complexGBsolve(3)

NAME
complexGBsolve - complex

SYNOPSIS

  Functions
subroutine cgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
subroutine cgbsvx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CGBSVX computes the solution to system of linear equations A * X = B for GB matrices
subroutine cgbsvxx (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, rwork, info)
CGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Detailed Description

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

Function Documentation

  subroutine cgbsv (integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

Purpose:

     CGBSV computes the solution to a complex 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 COMPLEX 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 COMPLEX 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.

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.

Definition at line 161 of file cgbsv.f.

  subroutine cgbsvx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, character equed, real, dimension( * ) r, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBSVX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

     CGBSVX uses the LU factorization to compute the solution to a complex
     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  (Conjugate 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 COMPLEX 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 COMPLEX 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 CGBTRF.  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 CGBTRF; 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (N)
              On exit, RWORK(1) contains the reciprocal pivot growth
              factor norm(A)/norm(U). The 'max absolute element' norm is
              used. If RWORK(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
              RWORK(1) contains the reciprocal pivot growth factor for the
              leading INFO columns of A.

    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
                           value of RCOND would suggest.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 367 of file cgbsvx.f.

  subroutine cgbsvxx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, character equed, real, dimension( * ) r, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, 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, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

        CGBSVXX uses the LU factorization to compute the solution to a
        complex 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. CGBSVXX 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.
        CGBSVXX 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
        CGBSVXX 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 CGBSVXX 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 COMPLEX 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 COMPLEX 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 CGBTRF.  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 CGETRF; 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 COMPLEX 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 COMPLEX 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 CGESVX, this quantity is
         returned in RWORK(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 < 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 <= 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 < 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 COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (2*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.

Definition at line 558 of file cgbsvxx.f.

Author

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