.TH "SRC/zgbsvxx.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/zgbsvxx.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBzgbsvxx\fP (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)"
.br
.RI "\fB ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine zgbsvxx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) r, double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx , * ) x, integer ldx, double precision rcond, double precision rpvgrw, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)"

.PP
\fB ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
    ZGBSVXX uses the LU factorization to compute the solution to a
    complex*16 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\&. ZGBSVXX 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\&.

    ZGBSVXX 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
    ZGBSVXX 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 ZGBSVXX would itself produce\&.
.fi
.PP
 
.RE
.PP
\fBDescription:\fP
.RS 4

.PP
.nf
    The following steps are performed:

    1\&. If FACT = 'E', double precision 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\&.
.fi
.PP
 
.PP
.nf
     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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIFACT\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fITRANS\fP 
.PP
.nf
          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)
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
     The number of linear equations, i\&.e\&., the order of the
     matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIKL\fP 
.PP
.nf
          KL is INTEGER
     The number of subdiagonals within the band of A\&.  KL >= 0\&.
.fi
.PP
.br
\fIKU\fP 
.PP
.nf
          KU is INTEGER
     The number of superdiagonals within the band of A\&.  KU >= 0\&.
.fi
.PP
.br
\fINRHS\fP 
.PP
.nf
          NRHS is INTEGER
     The number of right hand sides, i\&.e\&., the number of columns
     of the matrices B and X\&.  NRHS >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is COMPLEX*16 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)\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
     The leading dimension of the array AB\&.  LDAB >= KL+KU+1\&.
.fi
.PP
.br
\fIAFB\fP 
.PP
.nf
          AFB is COMPLEX*16 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 ZGBTRF\&.  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)\&.
.fi
.PP
.br
\fILDAFB\fP 
.PP
.nf
          LDAFB is INTEGER
     The leading dimension of the array AFB\&.  LDAFB >= 2*KL+KU+1\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          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 ZGETRF; 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\&.
.fi
.PP
.br
\fIEQUED\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIR\fP 
.PP
.nf
          R is DOUBLE PRECISION 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\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is DOUBLE PRECISION 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\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
     The leading dimension of the array B\&.  LDB >= max(1,N)\&.
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is COMPLEX*16 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'\&.
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
          LDX is INTEGER
     The leading dimension of the array X\&.  LDX >= max(1,N)\&.
.fi
.PP
.br
\fIRCOND\fP 
.PP
.nf
          RCOND is DOUBLE PRECISION
     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\&.
.fi
.PP
.br
\fIRPVGRW\fP 
.PP
.nf
          RPVGRW is DOUBLE PRECISION
     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 ZGESVX, this quantity is
     returned in RWORK(1)\&.
.fi
.PP
.br
\fIBERR\fP 
.PP
.nf
          BERR is DOUBLE PRECISION 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)\&.
.fi
.PP
.br
\fIN_ERR_BNDS\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIERR_BNDS_NORM\fP 
.PP
.nf
          ERR_BNDS_NORM is DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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\&.
.fi
.PP
.br
\fIERR_BNDS_COMP\fP 
.PP
.nf
          ERR_BNDS_COMP is DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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\&.
.fi
.PP
.br
\fINPARAMS\fP 
.PP
.nf
          NPARAMS is INTEGER
     Specifies the number of parameters set in PARAMS\&.  If <= 0, the
     PARAMS array is never referenced and default values are used\&.
.fi
.PP
.br
\fIPARAMS\fP 
.PP
.nf
          PARAMS is DOUBLE PRECISION 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\&.0D+0
            = 0\&.0:  No refinement is performed, and no error bounds are
                    computed\&.
            = 1\&.0:  Use the extra-precise refinement algorithm\&.
              (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)
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (2*N)
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (2*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB555\fP of file \fBzgbsvxx\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.