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

NAME

       LAPACK-3 - SPOSVXX use 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  and X and B are N-by-NRHS matrices

SYNOPSIS

       SUBROUTINE SPOSVXX( FACT,  UPLO,  N,  NRHS,  A,  LDA,  AF, LDAF, EQUED, S, B, LDB, X, LDX,
                           RCOND,  RPVGRW,  BERR,   N_ERR_BNDS,   ERR_BNDS_NORM,   ERR_BNDS_COMP,
                           NPARAMS, PARAMS, WORK, IWORK, INFO )

           IMPLICIT        NONE

           CHARACTER       EQUED, FACT, UPLO

           INTEGER         INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS

           REAL            RCOND, RPVGRW

           INTEGER         IWORK( * )

           REAL            A( LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX, * ), WORK( * )

           REAL            S(   *  ),  PARAMS(  *  ),  BERR(  *  ),  ERR_BNDS_NORM(  NRHS,  *  ),
                           ERR_BNDS_COMP( NRHS, * )

PURPOSE

          SPOSVXX 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
          and X and B are N-by-NRHS matrices.
           If requested, both normwise and maximum componentwise error bounds
           are returned. SPOSVXX will return a solution with a tiny
           guaranteed error (O(eps) where eps is the working machine
           precision) unless the matrix is very ill-conditioned, in which
           case a warning is returned. Relevant condition numbers also are
           calculated and returned.
           SPOSVXX accepts user-provided factorizations and equilibration
           factors; see the definitions of the FACT and EQUED options.
           Solving with refinement and using a factorization from a previous
           SPOSVXX call will also produce a solution with either O(eps)
           errors or warnings, but we cannot make that claim for general
           user-provided factorizations and equilibration factors if they
           differ from what SPOSVXX would itself produce.

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 (see argument RCOND).  If the reciprocal of the condition number
           is less than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
           the routine will use iterative refinement to try to get a small
           error and error bounds.  Refinement calculates the residual to at
           least twice the working precision.
           6. If equilibration was used, the matrix X is premultiplied by
           diag(S) so that it solves the original system before
           equilibration.

ARGUMENTS

        Some optional parameters are bundled in the PARAMS array.  These
        settings determine how refinement is performed, but often the
        defaults are acceptable.  If the defaults are acceptable, users
        can pass NPARAMS = 0 which prevents the source code from accessing
        the PARAMS argument.

        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, AF contains the factored form of A.
                If EQUED is not 'N', the matrix A has been
                equilibrated with scaling factors given by S.
                A and AF are not modified.
                = 'N':  The matrix A will be copied to AF and factored.
                = 'E':  The matrix A will be equilibrated if necessary, then
                copied to AF 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.

        A       (input/output) REAL array, dimension (LDA,N)
                On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
                'Y', then A must contain the equilibrated matrix
                diag(S)*A*diag(S).  If UPLO = 'U', the leading N-by-N upper
                triangular part of A contains the upper triangular part of the
                matrix A, and the strictly lower triangular part of A is not
                referenced.  If UPLO = 'L', the leading N-by-N lower triangular
                part of A contains the lower triangular part of the matrix A, and
                the strictly upper triangular part of A is not referenced.  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).

        LDA     (input) INTEGER
                The leading dimension of the array A.  LDA >= max(1,N).

        AF      (input or output) REAL array, dimension (LDAF,N)
                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, in the same storage
                format as A.  If EQUED .ne. 'N', then AF is the factored
                form of the equilibrated matrix diag(S)*A*diag(S).
                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 of the original
                matrix A.
                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) INTEGER
                The leading dimension of the array AF.  LDAF >= max(1,N).

        EQUED   (input or output) CHARACTER*1
                Specifies the form of equilibration that was done.
                = 'N':  No equilibration (always true if FACT = 'N').
                = 'Y':  Both row and column equilibration, 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 row scale factors for A.  If EQUED = 'Y', A is multiplied on
                the left and right by diag(S).  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.  If S is output, each
                element of S is a power of the radix. If S is input, each element
                of S should be a power of the radix to ensure a reliable solution
                and error estimates. Scaling by powers of the radix does not cause
                rounding errors unless the result underflows or overflows.
                Rounding errors during scaling lead to refining with a matrix that
                is not equivalent to the input matrix, producing error estimates
                that may not be reliable.

        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, the N-by-NRHS solution matrix X to the original
                system of equations.  Note that A and B are modified on exit if
                EQUED .ne. 'N', 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
                Reciprocal scaled condition number.  This is an estimate of the
                reciprocal Skeel condition number of the matrix A after
                equilibration (if done).  If this is less than the machine
                precision (in particular, if it is zero), the matrix is singular
                to working precision.  Note that the error may still be small even
                if this number is very small and the matrix appears ill-
                conditioned.

        RPVGRW  (output) REAL
                Reciprocal pivot growth.  On exit, this contains the reciprocal
                pivot growth factor norm(A)/norm(U). The "max absolute element"
                norm is used.  If this is much less than 1, then the stability of
                the LU factorization of the (equilibrated) matrix A could be poor.
                This also means that the solution X, estimated condition numbers,
                and error bounds could be unreliable. If factorization fails with
                0<INFO<=N, then this contains the reciprocal pivot growth factor
                for the leading INFO columns of A.

        BERR    (output) REAL array, dimension (NRHS)
                Componentwise relative backward error.  This is 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).
                N_ERR_BNDS (input) INTEGER
                Number of error bounds to return for each right hand side
                and each type (normwise or componentwise).  See ERR_BNDS_NORM and
                ERR_BNDS_COMP below.

        ERR_BNDS_NORM  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
                       For each right-hand side, this array contains information about
                       various error bounds and condition numbers corresponding to the
                       normwise relative error, which is defined as follows:
                       Normwise relative error in the ith solution vector:
                       max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                       max_j abs(X(j,i))
                       The array is indexed by the type of error information as described
                       below. There currently are up to three pieces of information
                       returned.
                       The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                       right-hand side.
                       The second index in ERR_BNDS_NORM(:,err) contains the following
                       three fields:
                       err = 1 "Trust/don't trust" boolean. Trust the answer if the
                       reciprocal condition number is less than the threshold
                       sqrt(n) * slamch('Epsilon').
                       err = 2 "Guaranteed" error bound: The estimated forward error,
                       almost certainly within a factor of 10 of the true error
                       so long as the next entry is greater than the threshold
                       sqrt(n) * slamch('Epsilon'). This error bound should only
                       be trusted if the previous boolean is true.
                       err = 3  Reciprocal condition number: Estimated normwise
                       reciprocal condition number.  Compared with the threshold
                       sqrt(n) * slamch('Epsilon') to determine if the error
                       estimate is "guaranteed". These reciprocal condition
                       numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                       appropriately scaled matrix Z.
                       Let Z = S*A, where S scales each row by a power of the
                       radix so all absolute row sums of Z are approximately 1.
                       See Lapack Working Note 165 for further details and extra
                       cautions.

        ERR_BNDS_COMP  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
                       For each right-hand side, this array contains information about
                       various error bounds and condition numbers corresponding to the
                       componentwise relative error, which is defined as follows:
                       Componentwise relative error in the ith solution vector:
                       abs(XTRUE(j,i) - X(j,i))
                       max_j ----------------------
                       abs(X(j,i))
                       The array is indexed by the right-hand side i (on which the
                       componentwise relative error depends), and the type of error
                       information as described below. There currently are up to three
                       pieces of information returned for each right-hand side. If
                       componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                       ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
                       the first (:,N_ERR_BNDS) entries are returned.
                       The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                       right-hand side.
                       The second index in ERR_BNDS_COMP(:,err) contains the following
                       three fields:
                       err = 1 "Trust/don't trust" boolean. Trust the answer if the
                       reciprocal condition number is less than the threshold
                       sqrt(n) * slamch('Epsilon').
                       err = 2 "Guaranteed" error bound: The estimated forward error,
                       almost certainly within a factor of 10 of the true error
                       so long as the next entry is greater than the threshold
                       sqrt(n) * slamch('Epsilon'). This error bound should only
                       be trusted if the previous boolean is true.
                       err = 3  Reciprocal condition number: Estimated componentwise
                       reciprocal condition number.  Compared with the threshold
                       sqrt(n) * slamch('Epsilon') to determine if the error
                       estimate is "guaranteed". These reciprocal condition
                       numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                       appropriately scaled matrix Z.
                       Let Z = S*(A*diag(x)), where x is the solution for the
                       current right-hand side and S scales each row of
                       A*diag(x) by a power of the radix so all absolute row
                       sums of Z are approximately 1.
                       See Lapack Working Note 165 for further details and extra
                       cautions.
                       NPARAMS (input) INTEGER
                       Specifies the number of parameters set in PARAMS.  If .LE. 0, the
                       PARAMS array is never referenced and default values are used.

        PARAMS  (input / output) REAL array, dimension NPARAMS
                Specifies algorithm parameters.  If an entry is .LT. 0.0, then
                that entry will be filled with default value used for that
                parameter.  Only positions up to NPARAMS are accessed; defaults
                are used for higher-numbered parameters.
                PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                refinement or not.
                Default: 1.0
                = 0.0 : No refinement is performed, and no error bounds are
                computed.
                = 1.0 : Use the double-precision refinement algorithm,
                possibly with doubled-single computations if the
                compilation environment does not support DOUBLE
                PRECISION.
                (other values are reserved for future use)
                PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                computations allowed for refinement.
                Default: 10
                Aggressive: Set to 100 to permit convergence using approximate
                factorizations or factorizations other than LU. If
                the factorization uses a technique other than
                Gaussian elimination, the guarantees in
                err_bnds_norm and err_bnds_comp may no longer be
                trustworthy.
                PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                will attempt to find a solution with small componentwise
                relative error in the double-precision algorithm.  Positive
                is true, 0.0 is false.
                Default: 1.0 (attempt componentwise convergence)

        WORK    (workspace) REAL array, dimension (4*N)

        IWORK   (workspace) INTEGER array, dimension (N)

        INFO    (output) INTEGER
                = 0:  Successful exit. The solution to every right-hand side is
                guaranteed.
                < 0:  If INFO = -i, the i-th argument had an illegal value
                > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
                has been completed, but the factor U is exactly singular, so
                the solution and error bounds could not be computed. RCOND = 0
                is returned.
                = N+J: The solution corresponding to the Jth right-hand side is
                not guaranteed. The solutions corresponding to other right-
                hand sides K with K > J may not be guaranteed as well, but
                only the first such right-hand side is reported. If a small
                componentwise error is not requested (PARAMS(3) = 0.0) then
                the Jth right-hand side is the first with a normwise error
                bound that is not guaranteed (the smallest J such
                that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
                the Jth right-hand side is the first with either a normwise or
                componentwise error bound that is not guaranteed (the smallest
                J such that either ERR_BNDS_NORM(J,1) = 0.0 or
                ERR_BNDS_COMP(J,1) = 0.0). See the definition of
                ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
                about all of the right-hand sides check ERR_BNDS_NORM or
                ERR_BNDS_COMP.

    LAPACK driver routine (version 3.2.2)   April 2011                           SPOSVXX(3lapack)