysv_rook()

subroutine csysv_rook ( character uplo,
integer n,
integer nrhs,
complex, dimension( lda, * ) a,
integer lda,
integer, dimension( * ) ipiv,
complex, dimension( ldb, * ) b,
integer ldb,
complex, dimension( * ) work,
integer lwork,
integer info )

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

Download CSYSV_ROOK + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> CSYSV_ROOK computes the solution to a complex 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 diagonal pivoting method is used to factor A as
!>    A = U * D * U**T,  if UPLO = 'U', or
!>    A = L * D * L**T,  if UPLO = 'L',
!> where U (or L) is a product of permutation and unit upper (lower)
!> triangular matrices, and D is symmetric and block diagonal with
!> 1-by-1 and 2-by-2 diagonal blocks.
!>
!> CSYTRF_ROOK is called to compute the factorization of a complex
!> symmetric matrix A using the bounded Bunch-Kaufman () diagonal
!> pivoting method.
!>
!> The factored form of A is then used to solve the system
!> of equations A * X = B by calling CSYTRS_ROOK.
!>
Parameters
[in]UPLO
!>          UPLO is CHARACTER*1
!>          = '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 COMPLEX 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, the block diagonal matrix D and the
!>          multipliers used to obtain the factor U or L from the
!>          factorization A = U*D*U**T or A = L*D*L**T as computed by
!>          CSYTRF_ROOK.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[out]IPIV
!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D,
!>          as determined by CSYTRF_ROOK.
!>
!>          If UPLO = 'U':
!>               If IPIV(k) > 0, then rows and columns k and IPIV(k)
!>               were interchanged and D(k,k) is a 1-by-1 diagonal block.
!>
!>               If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
!>               columns k and -IPIV(k) were interchanged and rows and
!>               columns k-1 and -IPIV(k-1) were inerchaged,
!>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
!>
!>          If UPLO = 'L':
!>               If IPIV(k) > 0, then rows and columns k and IPIV(k)
!>               were interchanged and D(k,k) is a 1-by-1 diagonal block.
!>
!>               If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
!>               columns k and -IPIV(k) were interchanged and rows and
!>               columns k+1 and -IPIV(k+1) were inerchaged,
!>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
!>
[in,out]B
!>          B is COMPLEX 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 COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= 1, and for best performance
!>          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
!>          CSYTRF_ROOK.
!>
!>          TRS will be done with Level 2 BLAS
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, 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 = -i, the i-th argument had an illegal value
!>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
!>               has been completed, but the block diagonal matrix D is
!>               exactly singular, so the solution could not be computed.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
!>
!>   April 2012, 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 202 of file csysv_rook.f.

204*
205* -- LAPACK driver routine --
206* -- LAPACK is a software package provided by Univ. of Tennessee, --
207* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208*
209* .. Scalar Arguments ..
210 CHARACTER UPLO
211 INTEGER INFO, LDA, LDB, LWORK, N, NRHS
212* ..
213* .. Array Arguments ..
214 INTEGER IPIV( * )
215 COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
216* ..
217*
218* =====================================================================
219*
220* .. Local Scalars ..
221 LOGICAL LQUERY
222 INTEGER LWKOPT
223* ..
224* .. External Functions ..
225 LOGICAL LSAME
226 EXTERNAL lsame
227* ..
228* .. External Subroutines ..
229 EXTERNAL xerbla, csytrf_rook, csytrs_rook
230* ..
231* .. Intrinsic Functions ..
232 INTRINSIC max
233* ..
234* .. Executable Statements ..
235*
236* Test the input parameters.
237*
238 info = 0
239 lquery = ( lwork.EQ.-1 )
240 IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
241 info = -1
242 ELSE IF( n.LT.0 ) THEN
243 info = -2
244 ELSE IF( nrhs.LT.0 ) THEN
245 info = -3
246 ELSE IF( lda.LT.max( 1, n ) ) THEN
247 info = -5
248 ELSE IF( ldb.LT.max( 1, n ) ) THEN
249 info = -8
250 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
251 info = -10
252 END IF
253*
254 IF( info.EQ.0 ) THEN
255 IF( n.EQ.0 ) THEN
256 lwkopt = 1
257 ELSE
258 CALL csytrf_rook( uplo, n, a, lda, ipiv, work, -1, info )
259 lwkopt = real( work(1) )
260 END IF
261 work( 1 ) = lwkopt
262 END IF
263*
264 IF( info.NE.0 ) THEN
265 CALL xerbla( 'CSYSV_ROOK ', -info )
266 RETURN
267 ELSE IF( lquery ) THEN
268 RETURN
269 END IF
270*
271* Compute the factorization A = U*D*U**T or A = L*D*L**T.
272*
273 CALL csytrf_rook( uplo, n, a, lda, ipiv, work, lwork, info )
274 IF( info.EQ.0 ) THEN
275*
276* Solve the system A*X = B, overwriting B with X.
277*
278* Solve with TRS_ROOK ( Use Level 2 BLAS)
279*
280 CALL csytrs_rook( uplo, n, nrhs, a, lda, ipiv, b, ldb, info )
281*
282 END IF
283*
284 work( 1 ) = lwkopt
285*
286 RETURN
287*
288* End of CSYSV_ROOK
289*
csytrs_rook
subroutine csytrs_rook(uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
CSYTRS_ROOK
Definition csytrs_rook.f:136
csytrf_rook
subroutine csytrf_rook(uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRF_ROOK
Definition csytrf_rook.f:208

◆ csysvx()

subroutine csysvx ( character fact,
character uplo,
integer n,
integer nrhs,
complex, dimension( lda, * ) a,
integer lda,
complex, dimension( ldaf, * ) af,
integer ldaf,
integer, dimension( * ) ipiv,
complex, dimension( ldb, * ) b,
integer ldb,
complex, dimension( ldx, * ) x,
integer ldx,
real rcond,
real, dimension( * ) ferr,
real, dimension( * ) berr,
complex, dimension( * ) work,
integer lwork,
real, dimension( * ) rwork,
integer info )

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

Download CSYSVX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> CSYSVX uses the diagonal pivoting factorization to compute the
!> solution to a complex 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.
!>
!> Error bounds on the solution and a condition estimate are also
!> provided.
!>
Description:
!>
!> The following steps are performed:
!>
!> 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
!>    The form of the factorization is
!>       A = U * D * U**T,  if UPLO = 'U', or
!>       A = L * D * L**T,  if UPLO = 'L',
!>    where U (or L) is a product of permutation and unit upper (lower)
!>    triangular matrices, and D is symmetric and block diagonal with
!>    1-by-1 and 2-by-2 diagonal blocks.
!>
!> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
!>    returns with INFO = i. Otherwise, the factored form of A is used
!>    to estimate the condition number of the matrix A.  If the
!>    reciprocal of the condition number is less than machine precision,
!>    INFO = N+1 is returned as a warning, but the routine still goes on
!>    to solve for X and compute error bounds as described below.
!>
!> 3. The system of equations is solved for X using the factored form
!>    of A.
!>
!> 4. Iterative refinement is applied to improve the computed solution
!>    matrix and calculate error bounds and backward error estimates
!>    for it.
!>
Parameters
[in]FACT
!>          FACT is CHARACTER*1
!>          Specifies whether or not the factored form of A has been
!>          supplied on entry.
!>          = 'F':  On entry, AF and IPIV contain the factored form
!>                  of A.  A, AF and IPIV will not be modified.
!>          = 'N':  The matrix A will be copied to AF and factored.
!>
[in]UPLO
!>          UPLO is CHARACTER*1
!>          = '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 matrices B and X.  NRHS >= 0.
!>
[in]A
!>          A is COMPLEX array, dimension (LDA,N)
!>          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.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[in,out]AF
!>          AF is COMPLEX array, dimension (LDAF,N)
!>          If FACT = 'F', then AF is an input argument and on entry
!>          contains the block diagonal matrix D and the multipliers used
!>          to obtain the factor U or L from the factorization
!>          A = U*D*U**T or A = L*D*L**T as computed by CSYTRF.
!>
!>          If FACT = 'N', then AF is an output argument and on exit
!>          returns the block diagonal matrix D and the multipliers used
!>          to obtain the factor U or L from the factorization
!>          A = U*D*U**T or A = L*D*L**T.
!>
[in]LDAF
!>          LDAF is INTEGER
!>          The leading dimension of the array AF.  LDAF >= max(1,N).
!>
[in,out]IPIV
!>          IPIV is INTEGER array, dimension (N)
!>          If FACT = 'F', then IPIV is an input argument and on entry
!>          contains details of the interchanges and the block structure
!>          of D, as determined by CSYTRF.
!>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
!>          interchanged and D(k,k) is a 1-by-1 diagonal block.
!>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
!>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
!>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
!>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
!>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
!>
!>          If FACT = 'N', then IPIV is an output argument and on exit
!>          contains details of the interchanges and the block structure
!>          of D, as determined by CSYTRF.
!>
[in]B
!>          B is COMPLEX array, dimension (LDB,NRHS)
!>          The N-by-NRHS right hand side matrix B.
!>
[in]LDB
!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!>
[out]X
!>          X is COMPLEX array, dimension (LDX,NRHS)
!>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
!>
[in]LDX
!>          LDX is INTEGER
!>          The leading dimension of the array X.  LDX >= max(1,N).
!>
[out]RCOND
!>          RCOND is REAL
!>          The estimate of the reciprocal condition number of the matrix
!>          A.  If RCOND is less than the machine precision (in
!>          particular, if RCOND = 0), the matrix is singular to working
!>          precision.  This condition is indicated by a return code of
!>          INFO > 0.
!>
[out]FERR
!>          FERR is REAL array, dimension (NRHS)
!>          The estimated forward error bound for each solution vector
!>          X(j) (the j-th column of the solution matrix X).
!>          If XTRUE is the true solution corresponding to X(j), FERR(j)
!>          is an estimated upper bound for the magnitude of the largest
!>          element in (X(j) - XTRUE) divided by the magnitude of the
!>          largest element in X(j).  The estimate is as reliable as
!>          the estimate for RCOND, and is almost always a slight
!>          overestimate of the true error.
!>
[out]BERR
!>          BERR is REAL array, dimension (NRHS)
!>          The componentwise relative backward error of each solution
!>          vector X(j) (i.e., the smallest relative change in
!>          any element of A or B that makes X(j) an exact solution).
!>
[out]WORK
!>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= max(1,2*N), and for best
!>          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where
!>          NB is the optimal blocksize for CSYTRF.
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA.
!>
[out]RWORK
!>          RWORK is REAL array, dimension (N)
!>
[out]INFO
!>          INFO is INTEGER
!>          = 0: successful exit
!>          < 0: if INFO = -i, the i-th argument had an illegal value
!>          > 0: if INFO = i, and i is
!>                <= N:  D(i,i) is exactly zero.  The factorization
!>                       has been completed but the factor D is exactly
!>                       singular, so the solution and error bounds could
!>                       not be computed. RCOND = 0 is returned.
!>                = N+1: D is nonsingular, but RCOND is less than machine
!>                       precision, meaning that the matrix is singular
!>                       to working precision.  Nevertheless, the
!>                       solution and error bounds are computed because
!>                       there are a number of situations where the
!>                       computed solution can be more accurate than the
!>                       value of RCOND would suggest.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 282 of file csysvx.f.

285*
286* -- LAPACK driver routine --
287* -- LAPACK is a software package provided by Univ. of Tennessee, --
288* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
289*
290* .. Scalar Arguments ..
291 CHARACTER FACT, UPLO
292 INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
293 REAL RCOND
294* ..
295* .. Array Arguments ..
296 INTEGER IPIV( * )
297 REAL BERR( * ), FERR( * ), RWORK( * )
298 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
299 $ WORK( * ), X( LDX, * )
300* ..
301*
302* =====================================================================
303*
304* .. Parameters ..
305 REAL ZERO
306 parameter( zero = 0.0e+0 )
307* ..
308* .. Local Scalars ..
309 LOGICAL LQUERY, NOFACT
310 INTEGER LWKOPT, NB
311 REAL ANORM
312* ..
313* .. External Functions ..
314 LOGICAL LSAME
315 INTEGER ILAENV
316 REAL CLANSY, SLAMCH
317 EXTERNAL ilaenv, lsame, clansy, slamch
318* ..
319* .. External Subroutines ..
320 EXTERNAL clacpy, csycon, csyrfs, csytrf, csytrs, xerbla
321* ..
322* .. Intrinsic Functions ..
323 INTRINSIC max
324* ..
325* .. Executable Statements ..
326*
327* Test the input parameters.
328*
329 info = 0
330 nofact = lsame( fact, 'N' )
331 lquery = ( lwork.EQ.-1 )
332 IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
333 info = -1
334 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
335 $ THEN
336 info = -2
337 ELSE IF( n.LT.0 ) THEN
338 info = -3
339 ELSE IF( nrhs.LT.0 ) THEN
340 info = -4
341 ELSE IF( lda.LT.max( 1, n ) ) THEN
342 info = -6
343 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
344 info = -8
345 ELSE IF( ldb.LT.max( 1, n ) ) THEN
346 info = -11
347 ELSE IF( ldx.LT.max( 1, n ) ) THEN
348 info = -13
349 ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
350 info = -18
351 END IF
352*
353 IF( info.EQ.0 ) THEN
354 lwkopt = max( 1, 2*n )
355 IF( nofact ) THEN
356 nb = ilaenv( 1, 'CSYTRF', uplo, n, -1, -1, -1 )
357 lwkopt = max( lwkopt, n*nb )
358 END IF
359 work( 1 ) = lwkopt
360 END IF
361*
362 IF( info.NE.0 ) THEN
363 CALL xerbla( 'CSYSVX', -info )
364 RETURN
365 ELSE IF( lquery ) THEN
366 RETURN
367 END IF
368*
369 IF( nofact ) THEN
370*
371* Compute the factorization A = U*D*U**T or A = L*D*L**T.
372*
373 CALL clacpy( uplo, n, n, a, lda, af, ldaf )
374 CALL csytrf( uplo, n, af, ldaf, ipiv, work, lwork, info )
375*
376* Return if INFO is non-zero.
377*
378 IF( info.GT.0 )THEN
379 rcond = zero
380 RETURN
381 END IF
382 END IF
383*
384* Compute the norm of the matrix A.
385*
386 anorm = clansy( 'I', uplo, n, a, lda, rwork )
387*
388* Compute the reciprocal of the condition number of A.
389*
390 CALL csycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work, info )
391*
392* Compute the solution vectors X.
393*
394 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
395 CALL csytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
396*
397* Use iterative refinement to improve the computed solutions and
398* compute error bounds and backward error estimates for them.
399*
400 CALL csyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
401 $ ldx, ferr, berr, work, rwork, info )
402*
403* Set INFO = N+1 if the matrix is singular to working precision.
404*
405 IF( rcond.LT.slamch( 'Epsilon' ) )
406 $ info = n + 1
407*
408 work( 1 ) = lwkopt
409*
410 RETURN
411*
412* End of CSYSVX
413*
clacpy
subroutine clacpy(uplo, m, n, a, lda, b, ldb)
CLACPY copies all or part of one two-dimensional array to another.
Definition clacpy.f:103
clansy
real function clansy(norm, uplo, n, a, lda, work)
CLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition clansy.f:123
csycon
subroutine csycon(uplo, n, a, lda, ipiv, anorm, rcond, work, info)
CSYCON
Definition csycon.f:125
csyrfs
subroutine csyrfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CSYRFS
Definition csyrfs.f:192
slamch
real function slamch(cmach)
SLAMCH
Definition slamch.f:68

◆ csysvxx()

subroutine csysvxx ( character fact,
character uplo,
integer n,
integer nrhs,
complex, dimension( lda, * ) a,
integer lda,
complex, dimension( ldaf, * ) af,
integer ldaf,
integer, dimension( * ) ipiv,
character equed,
real, dimension( * ) s,
complex, dimension( ldb, * ) b,
integer ldb,
complex, dimension( ldx, * ) x,
integer ldx,
real rcond,
real rpvgrw,
real, dimension( * ) berr,
integer n_err_bnds,
real, dimension( nrhs, * ) err_bnds_norm,
real, dimension( nrhs, * ) err_bnds_comp,
integer nparams,
real, dimension( * ) params,
complex, dimension( * ) work,
real, dimension( * ) rwork,
integer info )

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

Download CSYSVXX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!>    CSYSVXX uses the diagonal pivoting factorization to compute the
!>    solution to a complex 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.
!>
!>    If requested, both normwise and maximum componentwise error bounds
!>    are returned. CSYSVXX will return a solution with a tiny
!>    guaranteed error (O(eps) where eps is the working machine
!>    precision) unless the matrix is very ill-conditioned, in which
!>    case a warning is returned. Relevant condition numbers also are
!>    calculated and returned.
!>
!>    CSYSVXX accepts user-provided factorizations and equilibration
!>    factors; see the definitions of the FACT and EQUED options.
!>    Solving with refinement and using a factorization from a previous
!>    CSYSVXX call will also produce a solution with either O(eps)
!>    errors or warnings, but we cannot make that claim for general
!>    user-provided factorizations and equilibration factors if they
!>    differ from what CSYSVXX would itself produce.
!>
Description:
!>
!>    The following steps are performed:
!>
!>    1. If FACT = 'E', real scaling factors are computed to equilibrate
!>    the system:
!>
!>      diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B
!>
!>    Whether or not the system will be equilibrated depends on the
!>    scaling of the matrix A, but if equilibration is used, A is
!>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
!>
!>    2. If FACT = 'N' or 'E', the LU decomposition is used to factor
!>    the matrix A (after equilibration if FACT = 'E') as
!>
!>       A = U * D * U**T,  if UPLO = 'U', or
!>       A = L * D * L**T,  if UPLO = 'L',
!>
!>    where U (or L) is a product of permutation and unit upper (lower)
!>    triangular matrices, and D is symmetric and block diagonal with
!>    1-by-1 and 2-by-2 diagonal blocks.
!>
!>    3. If some D(i,i)=0, so that D is exactly singular, then the
!>    routine returns with INFO = i. Otherwise, the factored form of A
!>    is used to estimate the condition number of the matrix A (see
!>    argument RCOND).  If the reciprocal of the condition number is
!>    less than machine precision, the routine still goes on to solve
!>    for X and compute error bounds as described below.
!>
!>    4. The system of equations is solved for X using the factored form
!>    of A.
!>
!>    5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
!>    the routine will use iterative refinement to try to get a small
!>    error and error bounds.  Refinement calculates the residual to at
!>    least twice the working precision.
!>
!>    6. If equilibration was used, the matrix X is premultiplied by
!>    diag(R) so that it solves the original system before
!>    equilibration.
!> 
!>     Some optional parameters are bundled in the PARAMS array.  These
!>     settings determine how refinement is performed, but often the
!>     defaults are acceptable.  If the defaults are acceptable, users
!>     can pass NPARAMS = 0 which prevents the source code from accessing
!>     the PARAMS argument.
!>
Parameters
[in]FACT
!>          FACT is CHARACTER*1
!>     Specifies whether or not the factored form of the matrix A is
!>     supplied on entry, and if not, whether the matrix A should be
!>     equilibrated before it is factored.
!>       = 'F':  On entry, AF and IPIV contain the factored form of A.
!>               If EQUED is not 'N', the matrix A has been
!>               equilibrated with scaling factors given by S.
!>               A, AF, and IPIV are not modified.
!>       = 'N':  The matrix A will be copied to AF and factored.
!>       = 'E':  The matrix A will be equilibrated if necessary, then
!>               copied to AF and factored.
!>
[in]UPLO
!>          UPLO is CHARACTER*1
!>       = '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 matrices B and X.  NRHS >= 0.
!>
[in,out]A
!>          A is COMPLEX array, dimension (LDA,N)
!>     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 FACT = 'E' and EQUED = 'Y', A is overwritten by
!>     diag(S)*A*diag(S).
!>
[in]LDA
!>          LDA is INTEGER
!>     The leading dimension of the array A.  LDA >= max(1,N).
!>
[in,out]AF
!>          AF is COMPLEX array, dimension (LDAF,N)
!>     If FACT = 'F', then AF is an input argument and on entry
!>     contains the block diagonal matrix D and the multipliers
!>     used to obtain the factor U or L from the factorization A =
!>     U*D*U**T or A = L*D*L**T as computed by CSYTRF.
!>
!>     If FACT = 'N', then AF is an output argument and on exit
!>     returns the block diagonal matrix D and the multipliers
!>     used to obtain the factor U or L from the factorization A =
!>     U*D*U**T or A = L*D*L**T.
!>
[in]LDAF
!>          LDAF is INTEGER
!>     The leading dimension of the array AF.  LDAF >= max(1,N).
!>
[in,out]IPIV
!>          IPIV is INTEGER array, dimension (N)
!>     If FACT = 'F', then IPIV is an input argument and on entry
!>     contains details of the interchanges and the block
!>     structure of D, as determined by CSYTRF.  If IPIV(k) > 0,
!>     then rows and columns k and IPIV(k) were interchanged and
!>     D(k,k) is a 1-by-1 diagonal block.  If UPLO = 'U' and
!>     IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
!>     -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
!>     diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
!>     then rows and columns k+1 and -IPIV(k) were interchanged
!>     and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
!>
!>     If FACT = 'N', then IPIV is an output argument and on exit
!>     contains details of the interchanges and the block
!>     structure of D, as determined by CSYTRF.
!>
[in,out]EQUED
!>          EQUED is CHARACTER*1
!>     Specifies the form of equilibration that was done.
!>       = 'N':  No equilibration (always true if FACT = 'N').
!>       = 'Y':  Both row and column equilibration, i.e., A has been
!>               replaced by diag(S) * A * diag(S).
!>     EQUED is an input argument if FACT = 'F'; otherwise, it is an
!>     output argument.
!>
[in,out]S
!>          S is REAL array, dimension (N)
!>     The scale factors for A.  If EQUED = 'Y', A is multiplied on
!>     the left and right by diag(S).  S is an input argument if FACT =
!>     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
!>     = 'Y', each element of S must be positive.  If S is output, each
!>     element of S is a power of the radix. If S is input, each element
!>     of S should be a power of the radix to ensure a reliable solution
!>     and error estimates. Scaling by powers of the radix does not cause
!>     rounding errors unless the result underflows or overflows.
!>     Rounding errors during scaling lead to refining with a matrix that
!>     is not equivalent to the input matrix, producing error estimates
!>     that may not be reliable.
!>
[in,out]B
!>          B is COMPLEX array, dimension (LDB,NRHS)
!>     On entry, the N-by-NRHS right hand side matrix B.
!>     On exit,
!>     if EQUED = 'N', B is not modified;
!>     if EQUED = 'Y', B is overwritten by diag(S)*B;
!>
[in]LDB
!>          LDB is INTEGER
!>     The leading dimension of the array B.  LDB >= max(1,N).
!>
[out]X
!>          X is COMPLEX array, dimension (LDX,NRHS)
!>     If INFO = 0, the N-by-NRHS solution matrix X to the original
!>     system of equations.  Note that A and B are modified on exit if
!>     EQUED .ne. 'N', and the solution to the equilibrated system is
!>     inv(diag(S))*X.
!>
[in]LDX
!>          LDX is INTEGER
!>     The leading dimension of the array X.  LDX >= max(1,N).
!>
[out]RCOND
!>          RCOND is REAL
!>     Reciprocal scaled condition number.  This is an estimate of the
!>     reciprocal Skeel condition number of the matrix A after
!>     equilibration (if done).  If this is less than the machine
!>     precision (in particular, if it is zero), the matrix is singular
!>     to working precision.  Note that the error may still be small even
!>     if this number is very small and the matrix appears ill-
!>     conditioned.
!>
[out]RPVGRW
!>          RPVGRW is REAL
!>     Reciprocal pivot growth.  On exit, this contains the reciprocal
!>     pivot growth factor norm(A)/norm(U). The 
!>     norm is used.  If this is much less than 1, then the stability of
!>     the LU factorization of the (equilibrated) matrix A could be poor.
!>     This also means that the solution X, estimated condition numbers,
!>     and error bounds could be unreliable. If factorization fails with
!>     0<INFO<=N, then this contains the reciprocal pivot growth factor
!>     for the leading INFO columns of A.
!>
[out]BERR
!>          BERR is REAL array, dimension (NRHS)
!>     Componentwise relative backward error.  This is the
!>     componentwise relative backward error of each solution vector X(j)
!>     (i.e., the smallest relative change in any element of A or B that
!>     makes X(j) an exact solution).
!>
[in]N_ERR_BNDS
!>          N_ERR_BNDS is INTEGER
!>     Number of error bounds to return for each right hand side
!>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
!>     ERR_BNDS_COMP below.
!>
[out]ERR_BNDS_NORM
!>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
!>     For each right-hand side, this array contains information about
!>     various error bounds and condition numbers corresponding to the
!>     normwise relative error, which is defined as follows:
!>
!>     Normwise relative error in the ith solution vector:
!>             max_j (abs(XTRUE(j,i) - X(j,i)))
!>            ------------------------------
!>                  max_j abs(X(j,i))
!>
!>     The array is indexed by the type of error information as described
!>     below. There currently are up to three pieces of information
!>     returned.
!>
!>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
!>     right-hand side.
!>
!>     The second index in ERR_BNDS_NORM(:,err) contains the following
!>     three fields:
!>     err = 1  boolean. Trust the answer if the
!>              reciprocal condition number is less than the threshold
!>              sqrt(n) * slamch('Epsilon').
!>
!>     err = 2  error bound: The estimated forward error,
!>              almost certainly within a factor of 10 of the true error
!>              so long as the next entry is greater than the threshold
!>              sqrt(n) * slamch('Epsilon'). This error bound should only
!>              be trusted if the previous boolean is true.
!>
!>     err = 3  Reciprocal condition number: Estimated normwise
!>              reciprocal condition number.  Compared with the threshold
!>              sqrt(n) * slamch('Epsilon') to determine if the error
!>              estimate is . These reciprocal condition
!>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
!>              appropriately scaled matrix Z.
!>              Let Z = S*A, where S scales each row by a power of the
!>              radix so all absolute row sums of Z are approximately 1.
!>
!>     See Lapack Working Note 165 for further details and extra
!>     cautions.
!>
[out]ERR_BNDS_COMP
!>          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
!>     For each right-hand side, this array contains information about
!>     various error bounds and condition numbers corresponding to the
!>     componentwise relative error, which is defined as follows:
!>
!>     Componentwise relative error in the ith solution vector:
!>                    abs(XTRUE(j,i) - X(j,i))
!>             max_j ----------------------
!>                         abs(X(j,i))
!>
!>     The array is indexed by the right-hand side i (on which the
!>     componentwise relative error depends), and the type of error
!>     information as described below. There currently are up to three
!>     pieces of information returned for each right-hand side. If
!>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
!>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
!>     the first (:,N_ERR_BNDS) entries are returned.
!>
!>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
!>     right-hand side.
!>
!>     The second index in ERR_BNDS_COMP(:,err) contains the following
!>     three fields:
!>     err = 1  boolean. Trust the answer if the
!>              reciprocal condition number is less than the threshold
!>              sqrt(n) * slamch('Epsilon').
!>
!>     err = 2  error bound: The estimated forward error,
!>              almost certainly within a factor of 10 of the true error
!>              so long as the next entry is greater than the threshold
!>              sqrt(n) * slamch('Epsilon'). This error bound should only
!>              be trusted if the previous boolean is true.
!>
!>     err = 3  Reciprocal condition number: Estimated componentwise
!>              reciprocal condition number.  Compared with the threshold
!>              sqrt(n) * slamch('Epsilon') to determine if the error
!>              estimate is . These reciprocal condition
!>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
!>              appropriately scaled matrix Z.
!>              Let Z = S*(A*diag(x)), where x is the solution for the
!>              current right-hand side and S scales each row of
!>              A*diag(x) by a power of the radix so all absolute row
!>              sums of Z are approximately 1.
!>
!>     See Lapack Working Note 165 for further details and extra
!>     cautions.
!>
[in]NPARAMS
!>          NPARAMS is INTEGER
!>     Specifies the number of parameters set in PARAMS.  If <= 0, the
!>     PARAMS array is never referenced and default values are used.
!>
[in,out]PARAMS
!>          PARAMS is REAL array, dimension NPARAMS
!>     Specifies algorithm parameters.  If an entry is < 0.0, then
!>     that entry will be filled with default value used for that
!>     parameter.  Only positions up to NPARAMS are accessed; defaults
!>     are used for higher-numbered parameters.
!>
!>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
!>            refinement or not.
!>         Default: 1.0
!>            = 0.0:  No refinement is performed, and no error bounds are
!>                    computed.
!>            = 1.0:  Use the double-precision refinement algorithm,
!>                    possibly with doubled-single computations if the
!>                    compilation environment does not support DOUBLE
!>                    PRECISION.
!>              (other values are reserved for future use)
!>
!>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
!>            computations allowed for refinement.
!>         Default: 10
!>         Aggressive: Set to 100 to permit convergence using approximate
!>                     factorizations or factorizations other than LU. If
!>                     the factorization uses a technique other than
!>                     Gaussian elimination, the guarantees in
!>                     err_bnds_norm and err_bnds_comp may no longer be
!>                     trustworthy.
!>
!>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
!>            will attempt to find a solution with small componentwise
!>            relative error in the double-precision algorithm.  Positive
!>            is true, 0.0 is false.
!>         Default: 1.0 (attempt componentwise convergence)
!>
[out]WORK
!>          WORK is COMPLEX array, dimension (2*N)
!>
[out]RWORK
!>          RWORK is REAL array, dimension (2*N)
!>
[out]INFO
!>          INFO is INTEGER
!>       = 0:  Successful exit. The solution to every right-hand side is
!>         guaranteed.
!>       < 0:  If INFO = -i, the i-th argument had an illegal value
!>       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
!>         has been completed, but the factor U is exactly singular, so
!>         the solution and error bounds could not be computed. RCOND = 0
!>         is returned.
!>       = N+J: The solution corresponding to the Jth right-hand side is
!>         not guaranteed. The solutions corresponding to other right-
!>         hand sides K with K > J may not be guaranteed as well, but
!>         only the first such right-hand side is reported. If a small
!>         componentwise error is not requested (PARAMS(3) = 0.0) then
!>         the Jth right-hand side is the first with a normwise error
!>         bound that is not guaranteed (the smallest J such
!>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
!>         the Jth right-hand side is the first with either a normwise or
!>         componentwise error bound that is not guaranteed (the smallest
!>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
!>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
!>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
!>         about all of the right-hand sides check ERR_BNDS_NORM or
!>         ERR_BNDS_COMP.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 505 of file csysvxx.f.

509*
510* -- LAPACK driver routine --
511* -- LAPACK is a software package provided by Univ. of Tennessee, --
512* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
513*
514* .. Scalar Arguments ..
515 CHARACTER EQUED, FACT, UPLO
516 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
517 $ N_ERR_BNDS
518 REAL RCOND, RPVGRW
519* ..
520* .. Array Arguments ..
521 INTEGER IPIV( * )
522 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
523 $ X( LDX, * ), WORK( * )
524 REAL S( * ), PARAMS( * ), BERR( * ),
525 $ ERR_BNDS_NORM( NRHS, * ),
526 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
527* ..
528*
529* ==================================================================
530*
531* .. Parameters ..
532 REAL ZERO, ONE
533 parameter( zero = 0.0e+0, one = 1.0e+0 )
534 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
535 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
536 INTEGER CMP_ERR_I, PIV_GROWTH_I
537 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
538 $ berr_i = 3 )
539 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
540 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
541 $ piv_growth_i = 9 )
542* ..
543* .. Local Scalars ..
544 LOGICAL EQUIL, NOFACT, RCEQU
545 INTEGER INFEQU, J
546 REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
547* ..
548* .. External Functions ..
549 EXTERNAL lsame, slamch, cla_syrpvgrw
550 LOGICAL LSAME
551 REAL SLAMCH, CLA_SYRPVGRW
552* ..
553* .. External Subroutines ..
554 EXTERNAL csyequb, csytrf, csytrs, clacpy,
555 $ claqsy, xerbla, clascl2, csyrfsx
556* ..
557* .. Intrinsic Functions ..
558 INTRINSIC max, min
559* ..
560* .. Executable Statements ..
561*
562 info = 0
563 nofact = lsame( fact, 'N' )
564 equil = lsame( fact, 'E' )
565 smlnum = slamch( 'Safe minimum' )
566 bignum = one / smlnum
567 IF( nofact .OR. equil ) THEN
568 equed = 'N'
569 rcequ = .false.
570 ELSE
571 rcequ = lsame( equed, 'Y' )
572 ENDIF
573*
574* Default is failure. If an input parameter is wrong or
575* factorization fails, make everything look horrible. Only the
576* pivot growth is set here, the rest is initialized in CSYRFSX.
577*
578 rpvgrw = zero
579*
580* Test the input parameters. PARAMS is not tested until CSYRFSX.
581*
582 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
583 $ lsame( fact, 'F' ) ) THEN
584 info = -1
585 ELSE IF( .NOT.lsame(uplo, 'U') .AND.
586 $ .NOT.lsame(uplo, 'L') ) THEN
587 info = -2
588 ELSE IF( n.LT.0 ) THEN
589 info = -3
590 ELSE IF( nrhs.LT.0 ) THEN
591 info = -4
592 ELSE IF( lda.LT.max( 1, n ) ) THEN
593 info = -6
594 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
595 info = -8
596 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
597 $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
598 info = -10
599 ELSE
600 IF ( rcequ ) THEN
601 smin = bignum
602 smax = zero
603 DO 10 j = 1, n
604 smin = min( smin, s( j ) )
605 smax = max( smax, s( j ) )
606 10 CONTINUE
607 IF( smin.LE.zero ) THEN
608 info = -11
609 ELSE IF( n.GT.0 ) THEN
610 scond = max( smin, smlnum ) / min( smax, bignum )
611 ELSE
612 scond = one
613 END IF
614 END IF
615 IF( info.EQ.0 ) THEN
616 IF( ldb.LT.max( 1, n ) ) THEN
617 info = -13
618 ELSE IF( ldx.LT.max( 1, n ) ) THEN
619 info = -15
620 END IF
621 END IF
622 END IF
623*
624 IF( info.NE.0 ) THEN
625 CALL xerbla( 'CSYSVXX', -info )
626 RETURN
627 END IF
628*
629 IF( equil ) THEN
630*
631* Compute row and column scalings to equilibrate the matrix A.
632*
633 CALL csyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
634 IF( infequ.EQ.0 ) THEN
635*
636* Equilibrate the matrix.
637*
638 CALL claqsy( uplo, n, a, lda, s, scond, amax, equed )
639 rcequ = lsame( equed, 'Y' )
640 END IF
641
642 END IF
643*
644* Scale the right hand-side.
645*
646 IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
647*
648 IF( nofact .OR. equil ) THEN
649*
650* Compute the LDL^T or UDU^T factorization of A.
651*
652 CALL clacpy( uplo, n, n, a, lda, af, ldaf )
653 CALL csytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
654*
655* Return if INFO is non-zero.
656*
657 IF( info.GT.0 ) THEN
658*
659* Pivot in column INFO is exactly 0
660* Compute the reciprocal pivot growth factor of the
661* leading rank-deficient INFO columns of A.
662*
663 IF ( n.GT.0 )
664 $ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af,
665 $ ldaf, ipiv, rwork )
666 RETURN
667 END IF
668 END IF
669*
670* Compute the reciprocal pivot growth factor RPVGRW.
671*
672 IF ( n.GT.0 )
673 $ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
674 $ ipiv, rwork )
675*
676* Compute the solution matrix X.
677*
678 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
679 CALL csytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
680*
681* Use iterative refinement to improve the computed solution and
682* compute error bounds and backward error estimates for it.
683*
684 CALL csyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
685 $ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
686 $ err_bnds_comp, nparams, params, work, rwork, info )
687*
688* Scale solutions.
689*
690 IF ( rcequ ) THEN
691 CALL clascl2 (n, nrhs, s, x, ldx )
692 END IF
693*
694 RETURN
695*
696* End of CSYSVXX
697*
clascl2
subroutine clascl2(m, n, d, x, ldx)
CLASCL2 performs diagonal scaling on a vector.
Definition clascl2.f:91
claqsy
subroutine claqsy(uplo, n, a, lda, s, scond, amax, equed)
CLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition claqsy.f:134
csyrfsx
subroutine csyrfsx(uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CSYRFSX
Definition csyrfsx.f:402
csyequb
subroutine csyequb(uplo, n, a, lda, s, scond, amax, work, info)
CSYEQUB
Definition csyequb.f:132
cla_syrpvgrw
real function cla_syrpvgrw(uplo, n, info, a, lda, af, ldaf, ipiv, work)
CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite m...
Definition cla_syrpvgrw.f:123
min
#define min(a, b)
Definition macros.h:20