Provided by: liblapack-doc_3.8.0-2_all bug

NAME

       realGBsolve

SYNOPSIS

   Functions
       subroutine sgbsv (N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
            SGBSV computes the solution to system of linear equations A * X = B for GB matrices
           (simple driver)
       subroutine sgbsvx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C,
           B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
            SGBSVX computes the solution to system of linear equations A * X = B for GB matrices
       subroutine sgbsvxx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C,
           B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
           NPARAMS, PARAMS, WORK, IWORK, INFO)
            SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Detailed Description

       This is the group of real solve driver functions for GB matrices

Function Documentation

   subroutine sgbsv (integer N, integer KL, integer KU, integer NRHS, real, dimension( ldab, * )
       AB, integer LDAB, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB,
       integer INFO)
        SGBSV computes the solution to system of linear equations A * X = B for GB matrices
       (simple driver)

       Purpose:

            SGBSV computes the solution to a real system of linear equations
            A * X = B, where A is a band matrix of order N with KL subdiagonals
            and KU superdiagonals, and X and B are N-by-NRHS matrices.

            The LU decomposition with partial pivoting and row interchanges is
            used to factor A as A = L * U, where L is a product of permutation
            and unit lower triangular matrices with KL subdiagonals, and U is
            upper triangular with KL+KU superdiagonals.  The factored form of A
            is then used to solve the system of equations A * X = B.

       Parameters:
           N

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

           KL

                     KL is INTEGER
                     The number of subdiagonals within the band of A.  KL >= 0.

           KU

                     KU is INTEGER
                     The number of superdiagonals within the band of A.  KU >= 0.

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           AB

                     AB is REAL array, dimension (LDAB,N)
                     On entry, the matrix A in band storage, in rows KL+1 to
                     2*KL+KU+1; rows 1 to KL of the array need not be set.
                     The j-th column of A is stored in the j-th column of the
                     array AB as follows:
                     AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
                     On exit, details of the factorization: U is stored as an
                     upper triangular band matrix with KL+KU superdiagonals in
                     rows 1 to KL+KU+1, and the multipliers used during the
                     factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
                     See below for further details.

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     The pivot indices that define the permutation matrix P;
                     row i of the matrix was interchanged with row IPIV(i).

           B

                     B is REAL array, dimension (LDB,NRHS)
                     On entry, the N-by-NRHS right hand side matrix B.
                     On exit, if INFO = 0, the N-by-NRHS solution matrix X.

           LDB

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
                           has been completed, but the factor U is exactly
                           singular, and the solution has not been computed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:

             The band storage scheme is illustrated by the following example, when
             M = N = 6, KL = 2, KU = 1:

             On entry:                       On exit:

                 *    *    *    +    +    +       *    *    *   u14  u25  u36
                 *    *    +    +    +    +       *    *   u13  u24  u35  u46
                 *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
                a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
                a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
                a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

             Array elements marked * are not used by the routine; elements marked
             + need not be set on entry, but are required by the routine to store
             elements of U because of fill-in resulting from the row interchanges.

   subroutine sgbsvx (character FACT, character TRANS, integer N, integer KL, integer KU, integer
       NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB,
       integer LDAFB, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R,
       real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, *
       ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real,
       dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
        SGBSVX computes the solution to system of linear equations A * X = B for GB matrices

       Purpose:

            SGBSVX uses the LU factorization to compute the solution to a real
            system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
            where A is a band matrix of order N with KL subdiagonals and KU
            superdiagonals, 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 by this subroutine:

            1. If FACT = 'E', real scaling factors are computed to equilibrate
               the system:
                  TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
                  TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
                  TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
               or diag(C)*B (if TRANS = 'T' or 'C').

            2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
               matrix A (after equilibration if FACT = 'E') as
                  A = L * U,
               where L is a product of permutation and unit lower triangular
               matrices with KL subdiagonals, and U is upper triangular with
               KL+KU superdiagonals.

            3. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
               that it solves the original system before equilibration.

       Parameters:
           FACT

                     FACT is 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 and IPIV contain the factored form of
                             A.  If EQUED is not 'N', the matrix A has been
                             equilibrated with scaling factors given by R and C.
                             AB, AFB, and IPIV are not 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.

           TRANS

                     TRANS is CHARACTER*1
                     Specifies the form of the system of equations.
                     = 'N':  A * X = B     (No transpose)
                     = 'T':  A**T * X = B  (Transpose)
                     = 'C':  A**H * X = B  (Transpose)

           N

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

           KL

                     KL is INTEGER
                     The number of subdiagonals within the band of A.  KL >= 0.

           KU

                     KU is INTEGER
                     The number of superdiagonals within the band of A.  KU >= 0.

           NRHS

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

           AB

                     AB is REAL array, dimension (LDAB,N)
                     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
                     The j-th column of A is stored in the j-th column of the
                     array AB as follows:
                     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

                     If FACT = 'F' and EQUED is not 'N', then A must have been
                     equilibrated by the scaling factors in R and/or C.  AB is not
                     modified if FACT = 'F' or 'N', or if FACT = 'E' and
                     EQUED = 'N' on exit.

                     On exit, if EQUED .ne. 'N', A is scaled as follows:
                     EQUED = 'R':  A := diag(R) * A
                     EQUED = 'C':  A := A * diag(C)
                     EQUED = 'B':  A := diag(R) * A * diag(C).

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KL+KU+1.

           AFB

                     AFB is REAL array, dimension (LDAFB,N)
                     If FACT = 'F', then AFB is an input argument and on entry
                     contains details of the LU factorization of the band matrix
                     A, as computed by SGBTRF.  U is stored as an upper triangular
                     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
                     and the multipliers used during the factorization are stored
                     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
                     the factored form of the equilibrated matrix A.

                     If FACT = 'N', then AFB is an output argument and on exit
                     returns details of the LU factorization of A.

                     If FACT = 'E', then AFB is an output argument and on exit
                     returns details of the LU factorization of the equilibrated
                     matrix A (see the description of AB for the form of the
                     equilibrated matrix).

           LDAFB

                     LDAFB is INTEGER
                     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     If FACT = 'F', then IPIV is an input argument and on entry
                     contains the pivot indices from the factorization A = L*U
                     as computed by SGBTRF; row i of the matrix was interchanged
                     with row IPIV(i).

                     If FACT = 'N', then IPIV is an output argument and on exit
                     contains the pivot indices from the factorization A = L*U
                     of the original matrix A.

                     If FACT = 'E', then IPIV is an output argument and on exit
                     contains the pivot indices from the factorization A = L*U
                     of the equilibrated matrix A.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies the form of equilibration that was done.
                     = 'N':  No equilibration (always true if FACT = 'N').
                     = 'R':  Row equilibration, i.e., A has been premultiplied by
                             diag(R).
                     = 'C':  Column equilibration, i.e., A has been postmultiplied
                             by diag(C).
                     = 'B':  Both row and column equilibration, i.e., A has been
                             replaced by diag(R) * A * diag(C).
                     EQUED is an input argument if FACT = 'F'; otherwise, it is an
                     output argument.

           R

                     R is REAL array, dimension (N)
                     The row scale factors for A.  If EQUED = 'R' or 'B', A is
                     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                     is not accessed.  R is an input argument if FACT = 'F';
                     otherwise, R is an output argument.  If FACT = 'F' and
                     EQUED = 'R' or 'B', each element of R must be positive.

           C

                     C is REAL array, dimension (N)
                     The column scale factors for A.  If EQUED = 'C' or 'B', A is
                     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                     is not accessed.  C is an input argument if FACT = 'F';
                     otherwise, C is an output argument.  If FACT = 'F' and
                     EQUED = 'C' or 'B', each element of C must be positive.

           B

                     B is REAL array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit,
                     if EQUED = 'N', B is not modified;
                     if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
                     diag(R)*B;
                     if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
                     overwritten by diag(C)*B.

           LDB

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

           X

                     X is 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 A and B are
                     modified on exit if EQUED .ne. 'N', and the solution to the
                     equilibrated system is inv(diag(C))*X if TRANS = 'N' and
                     EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
                     and EQUED = 'R' or 'B'.

           LDX

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

           RCOND

                     RCOND is 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

                     FERR is 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

                     BERR is 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

                     WORK is REAL array, dimension (3*N)
                     On exit, WORK(1) contains the reciprocal pivot growth
                     factor norm(A)/norm(U). The "max absolute element" norm is
                     used. If WORK(1) 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, condition
                     estimator RCOND, and forward error bound FERR could be
                     unreliable. If factorization fails with 0<INFO<=N, then
                     WORK(1) contains the reciprocal pivot growth factor for the
                     leading INFO columns of A.

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, and i is
                           <= N:  U(i,i) 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+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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           April 2012

   subroutine sgbsvxx (character FACT, character TRANS, integer N, integer KL, integer KU,
       integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * )
       AFB, integer LDAFB, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R,
       real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , *
       ) X, integer LDX, real RCOND, real RPVGRW, real, dimension( * ) BERR, integer N_ERR_BNDS,
       real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP,
       integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer,
       dimension( * ) IWORK, integer INFO)
        SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

       Purpose:

               SGBSVXX uses the LU factorization to compute the solution to a
               real system of linear equations  A * X = B,  where A is an
               N-by-N matrix and X and B are N-by-NRHS matrices.

               If requested, both normwise and maximum componentwise error bounds
               are returned. SGBSVXX 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.

               SGBSVXX 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
               SGBSVXX 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 SGBSVXX would itself produce.

       Description:

               The following steps are performed:

               1. If FACT = 'E', real scaling factors are computed to equilibrate
               the system:

                 TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
                 TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
                 TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
               or diag(C)*B (if TRANS = 'T' or 'C').

               2. If FACT = 'N' or 'E', the LU decomposition is used to factor
               the matrix A (after equilibration if FACT = 'E') as

                 A = P * L * U,

               where P is a permutation matrix, L is a unit lower triangular
               matrix, and U is upper triangular.

               3. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
               that it solves the original system before equilibration.

                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.

       Parameters:
           FACT

                     FACT is 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 and IPIV contain the factored form of A.
                          If EQUED is not 'N', the matrix A has been
                          equilibrated with scaling factors given by R and C.
                          A, AF, and IPIV 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.

           TRANS

                     TRANS is CHARACTER*1
                Specifies the form of the system of equations:
                  = 'N':  A * X = B     (No transpose)
                  = 'T':  A**T * X = B  (Transpose)
                  = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)

           N

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

           KL

                     KL is INTEGER
                The number of subdiagonals within the band of A.  KL >= 0.

           KU

                     KU is INTEGER
                The number of superdiagonals within the band of A.  KU >= 0.

           NRHS

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

           AB

                     AB is REAL array, dimension (LDAB,N)
                On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
                The j-th column of A is stored in the j-th column of the
                array AB as follows:
                AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

                If FACT = 'F' and EQUED is not 'N', then AB must have been
                equilibrated by the scaling factors in R and/or C.  AB is not
                modified if FACT = 'F' or 'N', or if FACT = 'E' and
                EQUED = 'N' on exit.

                On exit, if EQUED .ne. 'N', A is scaled as follows:
                EQUED = 'R':  A := diag(R) * A
                EQUED = 'C':  A := A * diag(C)
                EQUED = 'B':  A := diag(R) * A * diag(C).

           LDAB

                     LDAB is INTEGER
                The leading dimension of the array AB.  LDAB >= KL+KU+1.

           AFB

                     AFB is REAL array, dimension (LDAFB,N)
                If FACT = 'F', then AFB is an input argument and on entry
                contains details of the LU factorization of the band matrix
                A, as computed by SGBTRF.  U is stored as an upper triangular
                band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
                and the multipliers used during the factorization are stored
                in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
                the factored form of the equilibrated matrix A.

                If FACT = 'N', then AF is an output argument and on exit
                returns the factors L and U from the factorization A = P*L*U
                of the original matrix A.

                If FACT = 'E', then AF is an output argument and on exit
                returns the factors L and U from the factorization A = P*L*U
                of the equilibrated matrix A (see the description of A for
                the form of the equilibrated matrix).

           LDAFB

                     LDAFB is INTEGER
                The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                If FACT = 'F', then IPIV is an input argument and on entry
                contains the pivot indices from the factorization A = P*L*U
                as computed by SGETRF; row i of the matrix was interchanged
                with row IPIV(i).

                If FACT = 'N', then IPIV is an output argument and on exit
                contains the pivot indices from the factorization A = P*L*U
                of the original matrix A.

                If FACT = 'E', then IPIV is an output argument and on exit
                contains the pivot indices from the factorization A = P*L*U
                of the equilibrated matrix A.

           EQUED

                     EQUED is CHARACTER*1
                Specifies the form of equilibration that was done.
                  = 'N':  No equilibration (always true if FACT = 'N').
                  = 'R':  Row equilibration, i.e., A has been premultiplied by
                          diag(R).
                  = 'C':  Column equilibration, i.e., A has been postmultiplied
                          by diag(C).
                  = 'B':  Both row and column equilibration, i.e., A has been
                          replaced by diag(R) * A * diag(C).
                EQUED is an input argument if FACT = 'F'; otherwise, it is an
                output argument.

           R

                     R is REAL array, dimension (N)
                The row scale factors for A.  If EQUED = 'R' or 'B', A is
                multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                is not accessed.  R is an input argument if FACT = 'F';
                otherwise, R is an output argument.  If FACT = 'F' and
                EQUED = 'R' or 'B', each element of R must be positive.
                If R is output, each element of R is a power of the radix.
                If R is input, each element of R 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.

           C

                     C is REAL array, dimension (N)
                The column scale factors for A.  If EQUED = 'C' or 'B', A is
                multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                is not accessed.  C is an input argument if FACT = 'F';
                otherwise, C is an output argument.  If FACT = 'F' and
                EQUED = 'C' or 'B', each element of C must be positive.
                If C is output, each element of C is a power of the radix.
                If C is input, each element of C 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

                     B is 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
                   diag(R)*B;
                if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
                   overwritten by diag(C)*B.

           LDB

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

           X

                     X is 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(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
                inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.

           LDX

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

           RCOND

                     RCOND is 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

                     RPVGRW is 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.  In SGESVX, this quantity is
                returned in WORK(1).

           BERR

                     BERR is 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

                     N_ERR_BNDS is 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

                     ERR_BNDS_NORM is 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

                     ERR_BNDS_COMP is 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

                     NPARAMS is INTEGER
                Specifies the number of parameters set in PARAMS.  If .LE. 0, the
                PARAMS array is never referenced and default values are used.

           PARAMS

                     PARAMS is 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

                     WORK is REAL array, dimension (4*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is 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.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           April 2012

Author

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