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CGBSVXX

Section: LAPACK driver routine (version 3.2) (1)
Updated: April 2011
Index

NAME

LAPACK-3 - CGBSVXX use 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

SYNOPSIS

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 )
: IMPLICITNONE
: CHARACTEREQUED, FACT, TRANS
: INTEGERINFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,N_ERR_BNDS
: REALRCOND, RPVGRW
: INTEGERIPIV( * )
: COMPLEXAB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),X( LDX , * ),WORK( * )
: REALR( * ), C( * ), PARAMS( * ), BERR( * ),ERR_BNDS_NORM( NRHS, * ),ERR_BNDS_COMP( NRHS, * ), RWORK( * )

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 = aqEaq, real scaling factors are computed to equilibrate
    the system:

      TRANS = aqNaq:  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
      TRANS = aqTaq: (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
      TRANS = aqCaq: (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=aqNaq)
    or diag(C)*B (if TRANS = aqTaq or aqCaq).

    2. If FACT = aqNaq or aqEaq, the LU decomposition is used to factor
    the matrix A (after equilibration if FACT = aqEaq) 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 = aqNaq) or diag(R) (if TRANS = aqTaq or aqCaq) so
    that it solves the original system before equilibration.

ARGUMENTS

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

FACT (input) CHARACTER*1: Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= aqFaq:  On entry, AF and IPIV contain the factored form of A.
If EQUED is not aqNaq, the matrix A has been
equilibrated with scaling factors given by R and C.
A, AF, and IPIV are not modified.
= aqNaq:  The matrix A will be copied to AF and factored.

= aqEaq:  The matrix A will be equilibrated if necessary, then
copied to AF and factored.
TRANS (input) CHARACTER*1: Specifies the form of the system of equations:

= aqNaq:  A * X = B     (No transpose)

= aqTaq:  A**T * X = B  (Transpose)

= aqCaq:  A**H * X = B  (Conjugate Transpose = Transpose)
N (input) INTEGER: The number of linear equations, i.e., the order of the
matrix A. N >= 0.
KL (input) INTEGER: The number of subdiagonals within the band of A. KL >= 0.
KU (input) INTEGER: The number of superdiagonals within the band of A. KU >= 0.
NRHS (input) INTEGER: The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AB (input/output) 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 = aqFaq and EQUED is not aqNaq, then AB must have been
equilibrated by the scaling factors in R and/or C.  AB is not
modified if FACT = aqFaq or aqNaq, or if FACT = aqEaq and
EQUED = aqNaq on exit.
On exit, if EQUED .ne. aqNaq, A is scaled as follows:
EQUED = aqRaq:  A := diag(R) * A

EQUED = aqCaq:  A := A * diag(C)

EQUED = aqBaq:  A := diag(R) * A * diag(C).
LDAB (input) INTEGER: The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB (input or output) REAL array, dimension (LDAFB,N): If FACT = aqFaq, 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. aqNaq, then AFB is
the factored form of the equilibrated matrix A.
If FACT = aqNaq, 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 = aqEaq, 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 (input) INTEGER: The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
IPIV (input or output) INTEGER array, dimension (N): If FACT = aqFaq, 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 = aqNaq, 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 = aqEaq, 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 (input or output) CHARACTER*1: Specifies the form of equilibration that was done.
= aqNaq:  No equilibration (always true if FACT = aqNaq).

= aqRaq:  Row equilibration, i.e., A has been premultiplied by
diag(R).
= aqCaq:  Column equilibration, i.e., A has been postmultiplied
by diag(C).
= aqBaq:  Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C).
EQUED is an input argument if FACT = aqFaq; otherwise, it is an
output argument.
R (input or output) REAL array, dimension (N): The row scale factors for A.  If EQUED = aqRaq or aqBaq, A is
multiplied on the left by diag(R); if EQUED = aqNaq or aqCaq, R
is not accessed.  R is an input argument if FACT = aqFaq;
otherwise, R is an output argument.  If FACT = aqFaq and
EQUED = aqRaq or aqBaq, 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 (input or output) REAL array, dimension (N): The column scale factors for A.  If EQUED = aqCaq or aqBaq, A is
multiplied on the right by diag(C); if EQUED = aqNaq or aqRaq, C
is not accessed.  C is an input argument if FACT = aqFaq;
otherwise, C is an output argument.  If FACT = aqFaq and
EQUED = aqCaq or aqBaq, 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 (input/output) REAL array, dimension (LDB,NRHS): On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = aqNaq, B is not modified;
if TRANS = aqNaq and EQUED = aqRaq or aqBaq, B is overwritten by
diag(R)*B;
if TRANS = aqTaq or aqCaq and EQUED = aqCaq or aqBaq, B is
overwritten by diag(C)*B.
LDB (input) INTEGER: The leading dimension of the array B. LDB >= max(1,N).
X (output) REAL array, dimension (LDX,NRHS): If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit
if EQUED .ne. aqNaq, and the solution to the equilibrated system is
inv(diag(C))*X if TRANS = aqNaq and EQUED = aqCaq or aqBaq, or
inv(diag(R))*X if TRANS = aqTaq or aqCaq and EQUED = aqRaq or aqBaq.
LDX (input) INTEGER: The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) REAL: Reciprocal scaled condition number.  This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done).  If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision.  Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
RPVGRW (output) REAL: Reciprocal pivot growth.  On exit, this contains the reciprocal
pivot growth factor norm(A)/norm(U). The "max absolute element"
norm is used.  If this is much less than 1, then the stability of
the LU factorization of the (equilibrated) matrix A could be poor.
This also means that the solution X, estimated condition numbers,
and error bounds could be unreliable. If factorization fails with
0<INFO<=N, then this contains the reciprocal pivot growth factor
for the leading INFO columns of A.  In SGESVX, this quantity is
returned in WORK(1).
BERR (output) REAL array, dimension (NRHS): Componentwise relative backward error. This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).
N_ERR_BNDS (input) INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise). See ERR_BNDS_NORM and
ERR_BNDS_COMP below.
ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS): For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/donaqt trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch(aqEpsilonaq).
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(aqEpsilonaq). 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(aqEpsilonaq) to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS): For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/donaqt trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch(aqEpsilonaq).
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(aqEpsilonaq). 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(aqEpsilonaq) to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
NPARAMS (input) INTEGER
Specifies the number of parameters set in PARAMS.  If .LE. 0, the
PARAMS array is never referenced and default values are used.
PARAMS (input / output) REAL array, dimension NPARAMS: Specifies algorithm parameters.  If an entry is .LT. 0.0, then
that entry will be filled with default value used for that
parameter.  Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0

= 0.0 : No refinement is performed, and no error bounds are
computed.
= 1.0 : Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10

Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm.  Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (2*N)
INFO (output) INTEGER: = 0:  Successful exit. The solution to every right-hand side is
guaranteed.
< 0:  If INFO = -i, the i-th argument had an illegal value

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

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