NAME

SYNOPSIS

  SUBROUTINE DPBSVX( FACT, UPLO, N, NDIAG, NRHS, A, LDA, AF, LDAF, 
 *      EQUED, SCALE, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, 
 *      INFO)
  CHARACTER * 1 FACT, UPLO, EQUED
  INTEGER N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER WORK2(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION A(LDA,*), AF(LDAF,*), SCALE(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
 
  SUBROUTINE DPBSVX_64( FACT, UPLO, N, NDIAG, NRHS, A, LDA, AF, LDAF, 
 *      EQUED, SCALE, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, 
 *      INFO)
  CHARACTER * 1 FACT, UPLO, EQUED
  INTEGER*8 N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER*8 WORK2(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION A(LDA,*), AF(LDAF,*), SCALE(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
 

F95 INTERFACE

  SUBROUTINE PBSVX( FACT, UPLO, [N], NDIAG, [NRHS], A, [LDA], AF, 
 *       [LDAF], EQUED, SCALE, B, [LDB], X, [LDX], RCOND, FERR, BERR, 
 *       [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, UPLO, EQUED
  INTEGER :: N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER, DIMENSION(:) :: WORK2
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: SCALE, FERR, BERR, WORK
  REAL(8), DIMENSION(:,:) :: A, AF, B, X
 
  SUBROUTINE PBSVX_64( FACT, UPLO, [N], NDIAG, [NRHS], A, [LDA], AF, 
 *       [LDAF], EQUED, SCALE, B, [LDB], X, [LDX], RCOND, FERR, BERR, 
 *       [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, UPLO, EQUED
  INTEGER(8) :: N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER(8), DIMENSION(:) :: WORK2
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: SCALE, FERR, BERR, WORK
  REAL(8), DIMENSION(:,:) :: A, AF, B, X
 

C INTERFACE

void dpbsvx(char fact, char uplo, int n, int ndiag, int nrhs, double *a, int lda, double *af, int ldaf, char equed, double *scale, double *b, int ldb, double *x, int ldx, double *rcond, double *ferr, double *berr, int *info);

void dpbsvx_64(char fact, char uplo, long n, long ndiag, long nrhs, double *a, long lda, double *af, long ldaf, char equed, double *scale, double *b, long ldb, double *x, long ldx, double *rcond, double *ferr, double *berr, long *info);

PURPOSE

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.

ARGUMENTS

* FACT (input)

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.

* UPLO (input)

* N (input)

The number of linear equations, i.e., the order of the matrix A. N >= 0.

* NDIAG (input)

The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. NDIAG >= 0.

* NRHS (input)

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

* A (input/output)

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first NDIAG+1 rows of the array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(SCALE)*A*diag(SCALE). The j-th column of A is stored in the j-th column of the array A as follows: if UPLO = 'U', A(NDIAG+1+i-j,j) = A(i,j) for max(1,j-NDIAG)<=i<=j; if UPLO = 'L', A(1+i-j,j) = A(i,j) for j<=i<=min(N,j+NDIAG). See below for further details.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SCALE)*A*diag(SCALE).

* LDA (input)

The leading dimension of the array A. LDA >= NDIAG+1.

* AF (input/output)

If FACT = 'F', then AF 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 A). If EQUED = 'Y', then AF is the factored form of the equilibrated matrix A.

If FACT = 'N', then AF 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 AF 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).

* LDAF (input)

The leading dimension of the array AF. LDAF >= NDIAG+1.

* EQUED (input)

Specifies the form of equilibration that was done.

* SCALE (input/output)

The scale factors for A; not accessed if EQUED = 'N'. SCALE is an input argument if FACT = 'F'; otherwise, SCALE is an output argument. If FACT = 'F' and EQUED = 'Y', each element of SCALE must be positive.

* B (input/output)

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(SCALE) * B.

* LDB (input)

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

* X (output)

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(SCALE))*X.

* LDX (input)

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

* RCOND (output)

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)

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)

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)

dimension(2*N)

* WORK2 (workspace)

* INFO (output)