LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
spbsvx.f
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00001 *> \brief <b> SPBSVX 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 *
00008 *> \htmlonly
00009 *> Download SPBSVX + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spbsvx.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/spbsvx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbsvx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
00022 *                          EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
00023 *                          WORK, IWORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          EQUED, FACT, UPLO
00027 *       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
00028 *       REAL               RCOND
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IWORK( * )
00032 *       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00033 *      $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
00034 *      $                   X( LDX, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
00044 *> compute the solution to a real system of linear equations
00045 *>    A * X = B,
00046 *> where A is an N-by-N symmetric positive definite band matrix and X
00047 *> and B are N-by-NRHS matrices.
00048 *>
00049 *> Error bounds on the solution and a condition estimate are also
00050 *> provided.
00051 *> \endverbatim
00052 *
00053 *> \par Description:
00054 *  =================
00055 *>
00056 *> \verbatim
00057 *>
00058 *> The following steps are performed:
00059 *>
00060 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
00061 *>    the system:
00062 *>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
00063 *>    Whether or not the system will be equilibrated depends on the
00064 *>    scaling of the matrix A, but if equilibration is used, A is
00065 *>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
00066 *>
00067 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
00068 *>    factor the matrix A (after equilibration if FACT = 'E') as
00069 *>       A = U**T * U,  if UPLO = 'U', or
00070 *>       A = L * L**T,  if UPLO = 'L',
00071 *>    where U is an upper triangular band matrix, and L is a lower
00072 *>    triangular band matrix.
00073 *>
00074 *> 3. If the leading i-by-i principal minor is not positive definite,
00075 *>    then the routine returns with INFO = i. Otherwise, the factored
00076 *>    form of A is used to estimate the condition number of the matrix
00077 *>    A.  If the reciprocal of the condition number is less than machine
00078 *>    precision, INFO = N+1 is returned as a warning, but the routine
00079 *>    still goes on to solve for X and compute error bounds as
00080 *>    described below.
00081 *>
00082 *> 4. The system of equations is solved for X using the factored form
00083 *>    of A.
00084 *>
00085 *> 5. Iterative refinement is applied to improve the computed solution
00086 *>    matrix and calculate error bounds and backward error estimates
00087 *>    for it.
00088 *>
00089 *> 6. If equilibration was used, the matrix X is premultiplied by
00090 *>    diag(S) so that it solves the original system before
00091 *>    equilibration.
00092 *> \endverbatim
00093 *
00094 *  Arguments:
00095 *  ==========
00096 *
00097 *> \param[in] FACT
00098 *> \verbatim
00099 *>          FACT is CHARACTER*1
00100 *>          Specifies whether or not the factored form of the matrix A is
00101 *>          supplied on entry, and if not, whether the matrix A should be
00102 *>          equilibrated before it is factored.
00103 *>          = 'F':  On entry, AFB contains the factored form of A.
00104 *>                  If EQUED = 'Y', the matrix A has been equilibrated
00105 *>                  with scaling factors given by S.  AB and AFB will not
00106 *>                  be modified.
00107 *>          = 'N':  The matrix A will be copied to AFB and factored.
00108 *>          = 'E':  The matrix A will be equilibrated if necessary, then
00109 *>                  copied to AFB and factored.
00110 *> \endverbatim
00111 *>
00112 *> \param[in] UPLO
00113 *> \verbatim
00114 *>          UPLO is CHARACTER*1
00115 *>          = 'U':  Upper triangle of A is stored;
00116 *>          = 'L':  Lower triangle of A is stored.
00117 *> \endverbatim
00118 *>
00119 *> \param[in] N
00120 *> \verbatim
00121 *>          N is INTEGER
00122 *>          The number of linear equations, i.e., the order of the
00123 *>          matrix A.  N >= 0.
00124 *> \endverbatim
00125 *>
00126 *> \param[in] KD
00127 *> \verbatim
00128 *>          KD is INTEGER
00129 *>          The number of superdiagonals of the matrix A if UPLO = 'U',
00130 *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
00131 *> \endverbatim
00132 *>
00133 *> \param[in] NRHS
00134 *> \verbatim
00135 *>          NRHS is INTEGER
00136 *>          The number of right-hand sides, i.e., the number of columns
00137 *>          of the matrices B and X.  NRHS >= 0.
00138 *> \endverbatim
00139 *>
00140 *> \param[in,out] AB
00141 *> \verbatim
00142 *>          AB is REAL array, dimension (LDAB,N)
00143 *>          On entry, the upper or lower triangle of the symmetric band
00144 *>          matrix A, stored in the first KD+1 rows of the array, except
00145 *>          if FACT = 'F' and EQUED = 'Y', then A must contain the
00146 *>          equilibrated matrix diag(S)*A*diag(S).  The j-th column of A
00147 *>          is stored in the j-th column of the array AB as follows:
00148 *>          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
00149 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
00150 *>          See below for further details.
00151 *>
00152 *>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
00153 *>          diag(S)*A*diag(S).
00154 *> \endverbatim
00155 *>
00156 *> \param[in] LDAB
00157 *> \verbatim
00158 *>          LDAB is INTEGER
00159 *>          The leading dimension of the array A.  LDAB >= KD+1.
00160 *> \endverbatim
00161 *>
00162 *> \param[in,out] AFB
00163 *> \verbatim
00164 *>          AFB is REAL array, dimension (LDAFB,N)
00165 *>          If FACT = 'F', then AFB is an input argument and on entry
00166 *>          contains the triangular factor U or L from the Cholesky
00167 *>          factorization A = U**T*U or A = L*L**T of the band matrix
00168 *>          A, in the same storage format as A (see AB).  If EQUED = 'Y',
00169 *>          then AFB is the factored form of the equilibrated matrix A.
00170 *>
00171 *>          If FACT = 'N', then AFB is an output argument and on exit
00172 *>          returns the triangular factor U or L from the Cholesky
00173 *>          factorization A = U**T*U or A = L*L**T.
00174 *>
00175 *>          If FACT = 'E', then AFB is an output argument and on exit
00176 *>          returns the triangular factor U or L from the Cholesky
00177 *>          factorization A = U**T*U or A = L*L**T of the equilibrated
00178 *>          matrix A (see the description of A for the form of the
00179 *>          equilibrated matrix).
00180 *> \endverbatim
00181 *>
00182 *> \param[in] LDAFB
00183 *> \verbatim
00184 *>          LDAFB is INTEGER
00185 *>          The leading dimension of the array AFB.  LDAFB >= KD+1.
00186 *> \endverbatim
00187 *>
00188 *> \param[in,out] EQUED
00189 *> \verbatim
00190 *>          EQUED is CHARACTER*1
00191 *>          Specifies the form of equilibration that was done.
00192 *>          = 'N':  No equilibration (always true if FACT = 'N').
00193 *>          = 'Y':  Equilibration was done, i.e., A has been replaced by
00194 *>                  diag(S) * A * diag(S).
00195 *>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
00196 *>          output argument.
00197 *> \endverbatim
00198 *>
00199 *> \param[in,out] S
00200 *> \verbatim
00201 *>          S is REAL array, dimension (N)
00202 *>          The scale factors for A; not accessed if EQUED = 'N'.  S is
00203 *>          an input argument if FACT = 'F'; otherwise, S is an output
00204 *>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
00205 *>          must be positive.
00206 *> \endverbatim
00207 *>
00208 *> \param[in,out] B
00209 *> \verbatim
00210 *>          B is REAL array, dimension (LDB,NRHS)
00211 *>          On entry, the N-by-NRHS right hand side matrix B.
00212 *>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
00213 *>          B is overwritten by diag(S) * B.
00214 *> \endverbatim
00215 *>
00216 *> \param[in] LDB
00217 *> \verbatim
00218 *>          LDB is INTEGER
00219 *>          The leading dimension of the array B.  LDB >= max(1,N).
00220 *> \endverbatim
00221 *>
00222 *> \param[out] X
00223 *> \verbatim
00224 *>          X is REAL array, dimension (LDX,NRHS)
00225 *>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
00226 *>          the original system of equations.  Note that if EQUED = 'Y',
00227 *>          A and B are modified on exit, and the solution to the
00228 *>          equilibrated system is inv(diag(S))*X.
00229 *> \endverbatim
00230 *>
00231 *> \param[in] LDX
00232 *> \verbatim
00233 *>          LDX is INTEGER
00234 *>          The leading dimension of the array X.  LDX >= max(1,N).
00235 *> \endverbatim
00236 *>
00237 *> \param[out] RCOND
00238 *> \verbatim
00239 *>          RCOND is REAL
00240 *>          The estimate of the reciprocal condition number of the matrix
00241 *>          A after equilibration (if done).  If RCOND is less than the
00242 *>          machine precision (in particular, if RCOND = 0), the matrix
00243 *>          is singular to working precision.  This condition is
00244 *>          indicated by a return code of INFO > 0.
00245 *> \endverbatim
00246 *>
00247 *> \param[out] FERR
00248 *> \verbatim
00249 *>          FERR is REAL array, dimension (NRHS)
00250 *>          The estimated forward error bound for each solution vector
00251 *>          X(j) (the j-th column of the solution matrix X).
00252 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00253 *>          is an estimated upper bound for the magnitude of the largest
00254 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00255 *>          largest element in X(j).  The estimate is as reliable as
00256 *>          the estimate for RCOND, and is almost always a slight
00257 *>          overestimate of the true error.
00258 *> \endverbatim
00259 *>
00260 *> \param[out] BERR
00261 *> \verbatim
00262 *>          BERR is REAL array, dimension (NRHS)
00263 *>          The componentwise relative backward error of each solution
00264 *>          vector X(j) (i.e., the smallest relative change in
00265 *>          any element of A or B that makes X(j) an exact solution).
00266 *> \endverbatim
00267 *>
00268 *> \param[out] WORK
00269 *> \verbatim
00270 *>          WORK is REAL array, dimension (3*N)
00271 *> \endverbatim
00272 *>
00273 *> \param[out] IWORK
00274 *> \verbatim
00275 *>          IWORK is INTEGER array, dimension (N)
00276 *> \endverbatim
00277 *>
00278 *> \param[out] INFO
00279 *> \verbatim
00280 *>          INFO is INTEGER
00281 *>          = 0:  successful exit
00282 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00283 *>          > 0:  if INFO = i, and i is
00284 *>                <= N:  the leading minor of order i of A is
00285 *>                       not positive definite, so the factorization
00286 *>                       could not be completed, and the solution has not
00287 *>                       been computed. RCOND = 0 is returned.
00288 *>                = N+1: U is nonsingular, but RCOND is less than machine
00289 *>                       precision, meaning that the matrix is singular
00290 *>                       to working precision.  Nevertheless, the
00291 *>                       solution and error bounds are computed because
00292 *>                       there are a number of situations where the
00293 *>                       computed solution can be more accurate than the
00294 *>                       value of RCOND would suggest.
00295 *> \endverbatim
00296 *
00297 *  Authors:
00298 *  ========
00299 *
00300 *> \author Univ. of Tennessee 
00301 *> \author Univ. of California Berkeley 
00302 *> \author Univ. of Colorado Denver 
00303 *> \author NAG Ltd. 
00304 *
00305 *> \date April 2012
00306 *
00307 *> \ingroup realOTHERsolve
00308 *
00309 *> \par Further Details:
00310 *  =====================
00311 *>
00312 *> \verbatim
00313 *>
00314 *>  The band storage scheme is illustrated by the following example, when
00315 *>  N = 6, KD = 2, and UPLO = 'U':
00316 *>
00317 *>  Two-dimensional storage of the symmetric matrix A:
00318 *>
00319 *>     a11  a12  a13
00320 *>          a22  a23  a24
00321 *>               a33  a34  a35
00322 *>                    a44  a45  a46
00323 *>                         a55  a56
00324 *>     (aij=conjg(aji))         a66
00325 *>
00326 *>  Band storage of the upper triangle of A:
00327 *>
00328 *>      *    *   a13  a24  a35  a46
00329 *>      *   a12  a23  a34  a45  a56
00330 *>     a11  a22  a33  a44  a55  a66
00331 *>
00332 *>  Similarly, if UPLO = 'L' the format of A is as follows:
00333 *>
00334 *>     a11  a22  a33  a44  a55  a66
00335 *>     a21  a32  a43  a54  a65   *
00336 *>     a31  a42  a53  a64   *    *
00337 *>
00338 *>  Array elements marked * are not used by the routine.
00339 *> \endverbatim
00340 *>
00341 *  =====================================================================
00342       SUBROUTINE SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
00343      $                   EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
00344      $                   WORK, IWORK, INFO )
00345 *
00346 *  -- LAPACK driver routine (version 3.4.1) --
00347 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00348 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00349 *     April 2012
00350 *
00351 *     .. Scalar Arguments ..
00352       CHARACTER          EQUED, FACT, UPLO
00353       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
00354       REAL               RCOND
00355 *     ..
00356 *     .. Array Arguments ..
00357       INTEGER            IWORK( * )
00358       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00359      $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
00360      $                   X( LDX, * )
00361 *     ..
00362 *
00363 *  =====================================================================
00364 *
00365 *     .. Parameters ..
00366       REAL               ZERO, ONE
00367       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00368 *     ..
00369 *     .. Local Scalars ..
00370       LOGICAL            EQUIL, NOFACT, RCEQU, UPPER
00371       INTEGER            I, INFEQU, J, J1, J2
00372       REAL               AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
00373 *     ..
00374 *     .. External Functions ..
00375       LOGICAL            LSAME
00376       REAL               SLAMCH, SLANSB
00377       EXTERNAL           LSAME, SLAMCH, SLANSB
00378 *     ..
00379 *     .. External Subroutines ..
00380       EXTERNAL           SCOPY, SLACPY, SLAQSB, SPBCON, SPBEQU, SPBRFS,
00381      $                   SPBTRF, SPBTRS, XERBLA
00382 *     ..
00383 *     .. Intrinsic Functions ..
00384       INTRINSIC          MAX, MIN
00385 *     ..
00386 *     .. Executable Statements ..
00387 *
00388       INFO = 0
00389       NOFACT = LSAME( FACT, 'N' )
00390       EQUIL = LSAME( FACT, 'E' )
00391       UPPER = LSAME( UPLO, 'U' )
00392       IF( NOFACT .OR. EQUIL ) THEN
00393          EQUED = 'N'
00394          RCEQU = .FALSE.
00395       ELSE
00396          RCEQU = LSAME( EQUED, 'Y' )
00397          SMLNUM = SLAMCH( 'Safe minimum' )
00398          BIGNUM = ONE / SMLNUM
00399       END IF
00400 *
00401 *     Test the input parameters.
00402 *
00403       IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
00404      $     THEN
00405          INFO = -1
00406       ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00407          INFO = -2
00408       ELSE IF( N.LT.0 ) THEN
00409          INFO = -3
00410       ELSE IF( KD.LT.0 ) THEN
00411          INFO = -4
00412       ELSE IF( NRHS.LT.0 ) THEN
00413          INFO = -5
00414       ELSE IF( LDAB.LT.KD+1 ) THEN
00415          INFO = -7
00416       ELSE IF( LDAFB.LT.KD+1 ) THEN
00417          INFO = -9
00418       ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
00419      $         ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
00420          INFO = -10
00421       ELSE
00422          IF( RCEQU ) THEN
00423             SMIN = BIGNUM
00424             SMAX = ZERO
00425             DO 10 J = 1, N
00426                SMIN = MIN( SMIN, S( J ) )
00427                SMAX = MAX( SMAX, S( J ) )
00428    10       CONTINUE
00429             IF( SMIN.LE.ZERO ) THEN
00430                INFO = -11
00431             ELSE IF( N.GT.0 ) THEN
00432                SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
00433             ELSE
00434                SCOND = ONE
00435             END IF
00436          END IF
00437          IF( INFO.EQ.0 ) THEN
00438             IF( LDB.LT.MAX( 1, N ) ) THEN
00439                INFO = -13
00440             ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00441                INFO = -15
00442             END IF
00443          END IF
00444       END IF
00445 *
00446       IF( INFO.NE.0 ) THEN
00447          CALL XERBLA( 'SPBSVX', -INFO )
00448          RETURN
00449       END IF
00450 *
00451       IF( EQUIL ) THEN
00452 *
00453 *        Compute row and column scalings to equilibrate the matrix A.
00454 *
00455          CALL SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
00456          IF( INFEQU.EQ.0 ) THEN
00457 *
00458 *           Equilibrate the matrix.
00459 *
00460             CALL SLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
00461             RCEQU = LSAME( EQUED, 'Y' )
00462          END IF
00463       END IF
00464 *
00465 *     Scale the right-hand side.
00466 *
00467       IF( RCEQU ) THEN
00468          DO 30 J = 1, NRHS
00469             DO 20 I = 1, N
00470                B( I, J ) = S( I )*B( I, J )
00471    20       CONTINUE
00472    30    CONTINUE
00473       END IF
00474 *
00475       IF( NOFACT .OR. EQUIL ) THEN
00476 *
00477 *        Compute the Cholesky factorization A = U**T *U or A = L*L**T.
00478 *
00479          IF( UPPER ) THEN
00480             DO 40 J = 1, N
00481                J1 = MAX( J-KD, 1 )
00482                CALL SCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
00483      $                     AFB( KD+1-J+J1, J ), 1 )
00484    40       CONTINUE
00485          ELSE
00486             DO 50 J = 1, N
00487                J2 = MIN( J+KD, N )
00488                CALL SCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
00489    50       CONTINUE
00490          END IF
00491 *
00492          CALL SPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
00493 *
00494 *        Return if INFO is non-zero.
00495 *
00496          IF( INFO.GT.0 )THEN
00497             RCOND = ZERO
00498             RETURN
00499          END IF
00500       END IF
00501 *
00502 *     Compute the norm of the matrix A.
00503 *
00504       ANORM = SLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
00505 *
00506 *     Compute the reciprocal of the condition number of A.
00507 *
00508       CALL SPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK,
00509      $             INFO )
00510 *
00511 *     Compute the solution matrix X.
00512 *
00513       CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
00514       CALL SPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
00515 *
00516 *     Use iterative refinement to improve the computed solution and
00517 *     compute error bounds and backward error estimates for it.
00518 *
00519       CALL SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
00520      $             LDX, FERR, BERR, WORK, IWORK, INFO )
00521 *
00522 *     Transform the solution matrix X to a solution of the original
00523 *     system.
00524 *
00525       IF( RCEQU ) THEN
00526          DO 70 J = 1, NRHS
00527             DO 60 I = 1, N
00528                X( I, J ) = S( I )*X( I, J )
00529    60       CONTINUE
00530    70    CONTINUE
00531          DO 80 J = 1, NRHS
00532             FERR( J ) = FERR( J ) / SCOND
00533    80    CONTINUE
00534       END IF
00535 *
00536 *     Set INFO = N+1 if the matrix is singular to working precision.
00537 *
00538       IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
00539      $   INFO = N + 1
00540 *
00541       RETURN
00542 *
00543 *     End of SPBSVX
00544 *
00545       END
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