CGBSVXX(1) LAPACK driver routine (version 3.2) CGBSVXX(1)NAME
CGBSVXX - 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 )
IMPLICIT NONE
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
N_ERR_BNDS
REAL RCOND, RPVGRW
INTEGER IPIV( * )
COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), X( LDX
, * ),WORK( * )
REAL R( * ), 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 = '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.
ARGUMENTS
Some optional parameters are bundled in the PARAMS array. These set‐
tings 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. = '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 (input) 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 (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 = '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 (input) INTEGER
The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB (input or output) REAL array, dimension (LDAFB,N)
If FACT = 'F', then AFB is an input argument and on entry con‐
tains 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 fac‐
tored 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 (input) INTEGER
The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry con‐
tains the pivot indices from the factorization A = P*L*U as
computed by SGETRF; row i of the matrix was interchanged with
row IPIV(i). If FACT = 'N', then IPIV is an output argument
and on exit contains the pivot indices from the factorization A
= P*L*U of the original matrix A. If FACT = 'E', then IPIV is
an output argument and on exit contains the pivot indices from
the factorization A = P*L*U of the equilibrated matrix A.
EQUED (input or output) 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 post‐
multiplied by diag(C). = 'B': Both row and column equilibra‐
tion, 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 (input or output) REAL array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
tiplied 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 ele‐
ment 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 solu‐
tion and error estimates. Scaling by powers of the radix does
not cause rounding errors unless the result underflows or over‐
flows. 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 = '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 ele‐
ment 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 solu‐
tion and error estimates. Scaling by powers of the radix does
not cause rounding errors unless the result underflows or over‐
flows. 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 = '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 (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. '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 (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 equili‐
bration (if done). If this is less than the machine precision
(in particular, if it is zero), the matrix is singular to work‐
ing precision. Note that the error may still be small even if
this number is very small and the matrix appears ill- condi‐
tioned.
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 component‐
wise 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 informa‐
tion 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 recipro‐
cal 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 abso‐
lute 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 informa‐
tion 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 informa‐
tion as described below. There currently are up to three
pieces of information returned for each right-hand side.
If componentwise accuracy is not requested (PARAMS(3) =
0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
.LT. 3, then at most the first (:,N_ERR_BNDS) entries
are returned. The first index in ERR_BNDS_COMP(i,:)
corresponds to the ith right-hand side. The second
index in ERR_BNDS_COMP(:,err) contains the following
three fields: err = 1 "Trust/don't trust" boolean. Trust
the answer if the reciprocal condition number is less
than the threshold sqrt(n) * slamch('Epsilon'). err = 2
"Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should
only be trusted if the previous boolean is true. err =
3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the thresh‐
old sqrt(n) * slamch('Epsilon') to determine if the
error estimate is "guaranteed". These reciprocal condi‐
tion 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 fur‐
ther details and extra cautions. NPARAMS (input) INTE‐
GER 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 algo‐
rithm, possibly with doubled-single computations if the compi‐
lation 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 factor‐
ization 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 compo‐
nentwise relative error in the double-precision algorithm.
Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
nentwise convergence)
WORK (workspace) REAL array, dimension (4*N)
IWORK (workspace) INTEGER array, dimension (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 ille‐
gal 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 compo‐
nentwise 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.
LAPACK driver routine (versioNovember 2008 CGBSVXX(1)
