Contents

# ?gbrfs

Refines the solution of a system of linear equations with a general band coefficient matrix and estimates its error.

## Syntax

Include Files
• mkl.h
Description
The routine performs an iterative refinement of the solution to a system of linear equations
A*X
=
B
or
A
T
*X
=
B
or
A
H
*X
=
B
with a band matrix
A
, with multiple right-hand sides. For each computed solution vector
x
, the routine computes the component-wise backward error
β
. This error is the smallest relative perturbation in elements of
A
and
b
such that
x
is the exact solution of the perturbed system:
|
δ
a
i
j
|
β
|
a
i
j
|, |
δ
b
i
|
β
|
b
i
|
such that
(
A
+
δ
A
)
x
= (
b
+
δ
b
)
.
Finally, the routine estimates the component-wise forward error in the computed solution
||
x
-
x
e
||
/||
x
||
(here
x
e
is the exact solution).
Before calling this routine:
• call the factorization routine
?gbtrf
• call the solver routine
?gbtrs
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
or
'T'
or
'C'
.
Indicates the form of the equations:
If
trans
=
'N'
, the system has the form
A*X
=
B
.
If
trans
=
'T'
, the system has the form
A
T
*X
=
B
.
If
trans
=
'C'
, the system has the form
A
H
*X
=
B
.
n
The order of the matrix
A
;
n
0.
kl
The number of sub-diagonals within the band of
A
;
kl
0.
ku
The number of super-diagonals within the band of
A
;
ku
0.
nrhs
The number of right-hand sides;
nrhs
0.
ab
,
afb
,
b
,
x
Arrays:
ab
(size max(1,
ldab
*
n
))
contains the original band matrix
A
, as supplied to
?gbtrf
, but stored in rows from
1 to
kl
+
ku
+ 1 for column major layout, and columns from 1 to
kl
+
ku
+ 1 for row major layout
.
afb
(size max(1,
ldafb
*
n
))
contains the factored band matrix
A
, as returned by
?gbtrf
.
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
contains the right-hand side matrix
B
.
x
of size max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
contains the solution matrix
X
.
ldab
ab
,
ldab
kl
+
ku
+ 1
.
ldafb
afb
,
ldafb
2*
kl
+
ku
+ 1
.
ldb
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ldx
x
;
ldx
max(1,
n
)
.
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
array, as returned by
?gbtrf
.
Output Parameters
x
The refined solution matrix
X
.
ferr
,
berr
Arrays, size at least
max(1,
nrhs
)
. Contain the component-wise forward and backward errors, respectively, for each solution vector.
Return Values
This function returns a value
info
.
If
info
=0, the execution is successful.
If
info
=
-i
, parameter
i
Application Notes
The bounds returned in
ferr
are not rigorous, but in practice they almost always overestimate the actual error.
For each right-hand side, computation of the backward error involves a minimum of 4
n
(
kl
+
ku
) floating-point operations (for real flavors) or 16
n
(
kl
+
ku
) operations (for complex flavors). In addition, each step of iterative refinement involves 2
n
(4
kl
+ 3
ku
) operations (for real flavors) or 8
n
(4
kl
+ 3
ku
) operations (for complex flavors); the number of iterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear equations
A
*
x
=
b
; the number is usually 4 or 5 and never more than 11. Each solution requires approximately
2
n
2
floating-point operations for real flavors or
8
n
2
for complex flavors.

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.