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version 1.8, 2011/07/22 07:38:09 version 1.9, 2011/11/21 20:43:01
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   *> \brief <b> DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DPBSV + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsv.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsv.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsv.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, KD, LDAB, LDB, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DPBSV computes the solution to a real system of linear equations
   *>    A * X = B,
   *> where A is an N-by-N symmetric positive definite band matrix and X
   *> and B are N-by-NRHS matrices.
   *>
   *> The Cholesky decomposition is used to factor A as
   *>    A = U**T * U,  if UPLO = 'U', or
   *>    A = L * L**T,  if UPLO = 'L',
   *> where U is an upper triangular band matrix, and L is a lower
   *> triangular band matrix, with the same number of superdiagonals or
   *> subdiagonals as A.  The factored form of A is then used to solve the
   *> system of equations A * X = B.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  Upper triangle of A is stored;
   *>          = 'L':  Lower triangle of A is stored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of linear equations, i.e., the order of the
   *>          matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] KD
   *> \verbatim
   *>          KD is INTEGER
   *>          The number of superdiagonals of the matrix A if UPLO = 'U',
   *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>          The number of right hand sides, i.e., the number of columns
   *>          of the matrix B.  NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] AB
   *> \verbatim
   *>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
   *>          On entry, the upper or lower triangle of the symmetric band
   *>          matrix A, stored in the first KD+1 rows of the array.  The
   *>          j-th column of A is stored in the j-th column of the array AB
   *>          as follows:
   *>          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
   *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
   *>          See below for further details.
   *>
   *>          On exit, if INFO = 0, the triangular factor U or L from the
   *>          Cholesky factorization A = U**T*U or A = L*L**T of the band
   *>          matrix A, in the same storage format as A.
   *> \endverbatim
   *>
   *> \param[in] LDAB
   *> \verbatim
   *>          LDAB is INTEGER
   *>          The leading dimension of the array AB.  LDAB >= KD+1.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B.  LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *>          > 0:  if INFO = i, the leading minor of order i of A is not
   *>                positive definite, so the factorization could not be
   *>                completed, and the solution has not been computed.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERsolve
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The band storage scheme is illustrated by the following example, when
   *>  N = 6, KD = 2, and UPLO = 'U':
   *>
   *>  On entry:                       On exit:
   *>
   *>      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
   *>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   *>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   *>
   *>  Similarly, if UPLO = 'L' the format of A is as follows:
   *>
   *>  On entry:                       On exit:
   *>
   *>     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
   *>     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
   *>     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
   *>
   *>  Array elements marked * are not used by the routine.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )        SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.3.1) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )        DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DPBSV computes the solution to a real system of linear equations  
 *     A * X = B,  
 *  where A is an N-by-N symmetric positive definite band matrix and X  
 *  and B are N-by-NRHS matrices.  
 *  
 *  The Cholesky decomposition is used to factor A as  
 *     A = U**T * U,  if UPLO = 'U', or  
 *     A = L * L**T,  if UPLO = 'L',  
 *  where U is an upper triangular band matrix, and L is a lower  
 *  triangular band matrix, with the same number of superdiagonals or  
 *  subdiagonals as A.  The factored form of A is then used to solve the  
 *  system of equations A * X = B.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  UPLO    (input) CHARACTER*1  
 *          = 'U':  Upper triangle of A is stored;  
 *          = 'L':  Lower triangle of A is stored.  
 *  
 *  N       (input) INTEGER  
 *          The number of linear equations, i.e., the order of the  
 *          matrix A.  N >= 0.  
 *  
 *  KD      (input) INTEGER  
 *          The number of superdiagonals of the matrix A if UPLO = 'U',  
 *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.  
 *  
 *  NRHS    (input) INTEGER  
 *          The number of right hand sides, i.e., the number of columns  
 *          of the matrix B.  NRHS >= 0.  
 *  
 *  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)  
 *          On entry, the upper or lower triangle of the symmetric band  
 *          matrix A, stored in the first KD+1 rows of the array.  The  
 *          j-th column of A is stored in the j-th column of the array AB  
 *          as follows:  
 *          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;  
 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).  
 *          See below for further details.  
 *  
 *          On exit, if INFO = 0, the triangular factor U or L from the  
 *          Cholesky factorization A = U**T*U or A = L*L**T of the band  
 *          matrix A, in the same storage format as A.  
 *  
 *  LDAB    (input) INTEGER  
 *          The leading dimension of the array AB.  LDAB >= KD+1.  
 *  
 *  B       (input/output) DOUBLE PRECISION 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.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *          > 0:  if INFO = i, the leading minor of order i of A is not  
 *                positive definite, so the factorization could not be  
 *                completed, and the solution has not been computed.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The band storage scheme is illustrated by the following example, when  
 *  N = 6, KD = 2, and UPLO = 'U':  
 *  
 *  On entry:                       On exit:  
 *  
 *      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46  
 *      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56  
 *     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66  
 *  
 *  Similarly, if UPLO = 'L' the format of A is as follows:  
 *  
 *  On entry:                       On exit:  
 *  
 *     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66  
 *     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *  
 *     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *  
 *  
 *  Array elements marked * are not used by the routine.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. External Functions ..  *     .. External Functions ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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