Provided by: liblapack-doc_3.12.0-3build1.1_all 
NAME
la_gbrfsx_extended - la_gbrfsx_extended: step in gbrfsx
SYNOPSIS
Functions
subroutine cla_gbrfsx_extended (prec_type, trans_type, n, kl, ku, nrhs, ab, ldab, afb,
ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm,
err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise,
info)
CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
subroutine dla_gbrfsx_extended (prec_type, trans_type, n, kl, ku, nrhs, ab, ldab, afb,
ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm,
err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise,
info)
DLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
subroutine sla_gbrfsx_extended (prec_type, trans_type, n, kl, ku, nrhs, ab, ldab, afb,
ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm,
err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise,
info)
SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
subroutine zla_gbrfsx_extended (prec_type, trans_type, n, kl, ku, nrhs, ab, ldab, afb,
ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm,
err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise,
info)
ZLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
Detailed Description
Function Documentation
subroutine cla_gbrfsx_extended (integer prec_type, integer trans_type, integer n, integer kl,
integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex,
dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, logical colequ,
real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension(
ldy, * ) y, integer ldy, real, dimension( * ) berr_out, integer n_norms, real, dimension(
nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, complex, dimension( * )
res, real, dimension(*) ayb, complex, dimension( * ) dy, complex, dimension( * ) y_tail,
real rcond, integer ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer info)
CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
Purpose:
CLA_GBRFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by CGBRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters
PREC_TYPE
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
TRANS_TYPE
TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= 'N': No transpose
= 'T': Transpose
= 'C': 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
matrix B.
AB
AB is COMPLEX array, dimension (LDAB,N)
On entry, the N-by-N matrix AB.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= max(1,N).
AFB
AFB is COMPLEX array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by CGBTRF.
LDAFB
LDAFB is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by CGBTRF; row i of the matrix was interchanged
with row IPIV(i).
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is REAL array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. 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)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is COMPLEX array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by CGBTRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is REAL array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by CLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
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.
This subroutine is only responsible for setting the second field
above.
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.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is COMPLEX array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is REAL array, dimension (N)
Workspace.
DY
DY is COMPLEX array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is COMPLEX array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
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.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For '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.
RTHRESH
RTHRESH is REAL
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is REAL
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to CGBTRS had an illegal
value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dla_gbrfsx_extended (integer prec_type, integer trans_type, integer n, integer kl,
integer ku, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double
precision, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, logical
colequ, double precision, dimension( * ) c, double precision, dimension( ldb, * ) b,
integer ldb, double precision, dimension( ldy, * ) y, integer ldy, double precision,
dimension(*) berr_out, integer n_norms, double precision, dimension( nrhs, * )
err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, double precision,
dimension(*) res, double precision, dimension(*) ayb, double precision, dimension(*) dy,
double precision, dimension(*) y_tail, double precision rcond, integer ithresh, double
precision rthresh, double precision dz_ub, logical ignore_cwise, integer info)
DLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
Purpose:
DLA_GBRFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by DGBRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters
PREC_TYPE
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
TRANS_TYPE
TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= 'N': No transpose
= 'T': Transpose
= 'C': 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
matrix B.
AB
AB is DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the N-by-N matrix AB.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDBA >= max(1,N).
AFB
AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
The factors L and U from the factorization
A = P*L*U as computed by DGBTRF.
LDAFB
LDAFB is INTEGER
The leading dimension of the array AF. LDAFB >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by DGBTRF; row i of the matrix was interchanged
with row IPIV(i).
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. 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 DOUBLE PRECISION array, dimension (LDB,NRHS)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by DGBTRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by DLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
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 '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.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
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 '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.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is DOUBLE PRECISION array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.
DY
DY is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is DOUBLE PRECISION array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
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.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For '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.
RTHRESH
RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to DGBTRS had an illegal
value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sla_gbrfsx_extended (integer prec_type, integer trans_type, integer n, integer kl,
integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension(
ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, logical colequ, real,
dimension( * ) c, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldy, * ) y,
integer ldy, real, dimension(*) berr_out, integer n_norms, real, dimension( nrhs, * )
err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, real, dimension(*) res, real,
dimension(*) ayb, real, dimension(*) dy, real, dimension(*) y_tail, real rcond, integer
ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer info)
SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
Purpose:
SLA_GBRFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by SGBRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters
PREC_TYPE
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
TRANS_TYPE
TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= 'N': No transpose
= 'T': Transpose
= 'C': 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
matrix B.
AB
AB is REAL array, dimension (LDAB,N)
On entry, the N-by-N matrix AB.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= max(1,N).
AFB
AFB is REAL array, dimension (LDAFB,N)
The factors L and U from the factorization
A = P*L*U as computed by SGBTRF.
LDAFB
LDAFB is INTEGER
The leading dimension of the array AF. LDAFB >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by SGBTRF; row i of the matrix was interchanged
with row IPIV(i).
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is REAL array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. 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 REAL array, dimension (LDB,NRHS)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is REAL array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by SGBTRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is REAL array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by SLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
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.
This subroutine is only responsible for setting the second field
above.
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.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is REAL array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is REAL array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.
DY
DY is REAL array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is REAL array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
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.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For '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.
RTHRESH
RTHRESH is REAL
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is REAL
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to SGBTRS had an illegal
value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zla_gbrfsx_extended (integer prec_type, integer trans_type, 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, logical colequ,
double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb,
complex*16, dimension( ldy, * ) y, integer ldy, double precision, dimension( * ) berr_out,
integer n_norms, double precision, dimension( nrhs, * ) err_bnds_norm, double precision,
dimension( nrhs, * ) err_bnds_comp, complex*16, dimension( * ) res, double precision,
dimension(*) ayb, complex*16, dimension( * ) dy, complex*16, dimension( * ) y_tail, double
precision rcond, integer ithresh, double precision rthresh, double precision dz_ub,
logical ignore_cwise, integer info)
ZLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for
general banded matrices by performing extra-precise iterative refinement and provides
error bounds and backward error estimates for the solution.
Purpose:
ZLA_GBRFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by ZGBRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters
PREC_TYPE
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
TRANS_TYPE
TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= 'N': No transpose
= 'T': Transpose
= 'C': 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
matrix B.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
On entry, the N-by-N matrix A.
LDAB
LDAB is INTEGER
The leading dimension of the array A. LDAB >= max(1,N).
AFB
AFB is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by ZGBTRF.
LDAFB
LDAFB is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by ZGBTRF; row i of the matrix was interchanged
with row IPIV(i).
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. 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*16 array, dimension (LDB,NRHS)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is COMPLEX*16 array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by ZGBTRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by ZLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
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 '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.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
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 '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.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is COMPLEX*16 array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is DOUBLE PRECISION array, dimension (N)
Workspace.
DY
DY is COMPLEX*16 array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is COMPLEX*16 array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
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.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For '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.
RTHRESH
RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to ZGBTRS had an illegal
value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Author
Generated automatically by Doxygen for LAPACK from the source code.