LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zspsvx.f
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00001 *> \brief <b> ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
00022 *                          LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          FACT, UPLO
00026 *       INTEGER            INFO, LDB, LDX, N, NRHS
00027 *       DOUBLE PRECISION   RCOND
00028 *       ..
00029 *       .. Array Arguments ..
00030 *       INTEGER            IPIV( * )
00031 *       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
00032 *       COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
00033 *      $                   X( LDX, * )
00034 *       ..
00035 *  
00036 *
00037 *> \par Purpose:
00038 *  =============
00039 *>
00040 *> \verbatim
00041 *>
00042 *> ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
00043 *> A = L*D*L**T to compute the solution to a complex system of linear
00044 *> equations A * X = B, where A is an N-by-N symmetric matrix stored
00045 *> in packed format and X and B are N-by-NRHS matrices.
00046 *>
00047 *> Error bounds on the solution and a condition estimate are also
00048 *> provided.
00049 *> \endverbatim
00050 *
00051 *> \par Description:
00052 *  =================
00053 *>
00054 *> \verbatim
00055 *>
00056 *> The following steps are performed:
00057 *>
00058 *> 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
00059 *>       A = U * D * U**T,  if UPLO = 'U', or
00060 *>       A = L * D * L**T,  if UPLO = 'L',
00061 *>    where U (or L) is a product of permutation and unit upper (lower)
00062 *>    triangular matrices and D is symmetric and block diagonal with
00063 *>    1-by-1 and 2-by-2 diagonal blocks.
00064 *>
00065 *> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
00066 *>    returns with INFO = i. Otherwise, the factored form of A is used
00067 *>    to estimate the condition number of the matrix A.  If the
00068 *>    reciprocal of the condition number is less than machine precision,
00069 *>    INFO = N+1 is returned as a warning, but the routine still goes on
00070 *>    to solve for X and compute error bounds as described below.
00071 *>
00072 *> 3. The system of equations is solved for X using the factored form
00073 *>    of A.
00074 *>
00075 *> 4. Iterative refinement is applied to improve the computed solution
00076 *>    matrix and calculate error bounds and backward error estimates
00077 *>    for it.
00078 *> \endverbatim
00079 *
00080 *  Arguments:
00081 *  ==========
00082 *
00083 *> \param[in] FACT
00084 *> \verbatim
00085 *>          FACT is CHARACTER*1
00086 *>          Specifies whether or not the factored form of A has been
00087 *>          supplied on entry.
00088 *>          = 'F':  On entry, AFP and IPIV contain the factored form
00089 *>                  of A.  AP, AFP and IPIV will not be modified.
00090 *>          = 'N':  The matrix A will be copied to AFP and factored.
00091 *> \endverbatim
00092 *>
00093 *> \param[in] UPLO
00094 *> \verbatim
00095 *>          UPLO is CHARACTER*1
00096 *>          = 'U':  Upper triangle of A is stored;
00097 *>          = 'L':  Lower triangle of A is stored.
00098 *> \endverbatim
00099 *>
00100 *> \param[in] N
00101 *> \verbatim
00102 *>          N is INTEGER
00103 *>          The number of linear equations, i.e., the order of the
00104 *>          matrix A.  N >= 0.
00105 *> \endverbatim
00106 *>
00107 *> \param[in] NRHS
00108 *> \verbatim
00109 *>          NRHS is INTEGER
00110 *>          The number of right hand sides, i.e., the number of columns
00111 *>          of the matrices B and X.  NRHS >= 0.
00112 *> \endverbatim
00113 *>
00114 *> \param[in] AP
00115 *> \verbatim
00116 *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
00117 *>          The upper or lower triangle of the symmetric matrix A, packed
00118 *>          columnwise in a linear array.  The j-th column of A is stored
00119 *>          in the array AP as follows:
00120 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00121 *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
00122 *>          See below for further details.
00123 *> \endverbatim
00124 *>
00125 *> \param[in,out] AFP
00126 *> \verbatim
00127 *>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
00128 *>          If FACT = 'F', then AFP is an input argument and on entry
00129 *>          contains the block diagonal matrix D and the multipliers used
00130 *>          to obtain the factor U or L from the factorization
00131 *>          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
00132 *>          a packed triangular matrix in the same storage format as A.
00133 *>
00134 *>          If FACT = 'N', then AFP is an output argument and on exit
00135 *>          contains the block diagonal matrix D and the multipliers used
00136 *>          to obtain the factor U or L from the factorization
00137 *>          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
00138 *>          a packed triangular matrix in the same storage format as A.
00139 *> \endverbatim
00140 *>
00141 *> \param[in,out] IPIV
00142 *> \verbatim
00143 *>          IPIV is INTEGER array, dimension (N)
00144 *>          If FACT = 'F', then IPIV is an input argument and on entry
00145 *>          contains details of the interchanges and the block structure
00146 *>          of D, as determined by ZSPTRF.
00147 *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
00148 *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
00149 *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
00150 *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
00151 *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
00152 *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
00153 *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
00154 *>
00155 *>          If FACT = 'N', then IPIV is an output argument and on exit
00156 *>          contains details of the interchanges and the block structure
00157 *>          of D, as determined by ZSPTRF.
00158 *> \endverbatim
00159 *>
00160 *> \param[in] B
00161 *> \verbatim
00162 *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
00163 *>          The N-by-NRHS right hand side matrix B.
00164 *> \endverbatim
00165 *>
00166 *> \param[in] LDB
00167 *> \verbatim
00168 *>          LDB is INTEGER
00169 *>          The leading dimension of the array B.  LDB >= max(1,N).
00170 *> \endverbatim
00171 *>
00172 *> \param[out] X
00173 *> \verbatim
00174 *>          X is COMPLEX*16 array, dimension (LDX,NRHS)
00175 *>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
00176 *> \endverbatim
00177 *>
00178 *> \param[in] LDX
00179 *> \verbatim
00180 *>          LDX is INTEGER
00181 *>          The leading dimension of the array X.  LDX >= max(1,N).
00182 *> \endverbatim
00183 *>
00184 *> \param[out] RCOND
00185 *> \verbatim
00186 *>          RCOND is DOUBLE PRECISION
00187 *>          The estimate of the reciprocal condition number of the matrix
00188 *>          A.  If RCOND is less than the machine precision (in
00189 *>          particular, if RCOND = 0), the matrix is singular to working
00190 *>          precision.  This condition is indicated by a return code of
00191 *>          INFO > 0.
00192 *> \endverbatim
00193 *>
00194 *> \param[out] FERR
00195 *> \verbatim
00196 *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
00197 *>          The estimated forward error bound for each solution vector
00198 *>          X(j) (the j-th column of the solution matrix X).
00199 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00200 *>          is an estimated upper bound for the magnitude of the largest
00201 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00202 *>          largest element in X(j).  The estimate is as reliable as
00203 *>          the estimate for RCOND, and is almost always a slight
00204 *>          overestimate of the true error.
00205 *> \endverbatim
00206 *>
00207 *> \param[out] BERR
00208 *> \verbatim
00209 *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
00210 *>          The componentwise relative backward error of each solution
00211 *>          vector X(j) (i.e., the smallest relative change in
00212 *>          any element of A or B that makes X(j) an exact solution).
00213 *> \endverbatim
00214 *>
00215 *> \param[out] WORK
00216 *> \verbatim
00217 *>          WORK is COMPLEX*16 array, dimension (2*N)
00218 *> \endverbatim
00219 *>
00220 *> \param[out] RWORK
00221 *> \verbatim
00222 *>          RWORK is DOUBLE PRECISION array, dimension (N)
00223 *> \endverbatim
00224 *>
00225 *> \param[out] INFO
00226 *> \verbatim
00227 *>          INFO is INTEGER
00228 *>          = 0: successful exit
00229 *>          < 0: if INFO = -i, the i-th argument had an illegal value
00230 *>          > 0:  if INFO = i, and i is
00231 *>                <= N:  D(i,i) is exactly zero.  The factorization
00232 *>                       has been completed but the factor D is exactly
00233 *>                       singular, so the solution and error bounds could
00234 *>                       not be computed. RCOND = 0 is returned.
00235 *>                = N+1: D is nonsingular, but RCOND is less than machine
00236 *>                       precision, meaning that the matrix is singular
00237 *>                       to working precision.  Nevertheless, the
00238 *>                       solution and error bounds are computed because
00239 *>                       there are a number of situations where the
00240 *>                       computed solution can be more accurate than the
00241 *>                       value of RCOND would suggest.
00242 *> \endverbatim
00243 *
00244 *  Authors:
00245 *  ========
00246 *
00247 *> \author Univ. of Tennessee 
00248 *> \author Univ. of California Berkeley 
00249 *> \author Univ. of Colorado Denver 
00250 *> \author NAG Ltd. 
00251 *
00252 *> \date April 2012
00253 *
00254 *> \ingroup complex16OTHERsolve
00255 *
00256 *> \par Further Details:
00257 *  =====================
00258 *>
00259 *> \verbatim
00260 *>
00261 *>  The packed storage scheme is illustrated by the following example
00262 *>  when N = 4, UPLO = 'U':
00263 *>
00264 *>  Two-dimensional storage of the symmetric matrix A:
00265 *>
00266 *>     a11 a12 a13 a14
00267 *>         a22 a23 a24
00268 *>             a33 a34     (aij = aji)
00269 *>                 a44
00270 *>
00271 *>  Packed storage of the upper triangle of A:
00272 *>
00273 *>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
00274 *> \endverbatim
00275 *>
00276 *  =====================================================================
00277       SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
00278      $                   LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
00279 *
00280 *  -- LAPACK driver routine (version 3.4.1) --
00281 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00282 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00283 *     April 2012
00284 *
00285 *     .. Scalar Arguments ..
00286       CHARACTER          FACT, UPLO
00287       INTEGER            INFO, LDB, LDX, N, NRHS
00288       DOUBLE PRECISION   RCOND
00289 *     ..
00290 *     .. Array Arguments ..
00291       INTEGER            IPIV( * )
00292       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
00293       COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
00294      $                   X( LDX, * )
00295 *     ..
00296 *
00297 *  =====================================================================
00298 *
00299 *     .. Parameters ..
00300       DOUBLE PRECISION   ZERO
00301       PARAMETER          ( ZERO = 0.0D+0 )
00302 *     ..
00303 *     .. Local Scalars ..
00304       LOGICAL            NOFACT
00305       DOUBLE PRECISION   ANORM
00306 *     ..
00307 *     .. External Functions ..
00308       LOGICAL            LSAME
00309       DOUBLE PRECISION   DLAMCH, ZLANSP
00310       EXTERNAL           LSAME, DLAMCH, ZLANSP
00311 *     ..
00312 *     .. External Subroutines ..
00313       EXTERNAL           XERBLA, ZCOPY, ZLACPY, ZSPCON, ZSPRFS, ZSPTRF,
00314      $                   ZSPTRS
00315 *     ..
00316 *     .. Intrinsic Functions ..
00317       INTRINSIC          MAX
00318 *     ..
00319 *     .. Executable Statements ..
00320 *
00321 *     Test the input parameters.
00322 *
00323       INFO = 0
00324       NOFACT = LSAME( FACT, 'N' )
00325       IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
00326          INFO = -1
00327       ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
00328      $          THEN
00329          INFO = -2
00330       ELSE IF( N.LT.0 ) THEN
00331          INFO = -3
00332       ELSE IF( NRHS.LT.0 ) THEN
00333          INFO = -4
00334       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00335          INFO = -9
00336       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00337          INFO = -11
00338       END IF
00339       IF( INFO.NE.0 ) THEN
00340          CALL XERBLA( 'ZSPSVX', -INFO )
00341          RETURN
00342       END IF
00343 *
00344       IF( NOFACT ) THEN
00345 *
00346 *        Compute the factorization A = U*D*U**T or A = L*D*L**T.
00347 *
00348          CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
00349          CALL ZSPTRF( UPLO, N, AFP, IPIV, INFO )
00350 *
00351 *        Return if INFO is non-zero.
00352 *
00353          IF( INFO.GT.0 )THEN
00354             RCOND = ZERO
00355             RETURN
00356          END IF
00357       END IF
00358 *
00359 *     Compute the norm of the matrix A.
00360 *
00361       ANORM = ZLANSP( 'I', UPLO, N, AP, RWORK )
00362 *
00363 *     Compute the reciprocal of the condition number of A.
00364 *
00365       CALL ZSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, INFO )
00366 *
00367 *     Compute the solution vectors X.
00368 *
00369       CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
00370       CALL ZSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
00371 *
00372 *     Use iterative refinement to improve the computed solutions and
00373 *     compute error bounds and backward error estimates for them.
00374 *
00375       CALL ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR,
00376      $             BERR, WORK, RWORK, INFO )
00377 *
00378 *     Set INFO = N+1 if the matrix is singular to working precision.
00379 *
00380       IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
00381      $   INFO = N + 1
00382 *
00383       RETURN
00384 *
00385 *     End of ZSPSVX
00386 *
00387       END
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