Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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,

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.

 LAPACK driver routine (version 3.3.1)      April 2011                            SPBSVX(3lapack)