DPBSVX(3)      LAPACK routine of NEC Numeric Library Collection      DPBSVX(3)



NAME
       DPBSVX

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



PURPOSE
            DPBSVX 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)
                     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.

           UPLO      (input)
                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N         (input)
                     N is INTEGER
                     The number of linear equations, i.e., the order of the
                     matrix A.  N >= 0.

           KD        (input)
                     KD is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           NRHS      (input)
                     NRHS is INTEGER
                     The number of right-hand sides, i.e., the number of columns
                     of the matrices B and X.  NRHS >= 0.

           AB        (input/output)
                     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, 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)
                     LDAB is INTEGER
                     The leading dimension of the array A.  LDAB >= KD+1.

           AFB       (input/output)
                     AFB is DOUBLE PRECISION 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)
                     LDAFB is INTEGER
                     The leading dimension of the array AFB.  LDAFB >= KD+1.

           EQUED     (input/output)
                     EQUED is 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/output)
                     S is DOUBLE PRECISION 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)
                     B is DOUBLE PRECISION 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)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           X         (output)
                     X is DOUBLE PRECISION 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)
                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(1,N).

           RCOND     (output)
                     RCOND is DOUBLE PRECISION
                     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)
                     FERR is DOUBLE PRECISION 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)
                     BERR is DOUBLE PRECISION 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      (output)
                     WORK is DOUBLE PRECISION array, dimension (3*N)

           IWORK     (output)
                     IWORK is INTEGER array, dimension (N)

           INFO      (output)
                     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.






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.



LAPACK routine                  31 October 2017                      DPBSVX(3)