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version 1.7, 2010/12/21 13:53:24 version 1.8, 2011/11/21 20:42:50
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   *> \brief <b> DGBSV computes the solution to system of linear equations A * X = B for GB matrices</b> (simple driver)
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DGBSV + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsv.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsv.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsv.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * )
   *       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGBSV computes the solution to a real system of linear equations
   *> A * X = B, where A is a band matrix of order N with KL subdiagonals
   *> and KU superdiagonals, and X and B are N-by-NRHS matrices.
   *>
   *> The LU decomposition with partial pivoting and row interchanges is
   *> used to factor A as A = L * U, where L is a product of permutation
   *> and unit lower triangular matrices with KL subdiagonals, and U is
   *> upper triangular with KL+KU superdiagonals.  The factored form of A
   *> is then used to solve the system of equations A * X = B.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \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] KL
   *> \verbatim
   *>          KL is INTEGER
   *>          The number of subdiagonals within the band of A.  KL >= 0.
   *> \endverbatim
   *>
   *> \param[in] KU
   *> \verbatim
   *>          KU is INTEGER
   *>          The number of superdiagonals within the band of A.  KU >= 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 matrix A in band storage, in rows KL+1 to
   *>          2*KL+KU+1; rows 1 to KL of the array need not be set.
   *>          The j-th column of A is stored in the j-th column of the
   *>          array AB as follows:
   *>          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
   *>          On exit, details of the factorization: U is stored as an
   *>          upper triangular band matrix with KL+KU superdiagonals in
   *>          rows 1 to KL+KU+1, and the multipliers used during the
   *>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
   *>          See below for further details.
   *> \endverbatim
   *>
   *> \param[in] LDAB
   *> \verbatim
   *>          LDAB is INTEGER
   *>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
   *> \endverbatim
   *>
   *> \param[out] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N)
   *>          The pivot indices that define the permutation matrix P;
   *>          row i of the matrix was interchanged with row IPIV(i).
   *> \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, U(i,i) is exactly zero.  The factorization
   *>                has been completed, but the factor U is exactly
   *>                singular, 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 doubleGBsolve
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The band storage scheme is illustrated by the following example, when
   *>  M = N = 6, KL = 2, KU = 1:
   *>
   *>  On entry:                       On exit:
   *>
   *>      *    *    *    +    +    +       *    *    *   u14  u25  u36
   *>      *    *    +    +    +    +       *    *   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
   *>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   *>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
   *>
   *>  Array elements marked * are not used by the routine; elements marked
   *>  + need not be set on entry, but are required by the routine to store
   *>  elements of U because of fill-in resulting from the row interchanges.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )        SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- 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..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS        INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
Line 13 Line 175
       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )        DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGBSV computes the solution to a real system of linear equations  
 *  A * X = B, where A is a band matrix of order N with KL subdiagonals  
 *  and KU superdiagonals, and X and B are N-by-NRHS matrices.  
 *  
 *  The LU decomposition with partial pivoting and row interchanges is  
 *  used to factor A as A = L * U, where L is a product of permutation  
 *  and unit lower triangular matrices with KL subdiagonals, and U is  
 *  upper triangular with KL+KU superdiagonals.  The factored form of A  
 *  is then used to solve the system of equations A * X = B.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The number of linear equations, i.e., the order of the  
 *          matrix A.  N >= 0.  
 *  
 *  KL      (input) INTEGER  
 *          The number of subdiagonals within the band of A.  KL >= 0.  
 *  
 *  KU      (input) INTEGER  
 *          The number of superdiagonals within the band of A.  KU >= 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 matrix A in band storage, in rows KL+1 to  
 *          2*KL+KU+1; rows 1 to KL of the array need not be set.  
 *          The j-th column of A is stored in the j-th column of the  
 *          array AB as follows:  
 *          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)  
 *          On exit, details of the factorization: U is stored as an  
 *          upper triangular band matrix with KL+KU superdiagonals in  
 *          rows 1 to KL+KU+1, and the multipliers used during the  
 *          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.  
 *          See below for further details.  
 *  
 *  LDAB    (input) INTEGER  
 *          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.  
 *  
 *  IPIV    (output) INTEGER array, dimension (N)  
 *          The pivot indices that define the permutation matrix P;  
 *          row i of the matrix was interchanged with row IPIV(i).  
 *  
 *  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, U(i,i) is exactly zero.  The factorization  
 *                has been completed, but the factor U is exactly  
 *                singular, and the solution has not been computed.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The band storage scheme is illustrated by the following example, when  
 *  M = N = 6, KL = 2, KU = 1:  
 *  
 *  On entry:                       On exit:  
 *  
 *      *    *    *    +    +    +       *    *    *   u14  u25  u36  
 *      *    *    +    +    +    +       *    *   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  
 *     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *  
 *     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *  
 *  
 *  Array elements marked * are not used by the routine; elements marked  
 *  + need not be set on entry, but are required by the routine to store  
 *  elements of U because of fill-in resulting from the row interchanges.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. External Subroutines ..  *     .. External Subroutines ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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