SPBSVX  - use 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,


SYNOPSIS

       SUBROUTINE SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB,
                          LDAFB, EQUED, S, B, LDB, X, LDX, RCOND,
                          FERR, BERR, WORK, IWORK, INFO )

           CHARACTER      EQUED, FACT, UPLO

           INTEGER        INFO,  KD,  LDAB,  LDAFB,  LDB, LDX, N,
                          NRHS

           REAL           RCOND

           INTEGER        IWORK( * )

           REAL           AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,
                          *  ),  BERR(  *  ),  FERR( * ), S( * ),
                          WORK( * ), 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  posi­
       tive definite band matrix and X and B are N-by-NRHS matri­
       ces.

       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',

       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  fac­
       tored 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 premulti­
       plied 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 fac­
               tored 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 nec­
               essary, 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.

               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  sym­
               metric  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 over­
               written 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').
               =  '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'; other­
               wise, 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 par­
               ticular, 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  solu­
               tion  vector X(j) (the j-th column of the solution
               matrix X).  If XTRUE is the true  solution  corre­
               sponding  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 reli­
               able  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

               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 ille­
               gal 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  com­
               puted. RCOND = 0 is returned.  = N+1: U is nonsin­
               gular, 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.


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