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 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.