# sla_gerfsx_extended.f - Man Page

SRC/sla_gerfsx_extended.f

## Synopsis

### Functions/Subroutines

subroutine sla_gerfsx_extended (prec_type, trans_type, n, nrhs, a, lda, af, ldaf, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, errs_n, errs_c, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

## Function/Subroutine Documentation

### subroutine sla_gerfsx_extended (integer prec_type, integer trans_type, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, 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, * ) errs_n, real, dimension( nrhs, * ) errs_c, 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_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

``` SLA_GERFSX_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 SGERFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERRS_N
and ERRS_C for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERRS_N and ERRS_C.```
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.```

NRHS

```          NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B.```

A

```          A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).```

AF

```          AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF.```

LDAF

```          LDAF 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 SGETRF; 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 SGETRS.
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 ERRS_N
and ERRS_C).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.```

ERRS_N

```          ERRS_N 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 ERRS_N(i,:) corresponds to the ith
right-hand side.

The second index in ERRS_N(:,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.```

ERRS_C

```          ERRS_C 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
ERRS_C is not accessed.  If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.

The first index in ERRS_C(i,:) corresponds to the ith
right-hand side.

The second index in ERRS_C(:,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
ERRS_N and ERRS_C 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 SGETRS had an illegal
value```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 391 of file sla_gerfsx_extended.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sla_gerfsx_extended(3) is an alias of sla_gerfsx_extended.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK