Integrating Doxygen in comments

1 files changed, 337 insertions, 221 deletions

diff --git a/SRC/spbsvx.f b/SRC/spbsvx.f
index 672fa6f4..48551040 100644
--- a/SRC/spbsvx.f
+++ b/SRC/spbsvx.f
@@ -1,11 +1,346 @@
+*> \brief <b> SPBSVX 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/ 
+*
+*  Definition
+*  ==========
+*
+*       SUBROUTINE SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
+*                          EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
+*                          WORK, IWORK, INFO )
+* 
+*       .. Scalar Arguments ..
+*       CHARACTER          EQUED, FACT, UPLO
+*       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
+*       REAL               RCOND
+*       ..
+*       .. Array Arguments ..
+*       INTEGER            IWORK( * )
+*       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
+*      $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
+*      $                   X( LDX, * )
+*       ..
+*  
+*  Purpose
+*  =======
+*
+*>\details \b Purpose:
+*>\verbatim
+*>
+*> SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
+*> compute 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.
+*>
+*> Error bounds on the solution and a condition estimate are also
+*> provided.
+*>
+*> 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 Cholesky decomposition is used to
+*>    factor the matrix A (after equilibration if FACT = 'E') 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.
+*>
+*> 3. If the leading i-by-i principal minor is not positive definite,
+*>    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.
+*>
+*> 4. The system of equations is solved for X using the factored form
+*>    of A.
+*>
+*> 5. Iterative refinement is applied to improve the computed solution
+*>    matrix and calculate error bounds and backward error estimates
+*>    for it.
+*>
+*> 6. If equilibration was used, the matrix X is premultiplied by
+*>    diag(S) so that it solves the original system before
+*>    equilibration.
+*>
+*>\endverbatim
+*
+*  Arguments
+*  =========
+*
+*> \param[in] FACT
+*> \verbatim
+*>          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, AFB contains the factored form of A.
+*>                  If EQUED = 'Y', the matrix A has been equilibrated
+*>                  with scaling factors given by S.  AB and AFB will not
+*>                  be modified.
+*>          = 'N':  The matrix A will be copied to AFB and factored.
+*>          = 'E':  The matrix A will be equilibrated if necessary, then
+*>                  copied to AFB and factored.
+*> \endverbatim
+*>
+*> \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 matrices B and X.  NRHS >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] AB
+*> \verbatim
+*>          AB is REAL 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, except
+*>          if FACT = 'F' and EQUED = 'Y', then A must contain the
+*>          equilibrated matrix diag(S)*A*diag(S).  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.
+*> \endverbatim
+*> \verbatim
+*>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
+*>          diag(S)*A*diag(S).
+*> \endverbatim
+*>
+*> \param[in] LDAB
+*> \verbatim
+*>          LDAB is INTEGER
+*>          The leading dimension of the array A.  LDAB >= KD+1.
+*> \endverbatim
+*>
+*> \param[in,out] AFB
+*> \verbatim
+*>          AFB is or output) REAL array, dimension (LDAFB,N)
+*>          If FACT = 'F', then AFB is an input argument and on entry
+*>          contains 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 (see AB).  If EQUED = 'Y',
+*>          then AFB is the factored form of the equilibrated matrix A.
+*> \endverbatim
+*> \verbatim
+*>          If FACT = 'N', then AFB is an output argument and on exit
+*>          returns the triangular factor U or L from the Cholesky
+*>          factorization A = U**T*U or A = L*L**T.
+*> \endverbatim
+*> \verbatim
+*>          If FACT = 'E', then AFB is an output argument and on exit
+*>          returns the triangular factor U or L from the Cholesky
+*>          factorization A = U**T*U or A = L*L**T of the equilibrated
+*>          matrix A (see the description of A for the form of the
+*>          equilibrated matrix).
+*> \endverbatim
+*>
+*> \param[in] LDAFB
+*> \verbatim
+*>          LDAFB is INTEGER
+*>          The leading dimension of the array AFB.  LDAFB >= KD+1.
+*> \endverbatim
+*>
+*> \param[in,out] EQUED
+*> \verbatim
+*>          EQUED is or output) CHARACTER*1
+*>          Specifies the form of equilibration that was done.
+*>          = 'N':  No equilibration (always true if FACT = 'N').
+*>          = 'Y':  Equilibration was done, 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.
+*> \endverbatim
+*>
+*> \param[in,out] S
+*> \verbatim
+*>          S is or output) REAL array, dimension (N)
+*>          The scale factors for A; not accessed if EQUED = 'N'.  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.
+*> \endverbatim
+*>
+*> \param[in,out] B
+*> \verbatim
+*>          B is REAL 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.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*>          LDB is INTEGER
+*>          The leading dimension of the array B.  LDB >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] X
+*> \verbatim
+*>          X is REAL array, dimension (LDX,NRHS)
+*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
+*>          the original system of equations.  Note that if EQUED = 'Y',
+*>          A and B are modified on exit, and the solution to the
+*>          equilibrated system is inv(diag(S))*X.
+*> \endverbatim
+*>
+*> \param[in] LDX
+*> \verbatim
+*>          LDX is INTEGER
+*>          The leading dimension of the array X.  LDX >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] RCOND
+*> \verbatim
+*>          RCOND is REAL
+*>          The estimate of the reciprocal condition number of the matrix
+*>          A after equilibration (if done).  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.
+*> \endverbatim
+*>
+*> \param[out] FERR
+*> \verbatim
+*>          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.
+*> \endverbatim
+*>
+*> \param[out] BERR
+*> \verbatim
+*>          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).
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*>          WORK is REAL array, dimension (3*N)
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*>          IWORK is INTEGER array, dimension (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, and i is
+*>                <= N:  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. RCOND = 0 is returned.
+*>                = N+1: U 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.
+*> \endverbatim
+*>
+*
+*  Authors
+*  =======
+*
+*> \author Univ. of Tennessee 
+*> \author Univ. of California Berkeley 
+*> \author Univ. of Colorado Denver 
+*> \author NAG Ltd. 
+*
+*> \date November 2011
+*
+*> \ingroup realOTHERsolve
+*
+*
+*  Further Details
+*  ===============
+*>\details \b Further \b Details
+*> \verbatim
+*>
+*>  The band storage scheme is illustrated by the following example, when
+*>  N = 6, KD = 2, and UPLO = 'U':
+*>
+*>  Two-dimensional storage of the symmetric matrix A:
+*>
+*>     a11  a12  a13
+*>          a22  a23  a24
+*>               a33  a34  a35
+*>                    a44  a45  a46
+*>                         a55  a56
+*>     (aij=conjg(aji))         a66
+*>
+*>  Band storage of the upper triangle of A:
+*>
+*>      *    *   a13  a24  a35  a46
+*>      *   a12  a23  a34  a45  a56
+*>     a11  a22  a33  a44  a55  a66
+*>
+*>  Similarly, if UPLO = 'L' the format of A is as follows:
+*>
+*>     a11  a22  a33  a44  a55  a66
+*>     a21  a32  a43  a54  a65   *
+*>     a31  a42  a53  a64   *    *
+*>
+*>  Array elements marked * are not used by the routine.
+*>
+*> \endverbatim
+*>
+*  =====================================================================
       SUBROUTINE SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
      $                   EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
      $                   WORK, IWORK, INFO )
 *
-*  -- LAPACK driver routine (version 3.3.1) --
+*  -- LAPACK solve routine (version 3.3.1) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-*  -- April 2011                                                      --
+*     November 2011
 *
 *     .. Scalar Arguments ..
       CHARACTER          EQUED, FACT, UPLO
@@ -19,225 +354,6 @@
      $                   X( LDX, * )
 *     ..
 *
-*  Purpose
-*  =======
-*
-*  SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
-*  compute 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.
-*
-*  Error bounds on the solution and a condition estimate are also
-*  provided.
-*
-*  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 Cholesky decomposition is used to
-*     factor the matrix A (after equilibration if FACT = 'E') 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.
-*
-*  3. If the leading i-by-i principal minor is not positive definite,
-*     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.
-*
-*  4. The system of equations is solved for X using the factored form
-*     of A.
-*
-*  5. Iterative refinement is applied to improve the computed solution
-*     matrix and calculate error bounds and backward error estimates
-*     for it.
-*
-*  6. If equilibration was used, the matrix X is premultiplied by
-*     diag(S) so that it solves the original system before
-*     equilibration.
-*
-*  Arguments
-*  =========
-*
-*  FACT    (input) 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, AFB contains the factored form of A.
-*                  If EQUED = 'Y', the matrix A has been equilibrated
-*                  with scaling factors given by S.  AB and AFB will not
-*                  be modified.
-*          = 'N':  The matrix A will be copied to AFB and factored.
-*          = 'E':  The matrix A will be equilibrated if necessary, then
-*                  copied to AFB and factored.
-*
-*  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 matrices B and X.  NRHS >= 0.
-*
-*  AB      (input/output) REAL 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, except
-*          if FACT = 'F' and EQUED = 'Y', then A must contain the
-*          equilibrated matrix diag(S)*A*diag(S).  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 FACT = 'E' and EQUED = 'Y', A is overwritten by
-*          diag(S)*A*diag(S).
-*
-*  LDAB    (input) INTEGER
-*          The leading dimension of the array A.  LDAB >= KD+1.
-*
-*  AFB     (input or output) REAL array, dimension (LDAFB,N)
-*          If FACT = 'F', then AFB is an input argument and on entry
-*          contains 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 (see AB).  If EQUED = 'Y',
-*          then AFB is the factored form of the equilibrated matrix A.
-*
-*          If FACT = 'N', then AFB is an output argument and on exit
-*          returns the triangular factor U or L from the Cholesky
-*          factorization A = U**T*U or A = L*L**T.
-*
-*          If FACT = 'E', then AFB is an output argument and on exit
-*          returns the triangular factor U or L from the Cholesky
-*          factorization A = U**T*U or A = L*L**T of the equilibrated
-*          matrix A (see the description of A for the form of the
-*          equilibrated matrix).
-*
-*  LDAFB   (input) INTEGER
-*          The leading dimension of the array AFB.  LDAFB >= KD+1.
-*
-*  EQUED   (input or output) CHARACTER*1
-*          Specifies the form of equilibration that was done.
-*          = 'N':  No equilibration (always true if FACT = 'N').
-*          = 'Y':  Equilibration was done, 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.
-*
-*  S       (input or output) REAL array, dimension (N)
-*          The scale factors for A; not accessed if EQUED = 'N'.  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.
-*
-*  B       (input/output) REAL 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.
-*
-*  LDB     (input) INTEGER
-*          The leading dimension of the array B.  LDB >= max(1,N).
-*
-*  X       (output) REAL array, dimension (LDX,NRHS)
-*          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
-*          the original system of equations.  Note that if EQUED = 'Y',
-*          A and B are modified on exit, and the solution to the
-*          equilibrated system is inv(diag(S))*X.
-*
-*  LDX     (input) INTEGER
-*          The leading dimension of the array X.  LDX >= max(1,N).
-*
-*  RCOND   (output) REAL
-*          The estimate of the reciprocal condition number of the matrix
-*          A after equilibration (if done).  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.
-*
-*  FERR    (output) 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.
-*
-*  BERR    (output) 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).
-*
-*  WORK    (workspace) REAL array, dimension (3*N)
-*
-*  IWORK   (workspace) INTEGER array, dimension (N)
-*
-*  INFO    (output) INTEGER
-*          = 0:  successful exit
-*          < 0:  if INFO = -i, the i-th argument had an illegal value
-*          > 0:  if INFO = i, and i is
-*                <= N:  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. RCOND = 0 is returned.
-*                = N+1: U 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.
-*
-*  Further Details
-*  ===============
-*
-*  The band storage scheme is illustrated by the following example, when
-*  N = 6, KD = 2, and UPLO = 'U':
-*
-*  Two-dimensional storage of the symmetric matrix A:
-*
-*     a11  a12  a13
-*          a22  a23  a24
-*               a33  a34  a35
-*                    a44  a45  a46
-*                         a55  a56
-*     (aij=conjg(aji))         a66
-*
-*  Band storage of the upper triangle of A:
-*
-*      *    *   a13  a24  a35  a46
-*      *   a12  a23  a34  a45  a56
-*     a11  a22  a33  a44  a55  a66
-*
-*  Similarly, if UPLO = 'L' the format of A is as follows:
-*
-*     a11  a22  a33  a44  a55  a66
-*     a21  a32  a43  a54  a65   *
-*     a31  a42  a53  a64   *    *
-*
-*  Array elements marked * are not used by the routine.
-*
 *  =====================================================================
 *
 *     .. Parameters ..