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## zgbsvxx()

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 )

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

Download ZGBSVXX + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !>    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.
      !>

Description:

      !>
      !>    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.
      !>

      !>     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

  [in]

  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.
      !>

  [in]

  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)
      !>

  [in]

  N

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

  [in]

  KL

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

  [in]

  KU

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

  [in]

  NRHS

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

  [in,out]

  AB

      !>          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).
      !>

  [in]

  LDAB

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

  [in,out]

  AFB

      !>          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).
      !>

  [in]

  LDAFB

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

  [in,out]

  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 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.
      !>

  [in,out]

  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.
      !>

  [in,out]

  R

      !>          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.
      !>

  [in,out]

  C

      !>          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.
      !>

  [in,out]

  B

      !>          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.
      !>

  [in]

  LDB

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

  [out]

  X

      !>          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'.
      !>

  [in]

  LDX

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

  [out]

  RCOND

      !>          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.
      !>

  [out]

  RPVGRW

      !>          RPVGRW is DOUBLE PRECISION
      !>     Reciprocal pivot growth.  On exit, this contains the reciprocal
      !>     pivot growth factor norm(A)/norm(U). The
      !>     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).
      !>

  [out]

  BERR

      !>          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).
      !>

  [in]

  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.
      !>

  [out]

  ERR_BNDS_NORM

      !>          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  boolean. Trust the answer if the
      !>              reciprocal condition number is less than the threshold
      !>              sqrt(n) * dlamch('Epsilon').
      !>
      !>     err = 2  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 . 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.
      !>

  [out]

  ERR_BNDS_COMP

      !>          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  boolean. Trust the answer if the
      !>              reciprocal condition number is less than the threshold
      !>              sqrt(n) * dlamch('Epsilon').
      !>
      !>     err = 2  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 . 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.
      !>

  [in]

  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.
      !>

  [in,out]

  PARAMS

      !>          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)
      !>

  [out]

  WORK

      !>          WORK is COMPLEX*16 array, dimension (2*N)
      !>

  [out]

  RWORK

      !>          RWORK is DOUBLE PRECISION array, dimension (2*N)
      !>

  [out]

  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.