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 DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X,  LDX,  RCOND,  FERR,
                          BERR, WORK, IWORK, INFO )

           CHARACTER      EQUED, FACT, UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           DOUBLE         PRECISION RCOND

           INTEGER        IWORK( * )

           DOUBLE         PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ), S( * ),
                          WORK( * ), X( LDX, * )

PURPOSE

       DPPSVX 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 matrix stored in
        packed format 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 matrix and L is a lower triangular
           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, AFP contains the factored form of A.
                If EQUED = 'Y', the matrix A has been equilibrated
                with scaling factors given by S.  AP and AFP will not
                be modified.
                = 'N':  The matrix A will be copied to AFP and factored.
                = 'E':  The matrix A will be equilibrated if necessary, then
                copied to AFP 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.

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

        AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
                On entry, the upper or lower triangle of the symmetric matrix
                A, packed columnwise in a linear 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
                array AP as follows:
                if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
                See below for further details.  A is not modified if
                FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
                On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
                diag(S)*A*diag(S).

        AFP     (input or output) DOUBLE PRECISION array, dimension
                (N*(N+1)/2)
                If FACT = 'F', then AFP 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, in the same storage
                format as A.  If EQUED .ne. 'N', then AFP is the factored
                form of the equilibrated matrix A.
                If FACT = 'N', then AFP 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 original
                matrix A.
                If FACT = 'E', then AFP 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 AP for the form of the
                equilibrated matrix).

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

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

        RCOND   (output) 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) 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) 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    (workspace) DOUBLE PRECISION 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 packed storage scheme is illustrated by the following example
        when N = 4, UPLO = 'U':
        Two-dimensional storage of the symmetric matrix A:
           a11 a12 a13 a14
               a22 a23 a24
                   a33 a34     (aij = conjg(aji))
                       a44
        Packed storage of the upper triangle of A:
        AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

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