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.

 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.

FACT

          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

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

KD

          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

          NRHS is INTEGER


          The number of right-hand sides, i.e., the number of columns


          of the matrices B and X.  NRHS >= 0.

AB

          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

          LDAB is INTEGER


          The leading dimension of the array A.  LDAB >= KD+1.

AFB

          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

          LDAFB is INTEGER


          The leading dimension of the array AFB.  LDAFB >= KD+1.

EQUED

          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

          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

          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

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          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

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(1,N).

RCOND

          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

          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

          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

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

April 2012

  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.