LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
complex
Collaboration diagram for complex:


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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 More...

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 More...

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 More...

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 More...

Detailed Description

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

Function Documentation

 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

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.```
Date
November 2011

Definition at line 173 of file csysv.f.

173 *
174 * -- LAPACK driver routine (version 3.4.0) --
175 * -- LAPACK is a software package provided by Univ. of Tennessee, --
176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177 * November 2011
178 *
179 * .. Scalar Arguments ..
180  CHARACTER uplo
181  INTEGER info, lda, ldb, lwork, n, nrhs
182 * ..
183 * .. Array Arguments ..
184  INTEGER ipiv( * )
185  COMPLEX a( lda, * ), b( ldb, * ), work( * )
186 * ..
187 *
188 * =====================================================================
189 *
190 * .. Local Scalars ..
191  LOGICAL lquery
192  INTEGER lwkopt
193 * ..
194 * .. External Functions ..
195  LOGICAL lsame
196  EXTERNAL lsame
197 * ..
198 * .. External Subroutines ..
199  EXTERNAL xerbla, csytrf, csytrs, csytrs2
200 * ..
201 * .. Intrinsic Functions ..
202  INTRINSIC max
203 * ..
204 * .. Executable Statements ..
205 *
206 * Test the input parameters.
207 *
208  info = 0
209  lquery = ( lwork.EQ.-1 )
210  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
211  info = -1
212  ELSE IF( n.LT.0 ) THEN
213  info = -2
214  ELSE IF( nrhs.LT.0 ) THEN
215  info = -3
216  ELSE IF( lda.LT.max( 1, n ) ) THEN
217  info = -5
218  ELSE IF( ldb.LT.max( 1, n ) ) THEN
219  info = -8
220  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
221  info = -10
222  END IF
223 *
224  IF( info.EQ.0 ) THEN
225  IF( n.EQ.0 ) THEN
226  lwkopt = 1
227  ELSE
228  CALL csytrf( uplo, n, a, lda, ipiv, work, -1, info )
229  lwkopt = work(1)
230  END IF
231  work( 1 ) = lwkopt
232  END IF
233 *
234  IF( info.NE.0 ) THEN
235  CALL xerbla( 'CSYSV ', -info )
236  RETURN
237  ELSE IF( lquery ) THEN
238  RETURN
239  END IF
240 *
241 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
242 *
243  CALL csytrf( uplo, n, a, lda, ipiv, work, lwork, info )
244  IF( info.EQ.0 ) THEN
245 *
246 * Solve the system A*X = B, overwriting B with X.
247 *
248  IF ( lwork.LT.n ) THEN
249 *
250 * Solve with TRS ( Use Level BLAS 2)
251 *
252  CALL csytrs( uplo, n, nrhs, a, lda, ipiv, b, ldb, info )
253 *
254  ELSE
255 *
256 * Solve with TRS2 ( Use Level BLAS 3)
257 *
258  CALL csytrs2( uplo,n,nrhs,a,lda,ipiv,b,ldb,work,info )
259 *
260  END IF
261 *
262  END IF
263 *
264  work( 1 ) = lwkopt
265 *
266  RETURN
267 *
268 * End of CSYSV
269 *
subroutine csytrs2(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)
CSYTRS2
Definition: csytrs2.f:134
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine csytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS
Definition: csytrs.f:122
subroutine csytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF
Definition: csytrf.f:184
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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 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

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 ("rook") 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.```
Date
April 2012
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 206 of file csysv_rook.f.

206 *
207 * -- LAPACK driver routine (version 3.4.1) --
208 * -- LAPACK is a software package provided by Univ. of Tennessee, --
209 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
210 * April 2012
211 *
212 * .. Scalar Arguments ..
213  CHARACTER uplo
214  INTEGER info, lda, ldb, lwork, n, nrhs
215 * ..
216 * .. Array Arguments ..
217  INTEGER ipiv( * )
218  COMPLEX a( lda, * ), b( ldb, * ), work( * )
219 * ..
220 *
221 * =====================================================================
222 *
223 * .. Local Scalars ..
224  LOGICAL lquery
225  INTEGER lwkopt
226 * ..
227 * .. External Functions ..
228  LOGICAL lsame
229  EXTERNAL lsame
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL xerbla, csytrf_rook, csytrs_rook
233 * ..
234 * .. Intrinsic Functions ..
235  INTRINSIC max
236 * ..
237 * .. Executable Statements ..
238 *
239 * Test the input parameters.
240 *
241  info = 0
242  lquery = ( lwork.EQ.-1 )
243  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
244  info = -1
245  ELSE IF( n.LT.0 ) THEN
246  info = -2
247  ELSE IF( nrhs.LT.0 ) THEN
248  info = -3
249  ELSE IF( lda.LT.max( 1, n ) ) THEN
250  info = -5
251  ELSE IF( ldb.LT.max( 1, n ) ) THEN
252  info = -8
253  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
254  info = -10
255  END IF
256 *
257  IF( info.EQ.0 ) THEN
258  IF( n.EQ.0 ) THEN
259  lwkopt = 1
260  ELSE
261  CALL csytrf_rook( uplo, n, a, lda, ipiv, work, -1, info )
262  lwkopt = work(1)
263  END IF
264  work( 1 ) = lwkopt
265  END IF
266 *
267  IF( info.NE.0 ) THEN
268  CALL xerbla( 'CSYSV_ROOK ', -info )
269  RETURN
270  ELSE IF( lquery ) THEN
271  RETURN
272  END IF
273 *
274 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
275 *
276  CALL csytrf_rook( uplo, n, a, lda, ipiv, work, lwork, info )
277  IF( info.EQ.0 ) THEN
278 *
279 * Solve the system A*X = B, overwriting B with X.
280 *
281 * Solve with TRS_ROOK ( Use Level 2 BLAS)
282 *
283  CALL csytrs_rook( uplo, n, nrhs, a, lda, ipiv, b, ldb, info )
284 *
285  END IF
286 *
287  work( 1 ) = lwkopt
288 *
289  RETURN
290 *
291 * End of CSYSV_ROOK
292 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine csytrs_rook(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS_ROOK
Definition: csytrs_rook.f:138
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine csytrf_rook(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF_ROOK
Definition: csytrf_rook.f:210

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 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

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.```
Date
April 2012

Definition at line 287 of file csysvx.f.

287 *
288 * -- LAPACK driver routine (version 3.4.1) --
289 * -- LAPACK is a software package provided by Univ. of Tennessee, --
290 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
291 * April 2012
292 *
293 * .. Scalar Arguments ..
294  CHARACTER fact, uplo
295  INTEGER info, lda, ldaf, ldb, ldx, lwork, n, nrhs
296  REAL rcond
297 * ..
298 * .. Array Arguments ..
299  INTEGER ipiv( * )
300  REAL berr( * ), ferr( * ), rwork( * )
301  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
302  \$ work( * ), x( ldx, * )
303 * ..
304 *
305 * =====================================================================
306 *
307 * .. Parameters ..
308  REAL zero
309  parameter( zero = 0.0e+0 )
310 * ..
311 * .. Local Scalars ..
312  LOGICAL lquery, nofact
313  INTEGER lwkopt, nb
314  REAL anorm
315 * ..
316 * .. External Functions ..
317  LOGICAL lsame
318  INTEGER ilaenv
319  REAL clansy, slamch
320  EXTERNAL ilaenv, lsame, clansy, slamch
321 * ..
322 * .. External Subroutines ..
323  EXTERNAL clacpy, csycon, csyrfs, csytrf, csytrs, xerbla
324 * ..
325 * .. Intrinsic Functions ..
326  INTRINSIC max
327 * ..
328 * .. Executable Statements ..
329 *
330 * Test the input parameters.
331 *
332  info = 0
333  nofact = lsame( fact, 'N' )
334  lquery = ( lwork.EQ.-1 )
335  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
336  info = -1
337  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
338  \$ THEN
339  info = -2
340  ELSE IF( n.LT.0 ) THEN
341  info = -3
342  ELSE IF( nrhs.LT.0 ) THEN
343  info = -4
344  ELSE IF( lda.LT.max( 1, n ) ) THEN
345  info = -6
346  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
347  info = -8
348  ELSE IF( ldb.LT.max( 1, n ) ) THEN
349  info = -11
350  ELSE IF( ldx.LT.max( 1, n ) ) THEN
351  info = -13
352  ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
353  info = -18
354  END IF
355 *
356  IF( info.EQ.0 ) THEN
357  lwkopt = max( 1, 2*n )
358  IF( nofact ) THEN
359  nb = ilaenv( 1, 'CSYTRF', uplo, n, -1, -1, -1 )
360  lwkopt = max( lwkopt, n*nb )
361  END IF
362  work( 1 ) = lwkopt
363  END IF
364 *
365  IF( info.NE.0 ) THEN
366  CALL xerbla( 'CSYSVX', -info )
367  RETURN
368  ELSE IF( lquery ) THEN
369  RETURN
370  END IF
371 *
372  IF( nofact ) THEN
373 *
374 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
375 *
376  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
377  CALL csytrf( uplo, n, af, ldaf, ipiv, work, lwork, info )
378 *
379 * Return if INFO is non-zero.
380 *
381  IF( info.GT.0 )THEN
382  rcond = zero
383  RETURN
384  END IF
385  END IF
386 *
387 * Compute the norm of the matrix A.
388 *
389  anorm = clansy( 'I', uplo, n, a, lda, rwork )
390 *
391 * Compute the reciprocal of the condition number of A.
392 *
393  CALL csycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work, info )
394 *
395 * Compute the solution vectors X.
396 *
397  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
398  CALL csytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
399 *
400 * Use iterative refinement to improve the computed solutions and
401 * compute error bounds and backward error estimates for them.
402 *
403  CALL csyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
404  \$ ldx, ferr, berr, work, rwork, info )
405 *
406 * Set INFO = N+1 if the matrix is singular to working precision.
407 *
408  IF( rcond.LT.slamch( 'Epsilon' ) )
409  \$ info = n + 1
410 *
411  work( 1 ) = lwkopt
412 *
413  RETURN
414 *
415 * End of CSYSVX
416 *
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine csytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS
Definition: csytrs.f:122
subroutine csytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF
Definition: csytrf.f:184
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine csyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CSYRFS
Definition: csyrfs.f:194
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine csycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON
Definition: csycon.f:127
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
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, or the element of largest absolute value of a complex symmetric matrix.
Definition: clansy.f:125

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 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

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 SSYTRF. 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 SSYTRF. 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 SSYTRF.``` [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 "max absolute element" 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 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.```
Date
April 2012

Definition at line 511 of file csysvxx.f.

511 *
512 * -- LAPACK driver routine (version 3.6.0) --
513 * -- LAPACK is a software package provided by Univ. of Tennessee, --
514 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
515 * April 2012
516 *
517 * .. Scalar Arguments ..
518  CHARACTER equed, fact, uplo
519  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
520  \$ n_err_bnds
521  REAL rcond, rpvgrw
522 * ..
523 * .. Array Arguments ..
524  INTEGER ipiv( * )
525  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
526  \$ x( ldx, * ), work( * )
527  REAL s( * ), params( * ), berr( * ),
528  \$ err_bnds_norm( nrhs, * ),
529  \$ err_bnds_comp( nrhs, * ), rwork( * )
530 * ..
531 *
532 * ==================================================================
533 *
534 * .. Parameters ..
535  REAL zero, one
536  parameter( zero = 0.0e+0, one = 1.0e+0 )
537  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
538  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
539  INTEGER cmp_err_i, piv_growth_i
540  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
541  \$ berr_i = 3 )
542  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
543  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
544  \$ piv_growth_i = 9 )
545 * ..
546 * .. Local Scalars ..
547  LOGICAL equil, nofact, rcequ
548  INTEGER infequ, j
549  REAL amax, bignum, smin, smax, scond, smlnum
550 * ..
551 * .. External Functions ..
552  EXTERNAL lsame, slamch, cla_syrpvgrw
553  LOGICAL lsame
554  REAL slamch, cla_syrpvgrw
555 * ..
556 * .. External Subroutines ..
557  EXTERNAL csycon, csyequb, csytrf, csytrs, clacpy,
559 * ..
560 * .. Intrinsic Functions ..
561  INTRINSIC max, min
562 * ..
563 * .. Executable Statements ..
564 *
565  info = 0
566  nofact = lsame( fact, 'N' )
567  equil = lsame( fact, 'E' )
568  smlnum = slamch( 'Safe minimum' )
569  bignum = one / smlnum
570  IF( nofact .OR. equil ) THEN
571  equed = 'N'
572  rcequ = .false.
573  ELSE
574  rcequ = lsame( equed, 'Y' )
575  ENDIF
576 *
577 * Default is failure. If an input parameter is wrong or
578 * factorization fails, make everything look horrible. Only the
579 * pivot growth is set here, the rest is initialized in CSYRFSX.
580 *
581  rpvgrw = zero
582 *
583 * Test the input parameters. PARAMS is not tested until CSYRFSX.
584 *
585  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
586  \$ lsame( fact, 'F' ) ) THEN
587  info = -1
588  ELSE IF( .NOT.lsame(uplo, 'U') .AND.
589  \$ .NOT.lsame(uplo, 'L') ) THEN
590  info = -2
591  ELSE IF( n.LT.0 ) THEN
592  info = -3
593  ELSE IF( nrhs.LT.0 ) THEN
594  info = -4
595  ELSE IF( lda.LT.max( 1, n ) ) THEN
596  info = -6
597  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
598  info = -8
599  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
600  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
601  info = -10
602  ELSE
603  IF ( rcequ ) THEN
604  smin = bignum
605  smax = zero
606  DO 10 j = 1, n
607  smin = min( smin, s( j ) )
608  smax = max( smax, s( j ) )
609  10 CONTINUE
610  IF( smin.LE.zero ) THEN
611  info = -11
612  ELSE IF( n.GT.0 ) THEN
613  scond = max( smin, smlnum ) / min( smax, bignum )
614  ELSE
615  scond = one
616  END IF
617  END IF
618  IF( info.EQ.0 ) THEN
619  IF( ldb.LT.max( 1, n ) ) THEN
620  info = -13
621  ELSE IF( ldx.LT.max( 1, n ) ) THEN
622  info = -15
623  END IF
624  END IF
625  END IF
626 *
627  IF( info.NE.0 ) THEN
628  CALL xerbla( 'CSYSVXX', -info )
629  RETURN
630  END IF
631 *
632  IF( equil ) THEN
633 *
634 * Compute row and column scalings to equilibrate the matrix A.
635 *
636  CALL csyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
637  IF( infequ.EQ.0 ) THEN
638 *
639 * Equilibrate the matrix.
640 *
641  CALL claqsy( uplo, n, a, lda, s, scond, amax, equed )
642  rcequ = lsame( equed, 'Y' )
643  END IF
644
645  END IF
646 *
647 * Scale the right hand-side.
648 *
649  IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
650 *
651  IF( nofact .OR. equil ) THEN
652 *
653 * Compute the LDL^T or UDU^T factorization of A.
654 *
655  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
656  CALL csytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
657 *
658 * Return if INFO is non-zero.
659 *
660  IF( info.GT.0 ) THEN
661 *
662 * Pivot in column INFO is exactly 0
663 * Compute the reciprocal pivot growth factor of the
664 * leading rank-deficient INFO columns of A.
665 *
666  IF ( n.GT.0 )
667  \$ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af,
668  \$ ldaf, ipiv, rwork )
669  RETURN
670  END IF
671  END IF
672 *
673 * Compute the reciprocal pivot growth factor RPVGRW.
674 *
675  IF ( n.GT.0 )
676  \$ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
677  \$ ipiv, rwork )
678 *
679 * Compute the solution matrix X.
680 *
681  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
682  CALL csytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
683 *
684 * Use iterative refinement to improve the computed solution and
685 * compute error bounds and backward error estimates for it.
686 *
687  CALL csyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
688  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
689  \$ err_bnds_comp, nparams, params, work, rwork, info )
690 *
691 * Scale solutions.
692 *
693  IF ( rcequ ) THEN
694  CALL clascl2 (n, nrhs, s, x, ldx )
695  END IF
696 *
697  RETURN
698 *
699 * End of CSYSVXX
700 *
subroutine clascl2(M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a vector.
Definition: clascl2.f:93
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:404
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:125
subroutine csytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS
Definition: csytrs.f:122
subroutine csytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF
Definition: csytrf.f:184
subroutine csyequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
CSYEQUB
Definition: csyequb.f:138
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine csycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON
Definition: csycon.f:127
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:136

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